From e7e53ce6823fe086e7413f855002bc7914258faf Mon Sep 17 00:00:00 2001 From: Emily Boudreaux Date: Sun, 20 Sep 2026 09:10:27 -0400 Subject: [PATCH] Consolidate numerical research and preserve a reproducible handoff Document the baseline, hypotheses, measured results, failed approaches, current policy defaults, and remaining rotating n=3 and virial limitations. Include the authoritative LaTeX report, Pandoc-generated Markdown, compiled PDF, source provenance, branch decisions, and consolidation validation. Organize raw results and miscellaneous artifacts under working/ without deleting them; version the directory guide, relocation hashes, archive verifier, and report build/conversion checks. Keep build caches at their original paths and clearly distinguish retained binaries from the validated research build. 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working/README.md diff --git a/.gitignore b/.gitignore index c94b5e0..322085d 100644 --- a/.gitignore +++ b/.gitignore @@ -5,3 +5,16 @@ build/ ._DS_Store .DSStore .DS_Store + +# Local build caches retain absolute source paths; do not relocate them. +cmake-build-*/ +__pycache__/ +*.pyc +test_summary.html +.cex/ +.codex/ + +# Preserved raw campaigns, scratch files and report build intermediates. +# The directory guide and audit manifests are versioned separately. +/working/* +!/working/README.md diff --git a/readme.md b/readme.md index 556215e..8383c87 100644 --- a/readme.md +++ b/readme.md @@ -17,7 +17,31 @@ This module is not intended for scientific use; rather, this will inform latter - We aim to develope numerical instrumentation which will be incorporated into SERiF including, but not limited to, preconditioners, integrators, and operators - We aim to develope utility infrastructure which will be incorporated into SERiF including, but not limited to, quadrature boost policies, domain mapping, and user experience tools. -# Current Status [as of July 15, 2026] +# Current research status (20 September 2026) + +The cumulative numerical investigations have been consolidated into `main`. +The preferred research surface is the coupled degree-four real spherical +harmonic prescription with all azimuthal orders (25 coefficients), explicitly +selected by the sandbox. Density remains independent DG Q2, enthalpy and +displacement remain H1 Q3, and the library's existing default policies remain +available unchanged. + +Some low-index, slowly rotating models meet their algebraic and virial targets. +A fully converged moderately rotating n=3 model, validation against ESTER, and +50% of critical rotation have **not** been demonstrated. The current research +virial acceptance target is 1e-5; the earlier 1e-6 statement above is the original +aspiration. Stellar LDU plus gravity-Schur is a promising experimental recipe, +not a promoted universal solver default. Log density and new positivity +enforcement remain deferred. + +Start with the [complete research report](reports/MeanField_research_handoff_2026-09-20.md), +the [branch disposition record](reports/branch_consolidation.md), and the +[report build instructions](reports/README.md). Raw results have been preserved +and organized under [working/](working/README.md). The +[model slice guide](experiments/MODEL_SLICES.md) explains physical-plane output +including the deformed stellar surface. + +# Historical status (15 July 2026) - Integrators for all mean field equations have been written. In other code these have been validated as self-consistent and consistent with analytic solutions generated with symbolic algebra tools. These tests will be incorporated into this module - Gravitational Potential calculations have been rigorously tested for a number of cases including uniform spherical densities, polynomial spherical densities, rational spherical densities, uniform ellipsoidal densities, and rational ellipsoidal densities. These potentials only apply the density profile and the potential at infinity boundary condition - We have shown that we can reach sufficient self-consistency and accuracy with the gravitational field such that the gravitational field representation will not prevent us achiveing the required virial ratio or mode identification accuracy. @@ -36,15 +60,25 @@ and no testing has been done to confirm if this works over a variety of targets. - ninja - cmake -once those are installed then the following commands will in theory build mean_field +Once those dependencies are installed, a fresh build on the development Mac uses: ```bash -mkdir build -cd build & cmake .. -Gninja && ninja -./build/tests +cmake -S . -B working/builds/release -G Ninja \ + -DCMAKE_BUILD_TYPE=Release \ + -DCMAKE_C_COMPILER=/opt/homebrew/opt/llvm/bin/clang \ + -DCMAKE_CXX_COMPILER=/opt/homebrew/opt/llvm/bin/clang++ +cmake --build working/builds/release --parallel 2 \ + --target tests sandbox plateau_diagnostic +OMPI_MCA_btl=self,sm ./working/builds/release/tests --catch2 ``` +Adjust compiler and dependency locations for another machine. The consolidation +was validated using the isolated research worktree's Release build. Older build +directories in this checkout were preserved, and their binaries may be stale: +rebuild from the current source before using them. Do not transplant a CMake +cache from a different checkout. + should you wish to build this and you run into issues (as you likely will) you are welcome to email Emily Boudreaux (emily.boudreaux@dartmouth.edu); however, maintenance and support of this module are not guaranteed. If SERiF has been released when you are reading this documentation we encourage you to look for that project on the 4D-STAR webpage (https://4d-star.org) as it inherits any and all scientific merit from this exploratory project. # Funding -> 4D-STAR is funded by European Research Council (ERC) under the Horizon Europe programme (Synergy Grant agreement No. 101071505: 4D-STAR). Work for this project is funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. \ No newline at end of file +> 4D-STAR is funded by European Research Council (ERC) under the Horizon Europe programme (Synergy Grant agreement No. 101071505: 4D-STAR). Work for this project is funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. diff --git a/reports/MeanField_research_handoff_2026-09-20.md b/reports/MeanField_research_handoff_2026-09-20.md new file mode 100644 index 0000000..2621a11 --- /dev/null +++ b/reports/MeanField_research_handoff_2026-09-20.md @@ -0,0 +1,1356 @@ +--- +author: +- | + Research record prepared with Codex + Source and numerical evidence: 16–20 September 2026 +date: 20 September 2026 +title: | + MeanField numerical research + and consolidation handoff + Formulation, experiments, evidence, architecture, and remaining work +--- + +- [Purpose, scope and current conclusion](#purpose-scope-and-current-conclusion) + - [Physical and software objectives](#physical-and-software-objectives) + - [What has actually been demonstrated](#what-has-actually-been-demonstrated) + - [How to use this record](#how-to-use-this-record) +- [Definitions and evidence discipline](#definitions-and-evidence-discipline) + - [Physical variables and virial diagnostics](#physical-variables-and-virial-diagnostics) + - [Algebraic quantities and cost accounting](#algebraic-quantities-and-cost-accounting) + - [Controls, provenance and limits of inference](#controls-provenance-and-limits-of-inference) +- [Starting point, problem definition, and architecture](#starting-point-problem-definition-and-architecture) + - [Continuum equations and independent discrete fields](#continuum-equations-and-independent-discrete-fields) + - [Discretization and model compilation](#discretization-and-model-compilation) + - [Physical acceptance and measurement conventions](#physical-acceptance-and-measurement-conventions) +- [Initial diagnosis: several different plateaus](#initial-diagnosis-several-different-plateaus) + - [Fresh control measurements](#fresh-control-measurements) + - [Normalization audit](#normalization-audit) + - [Central condition: a useful example correction](#central-condition-a-useful-example-correction) + - [Krylov window and warm-guess safeguard](#krylov-window-and-warm-guess-safeguard) + - [Rejected one-way gravity completion](#rejected-one-way-gravity-completion) +- [Separating pressure preparation from shape-related curvature](#separating-pressure-preparation-from-shape-related-curvature) + - [The rotating pressure obstruction was localized and real](#the-rotating-pressure-obstruction-was-localized-and-real) + - [Higher-index merit rejection had a different mechanism](#higher-index-merit-rejection-had-a-different-mechanism) +- [Three optional numerical strategies and their limits](#three-optional-numerical-strategies-and-their-limits) + - [Polynomial enthalpy certification and anchor repair](#polynomial-enthalpy-certification-and-anchor-repair) + - [Gravity-only curvature correction](#gravity-only-curvature-correction) + - [Gravity elimination](#gravity-elimination) + - [Production outcomes of the three strategies](#production-outcomes-of-the-three-strategies) +- [Preconditioner improvements measured at equal accuracy](#preconditioner-improvements-measured-at-equal-accuracy) + - [Baseline approximation and work accounting](#baseline-approximation-and-work-accounting) + - [Prepared cross-action optimization](#prepared-cross-action-optimization) + - [Mass-consistent gravity Schur refinement](#mass-consistent-gravity-schur-refinement) + - [Nonlinear transfer limited the recommendation](#nonlinear-transfer-limited-the-recommendation) +- [Why better linear solves did not remove the nonlinear stall](#why-better-linear-solves-did-not-remove-the-nonlinear-stall) + - [Tenfold tighter forcing did not improve the useful step](#tenfold-tighter-forcing-did-not-improve-the-useful-step) + - [A large compensated direction amplified moderate curvature](#a-large-compensated-direction-amplified-moderate-curvature) +- [Fully coupled second-order trial corrections](#fully-coupled-second-order-trial-corrections) + - [Bounded correction against the complete residual](#bounded-correction-against-the-complete-residual) + - [Damping and work-aware eligibility](#damping-and-work-aware-eligibility) + - [Matched six-update live test](#matched-six-update-live-test) +- [Shape freedom, field transport, and the displacement-order control](#shape-freedom-field-transport-and-the-displacement-order-control) + - [The isolated $Q_4$ displacement experiment](#the-isolated-q_4-displacement-experiment) + - [Two structural cautions](#two-structural-cautions) +- [The bounded smooth-surface study](#the-bounded-smooth-surface-study) + - [Frozen late-state result](#frozen-late-state-result) + - [Three cold reduced updates from the analytic seed](#three-cold-reduced-updates-from-the-analytic-seed) +- [Native spherical-harmonic surface prescription](#sec:recent-harmonics) + - [Architecture and normalization](#architecture-and-normalization) + - [Native coupled results](#native-coupled-results) +- [Negative density: a located discrete representation defect](#sec:recent-negative) + - [What the weak EOS equation determines](#what-the-weak-eos-equation-determines) + - [Spatial location and root-based integration](#spatial-location-and-root-based-integration) + - [Mechanism controls and their limits](#mechanism-controls-and-their-limits) +- [Independent density-order study and logarithmic assessment](#sec:recent-density-order) + - [Frozen approximation results](#frozen-approximation-results) + - [Coupled controls and the stalled $n=3$ trajectories](#coupled-controls-and-the-stalled-n3-trajectories) + - [What the logarithmic assessment established](#what-the-logarithmic-assessment-established) +- [Detailed coupled-convergence campaign](#sec:recent-coupled) + - [Initial nonlinear screen](#initial-nonlinear-screen) + - [Matched gravity-curvature trial-path control](#matched-gravity-curvature-trial-path-control) + - [Frozen inverse comparisons: rate, attained accuracy and total cost](#frozen-inverse-comparisons-rate-attained-accuracy-and-total-cost) + - [Weak coupled radial response](#weak-coupled-radial-response) +- [A physically compatible rotating $n=3$ family](#sec:recent-n3-family) + - [Matched live field-feedback result](#matched-live-field-feedback-result) +- [Virial consistency: pressure and gravity must be distinguished](#sec:recent-virial) + - [Pressure-moment identity and the density test space](#pressure-moment-identity-and-the-density-test-space) +- [Physical two-dimensional model slices](#sec:recent-slices) + - [Display-refinement measurements](#display-refinement-measurements) +- [What to carry forward and what remains unresolved](#sec:recent-next) + - [Recent implementation verification and provenance](#recent-implementation-verification-and-provenance) +- [Consolidated source, architecture, and handoff](#sec:consolidated-source) + - [What the consolidation changes](#what-the-consolidation-changes) + - [Architectural boundary between physics and numerics](#architectural-boundary-between-physics-and-numerics) + - [Fields, geometry, and the preferred surface representation](#fields-geometry-and-the-preferred-surface-representation) + - [Defaults, explicit controls, and experimental candidates](#defaults-explicit-controls-and-experimental-candidates) + - [Branch disposition and retained provenance](#branch-disposition-and-retained-provenance) + - [Adoption, retention, and exclusion rationale](#adoption-retention-and-exclusion-rationale) + - [A practical contract map for the rewrite](#sec:rewrite-contracts) +- [Evidence organization and reproducibility after consolidation](#evidence-organization-and-reproducibility-after-consolidation) + - [Rebuilding and reproducing results](#rebuilding-and-reproducing-results) +- [Consolidation validation on 20 September](#consolidation-validation-on-20-september) + - [Fresh physical smoke model](#fresh-physical-smoke-model) + - [Artifact and document checks](#artifact-and-document-checks) +- [Versioned study-report catalog](#versioned-study-report-catalog) + +
+ +Experimental three-dimensional FEM stellar equilibria +Independent density; coupled free surface; polytropic EOS +**The moderately rotating $n=3$ and full virial targets remain open.** +This document separates measured results from hypotheses, and retained research tools from promoted defaults. + +
+ +# Purpose, scope and current conclusion + +MeanField is an experimental three-dimensional finite-element laboratory for steady rotating stellar equilibria. Its immediate purpose is to establish reliable numerical choices for a later implementation, rather than to present a finished stellar-evolution or production equilibrium code. The user is independently rewriting the implementation in order to own and understand the eventual shipped software. The useful output of this investigation is therefore the formulation, verified contracts, measured successes and failures, and reproducible evidence that can guide that rewrite. + +This report consolidates the work recorded from 16 through 19 September 2026 and the source/evidence integration performed on 20 September. Earlier optimization and validation records are preserved as historical sources. The conversation is not treated as an experimental log: quantitative claims below come from committed reports, campaign manifests, saved vectors, diagnostic tables and recorded checks. The initial approximately tenfold performance gain and approximately 20-second Newton step for the nonrotating $n=1$ model were the user’s starting observation. They are not remeasured as a new speedup in this consolidation. At the outset, limited testing suggested plausible $n=1$ models at very low rotation, described by the user as roughly 1.5 percent of critical, while increasing index or rotation produced residual plateaus. That rough starting description is not a calibrated critical-rotation measurement for the later campaign inputs. + +## Physical and software objectives + +The immediate physical target is a moderately rotating $n=3$ polytrope, with an eventual comparison against ESTER. The operational rotation goal is at least 50 percent of critical rotation. The more distant ambition of exceeding 99 percent of critical rotation is outside the demonstrated regime and would require renewed attention to surface regularity and the vanishing effective gravity near mass shedding. No value of the dimensionless angular momentum used in these campaigns has been calibrated to a critical-rotation fraction. Neither a visually oblate model nor a small $J$ proves a particular fraction of critical rotation. + +The key global milestone is a relative virial error no larger than $10^{-5}$. The original July project README described an aspirational $10^{-6}$ target; the research acceptance target in this report is the subsequently specified $10^{-5}$. Internal pointwise correctness remains necessary. Exterior accuracy is required insofar as it supports the correct interior and boundary response; an independently accurate external field everywhere is not the immediate scientific objective. Nevertheless, the exterior formulation can influence interior gravity and cannot be disregarded merely because its own residual is small. + +The software objectives are equally explicit: retain a physics-oriented user API, express algorithm and field compatibility through types where practical, use runtime checks for mesh-dependent and state-dependent conditions, preserve independent density, and solve surface coefficients and interior fields in the same coupled system. Central density is an optional physical specification; it must not be silently imposed on every polytrope. Log density and new positivity enforcement are deferred. Existing optional feasibility experiments remain available for reproduction but are not enabled by consolidation. + +## What has actually been demonstrated + +There is substantial progress, but the central rotating $n=3$ goal is open. The strongest established results are: + +1. The native all-$m$, degree-four real spherical-harmonic surface prescription is integrated and tested against the corresponding nodal geometry, physical residuals and Jacobian actions. It reduces the surface to 25 independent coefficients without decoupling it from the interior solve. + +2. Some low-index, slowly rotating native-harmonic cases meet both their algebraic stopping criterion and the global virial target. Higher-index representation errors and unresolved pointwise checks prevent generalizing this to every algebraically converged case. + +3. At the same frozen state, gravity-Schur coupling reduced the measured total linear cost by 41.5 percent for $n=3,J=0$ and by 30.6 percent for $n=1.5,J=0.1$, at the stated true-defect gate. These are single matched comparisons, including setup, rather than statistical timing estimates. + +4. In the compatible rotating $n=3,J=0.035,K/K_0=0.99$ family, stellar LDU plus gravity-Schur reached a 12.687-times lower residual after ten updates than gravity-Schur alone, at 14.593 percent greater Newton time. This is an equal-update endpoint comparison, not a speedup to the same converged model. + +5. The best rotating $n=3$ endpoint still fails the full virial target: the enthalpy-pressure energy error is $1.81141\times10^{-5}$, and the density-pressure energy error is $3.51558\times10^{-5}$. Passing one force virial is insufficient. + +6. Physical-plane slices now show independent density and enthalpy on the actual deformed stellar geometry, with signed values and the undeformed outline available for comparison. Export refinement has been checked for selected $n=2$ and rotating $n=1.5$ states; it is not a mesh-convergence proof. + +Several attractive paths did not justify promotion: simple one-way material-to-gravity completion, increasing displacement order alone, expensive coarse surface/scalar corrections, and universal use of second-order trial corrections. Some improve an isolated metric while increasing total work or worsening the accepted nonlinear trajectory. Those negative results are preserved because they constrain the next design, not because the algorithms are impossible to improve. + +## How to use this record + +The report first reconstructs the baseline and sequence of hypotheses, then describes the current implementation, numerical evidence and interpretation. Later sections distinguish active defaults from selectable experimental policies and record the consolidation checks. The source reports under `experiments/` provide the most direct detailed companion records. `reports/archive/` contains branch and relocation inventories. `working/research/diagnostics/` contains the preserved raw campaigns. The large raw archive is local and intentionally not committed to Git; a source-only clone will not contain its saved models or copied executables. Versioned figure copies and this report remain usable without that archive. + +# Definitions and evidence discipline + +## Physical variables and virial diagnostics + +For the polytropic experiments, pressure obeys $$P(\rho)=K\rho^{1+1/n},\qquad + h(\rho)=(n+1)K\rho^{1/n},\qquad + \rho_\star(h)=\left(\frac{h}{(n+1)K}\right)^n + \label{eq:report-polytrope}$$ on their physical nonnegative domains. The discrete density and enthalpy are independent finite-element fields; the weak closure equation does not make them pointwise equal. Consequently an integral formed from $P(\rho_h)$ need not agree with one formed from $P(h_h)$ at the same discrete state. Undefined fractional powers of negative density must be reported as undefined, not silently replaced with a positive or clipped value. + +For a zero-pressure stellar boundary, define $$T=\frac12\int_{\mathcal{D}_\star}\rho \lvert \boldsymbol{v}\rvert^2\,dV, + \qquad \Pi=\int_{\mathcal{D}_\star}P\,dV, + \qquad W_\Phi=\frac12\int_{\mathcal{D}_\star}\rho\Phi\,dV.$$ With the usual negative gravitational potential, $W_\Phi<0$. The force binding integral $W_f$ uses the corresponding gravitational force moment and agrees with $W_\Phi$ for a consistent isolated equilibrium. The campaign observers report energy and force virials using the common normalization $\lvert W_\Phi\rvert$: $$v_\Phi=\frac{2T+3\Pi+W_\Phi}{\lvert W_\Phi\rvert},\qquad + v_f=\frac{2T+3\Pi+W_f}{\lvert W_\Phi\rvert}. + \label{eq:report-virials}$$ Tables generally show their absolute values unless signs are explicitly given. A nonzero external pressure requires the boundary-pressure term; the stated campaign comparisons use the zero-pressure surface convention. Both pressure reconstructions and both gravity forms are retained where they are defined. Agreement at quadrature orders 14 and 18 is an integration consistency check on a fixed discrete state, not evidence of spatial convergence or exact pointwise physics. + +The common campaign scale is $M=R_{\rm reference}=G=1$ unless a manifest states otherwise. Angular momentum $J$, angular velocity, $T/\lvert W\rvert$, radius and critical-rotation fraction are different quantities. The physical-family compatibility study later explains why this distinction is particularly important at $n=3$. + +## Algebraic quantities and cost accounting + +Let $x$ denote the complete normalized coupled coordinate vector and $F(x)$ its normalized residual. Write $A=DF(x)$ for its Jacobian; this notation avoids confusing the Jacobian with the stellar angular momentum $J$. A Newton correction $p$ is checked through the fresh full-system defect $$\eta_{\rm true}=\frac{\lVert F(x)+Ap\rVert}{\lVert F(x)\rVert}. + \label{eq:report-linear-defect}$$ An internal Krylov recurrence or a reduced-system residual cannot substitute for this full-system endpoint test. The nonlinear merit function is $\mathcal{M}(x)=\tfrac12\lVert F(x)\rVert^2$; its directional derivative is $F(x)^T A p$. Linear success and a negative initial merit derivative do not guarantee acceptance of a finite step on a curved coupled path. + +The state and residual norms are produced by the configured physical Riesz normalization. They should not be compared across a change of basis, field order or normalization as though they were the same physical norm. The experiments therefore retain physical residual checks and integrals in addition to normalized algebraic quantities. A block’s fraction of $\lVert F\rVert^2$ is an algebraic attribution; it is not an elementwise PDE error estimate. + +Frozen comparisons hold the complete base state, Jacobian, right-hand side, mesh and coordinates fixed. Their relevant linear cost includes construction, binding, setup, solve and a fresh endpoint verification. Live comparisons follow evolving nonlinear states and include preparation, accepted/rejected trials and auxiliary solves in the recorded Newton time. Process wall time can additionally include observers and export overhead. Setup amortization must therefore be demonstrated on an actual repeated-use path rather than assumed from a smaller outer iteration count. + +## Controls, provenance and limits of inference + +The adopted methodology used an isolated branch for each hypothesis or a clearly identified cumulative integration branch. A source revision and configuration were recorded before numerical work; campaign runners copied the executable, recorded its digest and retained the exact commands, mesh digest, saved state and block diagnostics. Builds, FEM runs and timing trials were serialized within the later controlled studies. Read-only analysis and source review could proceed independently. Single-run timings are reported as observations; no confidence interval or cross-machine performance claim is inferred from them. + +The pressure/shape investigations separated preparation feasibility, linear accuracy, initial descent and finite-step curvature. Derivative probes used directional or centered differences over a stated step grid and compared full versus component actions. Later snapshot loaders check dimensions, mesh, coordinate conventions and provenance rather than accepting a vector solely because its length matches. Unconverged restarts require an explicit diagnostic override. Saved-state initialization prepares and refreshes before committing, clears stale warm starts and rolls back failed initialization. + +The evidence checkers reject corrupt or inconsistent fixtures as well as accepting valid ones. Passing an artifact audit establishes consistency of the recorded experiment under that checker’s contract. It does not establish that the nonlinear model converged, that all interior samples are physical, or that a target virial was attained. Similarly, passing code assertions is not equivalent to obtaining a validated stellar model. + +All finite iteration budgets and diagnostic ceilings remain explicit. A late small-step trajectory can be described as a stall observed over the available window. It cannot be promoted to a mathematically established convergence floor. Dense quadrature samples can locate negativity but cannot certify its absence everywhere; interval subdivision and polynomial bounds have their own scope and unresolved cases. Conclusions in this report retain those distinctions. + +# Starting point, problem definition, and architecture + +The work began on 16 September 2026 with an already substantial experimental finite-element code. Its purpose was to discover a reliable numerical route to three-dimensional steady stellar equilibria before a later, independently written general-EOS implementation. The immediate physical model was a self-gravitating, rigidly rotating barotrope with a free stellar surface. The presence of older transport, Coriolis, viscosity, and continuity components did not mean that those equations participated in the active equilibrium solve. This distinction remains important when comparing the experimental code with the broader physics ultimately desired from a stellar structure application. + +The incoming Git HEAD was `75cc63873992984966ea20011cc04d9984f54be4`. It did not uniquely describe the audited source: 49 tracked files already differed, with 3,626 additions and 267 deletions, and additional untracked experiments and results were present. The later baseline commit `80cc3d8` on `research/baseline-20260916` preserved that optimized source. The initial architecture audit was read-only. The subsequent diagnosis used fresh Release builds rather than assuming that existing executables represented the modified source. This provenance distinction is essential: historical binaries and historical passing tests are evidence about their own source, not automatically about the current checkout. + +The user reported approximately a tenfold reduction in the cost of an ordinary nonrotating $n=1$ Newton step, to about twenty seconds. Fresh measurements found 22–24 seconds for the first three accepted steps and were consistent with that operating point. The old pre-optimization source was not rerun in a matched comparison; the full historical factor of ten was therefore not independently remeasured. The inherited 9 September report had separately measured factors of 1.53 in whole-process time, 1.70 in Newton time, and 2.02 in linear-solve time for its particular batch. Later tensor, cache, and compact-action changes had different baselines. These gains must not be multiplied together. + +## Continuum equations and independent discrete fields + +For a polytrope of index $n\geq1$ and constant $K>0$, $$\begin{aligned} + P&=K\rho^{1+1/n}, & + h&=(n+1)K\rho^{1/n}, & + \rho_{\mathrm{EOS}}(h)&=\left[\frac{h}{(n+1)K}\right]^n, + \qquad h\geq0. +\end{aligned}$$ With positive centrifugal potential $\Psi=\lvert\boldsymbol\Omega\mathbin{\times}\boldsymbol x\rvert^2/2$, the Bernoulli imbalance is $$B=h+\Phi-\Psi-C.$$ The gravity flux represents $\boldsymbol g=\nabla\Phi$, so physical gravitational acceleration is $-\boldsymbol g$. The active system combines a mixed gradient–potential gravity formulation, a weak density closure, weak Bernoulli balance, reduced mechanical virtual work, and integral constraints for total mass and axial angular momentum. Density, enthalpy, potential, and the gravity flux are independent finite-element fields. In particular, the mechanical pressure is evaluated from enthalpy while gravity and centrifugal forces use independent density. Continuum equivalence of these quantities does not force their pointwise equality after discretization. + +The surface condition replaces the relevant enthalpy test rows by the enthalpy value corresponding to prescribed surface pressure. At zero surface pressure this is an $h=0$ trace condition. The mechanical residual is first formed against volume displacements and then pulled back by the transpose of the surface-to-volume deformation derivative. If $q$ denotes independent surface parameters and $D(q)$ the generated displacement, its schematic form is $$R_s(q,u)=D'(q)^T R_{\mathrm{mech}}(D(q),u).$$ The Jacobian includes both the derivative of the mechanical force and the derivative of this pullback. A zero $R_s$ proves equilibrium only against the chosen extension-generated mechanical tests; it does not set every unrestricted volume mechanical row to zero. + +The usual initial example additionally imposed a central-density specification. Its actual row was a central enthalpy point condition, $$R_c=h(0)-h(\rho_{c,\mathrm{target}}),\qquad + R_h\longleftarrow R_h+\lambda_c e_c.$$ It was not a direct point condition on the independent discontinuous density. This distinction motivated measuring the unbordered residual and artificial source amplitude. The user clarified that the central condition should remain optional, principally useful for the $n=3$ homology, and should not be automatically inserted at other indices or hidden in a physics-facing API. + +## Discretization and model compilation + +The common mesh had 1,216 hexahedra: 832 stellar and 384 vacuum elements. Its SHA-256 was `ce11ae99e21e6c3bbfbee402acd2e191c1da0d8261d2227b4203f73f2f337a74`. Reference geometry was order four. The ordinary spaces were RT index two for gravity, discontinuous $Q_2$ for potential and stellar density, and continuous $Q_3$ for enthalpy and displacement. The original scalar nodal radial surface had 866 parameters. With its three scalar constraints the root had 178,074 unknowns; removing the central specification removed one scalar, yielding 178,073. Mesh geometry order and the approximation order of an individual physical field were distinct controls. + +| Initial root block | Number of unknowns | +|:-----------------------------|:-------------------| +| Gravity gradient | 98,928 | +| Potential | 32,832 | +| Stellar density | 22,464 | +| Stellar enthalpy | 22,981 | +| Nodal radial surface | 866 | +| Generated scalar coordinates | 3 | + +The architecture was already strongly compositional. A compile-time model specification pack generated the block layout, scalar coordinates, Jacobian couplings, normalization metadata, and prepared specification runtimes. The active aggregate was the variadic prepared equilibrium operator. C++23 module interfaces under `libmeanfield/interface` were separated from implementations under `libmeanfield/impl`. MFEM/MPI spaces used explicit maps between supported solver coordinates and full true degrees of freedom. The nonlinear context owned accepted and trial states, normalization, inverse preparation, and linear backend; refresh, stale-view rejection, rollback, and observer lifetime were explicit contracts rather than informal conventions. + +Prepared matrix-free actions cached reference basis tables, geometry, thermodynamics, and local contractions. Their dependency identities and revisions controlled invalidation. The numerical work therefore emphasized reusing established ownership and preparation interfaces rather than adding ad hoc mutable solver state. Static choices such as block identities, compatible policy combinations, and registered capabilities were checked at compile time. Mesh sizes, rank agreement, actual FE eligibility, values, normalization factors, and preparation generations necessarily required runtime validation. + +The geometry map initially used nodal radial surface variation, a logical radius-squared interior extension, and a fixed-infinity vacuum extension. Kelvin compactification represented the exterior through infinity. The root geometry unknown was the surface, not the much larger volume displacement field. Surface parameterization, interpolation of displacement, extension, and the physical map were therefore distinct architectural responsibilities. + +The seed initialized density and enthalpy independently by finite-element coefficient interpolation of a spherical Lane–Emden profile, using MFEM’s `ProjectCoefficient`, and solved mixed gravity separately. This initialization was distinct from solving the weak closure’s weighted $L^2$ projection relation; the method name must not be read as implying such a mass-matrix solve. It set shape to zero, initialized $C$ from $-GM/R$, and initialized $\Omega$ from the target $J$ divided by spherical inertia. A seed with nonzero target $J$ was not an already rotating equilibrium. Throughout the early controlled campaign, $M=R_{\mathrm{reference}}=G=1$; at $n=1$, $K=2/\pi$ and $\rho_c=\pi/4$. Other indices used the production Lane–Emden integrator with step $10^{-3}$ to calibrate the seed. That was a consistent seed reference, not an independent high-precision continuum solution. + +## Physical acceptance and measurement conventions + +The required milestone was virial balance within $10^{-5}$, together with credible interior fields. Exterior pointwise accuracy was not an independent goal; exterior errors mattered through their effect on stellar equilibrium. The campaign measured both gravitational energies $$W_E=\frac12\int_\star\rho\Phi\,dV, + \qquad + W_F=-\int_\star\rho\,\boldsymbol x\cdot\nabla\Phi\,dV,$$ and, for the zero-pressure surface, $$\epsilon_V(W,P)= + \frac{\left\lvert 2T+W+3\int_\star P\,dV\right\rvert}{\lvert W_E\rvert}.$$ Pressure from $h$ and pressure from $\rho$ were evaluated separately, at independent diagnostic quadrature orders 14 and 18. Signed density was retained. When the EOS could not define pressure at a negative density sample, the full density-pressure diagnostic was reported undefined, rather than manufactured by clipping. Quadrature agreement tested integration sensitivity; it did not constitute spatial convergence or global positivity certification. + +The first physical checker required recorded nonlinear convergence and all eight finite virial measurements to satisfy $10^{-5}$. This conservative combined screen was stricter than a statement that a particular enthalpy-based virial passed. Neither passing the checker nor a small scalar virial alone was a pointwise continuum-accuracy certificate. Useful accompanying observations were mass and angular-momentum errors, Bernoulli variation, EOS mismatch, gravity consistency, geometry minima, density sign, and surface shape. No early value of $J$ or reference-unit $\Omega$ was calibrated as a fraction of critical rotation. + +The inherited saved $n=1,J=0$ benchmark already provided a useful physical reference beyond the scalar residual: density, enthalpy, interior potential, and gravity-gradient relative volume $L^2$ errors were approximately $4.70\,10^{-4}$, $3.27\,10^{-4}$, $1.26\,10^{-4}$, and $5.98\,10^{-4}$; surface-radius RMS error divided by reference radius was $2.75\,10^{-4}$. Its density-pressure energy virial was $3.10\,10^{-7}$ and mass error $2.36\,10^{-11}$. These were the particular saved coarse model’s errors against its radial reference, not accuracy estimates for later rotating or higher-index cases. A one-level refined mesh and analytic geometry controls existed, but the corresponding directory did not contain completed refined equilibrium fields or an equilibrium convergence result. The audit did not infer completed mesh convergence from the presence of that directory. + +The historical full-suite XML contained 592 cases, 588 passing and four failing: centrifugal-virial refinement geometry, an $n=3$ projected-seed closure screen, and two- and four-rank MPI groups. Those failures had been reproduced in earlier preserved binaries. They were neither silently removed nor treated as fresh validation of the incoming modified source. At the initial audit, twelve production source files were newer than the available release sandbox binary and thirty newer than the test binary. This motivated the subsequent explicit rebuilds and targeted regression ledgers. A previous between-sample mesh fold was also documented, but the later saved $n=1$ mesh had healthier sampled geometry; it was not assumed to suffer the same demonstrated folding mechanism. + +#### Evidence. + +The original workspace reports were `reports/sources/original_audits/MeanField_audit_2026-09-16.md`, `reports/sources/original_audits/MeanField_solver_diagnosis_2026-09-16.md`, and `reports/sources/original_audits/formulation_audit_notes.md`. Their archival relocation is recorded in the consolidation manifest. Repository evidence includes `experiments/POLYTROPE_VALIDATION_FINDINGS_2026-09-09.md`, `experiments/POLYTROPE_H_CONVERGENCE_2026-09-09.md`, and `experiments/NUMERICAL_CAMPAIGN_2026-09-16.md`. + +# Initial diagnosis: several different plateaus + +## Fresh control measurements + +The diagnostic rebuilt the incoming optimized source in Release using Homebrew Clang 22.1.8, with IPO enabled, profiling disabled, and uniform field order increment zero. Cases ran serially on one MPI rank. Initial live controls used FGMRES restart 40, at most 200 linear iterations, relative forcing tolerance $\eta=0.03$, nonlinear relative tolerance $10^{-8}$ and zero absolute tolerance. The nonlinear threshold was relative to the initial residual, not an absolute $10^{-8}$ requirement. + +| Model | Accepted updates | Retained $\lVert F\rVert$ | $P(h)$ energy/force virials | +|:---|:---|:---|:---| +| $n=1,J=0$ | 3 full | $6.6053\,10^{-9}$ | $2.616\,10^{-7}/3.246\,10^{-7}$ | +| $n=1,J=0.01$ | 4 full | $5.0898\,10^{-10}$ | $2.610\,10^{-7}/3.246\,10^{-7}$ | +| $n=1,J=0.05$ | 5 full | $6.0406\,10^{-9}$ | $3.016\,10^{-7}/3.188\,10^{-7}$ | +| $n=1.5,J=0$ | $1,1/16$ | $1.6976\,10^{-4}$ | $2.245\,10^{-6}/2.318\,10^{-6}$ | +| $n=3,J=0$ | 0 | $3.1006\,10^{-3}$ | $5.326\,10^{-5}/8.341\,10^{-5}$ | + +All these controls stopped at a failed linear gate. The $n=3$ measurement was the unchanged seed. At $n=1,J=0$ the strict target was approximately $1.81\,10^{-12}$, whereas an earlier validation driver accepted the same three-step state under an absolute $10^{-8}$ threshold. These descriptions were consistent; they did not imply that a previously converged model had become physically invalid. Conversely, the $n=1$ seed itself already passed the virial threshold, directly demonstrating why virial alone was not a complete equilibrium test. + +The initial audit separated three mechanisms: loss of useful Krylov information late in otherwise accurate $n=1$ solves; stronger coupled linear difficulty at higher index; and severe nonlinear curvature or admissibility restrictions after a usable linear direction was obtained. It did not find evidence for one universal numerical floor. + +## Normalization audit + +The state and residual registrations supplied their own physical scales and reference Gram diagonals. Writing these as $s_x,d_{x,i}$ for a value block and $s_F,d_{F,i}$ for its residual/test block, ordinary coefficient factors were $$S_{ii}=\frac{\sqrt{d_{x,i}}}{s_x},\qquad + L_{ii}=\frac{1}{s_F\sqrt{d_{F,i}}}.$$ The Gram factors form paired primal and dual maps where the registered function-space topologies correspond, but the physical scales need not agree. For example, surface parameters use a length scale while mechanical shape residuals use a force scale; potential uses specific energy while its Poisson residual uses inverse time squared. Point replacement rows and scalar blocks have their declared special treatment. These distinctions are explicit in `libmeanfield/interface/normalization/physical_riesz.cppm`. + +The reference factors were frozen through a solve. Their characteristic scales came from $M,R_{\mathrm{reference}},G$ and did not divide by the requested angular momentum $J$; zero rotation was therefore not a normalization singularity. + +Let $\widehat x=Sx$ and $\widehat F=LF$. The consistent transformations were $$\widehat A=LAS^{-1},\qquad + \widehat P^{-1}=SP^{-1}L^{-1},\qquad + \widehat A\widehat P^{-1}=LAP^{-1}L^{-1}.$$ Consequently, changing only state scales cancels from a consistently right-preconditioned system. Direct probes multiplying block state scales by $10^{-3}$ through $10^3$ preserved the physical correction and preconditioned action to approximately $10^{-16}$. Wrapper/explicit actions and coordinate round trips agreed at roundoff. This ruled against a basic scaling-wrapper implementation error at the inspected states. It did not show that the chosen residual norm was optimal: changing $L$ changes what both Krylov and line search minimize. + +The same $M=R=G=1$ factors were used across indices despite changing stellar concentration. Large factor ranges alone did not diagnose a bug. Similarly, small normalized border matrices had raw condition numbers $2.78\,10^7$ and $9.69\,10^7$, dominated by a central-column scale near $7.3\,10^6$; diagnostic equilibration reduced these to about 1.46 and 2.39. Those were border-matrix observations, not the condition number of the full stellar Jacobian or proof of a singular physical system. + +## Central condition: a useful example correction + +At $n=1,J=0.05$, matched restart-40 runs with and without the central condition had identical first five accepted step lengths and linear iteration counts. Both then failed the sixth linear gate near relative defect 0.505. Removing the condition nevertheless markedly improved localized physical consistency: + +| Quadrature-18 measurement | With central condition | Without | +|:---|:---|:---| +| Peak absolute EOS density mismatch | $7.30451\,10^{-3}$ | $3.32568\,10^{-4}$ | +| Peak mismatch / sampled density scale | $9.33951\,10^{-3}$ | $4.30594\,10^{-4}$ | +| Bernoulli sampled range | $1.45751\,10^{-2}$ | $1.30398\,10^{-3}$ | +| Unbordered residual | $4.54227\,10^{-6}$ | $6.03609\,10^{-9}$ | +| $P(h)$ energy virial | $3.01617\,10^{-7}$ | $3.06913\,10^{-7}$ | + +The formerly fixed central enthalpy changed from 1 to 0.983383508. Mass, angular momentum, angular velocity, and volume RMS errors changed little. The result supported removing that artificial restriction from the $n=1$ example, while leaving explicit central constraints available. It did not establish that central constraints caused the general plateau. + +## Krylov window and warm-guess safeguard + +Frozen cold comparisons retained the same state, full Jacobian, physical right-hand side, and inverse. Increasing only the restart window from 40 to 160 gave: + +| Frozen state | Window 40: iterations / defect | Window 160: iterations / defect | +|:---|:---|:---| +| $n=1,J=0$ endpoint | $200/0.04442$ | $72/0.02914$ | +| $n=1,J=0.01$ endpoint | $200/0.35468$ | $125/0.02519$ | +| $n=1.5,J=0$ endpoint | $200/0.06491$ | $70/0.02694$ | +| $n=3,J=0$ seed | $200/0.05631$ | $200/0.04026$ | + +For the rotating $n=1$ state, histories coincided through iteration 40 and separated immediately after restart. Independently recomputed full residuals agreed with reported residuals. This established actual loss of useful Krylov information, not false convergence reporting. In a live unbordered $n=1,J=0.05$ comparison, restart 160 reached strict convergence in seven full steps with $F=3.38036\,10^{-11}$, versus five steps and $6.03609\,10^{-9}$ before linear failure with restart 40. Total linear iterations were 292 versus 370; complete diagnostic-process times were 271.95 versus 290.80 seconds. All eight virial checks were below $3.735\,10^{-7}$. These were single timing observations with increased Krylov storage, not a universal speedup claim. + +The solver reused $(1-\alpha)p$ after a damped update as the next initial linear guess. This was vector reuse, not Krylov recycling. At one $n=1.5$ state that guess had a true defect larger than the zero guess. A small backend safeguard now checks the current true defect of a supplied nonzero guess and discards it only if the norm is nonfinite or exceeds $\lVert b\rVert$. Nonfinite input coefficients remain an input-contract error. The guard adds one action for a nonzero guess and none for a zero guess, without a new public option. + +The matched $n=1$ output was bitwise unchanged. At $n=1.5$, the first shared guarded solve reduced the effective initial defect/RHS from 1.5097 to 1 and iterations from 71 to 65. Over eight attempts the control had seven successful linear solves followed by failure; the guard passed all eight. Total iterations fell from 672 to 561, mostly by avoiding that failed 200-iteration attempt. The endpoint changed only from $1.75135\,10^{-4}$ to $1.74946\,10^{-4}$, a 0.108 percent improvement, with final steps $1/2048$ and $1/8192$. This was a modest inner-solver robustness improvement, not a cure for nonlinear stagnation. + +## Rejected one-way gravity completion + +The first coupling hypothesis changed the structure inverse from independent material/gravity blocks to material-then-gravity triangular completion. It retained the material inverse, order-five Chebyshev flux-mass inverse, and three AMG cycles, and rebuilt the scalar border consistently. At the same centrally constrained $n=3$ seed, the restart-160 defect changed from 0.0402590 to 0.0381781 after 200 iterations; both failed $\eta=0.03$. Time increased from 98.92 to 117.71 seconds, correction norm from 0.75764 to 1.54083, and the sampled thermodynamic step limit worsened from 0.15244 to 0.09135. The candidate’s $\alpha=0.1$ trial was inadmissible. This particular one-way completion was not promoted. + +The baseline restart-40 and restart-160 corrections had cosine $-0.985$ despite the same matrix and right-hand side. Their difference had a response bound $\lVert A(p_2-p_1)\rVert/\lVert p_2-p_1\rVert\leq2.77\,10^{-4}$. This established a weak measured direction, not a singular value or an identification of homology, translation, or a surface mode. Later studies therefore examined the actual coupled directions instead of adding a gauge on analogy alone. + +#### Evidence. + +`experiments/NUMERICAL_CAMPAIGN_2026-09-16.md`, `experiments/CENTRAL_CONSTRAINT_HYPOTHESIS_2026-09-16.md`, `experiments/KRYLOV_WINDOW_HYPOTHESIS_2026-09-16.md`, and `experiments/KRYLOV_HYPOTHESIS_2026-09-16.md`. The combined nonrotating $n=1$ case subsequently reached $F=1.65157\,10^{-13}$ in six full steps with worst virial $3.73092\,10^{-7}$. The transfer case $n=1,J=0.1$ passed all twelve linear gates but ended at $6.25389278\,10^{-3}$, with final $\alpha=4.76837\,10^{-7}$ and failing enthalpy-based virials $4.75186\,10^{-4}/8.18118\,10^{-4}$. That transfer failure motivated the next investigation. + +# Separating pressure preparation from shape-related curvature + +The focused 17 September campaign made no production numerical changes. It captured exact trial states, identified rejection stages, and split the same Newton correction into its actual shape block $p_s$ and all remaining fields and scalars $p_f$. Frozen replays restored the base before each action; probing a trial never committed it as an accepted model. + +## The rotating pressure obstruction was localized and real + +At $n=1,J=0.1$, exact replays of the first rejected candidates at zero-based iterations 2, 3, 4, 7, and 11 all returned the production code `thermodynamic_domain / pressure_force / outside_domain`. The pressure source takes this branch for finite negative enthalpy. Thus these were enthalpy-domain failures, not inverted geometry. The limiting pressure point remained at local stellar element 459, quadrature point 50; the closure’s different rule limited at point 36 of the same element. + +| Iteration | Pressure-point $h$ | $\delta h$ | Pressure bound | Accepted $\alpha$ | +|:---|:---|:---|:---|:---| +| 2 | $9.63674\,10^{-4}$ | $-1.31441\,10^{-2}$ | $7.33159\,10^{-2}$ | $1/16$ | +| 3 | $1.42165\,10^{-4}$ | $-2.11119\,10^{-3}$ | $6.73390\,10^{-2}$ | $1/16$ | +| 4 | $1.02159\,10^{-5}$ | $-1.89704\,10^{-3}$ | $5.38520\,10^{-3}$ | $1/256$ | +| 7 | $7.04574\,10^{-8}$ | $-1.86569\,10^{-3}$ | $3.77648\,10^{-5}$ | $1/32768$ | +| 11 | $1.07227\,10^{-9}$ | $-1.86493\,10^{-3}$ | $5.74963\,10^{-7}$ | $1/2097152$ | + +The pressure step budget fell by a factor of 127,514 while the late negative direction remained finite. The limiting element touched the stellar surface; the pressure point’s reference Cartesian radius was 0.997553, near the negative rotation pole, approximately 8.66 degrees from the negative $z$ axis at onset. It was an interior quadrature point, not a boundary row. Topological adjacency did not measure distance to the deformed surface. + +At onset the closure bound, 0.0727489, was slightly tighter than the pressure bound; pressure was evaluated first and therefore reported first when both failed. A deliberately intermediate trial at $\alpha=0.07303236911639208$ passed pressure preparation and failed closure, with predicted closure $h=-5.01979\,10^{-6}$ and limiting pressure $h=+3.72619\,10^{-6}$. This showed why a first rejection stage must not be confused with the tightest thermodynamic constraint. + +All 866 surface enthalpy coefficients and their directions were exactly zero at onset and late capture. Boundary leakage was excluded for these trials. Nevertheless, positive nodal values did not imply a positive represented polynomial. A step just beyond the onset bound retained nonnegative values at all 64 local $Q_3$ nodes while making interior pressure and closure samples negative. At the late state even a unit step retained nonnegative nodes but gave $h=-1.86493\,10^{-3}$ at the limiting pressure point. + +An independent reconstruction of the local tensor polynomial reproduced the nodes exactly and limiting-point values within $6.94\,10^{-18}$. A much denser grid found that the late accepted base already had negative enthalpy: a sampled minimum $-2.87114\,10^{-5}$ at reference coordinates $(0.0130467,0.515625,0.01)$. A separate 60-digit reconstruction agreed with the negative value at another selected interior point. These findings proved negative values existed in that element; the reported minimum was not a certified global minimum. Passing production quadrature preparation was thus not a pointwise positivity certificate. They also did not prove that every low-rotation or nonrotating converged control shared this defect. + +## Higher-index merit rejection had a different mechanism + +For any directional piece $p_d$, the diagnostic measured $$R_d(\alpha)=F(x+\alpha p_d)-F(x)-\alpha Ap_d, + \qquad R_{\mathrm{mixed}}=R_{\mathrm{full}}-R_s-R_f.$$ The mixed vector was formed before taking norms. All pieces used common step lengths and the original, unrenormalized block amplitudes. With $L=F+\alpha Ap$, the exact merit identity was $$\lVert F(x+\alpha p)\rVert^2= + \lVert L\rVert^2+2\langle L,R_{\mathrm{full}}\rangle + +\lVert R_{\mathrm{full}}\rVert^2.$$ + +At $n=1.5,J=0$, captured iteration 1, the raw full trial was admissible with minimum sampled Jacobian 0.896116. Its linear prediction was only 0.02576812 of the base norm; the actual residual was 12.243161 times the base norm. At $\alpha=0.1$ the actual ratio remained 1.02052 rather than the predicted 0.900070; at 0.01 it decreased to 0.991209. This established nonlinear merit failure separately from the rotating thermodynamic wall. + +Gravity-gradient actions from shape and fields had norms 0.115959 and 0.116055, cosine $-0.999999831$, and combined norm $1.16990\,10^{-4}$. Poisson actions also nearly cancelled, with cosine $-0.999996055$. Damping shape alone would have destroyed this measured first-order balance. The full-step second-order remainder was $2.18903\,10^{-3}$, with 99.7747 percent of its squared norm in gravity. Fixed-shape gravity is affine in its fields in the source; measured fields-only gravity remainders remained at arithmetic scale. Resolved remainders divided by $\alpha^2$ stabilized, consistent with the directional Jacobian rather than a persistent first-order derivative error. + +At the constrained $n=3$ seed the original correction still failed the $0.03$ forcing gate, attaining 0.0402590; it was not silently replaced by a better direction. At its common cap $\alpha=0.12194828$, the actual norm was 1.0237652 of the base versus prediction 0.87826014. Gravity accounted for 90.7812 percent of the squared remainder. At one tenth of this cap the trial decreased the residual, and the net curvature contribution was slightly favorable. A large mixed term therefore did not imply that its signed merit projection was always adverse. + +The consequence was to keep feasibility and nonlinear path curvature as separate hypotheses. A sampled enthalpy cap could avoid futile trials but would not remove a wall where $h\to0$ and $\delta h<0$. Nor could exterior weighting be justified from whole-gravity-block norms alone: the original probes had not yet localized stellar versus exterior contributions. + +#### Evidence. + +`experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md`, `experiments/PRESSURE_OBSTRUCTION_METHODOLOGY_2026-09-17.md`, and `experiments/POLYNOMIAL_ENTHALPY_ADMISSIBILITY_2026-09-17.md`. The eight focused and two supplemental cases restored their base residuals exactly; direction partitions and signed-merit identities passed independent checks. No additional virial-qualified model was produced by these probes. + +# Three optional numerical strategies and their limits + +The next implementation added three interchangeable numerical approaches: polynomial enthalpy feasibility, gravity curvature correction, and gravity elimination. They were selected by concrete typed policies through `NewtonNumerics`, not added to the physics-facing model declarations. Unregistered or incompatible policies failed compile-time constraints; actual mesh and FE eligibility were checked collectively at runtime. The ordinary Newton path remained available and defaults were unchanged. + +## Polynomial enthalpy certification and anchor repair + +The diagnostic converted the represented nodal polynomial to a tensor Bernstein representation with outward-rounded interval arithmetic. Adaptive subdivision returned a nonnegativity certificate, a negative-value witness, or unresolved bounds. A negative Bernstein coefficient by itself was not misclassified as a negative polynomial. Exact zero traces were retained. Certification used physical enthalpy coordinates, because general diagonal normalization does not preserve the polynomial represented by the coefficients. + +For an admissible base, a fixed certified initial field supplied an anchor: $$p_h(\beta)=(1-\beta)p_h+\beta(h_{\mathrm{anchor}}-h), + \qquad0\leq\beta\leq1.$$ Sufficient Bernstein inequalities constrained the repair fraction and the full linear residual model selected a fraction. The actual normalized-and-added candidate was certified again, and its complete merit derivative had to remain negative. This was a conservative feasible-direction family, not a constrained Newton optimizer, and it did not make the independent density positive. + +At the rotating onset the base certified and the original full trial was proved negative. Whole-mesh certification cost about 3–4 ms. The sufficient repair fraction was 0.9152726503; repaired polynomials prepared successfully, but trial lengths 1, 0.5, and 0.125 all increased the full residual. The linear model’s best step was near 0.00825, with only about 0.46 percent predicted residual reduction. The already negative late base correctly returned `invalid_base` and left accepted buffers unchanged. This path supplied useful diagnostic and admissibility machinery, but was not established as an efficient route through the nonlinear obstacle. The later user decision to defer positivity enforcement was consistent with this mixed result. + +## Gravity-only curvature correction + +Partition the normalized state into retained variables $u$ and gravity variables $z$, with gravity equations $G(u,z)$. At a raw trial $y=x+\alpha p$, the correction solved $$G_z(y)\,\delta z= + -\bigl[G(y)-G(x)-\alpha G'(x)p\bigr].$$ Only gravity coordinates changed. This removes the nonlinear gravity remainder, not the entire gravity residual. It preserves the base tangent because the right-hand side is second order along a smooth path, and it vanishes on an affine gravity path even when the base residual is nonzero. The candidate inverse was refreshed, every residual was reevaluated, and ordinary full-merit acceptance remained authoritative. + +The first frozen $n=1.5$ comparison changed the full trial residual from $2.18913190375\,10^{-3}$ to $1.40702031008\,10^{-4}$ at base norm $1.78804469158\,10^{-4}$. The gravity remainder fell from $2.18656200516\,10^{-3}$ to $1.37482284578\,10^{-12}$, with observed correction order 1.999898 on the smallest resolved steps. This was direct evidence that the identified curvature could be removed from gravity. + +An initial production attempt was intentionally stopped after repeated auxiliary prepared-consistency rejection despite converged inner solves. The arithmetic allowance had omitted large cancelling gravity contributions. The revised check included separately measured gradient, potential, source, and correction actions; it used $256\epsilon$ times their norm sum as a heuristic allowance, not a rigorous error bound. The true inner gate, lifted full-Jacobian gate, nonlinear tolerance, and physical criteria were not relaxed. The interrupted run and correction history were preserved. + +## Gravity elimination + +Gravity elimination instead completed gravity at the initial state and each retained trial, then used the reduced Schur action $$S=F_u-F_zG_z^{-1}G_u.$$ Right-hand-side condensation included the residual gravity defect, and lifting reconstructed a full correction that still had to satisfy the original full normalized linear gate. Tolerance-stopped inner Krylov solves only approximated this linear action; flexible outer preconditioning did not magically make it exact. Tight inner gates, finite-difference checks, and lifted verification were therefore retained. + +The reference implementation restricted the full operator and inverse. A capability-checked compact path reused the existing prepared physical gravity actions with the same scaling. On three dense actual-FEM directions compact and full-restricted outputs agreed exactly. Gravity operator applications cost about 0.018 versus 0.244 seconds, around fourteen times faster, while inverse applications changed only from approximately 0.205 to 0.160 seconds. A matched tight inner solve retained forty iterations and the same result while falling from 18.36 to 7.25 seconds. The fourteenfold action gain was not a fourteenfold Newton speedup. + +## Production outcomes of the three strategies + +| Model and path | Accepted updates | Final $\lVert F\rVert$ | Newton seconds / outcome | +|:---|:---|:---|:---| +| $n=1,J=0$, curvature | 6 | $6.51136\,10^{-14}$ | 273.71 / converged | +| $n=1,J=0$, eliminated | 6 | $8.98743\,10^{-14}$ | 869.67 / converged | +| $n=1.5,J=0$, curvature | 6 | $1.48794\,10^{-5}$ | 711.72 / outer gate failure | +| $n=1.5,J=0$, eliminated | 5 | $4.47888\,10^{-6}$ | 1821.79 / completed updates only | +| $n=1,J=0.1$, polynomial | 3 | $7.13059\,10^{-3}$ | 265.03 / outer gate failure | +| Polynomial plus curvature | 1 | $8.40387\,10^{-3}$ | 127.05 / outer gate failure | +| Polynomial plus elimination | 1 | $2.72634\,10^{-2}$ | 149.83 / non-descent repair | + +The two nonrotating $n=1$ gravity paths and polynomial control reached the same discrete model to coefficient differences below $2.632\,10^{-11}$. All eight virials were below $3.73092\,10^{-7}$; independent enthalpy certificates covered all 832 stellar elements. The eliminated control used 107 inner solves and 3,875 inner iterations, versus six solves and 115 iterations for curvature correction. It was a verified alternative, but substantially more expensive. + +The $n=1.5$ curvature run reduced the initial norm about eighteenfold but failed its seventh outer solve at relative defect 0.059017 after 200 iterations. All fourteen auxiliary solves passed. The eliminated run was externally interrupted at 2,400 seconds during its sixth attempt, after five accepted steps of $1,1/2,1/4,1/128,1/2048$. Its last accepted gravity residual was about $3.27\,10^{-14}$ while retained equations remained of order $10^{-6}$; the last step improved the total residual only 0.00565 percent. This showed stagnation despite completed gravity. No final snapshot or complete physical diagnostics existed for the timed-out run, so these were not invented from its log. + +Rotating runs remained both unconverged and virially unacceptable. The polynomial-only fourth linear attempt attained 0.0331634 rather than the required 0.03. Adding curvature improved the first step with the same tangent but the second linear solve failed at 0.0826534. Polynomial plus elimination passed its reduced and lifted linear gates before a repaired direction failed descent. A reporting replay later established that this failure occurred before another gravity relift, correcting an earlier ambiguous interpretation of counters. All these models retained negative independent density samples. + +The implementation was covered by 43 targeted serial cases and 1,886 assertions at the final revision, with 37 supported cases on each of two MPI ranks. Full stellar multi-rank operation was not thereby certified: logical radial surface interpolation and per-rank material ownership restrictions remained. The frozen checks established algebra and lifecycle invariants; they did not constitute physical convergence. + +#### Evidence. + +`experiments/NUMERICAL_STRATEGIES_2026-09-17.md`, `experiments/GRAVITY_SUBSYSTEM_IMPLEMENTATION_2026-09-17.md`, `experiments/GRAVITY_REDUCTION_IMPLEMENTATION_2026-09-17.md`, `experiments/GRAVITY_COMPACT_APPLICATIONS_2026-09-17.md`, and `experiments/GRAVITY_PREPARED_ROUNDOFF_2026-09-17.md` retain the complete policy contracts, intermediate failed builds/runs, and replay history. + +# Preconditioner improvements measured at equal accuracy + +## Baseline approximation and work accounting + +The original inverse first treated surface and material through a triangular surface-then-enthalpy-then-density approximation. Surface used a reference mass diagonal; enthalpy used hydrostatic mass and identity treatment at replaced boundary rows; density included the live EOS enthalpy derivative. Material-to-surface feedback was omitted. Gravity used approximate mixed LDU with a degree-five Chebyshev mass inverse and three fixed AMG cycles for a potential surrogate. Structure-level material and gravity were independent, then a small dense scalar Schur complement completed specification borders relative to that approximate inverse. The live high-$n$ EOS derivative was present; there was no evidence that the baseline simply omitted it. + +Frozen benchmark roles each constructed fresh inverses and border responses at the same saved state, full Jacobian, normalization, and right-hand side. Reported complete verified cost included inverse construction, normalized binding, Krylov setup, solve, and an independent full-Jacobian endpoint action. Failed solves were retained but never used as finite baseline time-to-convergence measurements. Timing runs and large audits were serialized. + +## Prepared cross-action optimization + +Existing stellar LDU was initially unexpectedly expensive: at an $n=1.5$ plateau, 200 iterations cost 599.5473 seconds versus 87.9302 for the independent baseline, and both failed the gate. The gravity-to-material coupling used a legacy geometry-evaluation path. Replacing it with the already prepared complete gravity-force action, with zero density and displacement variations, preserved the same mathematical gravity-gradient column and subsequent deformation adjoint. + +The cached LDU reduced complete cost to 117.4845 seconds, a factor 5.1032 relative to the old LDU path, and mean inverse application time by 8.0425. The correction changed only $2.7051\,10^{-11}$ relatively and the gate still failed at 0.0600669. This was an execution optimization, not a conditioning improvement. It deserved retention independently of whether LDU was selected as a numerical policy. + +## Mass-consistent gravity Schur refinement + +Writing the mixed gravity block as $$D=\begin{bmatrix}M&B^T\\B&0\end{bmatrix},\qquad + D\begin{bmatrix}g\\\Phi\end{bmatrix}=\begin{bmatrix}f\\r_g\end{bmatrix},$$ let $R$ be the same approximate mass inverse used for flux recovery and $Q$ the existing AMG inverse on the diagonal-mass surrogate. The added policy used $$\begin{aligned} + S_R&=BRB^T, & r&=BRf-r_g, & \Phi_0&=Qr,\\ + \Phi_{j+1}&=\Phi_j+\omega Q(r-S_R\Phi_j), & + g&=Rf-RB^T\Phi_k. +\end{aligned}$$ The Poisson defect was thereby measured against the same approximate mass action as recovery. This was still not an exact inverse of the mixed block. Fixed work preserved stationarity only when the child actions themselves declared stationary linear behavior. Positive relaxation alone did not prove contraction. Each extra sweep cost one application of $R,Q,B,B^T$; zero sweeps retained the old arithmetic path for bitwise baseline comparison. + +One sweep with $\omega=0.5$ first crossed the original $0.03$ gate at the $n=1.5$ saved state. An equal-accuracy confirmation gave: + +| Inverse, frozen $n=1.5,J=0$ | Iterations | True relative defect | Complete cost (s) | +|:---|:---|:---|:---| +| Independent baseline | 231 | 0.0290033753 | 100.9943 | +| Mass-consistent gravity Schur | 132 | 0.0292091550 | 72.3727 | + +This was a measured 1.39548-fold speedup, 28.34 percent lower complete cost and 42.86 percent fewer iterations, without relaxing accuracy. At a saved $n=1,J=0.1$ system, the independent and gravity-only controls still failed after 600 iterations, at defects 0.0739197 and 0.0408506. Combining cached stellar LDU with gravity refinement passed in 142 iterations at defect 0.0299068971, complete cost 96.8236 seconds. It reduced Poisson and gradient defect norms by 63.33 and 94.47 percent relative to the matched 200-iteration independent control. Since the baseline had no finite observed successful solve, no rotating speedup ratio was asserted. + +A nine-mode coupled shape coarse inverse was a negative control. All modes were active, overlap/rank checks passed, and there was no fallback, but the 200-iteration defect worsened to 0.0668658. Setup added 8.6012 seconds and persistent vectors occupied 28,491,680 bytes. That failed candidate remained experimental; it did not rule out every possible coarse space. + +## Nonlinear transfer limited the recommendation + +For easy $n=1,J=0$, both independent and gravity-refined inverses converged in six full steps to the same physical state, with all virials below $3.74\,10^{-7}$. Linear iterations fell from 307 to 258, yet Newton time rose from 191.068 to 199.375 seconds, 4.35 percent. More expensive inverse applications outweighed iteration reduction. + +For $n=1.5,J=0$ with gravity curvature correction, the stronger inverse passed all twelve permitted outer linear gates and all 31 auxiliary gravity solves. It removed the earlier immediate linear failure but ended at $1.3966991\,10^{-5}$, only 6.13 percent below the old six-update endpoint and far above target $2.7365922\,10^{-12}$. All nineteen rejections were merit failures, with late accepted steps $1/64$ and $1/512$. Newton time was 1075.368 seconds, including 415.038 in outer linear solves and 590.003 in preparation; nested auxiliary time must not be added again. Enthalpy-based virials passed, but negative density persisted. + +The rotating combined-inverse follow-through accepted one repaired full step to $0.00685562$, then failed the next linear gate at 0.0339685. Its enthalpy energy virial, 0.00712711, was worse than the historical independent endpoint’s 0.000544539 despite its smaller nonlinear residual. Both failed physically. Frozen linear gains therefore did not justify claiming a solved nonlinear or physical problem, and the stronger policies remained optional. + +#### Evidence. + +`experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md`, `experiments/GRAVITY_CONSISTENT_SCHUR_2026-09-18.md`, and `experiments/PRECONDITIONER_CACHED_CROSS_ACTION_2026-09-18.md`. Focused serial coverage passed 20 cases and 441,978 assertions; two-rank algebra/coarse tests passed thirteen cases per rank. The actual multi-rank FEM test was blocked by a pre-existing partition where one rank had no vacuum element, required by mapped H(div) construction. Refining the mesh did not remove that ownership pattern. Serial FEM evidence must not be described as distributed FEM validation. + +# Why better linear solves did not remove the nonlinear stall + +The next frozen investigation used the saved $n=1.5,J=0$ endpoint with $\lVert F\rVert=1.3966991044\,10^{-5}$. All prior linear gates had passed; the final squared-residual shares were about 42.42 percent hydrostatic, 26.23 percent closure, 15.87 percent surface, 10.43 percent gravity, and 5.05 percent mass. It computed fresh cold directions rather than claiming to reconstruct raw directions that the earlier trajectory had not saved. + +For $\phi=\lVert F\rVert^2/2$ and a forcing condition $\lVert Ap+F\rVert\leq\eta\lVert F\rVert$, a consistent derivative gives $$\phi'(0)=\langle F,Ap\rangle\leq-(1-\eta)\lVert F\rVert^2.$$ Thus a gate-passing direction should be strongly locally descending, but this says nothing about how far the linear approximation remains useful. + +## Tenfold tighter forcing did not improve the useful step + +| Measurement | $\eta=0.03$ | $\eta=0.003$ | +|:------------------------------------|:------------|:-------------| +| Iterations | 84 | 143 | +| Cold linear seconds | 47.9374 | 81.6901 | +| Verified relative defect | 0.0295592 | 0.00294840 | +| Direction norm | 0.0794897 | 0.0796012 | +| Corrected residual/base at $1/64$ | 1.080323 | 1.080567 | +| Corrected residual/base at $1/256$ | 1.002025 | 1.002037 | +| Corrected residual/base at $1/1024$ | 0.999394139 | 0.999394272 | + +The tenfold stricter gate cost about seventy percent more linear work and left the useful sampled step essentially unchanged. The directions differed by only 0.8885 percent in norm and had cosine 0.99996157. A supplementary minimum-norm calculation in their two-dimensional span, subject to the original forcing gate, gained only 0.00058549 percent beyond ordinary scalar damping of the tighter direction. This was a saved-vector algebra experiment, not a fresh FEM trial, and did not rule out richer subspaces. + +Raw trials at $1,1/4,1/16$ failed pressure preparation for this new late direction. At $1/64$, raw and gravity-corrected residual ratios were 1.807268 and 1.080323; at $1/256$, 1.002510 and 1.002025. At $1/1024$ both decreased, to 0.999170 and 0.999394. The remainders were resolved and approximately quadratic over the small-step range. No persistent first-order error was detected in the tested tangent. + +## A large compensated direction amplified moderate curvature + +The separate shape and field action norms were 0.7730368 and 0.7730377, while their sum had norm $1.396089\,10^{-5}$. The image cosine was $-0.999999999838$, and the cancellation factor was approximately 110,743. At the seed, the corresponding factor had been about 332.1. Normalization factors were byte-identical across these states. + +The late direction was 17.35 times larger while the residual was 19.59 times smaller. The ratio $\lVert p\rVert/\lVert F\rVert$ grew from 16.7433 to 5691.26, about 340-fold. At the common step $1/64$, corrected curvature per squared direction norm grew only from 0.656881 to 0.926040, about 1.41-fold; curvature relative to the base residual grew about 8,313-fold. This pointed toward weak coupled response and large corrections as the amplification mechanism, rather than orders-of-magnitude growth of curvature per unit direction. It did not identify a continuum symmetry or estimate the full Hessian. + +Gravity correction reduced the small-step remainder norm by about 14.7-fold at $1/1024$, but increased its adverse projection onto the linear predictor 2.88-fold. Hydrostatic, closure, surface, and mass rows contributed 43.52, 32.33, 19.53, and 4.63 percent of that signed projection. A smaller remainder therefore need not produce better merit. The observed full/shape gain ratio also gave a condition-number lower bound of about $9.07\,10^4$ if the finite-dimensional Jacobian was nonsingular; it was not a measured global condition number or a license to add a central constraint at $n=1.5$. + +#### Evidence. + +`experiments/NONLINEAR_STEP_INVESTIGATION_2026-09-18.md`. Three immutable probes covered 144 path observations and sixty successful auxiliary gravity solves; final saved-artifact audits passed 20,899 checks. All states, factors, and restored actions were preserved. Those checks established provenance and arithmetic identities, not a new model. + +# Fully coupled second-order trial corrections + +## Bounded correction against the complete residual + +Because gravity-only correction moved adverse curvature into retained equations, a fully coupled trial correction was implemented. At a prepared trial $y$, it preserved the original affine target $L(\alpha)=F_0+\alpha A_0p$ and solved $$A(y)c=L(\alpha)-F(y).$$ The auxiliary backend had its own true full-Jacobian defect check, finite budget, and report. The correction was capped by $\lVert c\rVert\leq\alpha\lVert p\rVert$ by default. It was attempted only after a prepared candidate failed ordinary acceptance, with at most one solve per Newton iteration initially. A failed solve, cap, admissibility test, or merit comparison restored the original trial with fresh preparation. Accepted-state buffers were not mutated by the transaction. The policy composed statically with existing enthalpy/gravity choices and deliberately rejected the unimplemented eliminated-gravity combination at compile time. + +At the common late state, the original direction still passed with defect 0.02955918315. Independent frozen trials showed: + +| Path and $\alpha$ | Before/base | After/base | Auxiliary iterations | Added seconds | +|:---|:---|:---|:---|:---| +| Raw, $1/64$ | 1.80726810 | 1.03360191 | 34 | 27.33 | +| Raw, $1/256$ | 1.00250966 | 0.99746464 | 52 | 37.59 | +| Gravity first, $1/64$ | 1.08032296 | 1.09566246 | 55 | 47.60 | +| Gravity first, $1/256$ | 1.00202482 | 0.99810801 | 62 | 43.53 | + +All auxiliary solves passed their unchanged gate and cap. The $1/256$ trials became acceptable, while the gravity-first $1/64$ trial worsened and was restored. The already acceptable $1/1024$ and seed full-step controls incurred no coupled solve. Raw timing came from an uncontended repeat after a mistakenly overlapping artifact audit was stopped and fixed; fifty saved vector files matched the first run byte-for-byte, and the original timing was excluded. + +The correction’s own curvature remained significant. For $b=L-F(y)$, $\delta=A(y)c-b$, and $E=F(y+c)-F(y)-A(y)c$, $$F(y+c)-L=\delta+E.$$ At $1/256$, $\lVert E\rVert$ exceeded the verified auxiliary defect by about 11.28 for the raw path and 254 for the gravity-first path. Correction/direction cosines were $-0.9999330$ and $-0.9999942$: much of the correction reversed the original step. Actual displacement increases relative to the old $1/1024$ path were only 1.7596 and 1.2342, not the nominal factor four in $\alpha$. + +Fresh scalar controls matched projection along the original direction. They decreased residual 0.115623 and 0.096934 percent, whereas the coupled candidates decreased 0.253536 and 0.189199 percent. Small transverse components therefore had a measured benefit; the improvement was not just scalar damping. It remained a small absolute gain purchased with another coupled solve. Moreover, the independent frozen alphas each had a fresh correction budget, so this screen did not measure a live shared-budget line search. + +## Damping and work-aware eligibility + +The next optional policy reused one computed $c$ across $\beta=1,1/2,1/4,1/8$, always preparing $y+\beta c$ from the same $y$. The first ordinarily acceptable improvement ended the search; otherwise the best admissible improvement could be retained as a trial, with fresh replay. An eligibility rule reserved the single solve for a candidate near the affine prediction. For the configured deterministic merit $M$, it required positive finite $D=M(F_0)-M(L)$ and $$\frac{M(F(y))-M(F_0)}{D}\leq k,\qquad k=1\ \hbox{by default}.$$ This was a bounded-work heuristic, not a theorem of global convergence. + +A sequential halving screen exposed an important correction to the earlier interpretation. Ordinary raw and gravity paths already accepted $1/512$; the previous sparse grid had omitted it. The near-acceptance policy instead accepted $1/256$, a twofold nominal increase. Without eligibility deferral, the sole solve was spent at an unsuccessful $1/32$ and the final state was the baseline. Both successful frozen rescues used $\beta=1$, so eligibility, not damping, explained that particular improvement. + +In uncontended raw repeats, baseline search work cost 45.76 seconds for 0.116541 percent decrease; near-acceptance correction cost 74.90 seconds for 0.253536 percent decrease. The fractional decrease per measured operation-second was 1.329 times greater. Those timings excluded ordinary outer solve, commit refresh, initialization, and physical replay; they were not Newton speedups. A clock-anomalous original gravity/full run was retained and excluded, then repeated with independent UTC, monotonic, and child CPU measurements. + +## Matched six-update live test + +The live pair kept the seed, mesh, $n=1.5,J=0$, absent central constraint, gravity-Schur inverse, restart 160, and $\eta=0.03$ fixed. Only the trial policy changed. Both first updates reached $1.614841549\,10^{-5}$ with the same recorded diagnostics. + +| Endpoint after six updates | Gravity curvature control | Coupled policy | +|:-----------------------------|:--------------------------|:--------------------| +| Final residual | $1.497197\,10^{-5}$ | $1.245830\,10^{-5}$ | +| Newton iteration seconds | 502.21 | 613.09 | +| Outer Krylov iterations | 354 | 341 | +| Auxiliary coupled iterations | 0 | 163 | +| $P(h)$ energy virial | $4.22590\,10^{-6}$ | $3.78554\,10^{-6}$ | +| $P(h)$ force virial | $2.71367\,10^{-6}$ | $2.31894\,10^{-6}$ | +| Minimum sampled density | $-7.73185\,10^{-4}$ | $-7.85522\,10^{-4}$ | + +The new endpoint residual was 16.79 percent lower for 22.08 percent more Newton time. Excluding the common first update, residual decreases were 22.85 versus 7.29 percent, giving 2.53 times more decrease per iteration-second over that bounded interval. Both stopped at the six-update cap; neither converged. The new method initially crossed several modest residual thresholds later than the control before eventually overtaking it. + +Only one of three coupled searches was committed. At update two the full correction worsened merit but $\beta=1/2$ passed, with a very small 0.01795 percent residual decrease. Later ordinary steps benefited from the changed trajectory. Two other kept improvements remained unacceptable and were followed by ordinary backtracking; they were not accumulated into the model. This result supported retaining the policy as an experiment, not promoting it or treating every rescued step as an efficient update. Density-pressure virials remained undefined and local EOS extrema did not uniformly improve. + +#### Evidence. + +`experiments/COUPLED_TRIAL_CORRECTION_2026-09-18.md` and `experiments/DAMPED_COUPLED_TRIAL_2026-09-18.md`. The latter covered 58 serial numerical/solver cases, 23 correction cases on two MPI ranks, 56 Python checker tests, twelve frozen cases including repeats, and the matched live pair. Its tests covered fallback, exceptions, budgets, rank-local rejection, preserved outer reports, and typed composition. + +# Shape freedom, field transport, and the displacement-order control + +The next investigation targeted the weak coupled response itself, preserving the equations and avoiding an assumed gauge interpretation. At independent quadrature orders 10 and 14, scalar pullback changes were decomposed as $$\delta f_{\mathrm{Eulerian}}=\delta\widehat f-\nabla f\cdot v,$$ where $v$ was the generated physical displacement. Density and potential gradients were broken within-element gradients; facet jumps and RT transport were not included. Small Eulerian/pullback ratios therefore suggested transport compensation but could not certify a pure relabeling nullspace. + +At the old twelve-update $Q_3$ endpoint, these ratios were 0.05769 for density, 0.18870 for enthalpy, and 0.01288 for potential, versus seed values 0.3243, 0.3080, and 0.1993. A six-block sampled singular-response minimum fell from 0.0580574 at seed to 0.000143543 late, with largest sampled response near 22.1–22.6. The normalized block direction span itself changed with the state, so this was not a full-Jacobian spectrum or a rank proof. + +Despite an enthalpy energy virial of $4.06909\,10^{-6}$, the late nonrotating nodal surface radii spanned 0.997674–1.010093. Its direction was 99.9995107 percent cubic-invariant in sampled squared norm, while degree-eight spherical harmonics left 0.893679 of its nodal norm unexplained. This was fine angular structure consistent with mesh/discretization coupling, not evidence that all omitted modes were unphysical. The bulk and gravity discretizations also shared cubic symmetry. + +## The isolated $Q_4$ displacement experiment + +The original $Q_3$ displacement could not exactly represent the order-four reference coordinate map: the affine identity control gave relative value error $9.01277\,10^{-5}$ and gradient error $1.23472\,10^{-3}$. On an isolated branch, only `libmeanfield/interface/field/field_registry.cppm`, `Displacement::vectorOrder`, changed from $3+$ uniform increment to $4+$ uniform increment, with the increment fixed at zero. The rationale and date were recorded in source. The shared scalar surface collection therefore also changed from $Q_3$ to $Q_4$, increasing surface unknowns from 866 to 1538. Density/potential $Q_2$, enthalpy $Q_3$, RT2 gravity, geometry, and the radius-squared lift were retained. + +This was not a pure interpolation intervention: it added shape modes, changed surface normalization/preconditioning, increased derived pressure quadrature by one and gravity-force quadrature by two, and raised the geometry-inspection request from 18 to 24. Both variants started from analytic seeds; incompatible coefficient snapshots were not transferred. + +$Q_4$ reduced affine value and gradient errors to $2.11629\,10^{-16}$ and $3.24701\,10^{-15}$. Nevertheless, seed physical displacement direction norm grew 31.63-fold, from 0.00121967 to 0.0385749, and seed linear iterations grew from 26 to 67. Both initial linear gates passed, and small trial remainders remained quadratic. The fixed six-attempt live comparison was unfavorable: + +| Measurement | $Q_3$ displacement/surface | $Q_4$ displacement/surface | +|:---|:---|:---| +| Root unknowns | 178,073 | 178,745 | +| Accepted updates | 6 | 3 | +| First three accepted-step seconds | 269.31 | 505.65 | +| Residual after three / own initial | 0.056755 | 0.906188 | +| All attempted Newton seconds | 547.34 | 638.42 | +| Krylov iterations, including failure | 354 | 492 | +| $P(h)$ energy virial | $4.22590\,10^{-6}$ | $8.89666\,10^{-5}$ | +| $P(h)$ force virial | $2.71367\,10^{-6}$ | $8.27467\,10^{-6}$ | +| Hydrostatic-gradient RMS | 0.0106811 | 0.325413 | +| Nodal radius span / nodal mean | 1.09849% | 12.2559% | + +The fourth $Q_4$ solve and an independent cold endpoint solve both failed after 200 iterations at relative defect 0.0868564. Even its first three gate-passing corrections had already produced poor progress. The differing spaces and stopped endpoints prevented an equal-accuracy comparison; the result nevertheless ruled out matching displacement order alone as a sufficient cure in this configuration. The $Q_4$-only change was explicitly not recommended for promotion. It did not rule out compatible enrichment of multiple fields. + +## Two structural cautions + +A naive replacement of mechanical shape rows by surface Bernoulli moments was identified as rank-deficient at an ideal $B=0,J=0$ root for the current trace spaces. On the actual closed surface, continuous $Q_p$ had $96p^2+2$ degrees of freedom, versus $96p^2$ for broken $Q_{p-1}$ potential traces. After enforcing the zero enthalpy trace, $Q_3$ surface supplied 866 rows against 864 DG $Q_2$ trace coefficients; $C$ added no independent constant trace dimension and the direct shape-measure derivative vanished with $B$. This was an objection to that proposed replacement, not proof of rank deficiency in the existing mechanical operator. Any alternative boundary formulation needed compatible spaces and a stability analysis. + +Second, correcting ambient interpolation did not make the radius-squared extension generate a physical dilation. If $b$ was the boundary trace of reference position $X$, the existing lift $E_0(b)$ differed from $X$; at a logical radius $1/8$ vertex it gave one eighth of the desired affine value. A proposed but unimplemented rank-one correction was $$E(z)=E_0(z)+\bigl[X-E_0(b)\bigr]\ell(z),\qquad\ell(b)=1.$$ It would give $E(b)=X$ while retaining the supplied trace and admit an exact transpose. This could place physical dilation in the mechanical test space, but would not automatically enforce the potential-energy virial, which uses the independent mixed potential. It was kept separate from surface-space reduction and was not claimed as an implemented remedy. + +#### Evidence. + +`experiments/SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md`. The $Q_4$ production change was isolated at commit `4be1a23c3ec2650af6a2a49528a8f5248cb001c3`; 22 targeted cases and 380,856 assertions passed. That correctness coverage did not convert its negative numerical outcome into a recommendation. + +# The bounded smooth-surface study + +The user authorized a one-hour study, recorded as 19:30:15–20:30:15 UTC on 18 September. It retained $Q_3$ displacement, $Q_4$ reference geometry, $Q_2$ density/potential, $Q_3$ enthalpy, RT2 gravity, the same extension, $n=1.5,J=0$, and no central constraint. It changed only independent surface variation through a frozen affine embedding $$q=q_0+Ba.$$ The existing base $q_0$ was never projected. Real spherical harmonics through degree $L$ were evaluated at actual radial surface nodes and their interpolants orthonormalized in a quadrature-defined physical surface-area metric. All azimuthal components in that harmonic span were included. + +Correct primal and dual normalized maps were essential. For physical state scales $S$ and residual scales $R$, separate thin QR decompositions gave $$SB=Q_xT_x,\qquad R^{-1}B=Q_fT_f,$$ so that $$Q_f^T(RF_{\mathrm{physical}})=T_f^{-T}B^TF_{\mathrm{physical}}.$$ The reduced operator was $Q_f^T\widehat AQ_x$, not a blind transpose of a normalized state basis. Other fields and scalar equations remained active. The projected existing inverse was an approximation, not an exact reduced inverse. Both retained and original full residuals, and excluded shape force, were always exported. + +## Frozen late-state result + +At the common six-update $Q_3$ endpoint, $\lVert F\rVert=1.4971971\,10^{-5}$, cold FGMRES used restart 160, cap 200, $\eta=0.03$, and one gravity-Schur sweep with relaxation 0.5. + +| Shape space | Parameters | Iterations | Solve seconds | Unit-trial full norm | +|:-----------------|:-----------|:-----------|:--------------|:---------------------| +| Full nodal trace | 866 | 61 | 32.184 | Pressure rejection | +| $L=2$ | 9 | 12 | 6.447 | $2.263903\,10^{-6}$ | +| $L=4$ | 25 | 21 | 11.249 | $2.432960\,10^{-6}$ | +| $L=8$ | 81 | 29 | 15.745 | $2.964119\,10^{-6}$ | + +All reduced directions passed their retained linear gates, with relative defects 0.01719, 0.01608, and 0.01409. Their original full linear defects were 0.15121, 0.16187, and 0.19207: none passed the original full Newton gate. Nevertheless, all four sampled trial alphas prepared and decreased both norms, and full residual decreases at unit step were 84.88, 83.75, and 80.20 percent. This was a real improvement in the original residual, obtained by restricting the shape equations, not a more accurate solution of the original Newton system. + +The $L=4$ root had 177,232 unknowns, only 841 fewer than the full root; its less than 0.5 percent dimension reduction did not explain a factor 2–5 drop in cold solve time by itself. Fewer Krylov iterations supplied most of the observed gain. Physical displacement norm fell from $3.96028\,10^{-3}$ for the full direction to $9.14977\,10^{-5}$ at $L=4$, 43.28-fold. The $L=2$ direction was nearly field-only, while admitting higher modes increased coupled displacement and curvature. + +After the $L=4$ unit trial, excluded shape force accounted for 98.38 percent of the remaining original squared norm. The inherited fine-scale geometry also remained, since $q_0$ had not been projected. Enthalpy-based energy and force virials improved to $8.11913\,10^{-7}$ and $6.41398\,10^{-7}$, but local EOS/hydrostatic errors were almost unchanged and minimum sampled density remained $-7.27628\,10^{-4}$. + +## Three cold reduced updates from the analytic seed + +A conditional extension, recorded before launching, chained three saved unit candidates from the analytic seed. Each stage passed preparation, the retained independent linear gate, and both recorded norm-decrease screens. This was an externally replayed cold reduced-Newton chain, not a production warm-start trajectory or a fully converged model. + +| Updates | Full norm | Retained norm | Excluded shape norm | Energy virial $P(h)$ | +|:---|:---|:---|:---|:---| +| 0 | $2.736592\,10^{-4}$ | $2.590732\,10^{-4}$ | $8.815010\,10^{-5}$ | $2.410694\,10^{-5}$ | +| 1 | $8.624827\,10^{-5}$ | $9.387845\,10^{-6}$ | $8.573583\,10^{-5}$ | $2.180785\,10^{-6}$ | +| 2 | $8.473526\,10^{-5}$ | $3.147803\,10^{-6}$ | $8.467677\,10^{-5}$ | $8.660498\,10^{-7}$ | +| 3 | $8.465191\,10^{-5}$ | $7.357283\,10^{-8}$ | $8.465188\,10^{-5}$ | $6.769653\,10^{-7}$ | + +The retained norm fell about 3521-fold but the original norm only 3.23-fold. At the final trial, excluded shape force represented 99.999924 percent of the original squared norm. Cold solves took 18, 20, and 18 iterations and 10.574, 11.341, and 9.594 seconds. Full relative linear defects were 31.45, 98.18, and 99.90 percent; these were not full-system Newton solutions. + +The final force virial was $5.769064\,10^{-7}$ and mass error $3.79142\,10^{-9}$. Yet hydrostatic-gradient RMS rose from 0.001528098 at seed to 0.007255981, about 4.75-fold, while density remained negative at some samples. Nodal radius spread grew to 0.154699 percent, substantially less than unrestricted late distortion but not an exact spherical result. Small retained residual and excellent virial could therefore coexist with significant omitted mechanical force and imperfect interior fields. + +The result supported decoupling smooth independent surface parameters from the ambient displacement carrier, with nested configurable spaces and consistent dual restriction. It did not establish that a fixed degree-four space was sufficient at all rotation rates, that excluded force was irrelevant, or that the complete original mechanical system had converged. The subsequent native spherical-harmonic implementation and higher-index studies should be read as later developments, not retroactively attributed to this bounded experiment. + +All seven case audits passed 2,098 saved-artifact checks; a separate chain audit passed 107 checks with bitwise physical-state and residual handoffs. Actual builds, FEM cases, and expensive audits were serialized, and all seven FEM probes completed by 20:09:58 UTC. No production order, operator, or lift changed in this study. Optional $L=6$, rotating cases, higher indices, mesh/enrichment sweeps, and production integration were outside its scope. + +#### Evidence. + +`experiments/REDUCED_SURFACE_STUDY_2026-09-18.md`, producer commit `21cdbda22486e408adc52b65b25413f9380c7c30`, and the archived `working/research/diagnostics/reduced_surface_2026-09-18` campaigns. Together with the failed $Q_4$ experiment, this was the main reason to prefer a controlled smooth surface span over indiscriminate addition of nodal surface freedom. + + + +# Native spherical-harmonic surface prescription + +The reduced-surface experiments motivated making spherical harmonics a native, selectable surface prescription. The implementation on `research/spherical-harmonic-surface-20260919` preserved the simultaneous Newton solve: surface coefficients, material fields, gravity fields and global multipliers remain coordinates of one coupled nonlinear system. This was not a separate surface-relaxation iteration. The preferred research candidate became maximum degree $L=4$ with all real azimuthal orders, hence 25 independent coefficients. The unrestricted nodal radial prescription remained available as a control. The choice of $L=4$ was an experimentally useful starting point, not a demonstrated resolution requirement for 50% critical rotation. + +At reference boundary node $i$, with fixed radial direction $\widehat r_i$, the radial displacement is $$q_i=\sum_{\ell=0}^{L}\sum_{m=-\min(\ell,M)}^{\min(\ell,M)} + a_{\ell m}Y_{\ell m}(\widehat r_i).$$ Omitting the optional azimuthal cutoff retains $M\ge L$. The real harmonics are normalized to unit squared integral over solid angle, include the Condon–Shortley phase, and use negative $m$ for sine and positive $m$ for cosine. The catalog is ordered by increasing $\ell$ and then signed $m$. Coefficients have dimensions of length. Zero coefficients preserve the reference mesh; $\ell=0$ is a genuine monopole rather than a removed gauge. + +| Configuration | Independent coefficients | +|:----------------------------|:-------------------------| +| $L=4$, all $m$ | 25 | +| $L=4$, $\lvert m\rvert\le2$ | 19 | +| $L=4$, $m=0$ | 5 | +| $L=8$, all $m$ | 81 | + +No odd-degree or nonaxisymmetric modes are silently excluded. Choosing $m=0$ constrains the shape about the fixed global $z$ axis; it does not impose axisymmetry on the three-dimensional interior fields. The reference center is configurable, but the frame does not move as another Newton unknown. + +## Architecture and normalization + +The prescription implements the existing forward, transpose and pullback contracts, with a cached reference basis and zero derivative of that fixed pullback. It composes with the existing interior and fixed-infinity vacuum extensions. The numerical discretization now owns typed surface, interior and vacuum prescriptions, instead of the modern equilibrium compiler hardcoding a nodal surface. Existing two-argument discretization calls preserve their nodal behavior. The physics-facing EOS, pressure boundary, mass and angular-momentum specification is unchanged. + +Prescription/extension compatibility and coupled operator structure remain compile-time constrained. Requested degree, sampled rank and actual mesh resolution are runtime facts. Sampled aliasing is rejected rather than silently reducing the mode set. Modal normalization and the default material/surface inverse use a diagonal of the pulled-back reference vector trace mass matrix; state and residual factors retain their primal/dual relationship. Metric assembly checks FEM identity and a canonical trace-layout capability. The older nodal $H^1$ surface surrogate explicitly rejects harmonic coordinates. + +Geometry remained Q4 and displacement remained Q3. The harmonic samples feed the existing finite-element trace, so the resulting surface between nodes is not an exact analytic spherical-harmonic graph. Reference-mesh interpolation error remains. The native implementation also differs from the earlier frozen reduced-space experiment, which retained an arbitrary nodal base shape and used area whitening and QR maps. Physical residual and Jacobian equivalence is the appropriate comparison; identical normalized Krylov trajectories are not expected from those different coordinate metrics. + +The implementation was single-rank, consistent with the radial volume extension. Successful two-rank rejection tests do not establish distributed harmonic-solver support. + +## Native coupled results + +Five live cases used the same mesh, cold Lane–Emden initialization, no central constraint, a six-update budget, restart 160, at most 200 linear iterations and a true relative linear gate $\eta=0.03$. Nonlinear convergence required $\lVert F\rVert/\lVert F_0\rVert\le10^{-8}$, with zero absolute tolerance. All 29 accepted updates were full steps, with one trial each and passing linear gates. + +| Case | Updates | Final $\lVert F\rVert$ | $\lVert F\rVert/\lVert F_0\rVert$ | Newton s | +|:---|:---|:---|:---|:---| +| $n=1,J=0$, all $m$ | 5 | $1.52715\,10^{-12}$ | $9.64952\,10^{-9}$ | 146.00 | +| $n=1.5,J=0$, all $m$ | 6 | $1.29965\,10^{-12}$ | $5.01751\,10^{-9}$ | 221.40 | +| $n=1,J=.01$, all $m$ | 6 | $3.86299\,10^{-13}$ | $4.29274\,10^{-10}$ | 197.23 | +| $n=1,J=.01$, $m=0$ | 6 | $3.08442\,10^{-13}$ | $3.42757\,10^{-10}$ | 164.90 | +| $n=1.5,J=0$, gravity-Schur | 6 | $2.19823\,10^{-11}$ | $8.48662\,10^{-8}$ | 189.54 | + +The first four cases converged by their stated criterion. The gravity-Schur case exhausted its budget, although its last update still reduced the residual by approximately a factor of 40; calling that endpoint a plateau would be incorrect. Process wall times, including setup and observations, were 210.55, 290.70, 265.89, 232.54 and 257.49 seconds, respectively. Timings are single-run observations on the workstation, not replicated benchmark statistics. + +At q18, absolute energy/force virial errors formed with $P(h)$ were $3.87036\,10^{-7}$ / $2.65938\,10^{-7}$ for the nonrotating $n=1$ case, $6.55153\,10^{-7}$ / $6.00565\,10^{-7}$ for $n=1.5$, and approximately $3.87\,10^{-7}$ / $2.66\,10^{-7}$ for both low-rotation $n=1$ cases. They also passed $10^{-5}$ at q14. The corresponding density-pressure virials passed for the sampled $n=1$ cases, but were undefined for $n=1.5$, whose independent density was negative at some integration points. The later sign-region investigation also detected a thin $n=1$ negative layer missed by ordinary volume quadrature. + +Global balance did not establish pointwise accuracy. For nonrotating $n=1.5$, the normalized EOS-density RMS was $1.25551\,10^{-4}$ and the hydrostatic-gradient RMS increased from $1.52810\,10^{-3}$ in the seed to $7.27475\,10^{-3}$ at the root, a factor of 4.76. This gradient diagnostic has acceleration units in the model normalization; it is not the normalized weak residual. The four converged cases had mass errors below $5.3\,10^{-14}$ and angular-momentum errors below $3.6\,10^{-16}$. + +The full nodal mechanical force remained almost entirely outside the harmonic span: Euclidean nodal-dual norms were approximately $9.22\,10^{-6}$ for $n=1,J=0$ and $8.99\,10^{-6}$ for $n=1.5,J=0$. In the latter case the retained modal force norm was only $3.61\,10^{-14}$. This is convergence of the selected Galerkin shape problem, not convergence of the former unrestricted nodal shape problem. The excluded dimensional Euclidean norm cannot be directly compared with a normalized nonlinear norm or the dimensionless virial tolerance. + +Nonaxisymmetric coefficients persisted in nominally spherical models. The $(\ell,m)=(4,4)$ coefficients were $8.97263\,10^{-5}$ for $n=1$ and $9.23867\,10^{-4}$ for $n=1.5$, compared with $a_{40}=1.09313\,10^{-3}$ in the latter. These are evidence to examine mesh/discretization imprint, not evidence of a physical nonaxisymmetric branch. + +Total Krylov counts were 212, 335, 296, 219 and 206 in the table’s order. The $m=0$ rotating control used 26.0% fewer Krylov iterations and 16.4% less Newton time than the all-$m$ control, but changes the admissible shape and is not a pure preconditioner speedup. All $m$ remained the preferred three-dimensional candidate. Ten new harmonic tests and 64 selected existing regression cases passed; independent audits checked saved basis pullbacks, excluded force and physical/nodal equivalence. Those implementation checks do not validate the higher-rotation target. + + + +# Negative density: a located discrete representation defect + +The subsequent investigation was observational: it changed no equilibrium equation, field order, shape prescription, normalization, preconditioner or Newton policy, and solved no new equilibrium. It examined four frozen states: converged $n=1$ and $n=1.5$ nonrotating harmonic models, the original $n=1.5$ seed, and the $n=3$ nonrotating seed. Only the $n=3$ case had a central condition; it was not a converged $n=3$ model. + +## What the weak EOS equation determines + +Density and enthalpy are independent unknowns. Baseline density is DGQ2 and enthalpy is continuous Q3. On a stellar element the closure is $$R_i=\sum_q w_q\varphi_i(q)\left[\rho_h(q)-\rho_*(h_h(q))\right], + \qquad \rho_*(h)=\left[\frac{h}{(n+1)K}\right]^n,$$ with physical mapped weights. At zero closure residual, $$M\boldsymbol\rho=\boldsymbol b(h),\quad + M_{ij}=\sum_qw_q\varphi_i\varphi_j,\quad + b_i=\sum_qw_q\varphi_i\rho_*(h_h).$$ Thus the closure selects a quadrature-defined weighted $L^2$ projection, not pointwise EOS equality. Its unique polynomial can have negative regions even when the sampled EOS target is nonnegative. + +Independently assembled element mass matrices and Cholesky solutions reproduced the converged $n=1.5$ density coefficients within $1.40\,10^{-14}$ and the field within $2.25\,10^{-15}$ in physical $L^2$, while its negative density was of order $10^{-4}$. Independent/production closure-residual differences were approximately $1.2\,10^{-17}$ to $2.1\,10^{-17}$ across the four states. Raising projection quadrature to q26 changed coefficients by at most $2.36\,10^{-7}$ and did not remove the undershoot. These comparisons identify the defect at the measured discrete root; it is not explained by incomplete closure convergence or the small measured quadrature difference. + +The implemented dependency also corrects an earlier intuitive explanation: negative density does not directly generate the represented enthalpy. Pressure is evaluated from independently represented enthalpy; negative sampled enthalpy is rejected during preparation. All negative-density q18/q26 volume samples in this investigation had positive enthalpy. Some surface witnesses had exactly zero enthalpy. Density influences enthalpy indirectly through the coupled stellar equations. + +## Spatial location and root-based integration + +All 96 surface-adjacent stellar cells contained measured negative regions; no negative intervals were found on the sampled lines through the other 736 stellar cells. All stored density nodal values nevertheless remained positive. The nodes are interior Gauss–Legendre points, and positivity of their values does not imply positivity between them. + +Ordinary quadrature can miss thin sign regions: q18 and q26 missed the $n=1$ volume layer, while its sampled boundary trace was negative. For the same unchanged $n=1.5$ field, negative-mass estimates changed from $2.89\,10^{-5}$ at q18 to $2.18\,10^{-5}$ at q26 and the apparent negative volume nearly halved. The investigation therefore isolated zero crossings of the polynomial along reference-cell lines and integrated negative intervals. Define $M=\int\rho_h\,dV$ and $M_-=-\int\min(\rho_h,0)\,dV$. The mass denominator below is $\lvert M\rvert$, not $\int\lvert \rho_h\rvert\,dV$. + +| Frozen state | $M_-/\lvert M\rvert$ | $V_-/V$ | Measured physical radius range | +|:---|:---|:---|:---| +| $n=1$ equilibrium | $6.09585\,10^{-8}$ | $0.0327346\%$ | $.999643$–$.999886$ | +| $n=1.5$ seed | $3.66771\,10^{-5}$ | $1.83795\%$ | $.993813$–$1.000019$ | +| $n=1.5$ equilibrium | $2.24535\,10^{-5}$ | $1.49252\%$ | $.994917$–$1.001222$ | +| $n=3$ seed | $2.88372\,10^{-5}$ | $12.0835\%$ | $.934409$–$.978469$ | + +Radii are in reference-radius units and are not distances from the local, slightly deformed surface. The first three cases have intervals reaching the surface; the $n=3$ seed has an interior pocket in the same outer cell layer. Its surface trace is positive. Minimum-density witnesses were approximately $-1.14756\,10^{-4}$, $-9.26332\,10^{-4}$, $-7.21171\,10^{-4}$ and $-8.54870\,10^{-5}$, respectively; the $n=3$ witness had enthalpy $0.0455581$. These are sampled witnesses, not certified global minima. + +Tangential rules with 8, 12 and 16 points per direction agreed on the negative integrals to roundoff in these cases. The finest two negative masses differed by at most $10^{-14}$ relative, and whole mass/volume agreed with separate observations within $1.77\,10^{-13}$ relative. Eight points integrated each root-partitioned interval. Analytic controls, decimal-arithmetic audits and state/coefficient hash checks support the measurements. They are not rigorous three-dimensional sign certificates: unsampled angular pockets or roots below reconstruction uncertainty remain possible. + +## Mechanism controls and their limits + +The initial seed independently interpolates tabulated Lane–Emden density and enthalpy into their respective spaces. The converged closure instead projects the EOS image of represented enthalpy. Both operations can undershoot; neither Newton nor rotation is necessary. With inward surface-cell coordinate $t\in[0,1]$, a linear enthalpy gives target density proportional to $t^n$. Exact quadratic $L^2$ projections in a flat one-dimensional cell are $$\Pi_2t^{3/2}=-\frac{2}{105}+\frac{16}{35}t+\frac47t^2, + \qquad + \Pi_2t^3=\frac1{20}-\frac35t+\frac32t^2.$$ The fractional profile is negative for $0\le t<0.0396969$. The cubic profile has a negative pocket $0.118350 + +# Independent density-order study and logarithmic assessment + +The next controlled experiment varied density alone: DGQ2, DGQ3 and DGQ4 in the same Gauss–Legendre family. Enthalpy/displacement stayed Q3, gravity RT2/Q2, geometry Q4, and the surface all-$m$ $L=4$. There were 832 stellar elements, so the density block had 22,464, 53,248 or 104,000 coefficients. The default remained Q2. A compile-time registry option propagated order through typed field spaces, quadrature, adapters, normalization and block layouts; no extra physics-facing parameter was needed. + +The closure rule must cover both its density mass form and EOS source form. The selected rule is the greater resolved request, preserving policy overrides. For these cases Q2/Q3 used identical achieved orders: 17, 19 and 23 for $n=1,1.5,3$, with 729, 1000 and 1728 points per cell. Q4 used 19, 21 and 25, with 1000, 1331 and 2197 points. Thus the coupled Q4 comparison includes policy-selected quadrature enrichment; only the frozen common-rule comparison strictly isolates approximation order. + +## Frozen approximation results + +Saved enthalpy and geometry were fixed for the $n=1$ and $n=1.5$ equilibria and $n=3$ seed. Each density space received an independent physical-volume-weighted projection using common q26 quadrature. EOS error was checked with a separate q30 rule; no projection was installed in a solver state. Negative measures use the finest root-sliced tangential rule. The EOS $L^2$ error has model density and volume units, without further relative normalization. + +| Frozen target | Density | $M_-/\lvert M\rvert$ | $V_-/V$ | EOS $L^2$ error | +|:---|:---|:---|:---|:---| +| $n=1$ root | Q2 | $6.09585\,10^{-8}$ | $3.27346\,10^{-4}$ | $1.25266\,10^{-4}$ | +| | Q3 | $1.05\,10^{-31}$ | $2.49\,10^{-16}$ | $1.56\,10^{-15}$ | +| | Q4 | $1.25\,10^{-31}$ | $2.73\,10^{-16}$ | $1.60\,10^{-15}$ | +| $n=1.5$ root | Q2 | $2.24299\,10^{-5}$ | $.0149181$ | $3.67965\,10^{-4}$ | +| | Q3 | $9.33602\,10^{-6}$ | $.0104476$ | $1.12325\,10^{-4}$ | +| | Q4 | $3.18362\,10^{-6}$ | $.00683440$ | $4.80663\,10^{-5}$ | +| $n=3$ seed | Q2 | $2.92063\,10^{-5}$ | $.121064$ | $1.06409\,10^{-2}$ | +| | Q3 | $2.63486\,10^{-6}$ | $.0373200$ | $6.86223\,10^{-4}$ | +| | Q4 | $1.02066\,10^{-8}$ | $.0169628$ | $3.84980\,10^{-5}$ | + +The $n=1$ Q3/Q4 sign measures are unresolved roundoff, not measured physical negative regions. Q3 contains the linear EOS image of Q3 enthalpy, explaining its roundoff-level mismatch. For $n=3$, the generic target is Q9, and for $n=1.5$ it is nonpolynomial. Q4 reduces the $n=3$ projected negative mass by approximately 2862 times relative to Q2 but leaves measured negativity in about 1.70% of the volume. Enrichment is useful approximation evidence, not a positivity guarantee. + +For $n=1.5$ Q4, q26/q30 EOS norms differ by about 0.18%; the fixed-eight-point root-sliced norm differs from the unsliced q30 norm by about 2.2%. Stable negative masses under tangential refinement do not establish normal-direction integration accuracy for the fractional EOS error. This distinction motivated reporting the unsliced q30 norm. + +## Coupled controls and the stalled $n=3$ trajectories + +Twelve live cases retained the same default inverse, normalization, cold initialization, restart 160, 200-iteration linear cap, $\eta=.03$, eight-update budget and relative nonlinear tolerance $10^{-8}$. Only the $n=3$ cases used a central condition. All nine $n=1$ and $n=1.5$ cases converged with full steps and no rejected trials. All their $P(h)$ energy/force virials were below $6.6\,10^{-7}$. + +| Case | Density | Updates | Krylov iterations | Newton s | +|:------------|:--------|:--------|:------------------|:---------| +| $n=1,J=0$ | Q2 | 5 | 212 | 147.5 | +| | Q3 | 6 | 286 | 204.3 | +| | Q4 | 6 | 286 | 215.1 | +| $n=1,J=.01$ | Q2 | 6 | 296 | 196.1 | +| | Q3 | 6 | 305 | 217.8 | +| | Q4 | 6 | 304 | 229.1 | +| $n=1.5,J=0$ | Q2 | 6 | 335 | 222.0 | +| | Q3 | 6 | 329 | 241.7 | +| | Q4 | 7 | 397 | 303.7 | + +For $n=1$, Q3/Q4 removed resolved undershoot and reduced EOS mismatch to roundoff. Nonrotating total costs were 39%/46% above Q2, but the enriched runs needed one extra update because their fifth residuals narrowly missed their own thresholds. The equal-six-update rotating case cost approximately 11%/17% more. Total costs therefore cannot be read as per-action order overhead. + +For $n=1.5$, Q3 reduced negative mass and EOS error by factors 2.40 and 3.28 for 8.9% more Newton time; Q4 reduced them by 7.01 and 7.66 for 36.8% more time. Negative volume fractions remained 1.493%, 1.046% and 0.685%. The hydrostatic-gradient RMS values were $.00727475$, $.00731970$ and $.00729901$; density enrichment did not appreciably improve that local balance measure. + +No $n=3$ case converged within eight updates: + +| Density | $\lVert F\rVert/\lVert F_0\rVert$ | Min. $\alpha$ | Newton s | $\lvert v_E^h\rvert$ | $\lvert v_F^h\rvert$ | +|:---|:---|:---|:---|:---|:---| +| Q2 | $.430062$ | $1/64$ | 301.6 | $1.77163\,10^{-4}$ | $1.34597\,10^{-4}$ | +| Q3 | $.985301$ | $1/256$ | 478.8 | $8.07504\,10^{-6}$ | $2.28169\,10^{-5}$ | +| Q4 | $.987949$ | $1/512$ | 508.2 | $1.37586\,10^{-5}$ | $2.84372\,10^{-5}$ | + +Krylov totals were 221, 452 and 458. All reported true linear gates passed. Q2 had four pressure and four insufficient-decrease rejections; Q3/Q4 each had five pressure and nine insufficient-decrease rejections. Density order does not enter the pressure-force sampling rule, whose operands remain enthalpy and displacement gradient. These differences reflect reached states/directions, not changed pressure sampling. Q3’s passing energy virial alone does not make it successful: its force virial and nonlinear criterion fail. + +Unbordered residuals were $1.33327\,10^{-3}$, $7.18640\,10^{-4}$ and $7.26009\,10^{-4}$; normalized central forcing was $3.17039\,10^{-5}$, $1.27669\,10^{-5}$ and $1.20966\,10^{-5}$. The unbordered diagnostic removes artificial central forcing but retains scalar constraint rows. Neither a physical root nor vanishing artificial force was demonstrated. The existing central condition pins central enthalpy to the EOS value at requested central density, not the independent DG density point value. + +The production seed is $I_p(\rho_{\rm LE,tabulated})$, whereas the frozen controls are $I_p(\rho_*(h_h))$ and $\Pi_p(\rho_*(h_h))$. Nonlinear EOS evaluation does not commute with tabulation or interpolation. For example, the production Q4 $n=3$ seed had approximately $5.26\,10^{-4}$ EOS error at q18, versus $3.85\,10^{-5}$ for the frozen Q4 projection at q30. These are different fields and quadratures, not a controlled improvement ratio. A closure-consistent initial guess remains a distinct hypothesis compatible with independent density. + +Each order passed 48 focused contract tests (9766, 9822 and 9894 assertions), and the frozen/coupled artifact audits passed. Across-order residual norms belong to different discrete spaces and Riesz weights, so lower normalized residual does not rank physical accuracy. The endpoints are finite-budget outcomes, not proved floors or nonexistence results. + +## What the logarithmic assessment established + +The separate log-density branch contained derivations and one-dimensional controls only. No stellar log-density solve, production representation change, speedup or virial improvement was established. Three constructions must remain distinct: + +1. Exponentiating density nodal coefficients gives $\rho_h=\sum_j\rho_s e^{u_j}\varphi_j$. It changes coordinates on positive coefficient vectors but can retain the same between-node negative polynomial. + +2. A pointwise exponential $\rho=\rho_s e^{u_h(\xi)}$ is positive in exact arithmetic for finite $u_h$, with $\delta\rho=\rho\,\delta u_h$. It changes the approximation family and every density-dependent physical evaluation. + +3. A factored representation $\rho=\rho_{\rm ref}(\xi,q)e^{u_h}$ can preserve an exact vacuum trace if the reference is positive inside and vanishes at the surface. Its variation includes $e^{u_h}\delta\rho_{\rm ref}$, in addition to $\rho\delta u_h$. + +Raw logarithmic density for $\rho\sim A s^n$ behaves as $\log(A/\rho_s)+n\log s$ and is singular at an exact vacuum boundary. Interior nodes avoid evaluating $\log0$ but do not give a finite polynomial log field the correct zero trace. A reference $\rho_{\rm ref}\sim B s^m$ makes the log amplitude bounded only when $m=n$. Such an exponent is a polytropic boundary assumption, not a general-EOS prescription; taking the reference from live enthalpy would introduce an additional dependency that is not required by independent density. + +A Q2 pointwise exponential fitted to three exact moments of $t^n$ gave: + +| $n$ | Fitted surface density | Relative $L^2$ error | Tiny Newton updates | +|:----|:-----------------------|:---------------------|:--------------------| +| 1 | $.0673707$ | $.0578060$ | 3 | +| 1.5 | $.0229264$ | $.0517531$ | 4 | +| 3 | $.00141091$ | $.0333959$ | 5 | + +Verified moment defects were at most $2.38\,10^{-14}$. Thus positivity, mass conservation and weak closure can coexist with an incorrect positive vacuum trace. Exact Q2 polynomial projection errors for the same profiles were 0, $.00952381$ and $.05$, so the exponential family was not uniformly more accurate at the same degree. A matched factor $t^n e^0$ reproduces the toy target by construction, not by demonstrating a general reference-selection method. + +With equal test/log bases the local closure derivative is the weighted mass matrix $\int\varphi_i\rho\varphi_j\,dV$. For $\rho=A\Delta^n t^n$ it scales as $A\Delta^{n+1}H_n$, against $\Delta I$ for an orthonormal polynomial density mass matrix. Cell-matrix condition numbers in the Q2 one-dimensional exact-profile controls were approximately 7.873, 16.852 and 94.561. These are not stellar Jacobian condition numbers. Small density reduces the response scale and can increase disparities across cells/blocks; logarithms do not automatically improve conditioning or enthalpy admissibility. + +A future implementation would require a typed physical-density evaluator and exact directional action, consistent support in every operator, normalization, quadrature and cache. Zero inactive exterior coefficients must not be exponentiated into positive exterior density. Underflow/overflow needs an explicit policy rather than silent clipping. MESA’s zonewise use of $\exp(\log\rho)$ supports investigating independent density coordinates, but its atmosphere boundary and zone representation do not establish correctness for this high-order exact-vacuum FEM problem. Following user direction, positivity work was deferred in favor of coupled convergence. + + + +# Detailed coupled-convergence campaign + +The final campaign retained Q2 independent density, Q3 enthalpy/displacement, RT2/Q2 gravity, Q4 geometry, the same mesh/normalization and all-$m$ $L=4$ surface. No clipping, logarithmic field, derived density or positivity gate was introduced. Three kinds of evidence were deliberately separated: evolving nonlinear trajectories, same-state linear inverse comparisons, and unchanged saved-state visualization. Cost comparisons include the stated setup and verification work; a passed artifact audit is not equivalent to convergence. + +## Initial nonlinear screen + +Five cases used 12 updates, restart 160 and a 300-iteration linear cap. Only $n=3$ included the existing central phase condition. The two $n=3$ cases differed in forcing tolerance. Every case exhausted its update budget. + +| Case | Final $\lVert F\rVert$ | Newton s | Krylov total | $\lvert v_E^h\rvert/\lvert v_F^h\rvert$ | +|:---|:---|:---|:---|:---| +| $n=2,J=0,\eta=.03$ | $8.85263\,10^{-6}$ | 742.74 | 1009 | $3.25447\,10^{-7}$ / $2.24560\,10^{-6}$ | +| $n=3,J=0,\eta=.03$ | $8.24392\,10^{-4}$ | 499.89 | 429 | $9.46671\,10^{-5}$ / $6.20929\,10^{-5}$ | +| $n=3,J=0,\eta=.003$ | $5.50731\,10^{-4}$ | 734.09 | 821 | $2.23846\,10^{-4}$ / $1.98912\,10^{-4}$ | +| $n=1,J=.1,\eta=.03$ | $5.83293\,10^{-3}$ | 310.43 | 226 | $3.23282\,10^{-4}$ / $3.36216\,10^{-4}$ | +| $n=1.5,J=.1,\eta=.03$ | $7.22567\,10^{-5}$ | 550.70 | 717 | $5.10711\,10^{-5}$ / $5.20697\,10^{-5}$ | + +The actual nonlinear thresholds were $3.81795\,10^{-12}$ for $n=2$, $3.09931\,10^{-11}$ for $n=3$, $8.57344\,10^{-10}$ for rotating $n=1$ and $1.39488\,10^{-9}$ for rotating $n=1.5$. q14/q18 enthalpy virial differences in all nonrotating cases were below $8\,10^{-14}$ absolute; this is a quadrature check, not mesh convergence or proof of equilibrium. + +After $n=2$’s second update, ten more updates reduced its residual by only 2.112%. Its final step was $1/256$ and all rejections were insufficient decrease. Poisson and gravity-gradient blocks contributed 94.770% and 4.076% of the final squared normalized norm. Minimum q18 density was positive, $2.93907\,10^{-4}$, and density-pressure energy/force virials also passed: $2.49026\,10^{-6}$ / $5.70105\,10^{-7}$. This supplies a stalled control without an observed negative-density obstruction, although it is not a global positivity certificate. + +Tenfold tighter $n=3$ linear tolerance reduced its equal-update endpoint residual by 33.20%, but required 46.85% more Newton time and 91.38% more Krylov iterations, and worsened both virials. The final step decreased from $1/4$ to $1/32$. The ordinary run’s squared residual was 79.927% Poisson, 13.854% gravity gradient and 6.144% closure; the tight run was 71.329% closure and 25.876% Poisson. These are different nonlinear trajectories, not same-state inverse comparisons. Both progressed beyond the earlier eight-step endpoints, showing why those endpoints must not be called demonstrated floors. + +The rotating $n=1$ case ended at $\alpha=1/65536$, with 26 pressure rejections and no metric rejection. Rotating $n=1.5$ had seven of each and ended at $1/64$; four updates after its eighth reduced the residual only 4.97%. Both had negative sampled density, making density-pressure virials undefined. + +| Case | Scaled EOS RMS | Scaled Bernoulli RMS | Hydrostatic-gradient RMS | +|:---|:---|:---|:---| +| $n=2,J=0$ | $1.87717\,10^{-4}$ | $2.80766\,10^{-4}$ | $.0264341$ | +| $n=3,\eta=.03$ | $3.52140\,10^{-4}$ | $.00230017$ | $.398054$ | +| $n=3,\eta=.003$ | $3.69501\,10^{-4}$ | $.00265289$ | $.479759$ | +| $n=1,J=.1$ | $.00126849$ | $.00221774$ | $.104394$ | +| $n=1.5,J=.1$ | $1.69365\,10^{-4}$ | $6.32866\,10^{-4}$ | $.0366157$ | + +Scaled EOS and Bernoulli RMS use sampled density and enthalpy scales; hydrostatic-gradient RMS is unscaled in model acceleration units. These local observations remain distinct from weak residuals and virials. A source-derived row map placed 84.437% of saved Poisson norm squared inside the star for the $n=2$ update-seven capture, 99.9768%/99.9995% for ordinary/tight $n=3$, and over 99% for both rotating captures. The exterior rows alone therefore do not account for the stalled norm. Row shares are not pointwise PDE errors or proof that the exterior has no indirect influence on interior fields. + +## Matched gravity-curvature trial-path control + +Raw and gravity-curvature Newton restarted from the identical $n=2$ update-three capture, with initial $\lVert F\rVert=9.0134584624\,10^{-6}$, six updates, $\eta=.03$ and common absolute target $10^{-12}$. Every reported full linear gate passed. + +| Trial path | Final $\lVert F\rVert$ | Newton s | Final $\alpha$ | +|:------------------|:------------------------|:---------|:---------------| +| Raw Newton | $8.8762959137\,10^{-6}$ | 449.196 | $1/256$ | +| Gravity curvature | $8.6749533568\,10^{-6}$ | 887.543 | $1/2048$ | + +The corrected endpoint was only 2.2683% lower at 1.97585 times cost. Its first step accepted $1/16$, against raw $1/128$, but supplied 90.1683% of its eventual six-step gain; no durable recovery followed. Preparation time rose 3.72673-fold, with 22 auxiliary gravity solves and 866 auxiliary iterations. Their verified prepared defects passed, so the full stall was not a failed auxiliary solve. All rejections were residual-metric failures, not pressure failures. Both endpoints met the enthalpy virial target, but neither met the nonlinear target. This pair did not justify a default trial-path change. + +## Frozen inverse comparisons: rate, attained accuracy and total cost + +Each case below compares a freshly constructed baseline and candidate on the identical saved physical state, Jacobian and right-hand side. The fresh full endpoint defect is $\lVert Ap+F\rVert/\lVert F\rVert$. Ordinary tolerance is $.003$; the marked tight case requests $10^{-7}$. Every case had a 300-iteration cap. + +| State | Candidate | Iterations B/C | Total seconds B/C | Gate B/C | +|:------------------|:----------------|:---------------|:------------------|:----------| +| $n=2$ late | gravity | 105/83 | 51.1251/50.2194 | pass/pass | +| $n=2$ late | surface/scalars | 105/300 | 51.6185/244.6878 | pass/fail | +| $n=2$ late | combined | 105/109 | 51.4773/117.3489 | pass/pass | +| $n=2$ tight | combined | 300/300 | 145.5541/283.3130 | fail/fail | +| $n=3$ late | combined | 99/41 | 53.1688/65.6294 | pass/pass | +| $n=1,J=.1$ late | combined | 63/28 | 29.9344/44.3827 | pass/pass | +| $n=1.5,J=.1$ late | combined | 246/95 | 117.9250/101.7603 | pass/pass | +| $n=2$ late | LDU+gravity | 105/78 | 51.4910/60.0450 | pass/pass | +| $n=3$ late | LDU+gravity | 99/41 | 53.1904/35.2005 | pass/pass | +| $n=3$ late | gravity | 99/46 | 53.3147/31.1652 | pass/pass | +| $n=1.5,J=.1$ late | gravity | 246/137 | 118.6171/82.3113 | pass/pass | + +“Gravity” uses the gravity-Schur refinement with independent subsystem structure. “Surface/scalars” wraps the independent base in the new approximate coarse Schur inverse. “Combined” wraps the gravity-refined base. LDU adds existing two-way material/gravity feedback. The total cost includes inverse construction, normalization binding, Krylov setup/solve and one fresh full Jacobian check. It excludes initial model/context setup, separate diagnostic action checks, trial-grid observations and output. Internal Schur Jacobian work is already included in application time; it must not be counted twice. + +The strongest frozen improvements were gravity-only: 41.54% lower total cost for $n=3$ and 30.61% for rotating $n=1.5$. Their endpoint defects changed from $.00282583089$ to $.00259308753$ and from $.00299226031$ to $.00199325392$, respectively. The $n=2$ reduction was only 1.77%, too small for a robust timing claim from one pair. There were no repeated timing samples or confidence intervals, and these are cold one-right-hand-side comparisons, not measured benefits from amortized setup reuse. + +The surface-only $n=2$ candidate ended at $.00567029124$ and failed its gate. The tight $n=2$ pair attained $1.87757546\,10^{-4}$ and $1.21976551\,10^{-4}$, both failing $10^{-7}$. Those are attained accuracies within a finite budget, not arithmetic or discretization floors. All $n=2$ directions with raw-trial observations retained the same largest sampled acceptable step $1/256$; approximately 90.18% of their small-step squared Taylor remainder was Poisson. Quadratic scaling over the sampled range supports a finite-curvature mechanism, without proving its global behavior. + +The new coarse wrapper handles all 25 surface parameters and every generated scalar, with 27 or 28 coarse coordinates. It uses an explicitly unbordered field inverse, since restricting an already scalar-bordered inverse could spuriously constrain the field response. Typed maps and stationary-application contracts are checked at compile time; generation, dimensions and rank are checked after preparation. The wrapper is frozen-state only and must be reconstructed when the Jacobian changes. Coarse construction uses $2k$ full Jacobian actions and $k$ base-inverse actions; each application adds a full Jacobian action and base-inverse application. For the combined candidates, total preconditioner construction ranged from 22.311 to 27.717 seconds; the surface/scalar wrapper alone accounted for 20.515 to 25.578 seconds. These are distinct timing scopes. Reduced outer iterations therefore did not necessarily reduce total time, and the wrapper was not promoted to a live default. + +## Weak coupled radial response + +For an exact derivative and a passing linear gate, $$\lVert Ap+F\rVert\le\eta\lVert F\rVert + \quad\Longrightarrow\quad + F^T Ap\le-(1-\eta)\lVert F\rVert^2.$$ Hence $\eta=.03$ and $.003$ already guarantee infinitesimal descent in the stated norm. Tiny accepted steps motivate finite-step and inverse-response diagnostics, not the assumption that the direction fails to be a linear descent direction. + +At the common centrally constrained, nonrotating $n=3$ capture, baseline, combined and LDU directions all passed $.003$ and had merit slopes within $8\,10^{-6}$ relative of $-\lVert F\rVert^2$. Nevertheless: + +| Inverse | Normalized surface norm | Normalized signed monopole | Largest sampled $\alpha$ | Full-step $\lVert F_t\rVert/\lVert F\rVert$ | +|:---|:---|:---|:---|:---| +| Baseline | $.0681146$ | $-.057083$ | $1/4$ | 7.42157 | +| Combined | $.113440$ | $-.104359$ | $1/16$ | 14.7443 | +| LDU+gravity | $.0287899$ | $-.00681011$ | $1/4$ | 2.77979 | + +The surface norm and signed monopole in this table use normalized solver coordinates, not physical displacement units. The baseline and LDU quarter-step residual ratios were $.702686$ and $.674223$. The complete combined-minus-baseline and LDU-minus-baseline directions had normalized norms 58.44% and 61.89% of baseline, but their $A$ actions were only .3197% and .3364% of $\lVert F\rVert$. Response per unit difference norm was approximately 183–184 times smaller than for the baseline direction. This measures two weak directions, not a condition number, singular vector certificate or exact nullspace. + +Approximately 43% of each squared normalized direction-difference norm was surface, 34% potential and 14% density. Within the squared normalized surface difference norm alone, 97.67% and 96.77% were monopole. These percentages are coordinate-metric contributions, not physical displacement or energy fractions. The effect is a coupled radial-size/field response, not primarily translation or high-$m$ roughness. Freezing the monopole is unjustified: radius is physical, and this control already had a central phase condition. The combined inverse reduced the final surface/scalar defect from $4.64933\,10^{-4}$ to $1.07519\,10^{-6}$ relative to $\lVert F\rVert$, yet worsened finite trials. LDU permitted $9.45535\,10^{-4}$ there while improving the quarter-step trial. Coarse-row recovery alone is therefore not a reliable quality measure for a finite Krylov direction. The reported coarse reciprocal condition $1.31320\,10^{-9}$ belongs to the small approximate matrix, not full $A$. + +If $e_i=Ap_i+F$, the common gate bounds $\lVert A(p_c-p_b)\rVert\le\lVert e_c\rVert+\lVert e_b\rVert$, but does not bound $\lVert p_c-p_b\rVert$ without inverse information. This supplies a mathematical reason why preconditioners can produce comparable linear defects and different nonlinear behavior. The weak-mode magnitude and composition must still be remeasured in the unbordered rotating family; they were not demonstrated there by this frozen control. + + + +# A physically compatible rotating $n=3$ family + +The driver originally calibrated $K$ from a nonrotating Lane–Emden star with $M=R=1$ and then retained $M=1$ as $J$ changed. At $n=3$, nonrotating mass is independent of central density. This physical compatibility question is separate from the measured $n=2$ numerical stall. + +For nonnegative finite-energy density obeying the exact EOS $P=K\rho^{4/3}$, the sharp gravitational-energy inequality implies $$\lvert W\rvert\le3\Pi\left(\frac{M}{M_{\rm LE}(K)}\right)^{2/3}, + \qquad \Pi=\int P\,dV.$$ Together with isolated steady virial balance $2T+W+3\Pi=0$, this gives the necessary condition $$\beta=\frac{T}{\lvert W\rvert}\le + \frac12\left[1-\left(\frac{M_{\rm LE}(K)}M\right)^{2/3}\right].$$ In particular, exact $M=M_{\rm LE}(K)$ cannot support nonzero $T$. Nonzero fixed $J$ at finite inertia requires $T\ge J^2/(2I)>0$. An extra central-density condition cannot repair incompatible mass/EOS inputs; artificial forcing could instead conceal the mismatch. The energy inequality and its Lane–Emden mass relation are documented in equations (1.14)–(1.15) and Theorem 3.1 of Cheng, Cheng and Lin, *Expanding solutions near unstable Lane–Emden stars*; the rotation bound above is our consequence of that inequality and the virial identity. + +The numerical calibration is only approximately critical. Independent DOP853 Lane–Emden integrations using five tolerance/start/step controls found $M_{\rm LE}=.9999999918534539$ for recorded $K=.36393905995216302$, permitting only $\beta\lesssim2.71552\,10^{-9}$. The control spread in mass was $1.95\,10^{-14}$, a numerical sensitivity observation rather than a rigorous interval enclosure. This near-critical input cannot represent the intended moderate rotational support under the continuum assumptions. + +The discrete equations need not satisfy every assumption: independent $\rho/P(h)$, weak EOS closure, negative density, approximate gravity and central forcing all require separate scrutiny. The argument chooses a sensible target; it is not a density limiter or a proof that a discrete residual cannot vanish. + +A separate experiment reduced $K$ by 1%, kept target $M=1$, prescribed $J=.035$ and omitted the central condition. Its necessary bound became $\beta\le.00500000269$. The seed’s central density was multiplied by $.99^{3/2}$ to retain its reference radius, making its seed mass approximately $.99^{3/2}$ while target mass remained one. This is an explicitly off-constraint initial guess, not density rescaling during Newton. The seed Bernoulli scalar still used target mass; that mismatch remained in the initial residual. + +Nonzero prescribed $J$ already breaks the nonrotating homology, so an additional central condition was not added. At fixed $M,K$ an $n=3$ homology can change size and $J$ without specifying a new dimensionless shape/rotation fraction. Neither input $J$ nor an uncalibrated angular velocity should be called a fraction of critical rotation. + +## Matched live field-feedback result + +Two cases shared $n=3$, $J=.035$, $K$ factor $.99$, no central condition, all-$m$ L4/Q2, the gravity-curvature trial path, $\eta=.03$, ten updates and absolute target $10^{-12}$. Their initial completed norm was $.101333855231$. Only the inverse differed. Both exhausted the budget. + +| Quantity | Gravity-Schur | Stellar LDU + gravity-Schur | +|:---|:---|:---| +| Final normalized residual | $1.64730591\,10^{-4}$ | $1.29842469\,10^{-5}$ | +| Newton seconds | 918.881 | 1052.972 | +| Linear seconds | 348.441 | 335.399 | +| Krylov iterations | 534 | 410 | +| Trial preparation seconds | 477.771 | 621.957 | +| Final accepted $\alpha$ | $1/16$ | $1/64$ | +| Energy virial, $P(h)$ | $7.18429\,10^{-5}$ | $1.81141\,10^{-5}$ | +| Force virial, $P(h)$ | $5.23757\,10^{-5}$ | $2.90028\,10^{-6}$ | +| Energy virial, $P(\rho)$ | $6.96335\,10^{-5}$ | $3.51558\,10^{-5}$ | +| Force virial, $P(\rho)$ | $5.01663\,10^{-5}$ | $1.41415\,10^{-5}$ | +| Absolute mass error | $5.45536\,10^{-5}$ | $3.37148\,10^{-6}$ | +| EOS RMS / density scale | $3.99773\,10^{-4}$ | $4.00529\,10^{-4}$ | +| Bernoulli RMS / enthalpy scale | $.00408345$ | $.00372207$ | +| Hydrostatic-gradient RMS | $.656446$ | $.588702$ | +| $T/\lvert W\rvert$ | $.00497575$ | $.00497421$ | +| Angular velocity, model units | $.401764$ | $.401388$ | + +LDU reached a 12.687-times lower residual after the same update count, at 14.593% more Newton time. Linear time fell 3.743%, while preparation rose 30.179%. These are evolving trajectories, not a speedup to the same converged solution. All full linear gates passed, and every rejected trial failed sufficient decrease; neither trajectory had pressure or geometry rejection. q18 density samples were positive without repair, not a global positivity certificate. Thus forcing positivity is not supported as the immediate remedy for this measured live obstruction. + +The LDU trajectory reached $1.30261014\,10^{-5}$ after update seven with a full step; three subsequent $1/64$ steps improved it only .3213%. Its final squared residual was 36.500% Poisson, 34.432% hydrostatic, 20.234% surface and 6.742% mass. Gravity-only ended 80.254% hydrostatic and 10.967% mass. Field feedback is a substantive improvement on this trial path but does not remove small-step stagnation. Its merit over raw-Newton policies has not been established by this pair, since both shared the gravity-curvature strategy. + + + +# Virial consistency: pressure and gravity must be distinguished + +
+ +
Physical meridional density and enthalpy slices of the final rotating n = 3, J = .035, K-factor .99 LDU plus gravity-Schur endpoint. The image shows the actual surface deformation without magnification. The model remains unconverged and fails the overall virial milestone; only four display subdivisions were used for this image.
+
+ +The final rotating LDU model does not meet the overall $10^{-5}$ milestone. Only force virial formed with $P(h)$ passes. Its signed enthalpy virials are approximately $$v_E^h=+1.811408\,10^{-5},\qquad v_F^h=-2.900278\,10^{-6},$$ where both use $\lvert W_E\rvert$ as denominator. Their difference, approximately $2.10144\,10^{-5}$, agrees with the separately measured gravity-energy consistency discrepancy. Replacing $P(h)$ by $P(\rho)$ shifts both signed virials by approximately $+1.70418\,10^{-5}$, yielding $+3.515584\,10^{-5}$ and $+1.414148\,10^{-5}$. + +At one fixed state, pressure changes both signed virials equally: $$v_E^\rho-v_E^h=v_F^\rho-v_F^h + =\frac{3(\Pi_\rho-\Pi_h)}{\lvert W_E\rvert},\qquad + v_E-v_F=\frac{W_E-W_F}{\lvert W_E\rvert}.$$ Consequently, even an arbitrary common pressure-only shift leaves at least one absolute virial at or above half their separation, approximately $1.05072\,10^{-5}$. Pressure adjustment alone cannot make both pass in this frozen state. This is a fixed-state arithmetic observation, not a bound on a newly solved or refined model. + +q14/q18 energy virials agreed to below $8\,10^{-14}$ in both rotating runs. Quadrature stability must not be mistaken for mesh accuracy. The gravity gap remained about $2\,10^{-5}$, while EOS RMS stayed near $4\,10^{-4}$ despite the large algebraic gain. Since neither state solves the discrete equations and neither mesh nor order was varied, these observations do not establish a spatial error floor or identify an exterior-only cause. + +## Pressure-moment identity and the density test space + +For nonnegative $h,\rho$, set $\rho_*=[h/((n+1)K)]^n$ and $\delta\rho=\rho-\rho_*$. Convexity of $P(\rho)=K\rho^{1+1/n}$ gives $$P(\rho)-P(h)=\frac hn\delta\rho+D_P(\rho,\rho_*),\qquad D_P\ge0,$$ because $P'(\rho_*)=h/n$. For a represented density test $v=\sum_i c_i\varphi_i$ and the same mapped quadrature $Q$ used in closure, $$Q\left[\frac hn\delta\rho\right] + =\frac1nQ[(h-v)\delta\rho]+\frac1n\sum_i c_iR_i.$$ Current DGQ2 tests do not generally contain Q3 enthalpy. The first term can therefore remain even at zero weak closure. Compatible DGQ3 tests would permit $v=h$ elementwise, canceling the linear moment at exact closure under that same quadrature. Density would remain an independent field. At finite residual the remaining term is $c_h^TR/n$, which is not directly measured by a normalized Newton norm without undoing residual scaling. + +The convex remainder persists, and is not uniformly quadratic near vacuum: for $n>1$, $P''(\rho)$ diverges as density approaches zero; at $\rho_*=0$, $D_P=K\rho^{1+1/n}$, with exponent $4/3$ for $n=3$. Separate diagnostic quadrature can also retain a difference after assembled moment cancellation. The identity is conditional on nonnegative fields where evaluated and is specific to a constant-$K$ polytrope; a general EOS uses its own thermodynamic derivatives. The contribution of each term has not yet been measured in the final models. This is a consistency hypothesis and a diagnostic to implement, not proof that raising order, clipping density or changing pressure will solve convergence. + + + +# Physical two-dimensional model slices + +The added optional diagnostic exports physical $xz$ and $xy$ cuts, with $yz$ and offsets available, showing independent density and enthalpy and the actual material boundary. It changes no equations or field representation. The important geometric contract is a cut by a physical plane, not a reference plane deformed afterward. + +Each stellar hexahedron is subdivided in reference coordinates into a lattice and tetrahedra. Crossed edges solve the actual mapped plane equation using a safeguarded bracketed root calculation. Fields are evaluated at those reference coordinates on the owning element. Density traces are not averaged between DG elements; supplied connectivity prevents triangulation across vacuum. A plane coincident with an internal face uses the positive-normal-side trace. Boundary curves are separately evaluated from stellar/nonstellar faces, and the undeformed outline is independently cut by the same physical plane. + +The plotter uses equal physical axis scales, no deformation magnification, and signed field values without clipping, logarithms or EOS replacement. Scalar values are rasterized on a 512-by-512 grid before color mapping, avoiding color-interpolation artifacts under triangle refinement. Exterior pixels stay masked. Images distinguish solid deformed and optional dashed reference outlines and mark known unconverged endpoints. + +Straight triangles and boundary chords approximate the curved cut. Neighboring parent elements may choose different face diagonals; small gaps/overlaps can therefore occur on curved interfaces. A covered raster pixel uses the first supplied triangle, without cross-element averaging. Tangencies or multiple crossings between subdivision vertices can be missed. These limits are explicitly documented and prevent using the image as a pointwise positivity certificate or interpreting every seam as a physical feature. + +## Display-refinement measurements + +Four unchanged-state replays compared four/eight subdivisions for final $n=2$ and rotating $n=1.5,J=.1$ states, in both planes. Source buffers remained bitwise unchanged, and both resolutions used common axes and color ranges. + +| Model/plane | Area change | $\int\rho\,dA$ change | $\int h\,dA$ change | Perimeter change | +|:---|:---|:---|:---|:---| +| $n=2$, $xz$ | $.12333\%$ | $.07667\%$ | $.07036\%$ | $.03065\%$ | +| $n=2$, $xy$ | $.12333\%$ | $.07667\%$ | $.07036\%$ | $.03065\%$ | +| $n=1.5,J=.1$, $xz$ | $.12495\%$ | $.00319\%$ | $.08797\%$ | $.03165\%$ | +| $n=1.5,J=.1$, $xy$ | $.12395\%$ | $.01249\%$ | $.07838\%$ | $.03088\%$ | + +These are relative absolute changes of displayed straight-triangle measures, including overlaps; they are not curved-FEM integrals, three-dimensional mass, virial errors or a mesh convergence study. Sampled chord errors decreased about fourfold and axial extents agreed. The rotating meridional model was smoothly oblate, with equatorial extent approximately $\pm1.12084$ and polar extents $-.988807,+.988480$; the equatorial cut was approximately circular. Its sampled negative density minimum $-8.59\,10^{-4}$ was retained. The $n=2$ outline was nearly spherical at radius 1.0049. Only the latter met its recorded virial milestone, and both remained unconverged algebraically. + +
+ +
Same-state physical slice comparison for the unconverged rotating n = 1.5 endpoint at four and eight display subdivisions. Axes and color ranges are shared. This checks the displayed geometry and fields; it is not a change of the FEM mesh, a three-dimensional integral check or evidence of equilibrium.
+
+ +Final $n=3$ meridional/equatorial plots show strong central concentration and modest oblateness. Those exports used four subdivisions only. The separate $n=2/n=1.5$ display study must not be described as refinement verification of the $n=3$ images. Visual plausibility supplies an additional diagnostic, not physical validation. + + + +# What to carry forward and what remains unresolved + +The recent work establishes a usable native harmonic prescription, diagnoses a specific independent-density projection defect, measures order effects without changing density’s physical status, adds reproducible physical slices, and shows real frozen-linear and live-nonlinear gains. It does not establish a converged moderately rotating $n=3$ model, a calibrated 50%-critical sequence, or the ultimate 99%-critical ambition. No ESTER comparison was performed. Global virial balance at $10^{-5}$ remains a primary acceptance milestone, accompanied by internal EOS/hydrostatic checks; an attractive image or a small subset of residuals cannot replace it. + +The evidence supports the following staged continuation. + +1. Retain gravity-Schur as the strongest measured frozen linear-cost comparator and stellar LDU plus gravity as the better live candidate in the tested rotating $n=3$ configuration. The latter is conditional on the shared gravity-curvature trial path. Compare complete time to a common convergence and physical target before selecting a production default. Do not promote the new coarse surface/scalar wrapper merely because it lowers iterations. + +2. Remeasure coupled direction differences in the unbordered compatible rotating family, including true defects, field/modal composition and finite trial remainders. The centrally constrained nonrotating weak-monopole evidence is a motivated hypothesis, not a transferred diagnosis. A coupled trust-region or regularized step with a physical displacement scale is a reasonable swappable experiment. Radius must remain free, and surface/interior remain simultaneous. + +3. Strengthen the material response in controlled stages if its frozen block defects warrant it. Existing material preconditioning already contains the exact prepared enthalpy-to-density derivative; it is incorrect to claim that cross derivative is missing. Exact local mapped DG density mass inversion is a narrow candidate (27 Q2 coefficients per hexahedron). A stronger stationary enthalpy mass inverse must respect H1 connectivity and replaced boundary rows. Only then consider a more costly EOS-aware material/gravity Schur approximation. Such algebraic eliminations belong inside the inverse and need not make density a derived nonlinear field. + +4. Measure pressure-moment and gravity-energy consistency separately. Export the linear pressure moment, convex remainder, quadrature differences and both signed virials on identical states. Resolve the gravity relation/source and energy/force discrepancy with interior accuracy as the goal; the exterior need only be accurate insofar as it supports correct interior fields. The current observations do not justify allocating effort solely to exterior resolution. + +5. Define a physically compatible $n=3$ sequence in mass/EOS/rotation and calibrate a dimensionless rotation measure and critical reference. Input $J$ alone is not that calibration. Keep central homology handling optional and separate from incompatible physical specifications. + +6. After useful convergence is obtained, perform mesh and harmonic-degree studies, measure excluded surface force in a meaningful norm, and compare local profiles with analytic/nonrotating controls and eventually ESTER. Density-order results remain available as consistency controls; positivity intervention and logarithmic refactoring are deferred per the user’s direction. + +A closure-consistent seed and an affine-target closure correction are additional untested hypotheses, not completed remedies. At fixed admissible enthalpy and geometry, exact local density mass solves can populate independent density or correct its weak moments. Any trial correction must preserve the original Newton tangent and be freshly verified. If combined with gravity correction, density must be corrected first and gravity rebuilt against its original affine target afterward. That operation can change mass, inertia and mechanical balance, cannot repair unchanged inadmissible enthalpy, and must pass the full residual acceptance test. For fractional $1 + +# Consolidated source, architecture, and handoff + +## What the consolidation changes + +The consolidation uses the cumulative research revision `d7acb0d399aac5ac4c36d834a954c1c959042b81` as its numerical source baseline. This revision descends from the former local main, `75cc638`, through the baseline snapshot `80cc3d8` and the successive controlled investigations. Most topic branches were incorporated by cherry-pick during those investigations. Their tips therefore need not appear as ancestors of the cumulative branch for their changes to be present. The consolidation checks both ancestry and final source content; it does not indiscriminately merge every experimental tip. + +The integration branch is `integration/research-consolidation-20260920`. At the source checkpoint `b1fe359`, the additions after the numerical baseline are the preserved user formatting change, `6950dc6`, and six commits importing the historical log-density assessment. The latter begin at `549b1a7` and end at `b1fe359`; they add only `experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md` and `experiments/log_density_reference_model.py`. They do not implement log-density coordinates in the FEM solver. The consolidation itself adds no new numerical behavior. Its other changes organize evidence, document results, and retain useful source and tests. Local main is advanced to the validated consolidation descendant; there is no remote push and no deletion of historical research branches. The final integration commit and validation record identify the exact published local checkpoint. The final delivery is identified by the local Git tag `research-handoff-2026-09-20`; the tag records the complete report and validation as well as the integrated source. + +The original programming checkout was initially on `research/numerical-strategies-20260917` at `10adf7e`, rather than on main. Its only tracked modification was declaration formatting in `libmeanfield/interface/quadrature/mfem.cppm`; that modification is preserved in `6950dc6`. This distinction matters because replacing the original checkout with a research tree without first preserving the edit would lose user work. Research reports and raw evidence are also preserved, including unsuccessful experiments. Moving an artifact is not a revision of its original producer, parameters, or hash. + +## Architectural boundary between physics and numerics + +The intended public construction remains physics forward: an EOS, a surface condition, and explicit integral specifications describe the stellar model. A separately owned discretization describes the finite elements, normalization, surface coordinates, and displacement extensions. An owning preconditioner prescription and an owning linear-backend configuration are prepared only after the physical problem exists. The solve context then owns the accepted state, prepared operators, prepared inverse, and nonlinear transaction machinery. This separation is valuable in the rewrite: changing a numerical approximation must not silently add a central-density constraint, change an EOS relation, or select a different physical family. + +The current implementation uses C++23 modules, concepts, generated block manifests, and typed prescriptions. Compile-time checks cover structural facts: supported EOS relations, derivative availability, field and block roles, admissible prescription combinations, and backend application contracts. Mesh-dependent facts remain runtime checks: dimensions, rank of a sampled harmonic basis, actual topology, preparation generations, finite parameters, support layout, and distributed agreement. A type cannot prove that a sampled geometry is admissible or that a numerical inverse is accurate, so those checks remain explicit and measurable. + +There are three distinct levels of composition: + +1. A physical model compiles its field equations and specifications into value and residual blocks, including generated scalar coordinates, appended integral equations, and explicitly replaced carrier rows. + +2. A discretization supplies coordinates and maps for these blocks. Physical operators are prepared on the accepted geometry and state; their actions and cross derivatives remain the source of truth. + +3. A numerical recipe approximates the inverse or modifies a trial path, while the complete residual and physical diagnostics judge the resulting state. An exact coarse row in an approximate inverse is not an exact solution of the full problem. + +Important source boundaries are `libmeanfield/interface/models/`, `libmeanfield/interface/equilibrium/stellar_discretization.cppm`, `libmeanfield/interface/operators/root_manifest.cppm`, `libmeanfield/interface/operators/stellar_equilibrium_compiler.cppm`, `libmeanfield/interface/preconditioning/stellar_recipe.cppm`, and `libmeanfield/interface/solver/stellar_context.cppm`. The experiments directory owns case selection, historical studies, optional probe recipes, artifact verification, and visualization. It should not be mistaken for a minimal production API. + +## Fields, geometry, and the preferred surface representation + +The retained baseline has independent discontinuous Q2 density and independent continuous Q3 enthalpy. Gravity uses the existing RT2/Q2 mixed pair; displacement is Q3, while the reference mesh geometry is Q4. The density order is now independently configurable at compile time through `MEAN_FIELD_DENSITY_POLYNOMIAL_ORDER`; its default is two. A uniform order increment remains a separate existing option. The closure quadrature query now covers both density-mass and EOS-source contributions when density order changes. Neither this configuration mechanism nor the completed order study establishes that a higher density order cures the nonlinear stall. + +The preferred research surface is the native real spherical-harmonic prescription with maximum degree four and every azimuthal order: $$q_i = \sum_{\ell=0}^{4}\sum_{m=-\ell}^{\ell} + a_{\ell m}Y_{\ell m}(\widehat{\boldsymbol r}_i), + \qquad N_a=25.$$ The coefficients have dimensions of length. Negative real orders encode sine modes and positive orders encode cosine modes; neither odd degree nor nonzero $m$ is silently removed. A maximum absolute azimuthal order can be specified explicitly. For example, restricting the shape to $m=0$ leaves five coefficients at degree four, but does not by itself make the interior three-dimensional fields axisymmetric. + +The harmonic prescription supplies values at reference trace nodes. The existing Q3 trace and volume extension then realize the displacement in finite-element space. The resulting surface is therefore not an exact analytic harmonic graph between nodes. Q4 reference geometry and Q3 incremental displacement coexist deliberately; changing their orders is a separate approximation study. The higher displacement-order trial remains a recorded negative result, not the active default. + +Surface coefficients and interior fields participate in the same Newton system with their cross derivatives. The reference trace metric provides compatible primal and dual coefficient scaling. It does not approximate the complete physical response of the interior to a surface change. The current native harmonic implementation requires one MPI rank, consistent with the retained radial volume extension. Tests of collective rejection on two ranks do not demonstrate distributed harmonic solving. + +The sandbox selects the harmonic prescription explicitly. Existing *two-argument* calls to `makeStellarDiscretization` still select `NodalRadialSurface`; their behavior is preserved for compatibility. This is an intentional distinction between the preferred example and the backward-compatible factory default. + +## Defaults, explicit controls, and experimental candidates + +The following table records the source checkpoint rather than proposing new settings. A capability entering main is not automatically promoted to a default. + +| Layer | Retained behavior and interpretation | +|:---|:---| +| Model example | The sandbox is $n=1$, $J=0$, with fixed total mass and angular momentum. Central density initializes the Lane–Emden seed; it is not an additional $n=1$ equation. Central-density constraints remain explicit model specifications for studies that require them. | +| Surface | Sandbox: all-$m$, degree-four harmonics. Two-argument library factory: nodal radial surface. Both compose with the existing interior radial and fixed-infinity vacuum extensions. | +| Normalization | The sandbox selects `PhysicalRieszDiagonal` on reference geometry with a fixed branch mass/radius scale. Modal surface coordinates use their pulled-back reference trace metric. | +| Library linear options | `FGMRESOptions` has restart length 50. This default was not globally changed to 160. | +| Sandbox solve | Explicit restart 160; at most 30 Newton updates; relative nonlinear tolerance $10^{-8}$, absolute tolerance zero; linear relative tolerance $0.03$ and cap 200. Geometry validation/backtracking remain. | +| Diagnostic controls | Bare driver defaults retain nodal surface, restart 40, independent structure inverse, and baseline trial strategy. Campaign manifests explicitly override these. Reproduction requires the recorded manifest, not assumptions about the bare driver defaults. | +| Linear reuse | A nonzero incoming correction is checked with the current operator and RHS. It is reset if its true defect is nonfinite or larger than the cold-start RHS norm. Final convergence is checked against the fresh full-system true residual. | +| Default structure inverse | `IndependentStellarSubsystems` applies material/surface and gravity approximations independently at that level. Generated scalar-border completion is still present. | +| Default material inverse | `SurfaceThenMaterialTriangular` uses diagonal surface, enthalpy, and density inverses. The prepared EOS $h\rightarrow\rho$ derivative and surface-to-material cross actions are already included; the approximation is not fully field diagonal. | +| Default gravity inverse | `GravityApproximateLDU` uses a matrix-free Chebyshev mass inverse of order five with 20 power iterations, and three fixed AMG cycles for the potential surrogate. | +| Gravity refinement | `GravityConsistentSchurLDU` is an available typed policy. Its library default is one correction sweep with relaxation one; the diagnostic option defaults to relaxation $0.5$. Exact run settings must come from the manifest. | +| Live research recipes | `gravity-schur`, `stellar-ldu`, and `stellar-ldu-gravity-schur` are explicit experiment recipes. The last is the stronger measured live candidate for the tested compatible rotating $n=3$ family with a gravity-curvature trial path. It is not a demonstrated universal replacement for the default. | +| Default Newton path | `SampledEnthalpy`, `MonolithicGravity`, and `UncorrectedTrial`. No polynomial positivity repair, gravity elimination, or second-order correction is enabled implicitly. | +| Optional Newton paths | Polynomial admissibility/repair, gravity-curvature correction, reduced gravity, and coupled/damped trial correction remain explicit typed research choices with their checked contracts. Their presence does not imply successful high-rotation convergence. | +| Coarse surface/scalar inverse | Frozen experiment only. Its base inverse is explicitly unbordered and its state/generation binding is checked. Changed Jacobians require reconstruction. Setup and application expense prevent promotion on present evidence. | +| Continuation | Explicit diagnostic initialization validates coordinates, layout, mesh, and finite data; prepares and refreshes before commit; clears stale linear guesses; and rolls back on failure. Unconverged snapshots need an explicit experimental override. | +| Slices | Optional physical-plane output evaluates the independent fields inside their owning elements on the deformed stellar geometry. Signed values are retained. Display refinement is not mesh convergence. | + +The default material response can be written schematically, in physical block notation, as $$\begin{aligned} + \delta a &= D_a^{-1} b_a,\\ + \delta h &= D_h^{-1}(b_h-A_{ha}\delta a),\\ + \delta\rho &= D_\rho^{-1} + (b_\rho-A_{\rho a}\delta a-A_{\rho h}\delta h). +\end{aligned}$$ Here the diagonal inverses are approximate mass/trace inverses, whereas $A_{\rho h}$ is the actual prepared EOS cross action. At the next level the default structure inverse omits direct material–gravity feedback. The optional stellar LDU first obtains a material response, sends its source and geometry effects into gravity, and recovers gravity feedback into material. Gravity-only Schur refinement improves the gravity block; it does not itself add this EOS/material feedback. These distinctions explain why the later ablations compare multiple policies rather than treating all coupling changes as a single intervention. + +## Branch disposition and retained provenance + +The table below refers to the research tips retained from the investigation. “Included” means ancestry or inspected patch/content equivalence, not necessarily a merge commit with that branch tip as a parent. The six log-assessment commits were imported during consolidation. All named historical branches are retained; no branch deletion is part of this work. + +| Branch | Tip | Disposition | +|:---|:---|:---| +| `main` | `75cc638` | Former main checkpoint; ancestor of the consolidation baseline. | +| `research/baseline-20260916` | `80cc3d8` | Included baseline snapshot of user optimizations and initial diagnostics. | +| `research/central-constraint-20260916` | `2e19c2c` | Included equivalent commits: central density as seed data in the n1 example. | +| `research/combined-20260916` | `95d9fba` | Included cumulative first controlled campaign. | +| `research/experiment-harness-20260916` | `899cb18` | Included diagnostic and physical acceptance reporting. | +| `research/krylov-20260916` | `7127383` | Included warm-start true-residual safeguard and controls. | +| `research/krylov-window-20260916` | `1f9184b` | Included restart160 sandbox setting; report identical. | +| `research/gravity-coupling-20260916` | `db0ddd3` | Failed one-way completion remains historical; no old driver/default promotion. | +| `research/diagnostic-campaign-20260917` | `7a07cae` | Included pressure/shape attribution campaign. | +| `research/pressure-obstruction-20260917` | `c6572d3` | Included pressure, trace, and nodal diagnostic exports. | +| `research/shape-curvature-20260917` | `b24e6e0` | Included frozen shape/field curvature separation. | +| `research/numerical-strategies-20260917` | `10adf7e` | Included typed optional strategy integration and validation. | +| `research/polynomial-positive-newton-20260917` | `9aefb36` | Included explicit experimental certificate/repair; disabled by default. | +| `research/polynomial-endpoint-certification-20260917` | `cc6f4b0` | Included read-only polynomial endpoint certification. | +| `research/gravity-curvature-correction-20260917` | `3762c74` | Included checked auxiliary gravity correction; opt-in. | +| `research/gravity-reduced-newton-20260917` | `d25d141` | Included reduced formulation and full-system checks; opt-in. | +| `research/gravity-compact-application-20260917` | `ff1b889` | Included compact canonical gravity action and guarded fallback. | +| `research/gravity-prepared-roundoff-20260917` | `782827c` | Included prepared consistency arithmetic scale. | +| `research/gravity-trial-diagnostics-20260917` | `7f4b068` | Included current-attempt diagnostics and verified records. | +| `research/gravity-direction-diagnostics-20260917` | `47775a9` | Included relift accounting before descent rejection. | +| `research/preconditioner-benchmarks-20260918` | `dc1da62` | Included benchmark methodology, implementation, and results. | +| `research/gravity-consistent-schur-20260918` | `c10d019` | Included fixed-work mass-consistent gravity Schur policy/tests. | +| `research/preconditioner-cached-cross-actions-20260918` | `07769e7` | Included reuse of prepared gravity force in cross preconditioning. | +| `research/coupled-shape-coarse-20260918` | `4a6c1b5` | Included optional frozen coarse experiment and tests; not default. | +| `research/nonlinear-step-obstruction-20260918` | `2944dd5` | Included nonlinear path, direction, and remainder diagnostics. | +| `research/coupled-trial-correction-20260918` | `46249d0` | Included bounded coupled correction and measured limitations. | +| `research/damped-coupled-trial-20260918` | `db854c0` | Included bounded damping/eligibility controls; no default promotion. | +| `research/shape-field-weak-response-20260918` | `6ceda61` | Included weak-response study, preserving Q3 baseline. | +| `research/shape-field-displacement-q4-20260918` | `785a7ae` | Q4 source change excluded; final negative report/comparison tools included. | +| `research/reduced-smooth-surface-20260918` | `2171b8b` | Included frozen reduced-space study and provenance audit. | +| `research/spherical-harmonic-surface-20260919` | `1148c09` | Included native coupled harmonic prescription and metrics. | +| `research/negative-density-diagnosis-20260919` | `7ed4d23` | Included localization, independent projection, and interval evidence. | +| `research/density-order-study-20260919` | `ef76e0c` | Included independent order configuration/closure quadrature and study. | +| `research/log-density-assessment-20260919` | `90580b2` | Imported as deferred assessment/reference script through b1fe359; no FEM change. | +| `research/linear-coupling-study-20260919` | `0e6688f` | Included frozen surface/scalar Schur and unbordered adapter tests. | +| `research/model-slices-20260919` | `5da3f42` | Included physical slice exporter, plots, bounds fixes, and tests. | +| `research/n3-compatible-family-20260919` | `1d746b0` | Included explicit mass-compatible physical-family control. | +| `research/n3-linear-ablation-20260919` | `e78af98` | Included predeclared matched gravity-only n3 comparison. | +| `research/gravity-coupling-ablation-20260919` | `38c7184` | Included gravity-only attribution for rotating higher-index state. | +| `research/n3-live-field-feedback-20260919` | `1ed03f5` | Included matched live field-feedback control. | +| `research/coupled-convergence-study-20260919` | `d7acb0d` | Cumulative numerical baseline for consolidation. | +| `integration/research-consolidation-20260920` | `b1fe359` | Consolidation source checkpoint: preserved formatting and imported assessment; reports follow. | + +Several apparently unmerged commits are only isolated-branch registration changes. Commits `4a6c1b5`, `e599900`, `2c64a92`, `5952691`, and `a773c9a` add test or module registrations already present in the cumulative CMake and module exports. Reapplying them would add no capability. Similarly, the surface/scalar Schur implementation from `1d95602` is included as `2398118`, and the initial slice implementation from `966a4d4` is included as `9bac57a`. Final topic and integration source were compared directly where patch IDs differed. These are documented equivalences, not assumed equivalences from similar commit titles. + +## Adoption, retention, and exclusion rationale + +#### Adopted infrastructure. + +The consolidation retains the optimized prepared-operator baseline, reference-table and mapping caches, linear true-residual safeguards, checked operator/strategy interfaces, cache-aware preconditioner cross actions, consistent gravity Schur option, native harmonic surface, independent density-order configuration and corrected closure quadrature, transactional continuation, physical slice output, and artifact auditors. These are reusable capabilities with explicit contracts. In particular, the pre-existing performance work in `80cc3d8` predates many of the numerical experiments; the large diff from former main must not be attributed entirely to the later investigations. + +#### Retained as experiments. + +Polynomial admissibility/repair, gravity elimination, auxiliary curvature paths, coupled/damped trial corrections, and coarse surface inverses remain available for controlled comparison. Their type-level integration and tests are useful even when a particular campaign did not justify default promotion. The current user preference is to improve broad convergence without forcing density positivity. The baseline path therefore remains unchanged, and the newly imported log-density document is historical analysis, not an active implementation direction. + +#### Not adopted as active numerical behavior. + +The Q4 displacement change is excluded because the controlled trial did not supply a convergence improvement warranting the changed approximation and cost. Its report and analysis tools are retained with Q3 active. The early one-way material-to-gravity completion is also excluded from the current driver. At the matched $n=3$, $J=0$ seed with restart160 and cap200, it changed the true relative defect from $0.0402590$ to $0.0381781$, while both solves failed their $0.03$ gate and solve time rose from $98.915$ to $117.712$ s. The correction norm grew from $0.757644$ to $1.540825$ and the sampled thermodynamic step limit decreased from $0.152435$ to $0.0913461$. That counterexample concerns one particular triangular composition; it does not invalidate later two-way LDU with a stronger gravity approximation. + +#### Not implemented. + +A coupled trust region or regularized step, exact local DG density-mass inverse, stronger H1 enthalpy-mass inverse, EOS-aware field Schur inverse, and projected surface-equipotential equation are proposals, not completed solver features. Changing the physical $n=3$ family or calibrating a critical-rotation fraction is likewise distinct from improving an inverse. No consolidation wording should turn these recommendations into results. + + + +## A practical contract map for the rewrite + +The rewrite need not reproduce every experimental wrapper. It should reproduce the boundaries that made comparisons meaningful: explicit physical requirements, coordinates whose transformations are known, prepared operators whose validity is checkable, and a solver that reports what actually happened. The following contracts separate implementation responsibility from current algorithm choices. Source paths identify useful reference implementations, not a requirement to preserve every class name. + +| System | Contract to preserve and initial verification | +|:---|:---| +| EOS | Declare relations by input/output quantities and derivative support. An operator asks for the relation it needs rather than assuming every EOS has density-from-enthalpy. Type checks cover relation availability and units; evaluation checks cover finite inputs and the actual physical domain. Keep EOS variable choice separate from whether a field is independent. Verify derivatives on their valid domain and relation consistency for the polytrope. References: `interface/eos/` and `interface/material/thermodynamic_equations.cppm`. | +| Domain/support | Identify stellar, vacuum, interface, and outer-boundary support independently of coordinate storage. Density exists on stellar support; its zero exterior is not a vacuum density unknown. Generate restrictions/embeddings from support ownership, including shared trace DOFs. Verify coverage, no duplicated ownership, and interface consistency. References: `interface/utils/domain.cppm`, `interface/field/field_mfem.cppm`, and the deformation ownership layer. | +| Fields and blocks | Let typed field quantities carry approximation family, order, support, and units; let blocks distinguish value coordinates from residual duals. Generate layouts and equation contributions from these roles. Do not identify blocks only by historical numeric offsets or assume every field pair has equal dimensions. Verify rectangular coupling, restriction/injection identities, and generated specification borders. References: `interface/field/`, `interface/utils/blocks.cppm`, and `interface/operators/root_manifest.cppm`. | +| Quadrature | A form query declares the term, role, trial/test/coefficient orders, support, and mapping context. Runtime policy resolves actual rules and records them. A nonlinear EOS or curved map cannot be made polynomial merely by an order formula. Keep production, preconditioner, projection, and diagnostic roles distinct. Verify closure mass/source coverage and compare diagnostic rules independently. References: `interface/quadrature/` and field form descriptors. | +| Mapping | Consume mesh geometry, displacement data, and exterior-map policy; return physical coordinates, full map Jacobian/inverse/determinant, measures, normals, and coherent directional variations. Implement scalar pullback and H(div) Piola transformations consistently with the same map. Mapping must not depend on spherical harmonics. Verify identity/affine maps, transformation round trips, face orientation, flux/divergence identities, and finite-difference variations before caching. References: `interface/mapping/` and `impl/mapping/`. | +| Deformation | Compose a surface prescription with independent interior and vacuum extensions. Supply displacement, tangent, transpose pullback, and derivative of the pullback. A harmonic parameter basis and a nodal basis should produce equal physical residual/Jacobian actions when they represent the same displacement. Check trace continuity, adjoint identities, mode rank, ownership, and stale mesh generations. References: `interface/deformation/` and `interface/equilibrium/stellar_discretization.cppm`. | +| Physical operator | Separate mathematical residual/Jacobian actions from preparation and caching. Cache by the complete set of dependencies, including geometry, density, enthalpy, gravity, rotation, and relevant specifications. Use the same discrete equations and rules for residual and linearization. Verify isolated blocks, cross derivatives, and the full coupled directional derivative; include physical boundary row replacement. References: `interface/operators/` and `impl/operators/`. | +| Normalization | Keep explicit maps for state coordinates and residual duals. Derive physical scales from declared reference quantities and Gram factors from a documented geometry/basis. Every normalized Jacobian, inverse, restriction, and diagnostic must use the corresponding maps. Verify round trips, physical/normalized action equivalence, and pullback metric consistency, including harmonic coefficients. References: `interface/normalization/`. | +| Preconditioner | Separate an owning unbound recipe from its prepared, problem-bound inverse. State linear/stationary application guarantees in the type contract where possible; check dimensions and generations at runtime. Distinguish unbordered field approximation from generated scalar completion. Measure setup, refresh, actions, fresh true defect, and complete nonlinear cost. References: `interface/preconditioning/` and the experiment recipe `experiments/plateau_newton_preconditioner.hpp`. | +| Solver | Keep accepted state authoritative. Trial preparation, evaluation, optional correction, acceptance, commit, and rollback form a transaction. Invalidate stale caches/guesses explicitly. A backend recurrence residual is not the full-system true residual; a reduced solve report must retain its coordinate-space identity. Report failure stages, update budgets, step lengths, all attempted work, and actual stopping thresholds. References: `interface/solver/`. | +| Validation and output | Preserve input/source/mesh hashes and actual commands; separate numerical convergence, artifact integrity, and physical acceptance. Evaluate both pressure routes, both virial forms, mass/angular momentum, EOS, hydrostatic/Bernoulli observations, and resolution trends. Slices show the actual deformed star with signed fields and explicitly label unconverged states. References: `experiments/` auditors, physical diagnostics, and slice exporter. | + +In the source references inside this table, `interface/` and `impl/` are relative to `libmeanfield/`. The table describes contracts to retain; a proposed verification is not a statement that every possible backend, EOS, geometry, or MPI configuration has already passed it. + +### Mapping before deformation-specific optimization + +Given the systems already implemented in the rewrite—EOS, domains, fields/blocks, and quadrature configuration—mapping is a natural next foundation. Write it in terms of the complete element-to-physical map $\boldsymbol x=\mathcal T(\boldsymbol\xi;\boldsymbol a)$, with $J=\partial\mathcal T/\partial\boldsymbol\xi$ and determinant $j$. The ownership of each map factor must be explicit: reference element, computational mesh, displacement, and exterior compactification must each be applied exactly once. The current slice exporter specifically avoids applying displacement a second time when evaluating the physical model. For the orientation convention under which the map is admissible, the standard transformations include $$\nabla_x u = J^{-T}\nabla_\xi\widehat u, + \qquad + \boldsymbol v_x = j^{-1}J\widehat{\boldsymbol v}, + \qquad + \nabla_x\!\cdot\boldsymbol v_x + = j^{-1}\nabla_\xi\!\cdot\widehat{\boldsymbol v}.$$ Measures and oriented face transforms must use the same determinant and orientation convention. The rewrite should make this convention explicit rather than mix absolute integration weights with signed Piola factors. Useful derivative identities are $$\delta J^{-1}=-J^{-1}(\delta J)J^{-1}, + \qquad + \delta j=j\,\operatorname{tr}(J^{-1}\delta J).$$ These identities are small, independently testable building blocks for the shape derivatives of mass, gravity, pressure, and surface force. They are not a replacement for checking the derivative of the fully composed map. + +The surface prescription should sit above this layer and supply a consistent displacement and its variations. It is then possible to change nodal parameters to harmonics, or change an extension, without rebuilding the meaning of every physical operator. A reference metric for harmonic normalization likewise belongs to the parameter-to-trace map, not to an arbitrary interior displacement norm. + +Cache only after these contracts are established. The present mapper caches computational-mesh geometry separately from current displacement and compactification state. Mesh/node sequence changes invalidate cached mesh data; raw coordinate edits require the appropriate mesh-update signal or explicit clearing. Mesh lifetimes and thread ownership are part of this contract. Reusing a pointer or a quadrature index is not enough to prove that a prepared value is current. + +### Normalization and inverse composition + +Let $D_x$ map physical state coordinates to normalized coordinates and $D_F$ map physical residual duals to normalized residuals. With these maps fixed during a linearization, $$\widehat x=D_x x,\qquad + \widehat F(\widehat x)=D_F F(D_x^{-1}\widehat x),\qquad + \widehat J=D_F J D_x^{-1}.$$ A physical inverse approximation $B$ correspondingly acts as $$\widehat B=D_x B D_F^{-1}.$$ This explicit distinction prevents using the state scaling on residuals or accidentally applying a scalar-border completion twice inside a coarse Schur construction. If a future normalization changes with state, its lifecycle and any derivative interpretation need separate definition; the current fixed reference normalization should not be silently replaced by such a scheme. + +For a nonlinear surface map $\boldsymbol u=E(\boldsymbol a)$ and physical force dual $\boldsymbol f$, the parameter residual contains $E'(\boldsymbol a)^T\boldsymbol f$. Its directional derivative contains both the pullback of $\delta\boldsymbol f$ and $\delta[E'(\boldsymbol a)^T]\boldsymbol f$. The latter vanishes for a fixed linear parameterization but cannot be omitted from the general prescription contract. This is one reason the existing deformation interface exposes a pullback derivative separately from a forward Jacobian action. + +### Acceptance and evidence should remain separate + +The rewrite should report at least three distinct outcomes: whether a linear equation met its true-residual target, whether the nonlinear iteration met its stated stopping criterion, and whether the model met physical validation requirements. A campaign artifact can be internally consistent while recording failure of any of these. A single passing virial number is not sufficient when the energy and force forms, or the enthalpy- and density-based pressure routes, disagree. The current target remains a well-resolved, physically credible, rotating $n=3$ model with virial balance within $10^{-5}$; the consolidated code does not yet claim that milestone or a calibrated 50-percent-critical rotation rate. + +# Evidence organization and reproducibility after consolidation + +The consolidation preserved rather than deleted the accumulated working results. Seventy-one explicitly inventoried moves relocated 6,228 regular files containing 12,468,217,917 bytes. Every regular file was hashed before and after its move; symbolic links and directory structure were also compared. The complete inventory is `reports/archive/move_manifest.json`, with the original plan and organizing script preserved beside it. The verifier `reports/verify_archive.py` can recheck the archived contents on this machine or against a supplied repository location after transfer. + +The principal directory layout is: + + reports/ + MeanField_research_handoff_2026-09-20.tex + MeanField_research_handoff_2026-09-20.md + MeanField_research_handoff_2026-09-20.pdf + branch_consolidation.md + figures/ selected versioned scientific figures + sources/ original audits copied without modification + prior/ earlier optimization report and README + archive/ inventories, provenance, historical experiments + working/ + research/diagnostics/ complete September research campaigns + legacy_results/ earlier result directories and loose tables + legacy_misc/ ZIPs, patches, scripts, old test output + legacy_documents/ originals of earlier report files + report_builds/ LaTeX intermediates and rendering checks + consolidation_2026-09-20/ current validation records + +The raw working tree is ignored by Git, except its directory guide. This is a storage policy, not evidence deletion: the complete raw files remain local, including old binaries and unsuccessful trials. The smaller source, reports, selected figures, case manifests, audit code and relocation inventory are versioned. A recipient needs an explicit copy of `working/` to rerun saved-state analyses from this archive; a Git clone alone is insufficient. + +The original project-workspace `diagnostics` directory is now a symbolic link to `working/research/diagnostics/` in the main repository. Thus the absolute paths already recorded in case manifests and provenance continue to resolve on this machine. Original provenance was not rewritten. When moving the archive to another machine, produce a new execution manifest with rebased paths and retain the original manifest as provenance. + +Existing CMake build trees were intentionally left at their original paths: their caches and C++ module artifacts contain absolute source/build identities. They are now ignored by Git and documented as local caches. Validation used the isolated research worktree’s rebuilt binaries; retained binaries in the original checkout can be stale and must be rebuilt before use. Moving them would create an avoidable rebuild or stale-cache ambiguity. Research worktrees and branch references are retained. Local IDE and tool configuration remains in place. No synced file under the ChatGPT project’s `sources/` directory was modified or relocated. + +The only originally uncommitted tracked source change was quadrature declaration formatting. Its exact patch is archived and its content is in commit `6950dc6`. The original checkout’s edit is also retained in a named Git stash when that checkout is switched to main. The consolidated source already contains the edit, so replaying the stash on main is neither necessary nor advised. + +## Rebuilding and reproducing results + +The compiled implementation requires a C++23 compiler with module support, CMake, Ninja, MPI, MFEM and the other dependencies identified by `CMakeLists.txt`. The checked local build uses Homebrew LLVM rather than Apple’s default compiler, Release mode, interprocedural optimization, density order two and zero uniform order increment. Exact compiler/dependency paths remain in campaign provenance and the consolidation build record. A fresh build must be configured for its own source checkout; reusing a different checkout’s CMake cache is not a valid shortcut. + +The experiment entry point is `experiments/run_plateau_campaign.py`. Its arguments specify source, build, mesh, case manifest and a new output directory. It requires committed tracked source and refuses to overwrite an existing campaign directory. It copies the executable, records exact commands and input hashes, and runs cases serially. Case manifests state surface choice, constraint selection, tolerances, iteration budgets and preconditioner/trial policy. Starting the bare diagnostic executable without those options does not reproduce the preferred harmonic configuration. + +The reference recipes include `experiments/spherical_harmonic_surface_cases_2026-09-19.json`, `experiments/convergence_linear_cases_2026-09-19.json`, `experiments/convergence_linear_followup_cases_2026-09-19.json`, `experiments/convergence_screen_cases_2026-09-19.json`, and `experiments/n3_compatible_family_cases_2026-09-19.json`. The last file specifies a compatible physical control, not an interchangeable numerical tuning of the original $n=3$ model. Follow the saved-state chains and their exact coordinate metadata rather than manually transplanting a vector between discretizations. + +Saved campaign artifacts can be checked with the study-specific auditors before any new solve. The final combined study uses `experiments/summarize_coupled_convergence.py`; earlier studies have separate pressure, coupled-trial, density-order and harmonic auditors with their own schemas. A checker should not be used on another schema merely because the files have similar names. + +For visual output, use `experiments/MODEL_SLICES.md`. Export physical planes at the same saved state with two subdivision counts before interpreting small surface features. The existing comparison and convergence plot helpers require a fresh output path outside their active source checkout; render in external staging and then archive the outputs under `working/` with their sidecars. This consolidation preserves that output guard. The report builder is separate and deliberately writes its intermediates under `working/report_builds/research_handoff/`. + +To regenerate the delivered report, run + + python3 reports/build_report.py + +The script invokes Pandoc directly on the authoritative LaTeX source to produce GitHub-flavored Markdown, then compiles the PDF with `latexmk`. It records tool commands, exit status, source/output hashes and build time. `--markdown-only` does not require a TeX installation. The Markdown is generated, not separately edited. A narrowly scoped Pandoc filter removes only a repeated first table-body row identical to the heading; explicit norm and absolute-value delimiters avoid collision with Markdown table separators. Figure copies are bound to their original archived sources through `reports/archive/figure_sources.json`. + +# Consolidation validation on 20 September + +The integration was validated in the existing isolated research worktree, using the numerical source checkpoint `b1fe359`. Its active numerical implementation differs from `d7acb0d` only by the preserved user quadrature declaration formatting; the additional log-density files are offline reference work. Documentation and archival changes are recorded in the final delivery tag. A 474-file source/fixture SHA-256 inventory binds the checks to their actual inputs and is verified against the original checkout after integration. This section describes new consolidation checks, separately from the earlier study validations reported above. + +The Release build completed successfully for the main regression executable, sandbox, plateau diagnostic, density-order contracts, continuation contracts, surface/scalar Schur contracts and MPI test executable. Building the MPI test executable is not an execution of the distributed test suite. The checks below ran serially with the recorded local MPI shared-memory configuration; no new distributed harmonic-solver claim is made. + +| Check | Recorded result | Seconds | +|:---|:---|:---| +| `mapping_surface_normalization` | 150 cases; 7,224,720 assertions | 794.18 | +| `coupled_solver_preconditioner` | 118 cases; 1,369,031 assertions | 1432.04 | +| `density_order_contract_tests` | 52 cases; 722,405 assertions | 199.70 | +| `continuation_contract_tests` | 8 cases; 515 assertions | 610.43 | +| `surface_border_schur_contract_tests` | 9 cases; 70 assertions | 0.44 | +| `python_regressions` | 146 tests; 2 initially skipped | 12.32 | +| `coupled_artifact_auditor_selftests` | 11 tests | 0.09 | +| `slice_geometry_build` | completed | 0.88 | +| `slice_geometry` | completed | 0.25 | +| `n3_mass_compatibility` | completed | 0.93 | +| `n1_harmonic_smoke` | completed | 210.08 | +| `plot_render_contracts` | 2 tests | 11.04 | + +The selected C++ checks account for 337 test-case executions and 9,316,741 assertions; the Python checks account for 157 tests. The 2 plot-render checks initially skipped by their environment gate were subsequently enabled and passed in the separate serial rendering check. The Python total counts executed tests, excluding the initial skips. These are executions, not a count of distinct cases across overlapping tag selections. Large assertion counts include array-element comparisons and do not indicate an equivalent number of independent scientific tests. The full unrelated regression suite was not run. + +## Fresh physical smoke model + +The smoke case repeated the native all-$m$, $L=4$, $n=1,J=0$ control with no central-density constraint, restart 160, linear cap 200, true linear target 0.03, and a six-update budget. The nonlinear relative target was $10^{-8}$ with zero absolute tolerance. It also exported physical model slices at four display subdivisions. This is a baseline regression check, not another high-index investigation or a claim about the unresolved rotating target. + +The model converged in 5 accepted updates. Its final normalized residual was $1.5271452\times10^{-12}$. The four q18 absolute virials, in order energy/force from enthalpy pressure and energy/force from density pressure, were $3.8703561\times10^{-7}$, $2.6593819\times10^{-7}$, $4.2699365\times10^{-7}$, $3.0589623\times10^{-7}$. All four also passed $10^{-5}$ at q14. The result reproduces the modest baseline capability; it does not resolve the rotating $n=3$ deficit. + +## Artifact and document checks + +All 6,228 relocated files were independently rechecked against their original content hashes with no mismatch. The report compiled to PDF and was converted from its authoritative LaTeX source with Pandoc. Conversion checks compared table cells, mathematical expressions, paths and figure captions, and verified the contents links. Repeated PDF table headings are removed from Markdown without removing data rows. PDF pages were rendered and inspected for table width, equation clipping, figure labels and page layout. + +The full validation commands, durations, logs, source/fixture inventory and smoke artifacts are preserved under `working/consolidation_2026-09-20/validation/`. Compact versioned copies of the build record, validation results and source inventory are in `reports/validation/`. This provides both a readable handoff and a record that can be checked independently. Successful build, regression and artifact checks do not upgrade any unconverged historical model to a physical success. + +# Versioned study-report catalog + +The following reports preserve the detailed methods and original interpretation. Their source filenames are repository-relative; raw campaign locations are recorded within the reports and in the archive index. + +| | +|:---------------------------------------------------------------| +| `experiments/CENTRAL_CONSTRAINT_HYPOTHESIS_2026-09-16.md` | +| `experiments/COUPLED_CONVERGENCE_STUDY_2026-09-19.md` | +| `experiments/COUPLED_TRIAL_CORRECTION_2026-09-18.md` | +| `experiments/DAMPED_COUPLED_TRIAL_2026-09-18.md` | +| `experiments/DENSITY_ORDER_STUDY_2026-09-19.md` | +| `experiments/FIELD_COUPLING_REVIEW_2026-09-19.md` | +| `experiments/GEOMETRY_QUALITY_EXPERIMENT.md` | +| `experiments/GEOMETRY_QUALITY_FINDINGS_2026-09-08.md` | +| `experiments/GRAVITY_COMPACT_APPLICATIONS_2026-09-17.md` | +| `experiments/GRAVITY_CONSISTENT_SCHUR_2026-09-18.md` | +| `experiments/GRAVITY_DIRECTION_DIAGNOSTICS_2026-09-17.md` | +| `experiments/GRAVITY_PREPARED_ROUNDOFF_2026-09-17.md` | +| `experiments/GRAVITY_REDUCTION_IMPLEMENTATION_2026-09-17.md` | +| `experiments/GRAVITY_SUBSYSTEM_IMPLEMENTATION_2026-09-17.md` | +| `experiments/KRYLOV_HYPOTHESIS_2026-09-16.md` | +| `experiments/KRYLOV_WINDOW_HYPOTHESIS_2026-09-16.md` | +| `experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md` | +| `experiments/MODEL_SLICES.md` | +| `experiments/N3_COMPATIBLE_FAMILY_2026-09-19.md` | +| `experiments/NEGATIVE_DENSITY_INVESTIGATION_2026-09-19.md` | +| `experiments/NONLINEAR_STEP_INVESTIGATION_2026-09-18.md` | +| `experiments/NUMERICAL_CAMPAIGN_2026-09-16.md` | +| `experiments/NUMERICAL_STRATEGIES_2026-09-17.md` | +| `experiments/POLYNOMIAL_ENTHALPY_ADMISSIBILITY_2026-09-17.md` | +| `experiments/POLYTROPE_H_CONVERGENCE_2026-09-09.md` | +| `experiments/POLYTROPE_VALIDATION.md` | +| `experiments/POLYTROPE_VALIDATION_FINDINGS_2026-09-09.md` | +| `experiments/PRECONDITIONER_CACHED_CROSS_ACTION_2026-09-18.md` | +| `experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md` | +| `experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md` | +| `experiments/PRESSURE_OBSTRUCTION_METHODOLOGY_2026-09-17.md` | +| `experiments/README.md` | +| `experiments/REDUCED_SURFACE_STUDY_2026-09-18.md` | +| `experiments/SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md` | +| `experiments/SPHERICAL_HARMONIC_SURFACE_2026-09-19.md` | +| `experiments/SYSTEM_ISOLATION_PROFILE_HARNESS.md` | +| `experiments/check_numerical_strategies_README.md` | +| `experiments/check_plateau_physics_README.md` | +| `experiments/plateau_diagnostic_README.md` | +| `experiments/plateau_shape_curvature_README.md` | diff --git a/reports/MeanField_research_handoff_2026-09-20.pdf b/reports/MeanField_research_handoff_2026-09-20.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2791c0df3cfa5f93bbcebb2a989311b36345d590 GIT binary patch literal 1513961 zcma&MQ?O{=wxzjjuf1&Bwr$(Cy_ap+aNt@W3Ih*4%GO^L)|L^a=POY6y9P#Nytqq(_giVa>j7^|;d7+$~9Zd{u zpxif_HMi|HSrL3+_4-Fq%rzAXge%CQoG({k2U~1u>QZ~yTe8|3t>y3}8hd@j3JIK! 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+1,3553 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=23mm]{geometry} +\usepackage{amsmath,amssymb} +\usepackage{booktabs,longtable,array} +\usepackage{graphicx} +\usepackage{xcolor} +\usepackage{xurl} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,urlcolor=blue!55!black, + pdftitle={MeanField numerical research and consolidation handoff}, + pdfauthor={MeanField research record}} +\setlength{\emergencystretch}{3em} +\setlength{\parskip}{0.35em} +\setlength{\parindent}{0pt} +\setcounter{tocdepth}{2} +\setcounter{secnumdepth}{3} +\setlist{itemsep=0.25em,topsep=0.35em} +\title{MeanField numerical research\\and consolidation handoff\\ +\large Formulation, experiments, evidence, architecture, and remaining work} +\author{Research record prepared with Codex\\ +\small Source and numerical evidence: 16--20 September 2026} +\date{20 September 2026} +\begin{document} +\maketitle +\thispagestyle{empty} +\vfill +\begin{center} +Experimental three-dimensional FEM stellar equilibria\\ +Independent density; coupled free surface; polytropic EOS\\[1em] +\textbf{The moderately rotating $n=3$ and full virial targets remain open.}\\[1em] +This document separates measured results from hypotheses, and retained +research tools from promoted defaults. +\end{center} +\clearpage +\tableofcontents +\clearpage + +\section{Purpose, scope and current conclusion} + +MeanField is an experimental three-dimensional finite-element laboratory for +steady rotating stellar equilibria. Its immediate purpose is to establish +reliable numerical choices for a later implementation, rather than to present +a finished stellar-evolution or production equilibrium code. The user is +independently rewriting the implementation in order to own and understand the +eventual shipped software. The useful output of this investigation is therefore +the formulation, verified contracts, measured successes and failures, and +reproducible evidence that can guide that rewrite. + +This report consolidates the work recorded from 16 through 19 September 2026 +and the source/evidence integration performed on 20 September. Earlier +optimization and validation records are preserved as historical sources. The +conversation is not treated as an experimental log: quantitative claims below +come from committed reports, campaign manifests, saved vectors, diagnostic +tables and recorded checks. The initial approximately tenfold performance gain +and approximately 20-second Newton step for the nonrotating $n=1$ model were +the user's starting observation. They are not remeasured as a new speedup in +this consolidation. At the outset, limited testing suggested plausible $n=1$ +models at very low rotation, described by the user as roughly 1.5 percent of +critical, while increasing index or rotation produced residual plateaus. That +rough starting description is not a calibrated critical-rotation measurement +for the later campaign inputs. + +\subsection{Physical and software objectives} + +The immediate physical target is a moderately rotating $n=3$ polytrope, with +an eventual comparison against ESTER. The operational rotation goal is at least +50 percent of critical rotation. The more distant ambition of exceeding +99 percent of critical rotation is outside the demonstrated regime and would +require renewed attention to surface regularity and the vanishing effective +gravity near mass shedding. No value of the dimensionless angular momentum +used in these campaigns has been calibrated to a critical-rotation fraction. +Neither a visually oblate model nor a small $J$ proves a particular fraction +of critical rotation. + +The key global milestone is a relative virial error no larger than $10^{-5}$. +The original July project README described an aspirational $10^{-6}$ target; +the research acceptance target in this report is the subsequently specified +$10^{-5}$. Internal pointwise correctness remains necessary. Exterior accuracy +is required insofar as it supports the correct interior and boundary response; +an independently accurate external field everywhere is not the immediate +scientific objective. Nevertheless, the exterior formulation can influence +interior gravity and cannot be disregarded merely because its own residual is +small. + +The software objectives are equally explicit: retain a physics-oriented user +API, express algorithm and field compatibility through types where practical, +use runtime checks for mesh-dependent and state-dependent conditions, preserve +independent density, and solve surface coefficients and interior fields in the +same coupled system. Central density is an optional physical specification; +it must not be silently imposed on every polytrope. Log density and new +positivity enforcement are deferred. Existing optional feasibility experiments +remain available for reproduction but are not enabled by consolidation. + +\subsection{What has actually been demonstrated} + +There is substantial progress, but the central rotating $n=3$ goal is open. +The strongest established results are: +\begin{enumerate} +\item The native all-$m$, degree-four real spherical-harmonic surface +prescription is integrated and tested against the corresponding nodal geometry, +physical residuals and Jacobian actions. It reduces the surface to 25 +independent coefficients without decoupling it from the interior solve. +\item Some low-index, slowly rotating native-harmonic cases meet both their +algebraic stopping criterion and the global virial target. Higher-index +representation errors and unresolved pointwise checks prevent generalizing +this to every algebraically converged case. +\item At the same frozen state, gravity-Schur coupling reduced the measured +total linear cost by 41.5 percent for $n=3,J=0$ and by 30.6 percent for +$n=1.5,J=0.1$, at the stated true-defect gate. These are single matched +comparisons, including setup, rather than statistical timing estimates. +\item In the compatible rotating $n=3,J=0.035,K/K_0=0.99$ family, stellar LDU +plus gravity-Schur reached a 12.687-times lower residual after ten updates than +gravity-Schur alone, at 14.593 percent greater Newton time. This is an +equal-update endpoint comparison, not a speedup to the same converged model. +\item The best rotating $n=3$ endpoint still fails the full virial target: +the enthalpy-pressure energy error is $1.81141\times10^{-5}$, and the +density-pressure energy error is $3.51558\times10^{-5}$. Passing one force +virial is insufficient. +\item Physical-plane slices now show independent density and enthalpy on the +actual deformed stellar geometry, with signed values and the undeformed +outline available for comparison. Export refinement has been checked for +selected $n=2$ and rotating $n=1.5$ states; it is not a mesh-convergence proof. +\end{enumerate} + +Several attractive paths did not justify promotion: simple one-way +material-to-gravity completion, increasing displacement order alone, expensive +coarse surface/scalar corrections, and universal use of second-order trial +corrections. Some improve an isolated metric while increasing total work or +worsening the accepted nonlinear trajectory. Those negative results are +preserved because they constrain the next design, not because the algorithms +are impossible to improve. + +\subsection{How to use this record} + +The report first reconstructs the baseline and sequence of hypotheses, then +describes the current implementation, numerical evidence and interpretation. +Later sections distinguish active defaults from selectable experimental +policies and record the consolidation checks. The source reports under +\nolinkurl{experiments/} provide the most direct detailed companion records. +\nolinkurl{reports/archive/} contains branch and relocation inventories. +\nolinkurl{working/research/diagnostics/} contains the preserved raw campaigns. +The large raw archive is local and intentionally not committed to Git; a +source-only clone will not contain its saved models or copied executables. +Versioned figure copies and this report remain usable without that archive. + +\section{Definitions and evidence discipline} + +\subsection{Physical variables and virial diagnostics} + +For the polytropic experiments, pressure obeys +\begin{equation} + P(\rho)=K\rho^{1+1/n},\qquad + h(\rho)=(n+1)K\rho^{1/n},\qquad + \rho_\star(h)=\left(\frac{h}{(n+1)K}\right)^n + \label{eq:report-polytrope} +\end{equation} +on their physical nonnegative domains. The discrete density and enthalpy are +independent finite-element fields; the weak closure equation does not make +them pointwise equal. Consequently an integral formed from $P(\rho_h)$ need +not agree with one formed from $P(h_h)$ at the same discrete state. Undefined +fractional powers of negative density must be reported as undefined, not +silently replaced with a positive or clipped value. + +For a zero-pressure stellar boundary, define +\begin{equation} + T=\frac12\int_{\mathcal{D}_\star}\rho \lvert \boldsymbol{v}\rvert^2\,dV, + \qquad \Pi=\int_{\mathcal{D}_\star}P\,dV, + \qquad W_\Phi=\frac12\int_{\mathcal{D}_\star}\rho\Phi\,dV. +\end{equation} +With the usual negative gravitational potential, $W_\Phi<0$. The force +binding integral $W_f$ uses the corresponding gravitational force moment and +agrees with $W_\Phi$ for a consistent isolated equilibrium. The campaign +observers report energy and force virials using the common normalization +$\lvert W_\Phi\rvert$: +\begin{equation} + v_\Phi=\frac{2T+3\Pi+W_\Phi}{\lvert W_\Phi\rvert},\qquad + v_f=\frac{2T+3\Pi+W_f}{\lvert W_\Phi\rvert}. + \label{eq:report-virials} +\end{equation} +Tables generally show their absolute values unless signs are explicitly +given. A nonzero external pressure requires the boundary-pressure term; +the stated campaign comparisons use the zero-pressure surface convention. +Both pressure reconstructions and both gravity forms are retained where +they are defined. Agreement at quadrature orders 14 and 18 is an integration +consistency check on a fixed discrete state, not evidence of spatial +convergence or exact pointwise physics. + +The common campaign scale is $M=R_{\rm reference}=G=1$ unless a manifest +states otherwise. Angular momentum $J$, angular velocity, $T/\lvert W\rvert$, radius +and critical-rotation fraction are different quantities. The physical-family +compatibility study later explains why this distinction is particularly +important at $n=3$. + +\subsection{Algebraic quantities and cost accounting} + +Let $x$ denote the complete normalized coupled coordinate vector and $F(x)$ its +normalized residual. Write $A=DF(x)$ for its Jacobian; this notation avoids +confusing the Jacobian with the stellar angular momentum $J$. A Newton +correction $p$ is checked through the fresh full-system defect +\begin{equation} + \eta_{\rm true}=\frac{\lVert F(x)+Ap\rVert}{\lVert F(x)\rVert}. + \label{eq:report-linear-defect} +\end{equation} +An internal Krylov recurrence or a reduced-system residual cannot substitute +for this full-system endpoint test. The nonlinear merit function is +$\mathcal{M}(x)=\tfrac12\lVert F(x)\rVert^2$; its directional derivative is +$F(x)^T A p$. Linear success and a negative initial merit derivative do not +guarantee acceptance of a finite step on a curved coupled path. + +The state and residual norms are produced by the configured physical Riesz +normalization. They should not be compared across a change of basis, field +order or normalization as though they were the same physical norm. The +experiments therefore retain physical residual checks and integrals in +addition to normalized algebraic quantities. A block's fraction of $\lVert F\rVert^2$ +is an algebraic attribution; it is not an elementwise PDE error estimate. + +Frozen comparisons hold the complete base state, Jacobian, right-hand side, +mesh and coordinates fixed. Their relevant linear cost includes +construction, binding, setup, solve and a fresh endpoint verification. Live +comparisons follow evolving nonlinear states and include preparation, +accepted/rejected trials and auxiliary solves in the recorded Newton time. +Process wall time can additionally include observers and export overhead. +Setup amortization must therefore be demonstrated on an actual repeated-use +path rather than assumed from a smaller outer iteration count. + +\subsection{Controls, provenance and limits of inference} + +The adopted methodology used an isolated branch for each hypothesis or a +clearly identified cumulative integration branch. A source revision and +configuration were recorded before numerical work; campaign runners copied +the executable, recorded its digest and retained the exact commands, mesh +digest, saved state and block diagnostics. Builds, FEM runs and timing trials +were serialized within the later controlled studies. Read-only analysis and +source review could proceed independently. Single-run timings are reported +as observations; no confidence interval or cross-machine performance claim +is inferred from them. + +The pressure/shape investigations separated preparation feasibility, +linear accuracy, initial descent and finite-step curvature. Derivative probes +used directional or centered differences over a stated step grid and compared +full versus component actions. Later snapshot loaders check dimensions, +mesh, coordinate conventions and provenance rather than accepting a vector +solely because its length matches. Unconverged restarts require an explicit +diagnostic override. Saved-state initialization prepares and refreshes before +committing, clears stale warm starts and rolls back failed initialization. + +The evidence checkers reject corrupt or inconsistent fixtures as well as +accepting valid ones. Passing an artifact audit establishes consistency of +the recorded experiment under that checker's contract. It does not establish +that the nonlinear model converged, that all interior samples are physical, +or that a target virial was attained. Similarly, passing code assertions is +not equivalent to obtaining a validated stellar model. + +All finite iteration budgets and diagnostic ceilings remain explicit. +A late small-step trajectory can be described as a stall observed over the +available window. It cannot be promoted to a mathematically established +convergence floor. Dense quadrature samples can locate negativity but cannot +certify its absence everywhere; interval subdivision and polynomial bounds +have their own scope and unresolved cases. Conclusions in this report retain +those distinctions. + +\section{Starting point, problem definition, and architecture} + +The work began on 16 September 2026 with an already substantial experimental +finite-element code. Its purpose was to discover a reliable numerical route to +three-dimensional steady stellar equilibria before a later, independently +written general-EOS implementation. The immediate physical model was a +self-gravitating, rigidly rotating barotrope with a free stellar surface. The +presence of older transport, Coriolis, viscosity, and continuity components did +not mean that those equations participated in the active equilibrium solve. +This distinction remains important when comparing the experimental code with +the broader physics ultimately desired from a stellar structure application. + +The incoming Git HEAD was \texttt{75cc63873992984966ea20011cc04d9984f54be4}. +It did not uniquely describe the audited source: 49 tracked files already +differed, with 3,626 additions and 267 deletions, and additional untracked +experiments and results were present. The later baseline commit +\texttt{80cc3d8} on \nolinkurl{research/baseline-20260916} preserved that optimized +source. The initial architecture audit was read-only. The subsequent diagnosis +used fresh Release builds rather than assuming that existing executables +represented the modified source. This provenance distinction is essential: +historical binaries and historical passing tests are evidence about their own +source, not automatically about the current checkout. + +The user reported approximately a tenfold reduction in the cost of an ordinary +nonrotating $n=1$ Newton step, to about twenty seconds. Fresh measurements found +22--24 seconds for the first three accepted steps and were consistent with that +operating point. The old pre-optimization source was not rerun in a matched +comparison; the full historical factor of ten was therefore not independently +remeasured. The inherited 9 September report had separately measured factors +of 1.53 in whole-process time, 1.70 in Newton time, and 2.02 in linear-solve time +for its particular batch. Later tensor, cache, and compact-action changes had +different baselines. These gains must not be multiplied together. + +\subsection{Continuum equations and independent discrete fields} + +For a polytrope of index $n\geq1$ and constant $K>0$, +\begin{align} + P&=K\rho^{1+1/n}, & + h&=(n+1)K\rho^{1/n}, & + \rho_{\mathrm{EOS}}(h)&=\left[\frac{h}{(n+1)K}\right]^n, + \qquad h\geq0. +\end{align} +With positive centrifugal potential +$\Psi=\lvert\boldsymbol\Omega\mathbin{\times}\boldsymbol x\rvert^2/2$, +the Bernoulli imbalance is +\begin{equation} + B=h+\Phi-\Psi-C. +\end{equation} +The gravity flux represents $\boldsymbol g=\nabla\Phi$, so physical +gravitational acceleration is $-\boldsymbol g$. The active system combines a +mixed gradient--potential gravity formulation, a weak density closure, weak +Bernoulli balance, reduced mechanical virtual work, and integral constraints +for total mass and axial angular momentum. Density, enthalpy, potential, and +the gravity flux are independent finite-element fields. In particular, the +mechanical pressure is evaluated from enthalpy while gravity and centrifugal +forces use independent density. Continuum equivalence of these quantities +does not force their pointwise equality after discretization. + +The surface condition replaces the relevant enthalpy test rows by the +enthalpy value corresponding to prescribed surface pressure. At zero surface +pressure this is an $h=0$ trace condition. The mechanical residual is first +formed against volume displacements and then pulled back by the transpose of +the surface-to-volume deformation derivative. If $q$ denotes independent +surface parameters and $D(q)$ the generated displacement, its schematic form is +\begin{equation} + R_s(q,u)=D'(q)^T R_{\mathrm{mech}}(D(q),u). +\end{equation} +The Jacobian includes both the derivative of the mechanical force and the +derivative of this pullback. A zero $R_s$ proves equilibrium only against the +chosen extension-generated mechanical tests; it does not set every unrestricted +volume mechanical row to zero. + +The usual initial example additionally imposed a central-density specification. +Its actual row was a central enthalpy point condition, +\begin{equation} + R_c=h(0)-h(\rho_{c,\mathrm{target}}),\qquad + R_h\longleftarrow R_h+\lambda_c e_c. +\end{equation} +It was not a direct point condition on the independent discontinuous density. +This distinction motivated measuring the unbordered residual and artificial +source amplitude. The user clarified that the central condition should remain +optional, principally useful for the $n=3$ homology, and should not be +automatically inserted at other indices or hidden in a physics-facing API. + +\subsection{Discretization and model compilation} + +The common mesh had 1,216 hexahedra: 832 stellar and 384 vacuum elements. Its +SHA-256 was +\nolinkurl{ce11ae99e21e6c3bbfbee402acd2e191c1da0d8261d2227b4203f73f2f337a74}. +Reference geometry was order four. The ordinary spaces were RT index two for +gravity, discontinuous $Q_2$ for potential and stellar density, and continuous +$Q_3$ for enthalpy and displacement. The original scalar nodal radial surface +had 866 parameters. With its three scalar constraints the root had 178,074 +unknowns; removing the central specification removed one scalar, yielding +178,073. Mesh geometry order and the approximation order of an individual +physical field were distinct controls. + +{\small +\begin{longtable}{@{}p{105mm}p{40mm}@{}} +\hline +Initial root block & Number of unknowns \\ +\hline +Gravity gradient & 98,928 \\ +Potential & 32,832 \\ +Stellar density & 22,464 \\ +Stellar enthalpy & 22,981 \\ +Nodal radial surface & 866 \\ +Generated scalar coordinates & 3 \\ +\hline +\end{longtable} +} + +The architecture was already strongly compositional. A compile-time model +specification pack generated the block layout, scalar coordinates, Jacobian +couplings, normalization metadata, and prepared specification runtimes. The +active aggregate was the variadic prepared equilibrium operator. C++23 module +interfaces under \nolinkurl{libmeanfield/interface} were separated from +implementations under \nolinkurl{libmeanfield/impl}. MFEM/MPI spaces used explicit +maps between supported solver coordinates and full true degrees of freedom. +The nonlinear context owned accepted and trial states, normalization, inverse +preparation, and linear backend; refresh, stale-view rejection, rollback, and +observer lifetime were explicit contracts rather than informal conventions. + +Prepared matrix-free actions cached reference basis tables, geometry, +thermodynamics, and local contractions. Their dependency identities and +revisions controlled invalidation. The numerical work therefore emphasized +reusing established ownership and preparation interfaces rather than adding +ad hoc mutable solver state. Static choices such as block identities, +compatible policy combinations, and registered capabilities were checked at +compile time. Mesh sizes, rank agreement, actual FE eligibility, values, +normalization factors, and preparation generations necessarily required +runtime validation. + +The geometry map initially used nodal radial surface variation, a logical +radius-squared interior extension, and a fixed-infinity vacuum extension. +Kelvin compactification represented the exterior through infinity. The root +geometry unknown was the surface, not the much larger volume displacement +field. Surface parameterization, interpolation of displacement, extension, +and the physical map were therefore distinct architectural responsibilities. + +The seed initialized density and enthalpy independently by finite-element +coefficient interpolation of a spherical Lane--Emden profile, using MFEM's +\texttt{ProjectCoefficient}, and solved mixed gravity separately. This +initialization was distinct from solving the weak closure's weighted $L^2$ +projection relation; the method name must not be read as implying such a +mass-matrix solve. It set shape to zero, +initialized $C$ from $-GM/R$, and initialized $\Omega$ from the target $J$ +divided by spherical inertia. A seed with nonzero target $J$ was not an +already rotating equilibrium. Throughout the early controlled campaign, +$M=R_{\mathrm{reference}}=G=1$; at $n=1$, $K=2/\pi$ and +$\rho_c=\pi/4$. Other indices used the production Lane--Emden integrator +with step $10^{-3}$ to calibrate the seed. That was a consistent seed +reference, not an independent high-precision continuum solution. + +\subsection{Physical acceptance and measurement conventions} + +The required milestone was virial balance within $10^{-5}$, together with +credible interior fields. Exterior pointwise accuracy was not an independent +goal; exterior errors mattered through their effect on stellar equilibrium. +The campaign measured both gravitational energies +\begin{equation} + W_E=\frac12\int_\star\rho\Phi\,dV, + \qquad + W_F=-\int_\star\rho\,\boldsymbol x\cdot\nabla\Phi\,dV, +\end{equation} +and, for the zero-pressure surface, +\begin{equation} + \epsilon_V(W,P)= + \frac{\left\lvert 2T+W+3\int_\star P\,dV\right\rvert}{\lvert W_E\rvert}. +\end{equation} +Pressure from $h$ and pressure from $\rho$ were evaluated separately, at +independent diagnostic quadrature orders 14 and 18. Signed density was retained. +When the EOS could not define pressure at a negative density sample, the full +density-pressure diagnostic was reported undefined, rather than manufactured +by clipping. Quadrature agreement tested integration sensitivity; it did not +constitute spatial convergence or global positivity certification. + +The first physical checker required recorded nonlinear convergence and all +eight finite virial measurements to satisfy $10^{-5}$. This conservative +combined screen was stricter than a statement that a particular enthalpy-based +virial passed. Neither passing the checker nor a small scalar virial alone was +a pointwise continuum-accuracy certificate. Useful accompanying observations +were mass and angular-momentum errors, Bernoulli variation, EOS mismatch, +gravity consistency, geometry minima, density sign, and surface shape. +No early value of $J$ or reference-unit $\Omega$ was calibrated as a fraction +of critical rotation. + +The inherited saved $n=1,J=0$ benchmark already provided a useful physical +reference beyond the scalar residual: density, enthalpy, interior potential, +and gravity-gradient relative volume $L^2$ errors were approximately +$4.70\,10^{-4}$, $3.27\,10^{-4}$, $1.26\,10^{-4}$, and +$5.98\,10^{-4}$; surface-radius RMS error divided by reference radius was +$2.75\,10^{-4}$. Its density-pressure energy virial was +$3.10\,10^{-7}$ and mass error $2.36\,10^{-11}$. These were the +particular saved coarse model's errors against its radial reference, not +accuracy estimates for later rotating or higher-index cases. A one-level +refined mesh and analytic geometry controls existed, but the corresponding +directory did not contain completed refined equilibrium fields or an +equilibrium convergence result. The audit did not infer completed mesh +convergence from the presence of that directory. + +The historical full-suite XML contained 592 cases, 588 passing and four +failing: centrifugal-virial refinement geometry, an $n=3$ projected-seed +closure screen, and two- and four-rank MPI groups. Those failures had been +reproduced in earlier preserved binaries. They were neither silently removed +nor treated as fresh validation of the incoming modified source. At the +initial audit, twelve production source files were newer than the available +release sandbox binary and thirty newer than the test binary. This motivated +the subsequent explicit rebuilds and targeted regression ledgers. A +previous between-sample mesh fold was also documented, but the later saved +$n=1$ mesh had healthier sampled geometry; it was not assumed to suffer the +same demonstrated folding mechanism. + +\paragraph{Evidence.} +The original workspace reports were +\nolinkurl{reports/sources/original_audits/MeanField_audit_2026-09-16.md}, +\nolinkurl{reports/sources/original_audits/MeanField_solver_diagnosis_2026-09-16.md}, and +\nolinkurl{reports/sources/original_audits/formulation_audit_notes.md}. Their archival relocation is recorded in +the consolidation manifest. Repository evidence includes +\nolinkurl{experiments/POLYTROPE_VALIDATION_FINDINGS_2026-09-09.md}, +\nolinkurl{experiments/POLYTROPE_H_CONVERGENCE_2026-09-09.md}, and +\nolinkurl{experiments/NUMERICAL_CAMPAIGN_2026-09-16.md}. + +\section{Initial diagnosis: several different plateaus} + +\subsection{Fresh control measurements} + +The diagnostic rebuilt the incoming optimized source in Release using +Homebrew Clang 22.1.8, with IPO enabled, profiling disabled, and uniform field +order increment zero. Cases ran serially on one MPI rank. Initial live controls +used FGMRES restart 40, at most 200 linear iterations, relative forcing +tolerance $\eta=0.03$, nonlinear relative tolerance $10^{-8}$ and zero absolute +tolerance. The nonlinear threshold was relative to the initial residual, not +an absolute $10^{-8}$ requirement. + +{\small +\begin{longtable}{@{}p{26mm}p{24mm}p{35mm}p{59mm}@{}} +\hline +Model & Accepted updates & Retained $\lVert F\rVert$ & $P(h)$ energy/force virials \\ +\hline +$n=1,J=0$ & 3 full & $6.6053\,10^{-9}$ & $2.616\,10^{-7}/3.246\,10^{-7}$ \\ +$n=1,J=0.01$ & 4 full & $5.0898\,10^{-10}$ & $2.610\,10^{-7}/3.246\,10^{-7}$ \\ +$n=1,J=0.05$ & 5 full & $6.0406\,10^{-9}$ & $3.016\,10^{-7}/3.188\,10^{-7}$ \\ +$n=1.5,J=0$ & $1,1/16$ & $1.6976\,10^{-4}$ & $2.245\,10^{-6}/2.318\,10^{-6}$ \\ +$n=3,J=0$ & 0 & $3.1006\,10^{-3}$ & $5.326\,10^{-5}/8.341\,10^{-5}$ \\ +\hline +\end{longtable} +} + +All these controls stopped at a failed linear gate. The $n=3$ measurement was +the unchanged seed. At $n=1,J=0$ the strict target was approximately +$1.81\,10^{-12}$, whereas an earlier validation driver accepted the same +three-step state under an absolute $10^{-8}$ threshold. These descriptions +were consistent; they did not imply that a previously converged model had +become physically invalid. Conversely, the $n=1$ seed itself already passed +the virial threshold, directly demonstrating why virial alone was not a +complete equilibrium test. + +The initial audit separated three mechanisms: loss of useful Krylov +information late in otherwise accurate $n=1$ solves; stronger coupled linear +difficulty at higher index; and severe nonlinear curvature or admissibility +restrictions after a usable linear direction was obtained. It did not find +evidence for one universal numerical floor. + +\subsection{Normalization audit} + +The state and residual registrations supplied their own physical scales and +reference Gram diagonals. Writing these as $s_x,d_{x,i}$ for a value block and +$s_F,d_{F,i}$ for its residual/test block, ordinary coefficient factors were +\begin{equation} + S_{ii}=\frac{\sqrt{d_{x,i}}}{s_x},\qquad + L_{ii}=\frac{1}{s_F\sqrt{d_{F,i}}}. +\end{equation} +The Gram factors form paired primal and dual maps where the registered +function-space topologies correspond, but the physical scales need not agree. +For example, surface parameters use a length scale while mechanical shape +residuals use a force scale; potential uses specific energy while its Poisson +residual uses inverse time squared. Point replacement rows and scalar blocks +have their declared special treatment. These distinctions are explicit in +\nolinkurl{libmeanfield/interface/normalization/physical_riesz.cppm}. + +The reference factors were frozen through a solve. Their characteristic +scales came from $M,R_{\mathrm{reference}},G$ and did not divide by the +requested angular momentum $J$; zero rotation was therefore not a +normalization singularity. + +Let $\widehat x=Sx$ and $\widehat F=LF$. The consistent transformations were +\begin{equation} + \widehat A=LAS^{-1},\qquad + \widehat P^{-1}=SP^{-1}L^{-1},\qquad + \widehat A\widehat P^{-1}=LAP^{-1}L^{-1}. +\end{equation} +Consequently, changing only state scales cancels from a consistently +right-preconditioned system. Direct probes multiplying block state scales by +$10^{-3}$ through $10^3$ preserved the physical correction and preconditioned +action to approximately $10^{-16}$. Wrapper/explicit actions and coordinate +round trips agreed at roundoff. This ruled against a basic scaling-wrapper +implementation error at the inspected states. It did not show that the chosen +residual norm was optimal: changing $L$ changes what both Krylov and line +search minimize. + +The same $M=R=G=1$ factors were used across indices despite changing stellar +concentration. Large factor ranges alone did not diagnose a bug. Similarly, +small normalized border matrices had raw condition numbers +$2.78\,10^7$ and $9.69\,10^7$, dominated by a central-column scale near +$7.3\,10^6$; diagnostic equilibration reduced these to about 1.46 and 2.39. +Those were border-matrix observations, not the condition number of the full +stellar Jacobian or proof of a singular physical system. + +\subsection{Central condition: a useful example correction} + +At $n=1,J=0.05$, matched restart-40 runs with and without the central +condition had identical first five accepted step lengths and linear iteration +counts. Both then failed the sixth linear gate near relative defect 0.505. +Removing the condition nevertheless markedly improved localized physical +consistency: + +{\small +\begin{longtable}{@{}p{70mm}p{38mm}p{38mm}@{}} +\hline +Quadrature-18 measurement & With central condition & Without \\ +\hline +Peak absolute EOS density mismatch & $7.30451\,10^{-3}$ & $3.32568\,10^{-4}$ \\ +Peak mismatch / sampled density scale & $9.33951\,10^{-3}$ & $4.30594\,10^{-4}$ \\ +Bernoulli sampled range & $1.45751\,10^{-2}$ & $1.30398\,10^{-3}$ \\ +Unbordered residual & $4.54227\,10^{-6}$ & $6.03609\,10^{-9}$ \\ +$P(h)$ energy virial & $3.01617\,10^{-7}$ & $3.06913\,10^{-7}$ \\ +\hline +\end{longtable} +} + +The formerly fixed central enthalpy changed from 1 to 0.983383508. Mass, +angular momentum, angular velocity, and volume RMS errors changed little. +The result supported removing that artificial restriction from the $n=1$ +example, while leaving explicit central constraints available. It did not +establish that central constraints caused the general plateau. + +\subsection{Krylov window and warm-guess safeguard} + +Frozen cold comparisons retained the same state, full Jacobian, physical +right-hand side, and inverse. Increasing only the restart window from 40 to +160 gave: + +{\small +\begin{longtable}{@{}p{70mm}p{38mm}p{38mm}@{}} +\hline +Frozen state & Window 40: iterations / defect & Window 160: iterations / defect \\ +\hline +$n=1,J=0$ endpoint & $200/0.04442$ & $72/0.02914$ \\ +$n=1,J=0.01$ endpoint & $200/0.35468$ & $125/0.02519$ \\ +$n=1.5,J=0$ endpoint & $200/0.06491$ & $70/0.02694$ \\ +$n=3,J=0$ seed & $200/0.05631$ & $200/0.04026$ \\ +\hline +\end{longtable} +} + +For the rotating $n=1$ state, histories coincided through iteration 40 and +separated immediately after restart. Independently recomputed full residuals +agreed with reported residuals. This established actual loss of useful Krylov +information, not false convergence reporting. In a live unbordered +$n=1,J=0.05$ comparison, restart 160 reached strict convergence in seven full +steps with $F=3.38036\,10^{-11}$, versus five steps and +$6.03609\,10^{-9}$ before linear failure with restart 40. Total linear +iterations were 292 versus 370; complete diagnostic-process times were 271.95 +versus 290.80 seconds. All eight virial checks were below +$3.735\,10^{-7}$. These were single timing observations with increased +Krylov storage, not a universal speedup claim. + +The solver reused $(1-\alpha)p$ after a damped update as the next initial +linear guess. This was vector reuse, not Krylov recycling. At one $n=1.5$ +state that guess had a true defect larger than the zero guess. A small +backend safeguard now checks the current true defect of a supplied nonzero +guess and discards it only if the norm is nonfinite or exceeds $\lVert b\rVert$. +Nonfinite input coefficients remain an input-contract error. The guard adds +one action for a nonzero guess and none for a zero guess, without a new public +option. + +The matched $n=1$ output was bitwise unchanged. At $n=1.5$, the first shared +guarded solve reduced the effective initial defect/RHS from 1.5097 to 1 and +iterations from 71 to 65. Over eight attempts the control had seven successful +linear solves followed by failure; the guard passed all eight. Total iterations +fell from 672 to 561, mostly by avoiding that failed 200-iteration attempt. +The endpoint changed only from $1.75135\,10^{-4}$ to +$1.74946\,10^{-4}$, a 0.108 percent improvement, with final steps +$1/2048$ and $1/8192$. This was a modest inner-solver robustness improvement, +not a cure for nonlinear stagnation. + +\subsection{Rejected one-way gravity completion} + +The first coupling hypothesis changed the structure inverse from independent +material/gravity blocks to material-then-gravity triangular completion. It +retained the material inverse, order-five Chebyshev flux-mass inverse, and +three AMG cycles, and rebuilt the scalar border consistently. At the same +centrally constrained $n=3$ seed, the restart-160 defect changed from +0.0402590 to 0.0381781 after 200 iterations; both failed $\eta=0.03$. +Time increased from 98.92 to 117.71 seconds, correction norm from 0.75764 +to 1.54083, and the sampled thermodynamic step limit worsened from 0.15244 +to 0.09135. The candidate's $\alpha=0.1$ trial was inadmissible. +This particular one-way completion was not promoted. + +The baseline restart-40 and restart-160 corrections had cosine $-0.985$ +despite the same matrix and right-hand side. Their difference had a response +bound $\lVert A(p_2-p_1)\rVert/\lVert p_2-p_1\rVert\leq2.77\,10^{-4}$. +This established a weak measured direction, not a singular value or an +identification of homology, translation, or a surface mode. Later studies +therefore examined the actual coupled directions instead of adding a gauge +on analogy alone. + +\paragraph{Evidence.} +\nolinkurl{experiments/NUMERICAL_CAMPAIGN_2026-09-16.md}, +\nolinkurl{experiments/CENTRAL_CONSTRAINT_HYPOTHESIS_2026-09-16.md}, +\nolinkurl{experiments/KRYLOV_WINDOW_HYPOTHESIS_2026-09-16.md}, and +\nolinkurl{experiments/KRYLOV_HYPOTHESIS_2026-09-16.md}. The combined nonrotating +$n=1$ case subsequently reached $F=1.65157\,10^{-13}$ in six full steps +with worst virial $3.73092\,10^{-7}$. The transfer case $n=1,J=0.1$ +passed all twelve linear gates but ended at $6.25389278\,10^{-3}$, with +final $\alpha=4.76837\,10^{-7}$ and failing enthalpy-based virials +$4.75186\,10^{-4}/8.18118\,10^{-4}$. That transfer failure motivated +the next investigation. + +\section{Separating pressure preparation from shape-related curvature} + +The focused 17 September campaign made no production numerical changes. It +captured exact trial states, identified rejection stages, and split the same +Newton correction into its actual shape block $p_s$ and all remaining fields +and scalars $p_f$. Frozen replays restored the base before each action; +probing a trial never committed it as an accepted model. + +\subsection{The rotating pressure obstruction was localized and real} + +At $n=1,J=0.1$, exact replays of the first rejected candidates at zero-based +iterations 2, 3, 4, 7, and 11 all returned the production code +\texttt{thermodynamic\_domain / pressure\_force / outside\_domain}. +The pressure source takes this branch for finite negative enthalpy. +Thus these were enthalpy-domain failures, not inverted geometry. The limiting +pressure point remained at local stellar element 459, quadrature point 50; +the closure's different rule limited at point 36 of the same element. + +{\small +\begin{longtable}{@{}p{14mm}p{30mm}p{30mm}p{30mm}p{30mm}@{}} +\hline +Iteration & Pressure-point $h$ & $\delta h$ & Pressure bound & Accepted $\alpha$ \\ +\hline +2 & $9.63674\,10^{-4}$ & $-1.31441\,10^{-2}$ & $7.33159\,10^{-2}$ & $1/16$ \\ +3 & $1.42165\,10^{-4}$ & $-2.11119\,10^{-3}$ & $6.73390\,10^{-2}$ & $1/16$ \\ +4 & $1.02159\,10^{-5}$ & $-1.89704\,10^{-3}$ & $5.38520\,10^{-3}$ & $1/256$ \\ +7 & $7.04574\,10^{-8}$ & $-1.86569\,10^{-3}$ & $3.77648\,10^{-5}$ & $1/32768$ \\ +11 & $1.07227\,10^{-9}$ & $-1.86493\,10^{-3}$ & $5.74963\,10^{-7}$ & $1/2097152$ \\ +\hline +\end{longtable} +} + +The pressure step budget fell by a factor of 127,514 while the late negative +direction remained finite. The limiting element touched the stellar surface; +the pressure point's reference Cartesian radius was 0.997553, near the negative +rotation pole, approximately 8.66 degrees from the negative $z$ axis at onset. +It was an interior quadrature point, not a boundary row. Topological adjacency +did not measure distance to the deformed surface. + +At onset the closure bound, 0.0727489, was slightly tighter than the pressure +bound; pressure was evaluated first and therefore reported first when both +failed. A deliberately intermediate trial at +$\alpha=0.07303236911639208$ passed pressure preparation and failed closure, +with predicted closure $h=-5.01979\,10^{-6}$ and limiting pressure +$h=+3.72619\,10^{-6}$. This showed why a first rejection stage must not be +confused with the tightest thermodynamic constraint. + +All 866 surface enthalpy coefficients and their directions were exactly zero +at onset and late capture. Boundary leakage was excluded for these trials. +Nevertheless, positive nodal values did not imply a positive represented +polynomial. A step just beyond the onset bound retained nonnegative values +at all 64 local $Q_3$ nodes while making interior pressure and closure samples +negative. At the late state even a unit step retained nonnegative nodes but +gave $h=-1.86493\,10^{-3}$ at the limiting pressure point. + +An independent reconstruction of the local tensor polynomial reproduced the +nodes exactly and limiting-point values within $6.94\,10^{-18}$. A much +denser grid found that the late accepted base already had negative enthalpy: +a sampled minimum $-2.87114\,10^{-5}$ at reference coordinates +$(0.0130467,0.515625,0.01)$. A separate 60-digit reconstruction agreed with +the negative value at another selected interior point. These findings proved +negative values existed in that element; the reported minimum was not a +certified global minimum. Passing production quadrature preparation was thus +not a pointwise positivity certificate. They also did not prove that every +low-rotation or nonrotating converged control shared this defect. + +\subsection{Higher-index merit rejection had a different mechanism} + +For any directional piece $p_d$, the diagnostic measured +\begin{equation} + R_d(\alpha)=F(x+\alpha p_d)-F(x)-\alpha Ap_d, + \qquad R_{\mathrm{mixed}}=R_{\mathrm{full}}-R_s-R_f. +\end{equation} +The mixed vector was formed before taking norms. All pieces used common step +lengths and the original, unrenormalized block amplitudes. With +$L=F+\alpha Ap$, the exact merit identity was +\begin{equation} + \lVert F(x+\alpha p)\rVert^2= + \lVert L\rVert^2+2\langle L,R_{\mathrm{full}}\rangle + +\lVert R_{\mathrm{full}}\rVert^2. +\end{equation} + +At $n=1.5,J=0$, captured iteration 1, the raw full trial was admissible with +minimum sampled Jacobian 0.896116. Its linear prediction was only 0.02576812 +of the base norm; the actual residual was 12.243161 times the base norm. +At $\alpha=0.1$ the actual ratio remained 1.02052 rather than the predicted +0.900070; at 0.01 it decreased to 0.991209. This established nonlinear merit +failure separately from the rotating thermodynamic wall. + +Gravity-gradient actions from shape and fields had norms 0.115959 and +0.116055, cosine $-0.999999831$, and combined norm +$1.16990\,10^{-4}$. Poisson actions also nearly cancelled, with cosine +$-0.999996055$. Damping shape alone would have destroyed this measured +first-order balance. The full-step second-order remainder was +$2.18903\,10^{-3}$, with 99.7747 percent of its squared norm in gravity. +Fixed-shape gravity is affine in its fields in the source; measured +fields-only gravity remainders remained at arithmetic scale. Resolved +remainders divided by $\alpha^2$ stabilized, consistent with the directional +Jacobian rather than a persistent first-order derivative error. + +At the constrained $n=3$ seed the original correction still failed the +$0.03$ forcing gate, attaining 0.0402590; it was not silently replaced by +a better direction. At its common cap $\alpha=0.12194828$, the actual norm +was 1.0237652 of the base versus prediction 0.87826014. Gravity accounted +for 90.7812 percent of the squared remainder. At one tenth of this cap the +trial decreased the residual, and the net curvature contribution was slightly +favorable. A large mixed term therefore did not imply that its signed merit +projection was always adverse. + +The consequence was to keep feasibility and nonlinear path curvature as +separate hypotheses. A sampled enthalpy cap could avoid futile trials but +would not remove a wall where $h\to0$ and $\delta h<0$. Nor could exterior +weighting be justified from whole-gravity-block norms alone: the original +probes had not yet localized stellar versus exterior contributions. + +\paragraph{Evidence.} +\nolinkurl{experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md}, +\nolinkurl{experiments/PRESSURE_OBSTRUCTION_METHODOLOGY_2026-09-17.md}, and +\nolinkurl{experiments/POLYNOMIAL_ENTHALPY_ADMISSIBILITY_2026-09-17.md}. +The eight focused and two supplemental cases restored their base residuals +exactly; direction partitions and signed-merit identities passed independent +checks. No additional virial-qualified model was produced by these probes. + +\section{Three optional numerical strategies and their limits} + +The next implementation added three interchangeable numerical approaches: +polynomial enthalpy feasibility, gravity curvature correction, and gravity +elimination. They were selected by concrete typed policies through +\texttt{NewtonNumerics}, not added to the physics-facing model declarations. +Unregistered or incompatible policies failed compile-time constraints; +actual mesh and FE eligibility were checked collectively at runtime. The +ordinary Newton path remained available and defaults were unchanged. + +\subsection{Polynomial enthalpy certification and anchor repair} + +The diagnostic converted the represented nodal polynomial to a tensor +Bernstein representation with outward-rounded interval arithmetic. Adaptive +subdivision returned a nonnegativity certificate, a negative-value witness, +or unresolved bounds. A negative Bernstein coefficient by itself was not +misclassified as a negative polynomial. Exact zero traces were retained. +Certification used physical enthalpy coordinates, because general diagonal +normalization does not preserve the polynomial represented by the coefficients. + +For an admissible base, a fixed certified initial field supplied an anchor: +\begin{equation} + p_h(\beta)=(1-\beta)p_h+\beta(h_{\mathrm{anchor}}-h), + \qquad0\leq\beta\leq1. +\end{equation} +Sufficient Bernstein inequalities constrained the repair fraction and the full +linear residual model selected a fraction. The actual normalized-and-added +candidate was certified again, and its complete merit derivative had to remain +negative. This was a conservative feasible-direction family, not a constrained +Newton optimizer, and it did not make the independent density positive. + +At the rotating onset the base certified and the original full trial was +proved negative. Whole-mesh certification cost about 3--4 ms. The sufficient +repair fraction was 0.9152726503; repaired polynomials prepared successfully, +but trial lengths 1, 0.5, and 0.125 all increased the full residual. The linear +model's best step was near 0.00825, with only about 0.46 percent predicted +residual reduction. The already negative late base correctly returned +\texttt{invalid\_base} and left accepted buffers unchanged. This path supplied +useful diagnostic and admissibility machinery, but was not established as an +efficient route through the nonlinear obstacle. The later user decision to +defer positivity enforcement was consistent with this mixed result. + +\subsection{Gravity-only curvature correction} + +Partition the normalized state into retained variables $u$ and gravity +variables $z$, with gravity equations $G(u,z)$. At a raw trial +$y=x+\alpha p$, the correction solved +\begin{equation} + G_z(y)\,\delta z= + -\bigl[G(y)-G(x)-\alpha G'(x)p\bigr]. +\end{equation} +Only gravity coordinates changed. This removes the nonlinear gravity +remainder, not the entire gravity residual. It preserves the base tangent +because the right-hand side is second order along a smooth path, and it +vanishes on an affine gravity path even when the base residual is nonzero. +The candidate inverse was refreshed, every residual was reevaluated, and +ordinary full-merit acceptance remained authoritative. + +The first frozen $n=1.5$ comparison changed the full trial residual from +$2.18913190375\,10^{-3}$ to $1.40702031008\,10^{-4}$ at base norm +$1.78804469158\,10^{-4}$. The gravity remainder fell from +$2.18656200516\,10^{-3}$ to $1.37482284578\,10^{-12}$, with observed +correction order 1.999898 on the smallest resolved steps. This was direct +evidence that the identified curvature could be removed from gravity. + +An initial production attempt was intentionally stopped after repeated +auxiliary prepared-consistency rejection despite converged inner solves. +The arithmetic allowance had omitted large cancelling gravity contributions. +The revised check included separately measured gradient, potential, source, +and correction actions; it used $256\epsilon$ times their norm sum as a +heuristic allowance, not a rigorous error bound. The true inner gate, lifted +full-Jacobian gate, nonlinear tolerance, and physical criteria were not relaxed. +The interrupted run and correction history were preserved. + +\subsection{Gravity elimination} + +Gravity elimination instead completed gravity at the initial state and each +retained trial, then used the reduced Schur action +\begin{equation} + S=F_u-F_zG_z^{-1}G_u. +\end{equation} +Right-hand-side condensation included the residual gravity defect, and lifting +reconstructed a full correction that still had to satisfy the original full +normalized linear gate. Tolerance-stopped inner Krylov solves only approximated +this linear action; flexible outer preconditioning did not magically make it +exact. Tight inner gates, finite-difference checks, and lifted verification +were therefore retained. + +The reference implementation restricted the full operator and inverse. A +capability-checked compact path reused the existing prepared physical gravity +actions with the same scaling. On three dense actual-FEM directions compact +and full-restricted outputs agreed exactly. Gravity operator applications +cost about 0.018 versus 0.244 seconds, around fourteen times faster, while +inverse applications changed only from approximately 0.205 to 0.160 seconds. +A matched tight inner solve retained forty iterations and the same result +while falling from 18.36 to 7.25 seconds. The fourteenfold action gain was +not a fourteenfold Newton speedup. + +\subsection{Production outcomes of the three strategies} + +{\small +\begin{longtable}{@{}p{55mm}p{24mm}p{32mm}p{35mm}@{}} +\hline +Model and path & Accepted updates & Final $\lVert F\rVert$ & Newton seconds / outcome \\ +\hline +$n=1,J=0$, curvature & 6 & $6.51136\,10^{-14}$ & 273.71 / converged \\ +$n=1,J=0$, eliminated & 6 & $8.98743\,10^{-14}$ & 869.67 / converged \\ +$n=1.5,J=0$, curvature & 6 & $1.48794\,10^{-5}$ & 711.72 / outer gate failure \\ +$n=1.5,J=0$, eliminated & 5 & $4.47888\,10^{-6}$ & 1821.79 / completed updates only \\ +$n=1,J=0.1$, polynomial & 3 & $7.13059\,10^{-3}$ & 265.03 / outer gate failure \\ +Polynomial plus curvature & 1 & $8.40387\,10^{-3}$ & 127.05 / outer gate failure \\ +Polynomial plus elimination & 1 & $2.72634\,10^{-2}$ & 149.83 / non-descent repair \\ +\hline +\end{longtable} +} + +The two nonrotating $n=1$ gravity paths and polynomial control reached the +same discrete model to coefficient differences below $2.632\,10^{-11}$. +All eight virials were below $3.73092\,10^{-7}$; independent enthalpy +certificates covered all 832 stellar elements. The eliminated control used +107 inner solves and 3,875 inner iterations, versus six solves and 115 +iterations for curvature correction. It was a verified alternative, but +substantially more expensive. + +The $n=1.5$ curvature run reduced the initial norm about eighteenfold but +failed its seventh outer solve at relative defect 0.059017 after 200 +iterations. All fourteen auxiliary solves passed. The eliminated run was +externally interrupted at 2,400 seconds during its sixth attempt, after five +accepted steps of $1,1/2,1/4,1/128,1/2048$. Its last accepted gravity residual +was about $3.27\,10^{-14}$ while retained equations remained of order +$10^{-6}$; the last step improved the total residual only 0.00565 percent. +This showed stagnation despite completed gravity. No final snapshot or +complete physical diagnostics existed for the timed-out run, so these were +not invented from its log. + +Rotating runs remained both unconverged and virially unacceptable. The +polynomial-only fourth linear attempt attained 0.0331634 rather than the +required 0.03. Adding curvature improved the first step with the same tangent +but the second linear solve failed at 0.0826534. Polynomial plus elimination +passed its reduced and lifted linear gates before a repaired direction failed +descent. A reporting replay later established that this failure occurred +before another gravity relift, correcting an earlier ambiguous interpretation +of counters. All these models retained negative independent density samples. + +The implementation was covered by 43 targeted serial cases and 1,886 +assertions at the final revision, with 37 supported cases on each of two +MPI ranks. Full stellar multi-rank operation was not thereby certified: +logical radial surface interpolation and per-rank material ownership +restrictions remained. The frozen checks established algebra and lifecycle +invariants; they did not constitute physical convergence. + +\paragraph{Evidence.} +\nolinkurl{experiments/NUMERICAL_STRATEGIES_2026-09-17.md}, +\nolinkurl{experiments/GRAVITY_SUBSYSTEM_IMPLEMENTATION_2026-09-17.md}, +\nolinkurl{experiments/GRAVITY_REDUCTION_IMPLEMENTATION_2026-09-17.md}, +\nolinkurl{experiments/GRAVITY_COMPACT_APPLICATIONS_2026-09-17.md}, and +\nolinkurl{experiments/GRAVITY_PREPARED_ROUNDOFF_2026-09-17.md} retain the +complete policy contracts, intermediate failed builds/runs, and replay history. + +\section{Preconditioner improvements measured at equal accuracy} + +\subsection{Baseline approximation and work accounting} + +The original inverse first treated surface and material through a triangular +surface-then-enthalpy-then-density approximation. Surface used a reference +mass diagonal; enthalpy used hydrostatic mass and identity treatment at +replaced boundary rows; density included the live EOS enthalpy derivative. +Material-to-surface feedback was omitted. Gravity used approximate mixed LDU +with a degree-five Chebyshev mass inverse and three fixed AMG cycles for a +potential surrogate. Structure-level material and gravity were independent, +then a small dense scalar Schur complement completed specification borders +relative to that approximate inverse. The live high-$n$ EOS derivative was +present; there was no evidence that the baseline simply omitted it. + +Frozen benchmark roles each constructed fresh inverses and border responses +at the same saved state, full Jacobian, normalization, and right-hand side. +Reported complete verified cost included inverse construction, normalized +binding, Krylov setup, solve, and an independent full-Jacobian endpoint action. +Failed solves were retained but never used as finite baseline +time-to-convergence measurements. Timing runs and large audits were serialized. + +\subsection{Prepared cross-action optimization} + +Existing stellar LDU was initially unexpectedly expensive: at an $n=1.5$ +plateau, 200 iterations cost 599.5473 seconds versus 87.9302 for the +independent baseline, and both failed the gate. The gravity-to-material +coupling used a legacy geometry-evaluation path. Replacing it with the +already prepared complete gravity-force action, with zero density and +displacement variations, preserved the same mathematical gravity-gradient +column and subsequent deformation adjoint. + +The cached LDU reduced complete cost to 117.4845 seconds, a factor 5.1032 +relative to the old LDU path, and mean inverse application time by 8.0425. +The correction changed only $2.7051\,10^{-11}$ relatively and the gate still +failed at 0.0600669. This was an execution optimization, not a conditioning +improvement. It deserved retention independently of whether LDU was selected +as a numerical policy. + +\subsection{Mass-consistent gravity Schur refinement} + +Writing the mixed gravity block as +\begin{equation} + D=\begin{bmatrix}M&B^T\\B&0\end{bmatrix},\qquad + D\begin{bmatrix}g\\\Phi\end{bmatrix}=\begin{bmatrix}f\\r_g\end{bmatrix}, +\end{equation} +let $R$ be the same approximate mass inverse used for flux recovery and $Q$ +the existing AMG inverse on the diagonal-mass surrogate. The added policy +used +\begin{align} + S_R&=BRB^T, & r&=BRf-r_g, & \Phi_0&=Qr,\\ + \Phi_{j+1}&=\Phi_j+\omega Q(r-S_R\Phi_j), & + g&=Rf-RB^T\Phi_k. +\end{align} +The Poisson defect was thereby measured against the same approximate mass +action as recovery. This was still not an exact inverse of the mixed block. +Fixed work preserved stationarity only when the child actions themselves +declared stationary linear behavior. Positive relaxation alone did not prove +contraction. Each extra sweep cost one application of $R,Q,B,B^T$; zero +sweeps retained the old arithmetic path for bitwise baseline comparison. + +One sweep with $\omega=0.5$ first crossed the original $0.03$ gate at the +$n=1.5$ saved state. An equal-accuracy confirmation gave: + +{\small +\begin{longtable}{@{}p{55mm}p{22mm}p{34mm}p{29mm}@{}} +\hline +Inverse, frozen $n=1.5,J=0$ & Iterations & True relative defect & Complete cost (s) \\ +\hline +Independent baseline & 231 & 0.0290033753 & 100.9943 \\ +Mass-consistent gravity Schur & 132 & 0.0292091550 & 72.3727 \\ +\hline +\end{longtable} +} + +This was a measured 1.39548-fold speedup, 28.34 percent lower complete cost +and 42.86 percent fewer iterations, without relaxing accuracy. At a saved +$n=1,J=0.1$ system, the independent and gravity-only controls still failed +after 600 iterations, at defects 0.0739197 and 0.0408506. Combining cached +stellar LDU with gravity refinement passed in 142 iterations at defect +0.0299068971, complete cost 96.8236 seconds. It reduced Poisson and gradient +defect norms by 63.33 and 94.47 percent relative to the matched 200-iteration +independent control. Since the baseline had no finite observed successful +solve, no rotating speedup ratio was asserted. + +A nine-mode coupled shape coarse inverse was a negative control. All modes +were active, overlap/rank checks passed, and there was no fallback, but the +200-iteration defect worsened to 0.0668658. Setup added 8.6012 seconds and +persistent vectors occupied 28,491,680 bytes. That failed candidate remained +experimental; it did not rule out every possible coarse space. + +\subsection{Nonlinear transfer limited the recommendation} + +For easy $n=1,J=0$, both independent and gravity-refined inverses converged +in six full steps to the same physical state, with all virials below +$3.74\,10^{-7}$. Linear iterations fell from 307 to 258, yet Newton time +rose from 191.068 to 199.375 seconds, 4.35 percent. More expensive inverse +applications outweighed iteration reduction. + +For $n=1.5,J=0$ with gravity curvature correction, the stronger inverse +passed all twelve permitted outer linear gates and all 31 auxiliary gravity +solves. It removed the earlier immediate linear failure but ended at +$1.3966991\,10^{-5}$, only 6.13 percent below the old six-update endpoint +and far above target $2.7365922\,10^{-12}$. All nineteen rejections were +merit failures, with late accepted steps $1/64$ and $1/512$. Newton time +was 1075.368 seconds, including 415.038 in outer linear solves and 590.003 +in preparation; nested auxiliary time must not be added again. Enthalpy-based +virials passed, but negative density persisted. + +The rotating combined-inverse follow-through accepted one repaired full step +to $0.00685562$, then failed the next linear gate at 0.0339685. Its +enthalpy energy virial, 0.00712711, was worse than the historical independent +endpoint's 0.000544539 despite its smaller nonlinear residual. Both failed +physically. Frozen linear gains therefore did not justify claiming a solved +nonlinear or physical problem, and the stronger policies remained optional. + +\paragraph{Evidence.} +\nolinkurl{experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md}, +\nolinkurl{experiments/GRAVITY_CONSISTENT_SCHUR_2026-09-18.md}, and +\nolinkurl{experiments/PRECONDITIONER_CACHED_CROSS_ACTION_2026-09-18.md}. +Focused serial coverage passed 20 cases and 441,978 assertions; two-rank +algebra/coarse tests passed thirteen cases per rank. The actual multi-rank +FEM test was blocked by a pre-existing partition where one rank had no +vacuum element, required by mapped H(div) construction. Refining the mesh +did not remove that ownership pattern. Serial FEM evidence must not be +described as distributed FEM validation. + +\section{Why better linear solves did not remove the nonlinear stall} + +The next frozen investigation used the saved $n=1.5,J=0$ endpoint with +$\lVert F\rVert=1.3966991044\,10^{-5}$. All prior linear gates had passed; the final +squared-residual shares were about 42.42 percent hydrostatic, 26.23 percent +closure, 15.87 percent surface, 10.43 percent gravity, and 5.05 percent mass. +It computed fresh cold directions rather than claiming to reconstruct raw +directions that the earlier trajectory had not saved. + +For $\phi=\lVert F\rVert^2/2$ and a forcing condition +$\lVert Ap+F\rVert\leq\eta\lVert F\rVert$, a consistent derivative gives +\begin{equation} + \phi'(0)=\langle F,Ap\rangle\leq-(1-\eta)\lVert F\rVert^2. +\end{equation} +Thus a gate-passing direction should be strongly locally descending, but +this says nothing about how far the linear approximation remains useful. + +\subsection{Tenfold tighter forcing did not improve the useful step} + +{\small +\begin{longtable}{@{}p{70mm}p{38mm}p{38mm}@{}} +\hline +Measurement & $\eta=0.03$ & $\eta=0.003$ \\ +\hline +Iterations & 84 & 143 \\ +Cold linear seconds & 47.9374 & 81.6901 \\ +Verified relative defect & 0.0295592 & 0.00294840 \\ +Direction norm & 0.0794897 & 0.0796012 \\ +Corrected residual/base at $1/64$ & 1.080323 & 1.080567 \\ +Corrected residual/base at $1/256$ & 1.002025 & 1.002037 \\ +Corrected residual/base at $1/1024$ & 0.999394139 & 0.999394272 \\ +\hline +\end{longtable} +} + +The tenfold stricter gate cost about seventy percent more linear work and +left the useful sampled step essentially unchanged. The directions differed +by only 0.8885 percent in norm and had cosine 0.99996157. A supplementary +minimum-norm calculation in their two-dimensional span, subject to the +original forcing gate, gained only 0.00058549 percent beyond ordinary scalar +damping of the tighter direction. This was a saved-vector algebra experiment, +not a fresh FEM trial, and did not rule out richer subspaces. + +Raw trials at $1,1/4,1/16$ failed pressure preparation for this new late +direction. At $1/64$, raw and gravity-corrected residual ratios were 1.807268 +and 1.080323; at $1/256$, 1.002510 and 1.002025. At $1/1024$ both decreased, +to 0.999170 and 0.999394. The remainders were resolved and approximately +quadratic over the small-step range. No persistent first-order error was +detected in the tested tangent. + +\subsection{A large compensated direction amplified moderate curvature} + +The separate shape and field action norms were 0.7730368 and 0.7730377, +while their sum had norm $1.396089\,10^{-5}$. The image cosine was +$-0.999999999838$, and the cancellation factor was approximately 110,743. +At the seed, the corresponding factor had been about 332.1. Normalization +factors were byte-identical across these states. + +The late direction was 17.35 times larger while the residual was 19.59 times +smaller. The ratio $\lVert p\rVert/\lVert F\rVert$ grew from 16.7433 to 5691.26, about +340-fold. At the common step $1/64$, corrected curvature per squared +direction norm grew only from 0.656881 to 0.926040, about 1.41-fold; curvature +relative to the base residual grew about 8,313-fold. This pointed toward +weak coupled response and large corrections as the amplification mechanism, +rather than orders-of-magnitude growth of curvature per unit direction. +It did not identify a continuum symmetry or estimate the full Hessian. + +Gravity correction reduced the small-step remainder norm by about 14.7-fold +at $1/1024$, but increased its adverse projection onto the linear predictor +2.88-fold. Hydrostatic, closure, surface, and mass rows contributed 43.52, +32.33, 19.53, and 4.63 percent of that signed projection. A smaller remainder +therefore need not produce better merit. The observed full/shape gain ratio +also gave a condition-number lower bound of about $9.07\,10^4$ if the +finite-dimensional Jacobian was nonsingular; it was not a measured global +condition number or a license to add a central constraint at $n=1.5$. + +\paragraph{Evidence.} +\nolinkurl{experiments/NONLINEAR_STEP_INVESTIGATION_2026-09-18.md}. Three +immutable probes covered 144 path observations and sixty successful +auxiliary gravity solves; final saved-artifact audits passed 20,899 checks. +All states, factors, and restored actions were preserved. Those checks +established provenance and arithmetic identities, not a new model. + +\section{Fully coupled second-order trial corrections} + +\subsection{Bounded correction against the complete residual} + +Because gravity-only correction moved adverse curvature into retained +equations, a fully coupled trial correction was implemented. At a prepared +trial $y$, it preserved the original affine target +$L(\alpha)=F_0+\alpha A_0p$ and solved +\begin{equation} + A(y)c=L(\alpha)-F(y). +\end{equation} +The auxiliary backend had its own true full-Jacobian defect check, finite budget, +and report. The correction was capped by $\lVert c\rVert\leq\alpha\lVert p\rVert$ by +default. It was attempted only after a prepared candidate failed ordinary +acceptance, with at most one solve per Newton iteration initially. A failed +solve, cap, admissibility test, or merit comparison restored the original +trial with fresh preparation. Accepted-state buffers were not mutated by the +transaction. The policy composed statically with existing enthalpy/gravity +choices and deliberately rejected the unimplemented eliminated-gravity +combination at compile time. + +At the common late state, the original direction still passed with defect +0.02955918315. Independent frozen trials showed: + +{\small +\begin{longtable}{@{}p{40mm}p{22mm}p{22mm}p{22mm}p{30mm}@{}} +\hline +Path and $\alpha$ & Before/base & After/base & Auxiliary iterations & Added seconds \\ +\hline +Raw, $1/64$ & 1.80726810 & 1.03360191 & 34 & 27.33 \\ +Raw, $1/256$ & 1.00250966 & 0.99746464 & 52 & 37.59 \\ +Gravity first, $1/64$ & 1.08032296 & 1.09566246 & 55 & 47.60 \\ +Gravity first, $1/256$ & 1.00202482 & 0.99810801 & 62 & 43.53 \\ +\hline +\end{longtable} +} + +All auxiliary solves passed their unchanged gate and cap. The $1/256$ +trials became acceptable, while the gravity-first $1/64$ trial worsened and +was restored. The already acceptable $1/1024$ and seed full-step controls +incurred no coupled solve. Raw timing came from an uncontended repeat after +a mistakenly overlapping artifact audit was stopped and fixed; fifty saved +vector files matched the first run byte-for-byte, and the original timing +was excluded. + +The correction's own curvature remained significant. For +$b=L-F(y)$, $\delta=A(y)c-b$, and +$E=F(y+c)-F(y)-A(y)c$, +\begin{equation} + F(y+c)-L=\delta+E. +\end{equation} +At $1/256$, $\lVert E\rVert$ exceeded the verified auxiliary defect by about 11.28 +for the raw path and 254 for the gravity-first path. Correction/direction +cosines were $-0.9999330$ and $-0.9999942$: much of the correction reversed +the original step. Actual displacement increases relative to the old +$1/1024$ path were only 1.7596 and 1.2342, not the nominal factor four in +$\alpha$. + +Fresh scalar controls matched projection along the original direction. +They decreased residual 0.115623 and 0.096934 percent, whereas the coupled +candidates decreased 0.253536 and 0.189199 percent. Small transverse components +therefore had a measured benefit; the improvement was not just scalar damping. +It remained a small absolute gain purchased with another coupled solve. +Moreover, the independent frozen alphas each had a fresh correction budget, +so this screen did not measure a live shared-budget line search. + +\subsection{Damping and work-aware eligibility} + +The next optional policy reused one computed $c$ across +$\beta=1,1/2,1/4,1/8$, always preparing $y+\beta c$ from the same $y$. +The first ordinarily acceptable improvement ended the search; otherwise the +best admissible improvement could be retained as a trial, with fresh replay. +An eligibility rule reserved the single solve for a candidate near the affine +prediction. For the configured deterministic merit $M$, it required positive +finite $D=M(F_0)-M(L)$ and +\begin{equation} + \frac{M(F(y))-M(F_0)}{D}\leq k,\qquad k=1\ \hbox{by default}. +\end{equation} +This was a bounded-work heuristic, not a theorem of global convergence. + +A sequential halving screen exposed an important correction to the earlier +interpretation. Ordinary raw and gravity paths already accepted $1/512$; +the previous sparse grid had omitted it. The near-acceptance policy instead +accepted $1/256$, a twofold nominal increase. Without eligibility deferral, +the sole solve was spent at an unsuccessful $1/32$ and the final state was +the baseline. Both successful frozen rescues used $\beta=1$, so eligibility, +not damping, explained that particular improvement. + +In uncontended raw repeats, baseline search work cost 45.76 seconds for +0.116541 percent decrease; near-acceptance correction cost 74.90 seconds +for 0.253536 percent decrease. The fractional decrease per measured +operation-second was 1.329 times greater. Those timings excluded ordinary +outer solve, commit refresh, initialization, and physical replay; they were +not Newton speedups. A clock-anomalous original gravity/full run was retained +and excluded, then repeated with independent UTC, monotonic, and child CPU +measurements. + +\subsection{Matched six-update live test} + +The live pair kept the seed, mesh, $n=1.5,J=0$, absent central constraint, +gravity-Schur inverse, restart 160, and $\eta=0.03$ fixed. Only the trial +policy changed. Both first updates reached +$1.614841549\,10^{-5}$ with the same recorded diagnostics. + +{\small +\begin{longtable}{@{}p{70mm}p{38mm}p{38mm}@{}} +\hline +Endpoint after six updates & Gravity curvature control & Coupled policy \\ +\hline +Final residual & $1.497197\,10^{-5}$ & $1.245830\,10^{-5}$ \\ +Newton iteration seconds & 502.21 & 613.09 \\ +Outer Krylov iterations & 354 & 341 \\ +Auxiliary coupled iterations & 0 & 163 \\ +$P(h)$ energy virial & $4.22590\,10^{-6}$ & $3.78554\,10^{-6}$ \\ +$P(h)$ force virial & $2.71367\,10^{-6}$ & $2.31894\,10^{-6}$ \\ +Minimum sampled density & $-7.73185\,10^{-4}$ & $-7.85522\,10^{-4}$ \\ +\hline +\end{longtable} +} + +The new endpoint residual was 16.79 percent lower for 22.08 percent more +Newton time. Excluding the common first update, residual decreases were +22.85 versus 7.29 percent, giving 2.53 times more decrease per iteration-second +over that bounded interval. Both stopped at the six-update cap; neither +converged. The new method initially crossed several modest residual thresholds +later than the control before eventually overtaking it. + +Only one of three coupled searches was committed. At update two the full +correction worsened merit but $\beta=1/2$ passed, with a very small 0.01795 +percent residual decrease. Later ordinary steps benefited from the changed +trajectory. Two other kept improvements remained unacceptable and were +followed by ordinary backtracking; they were not accumulated into the model. +This result supported retaining the policy as an experiment, not promoting it +or treating every rescued step as an efficient update. Density-pressure +virials remained undefined and local EOS extrema did not uniformly improve. + +\paragraph{Evidence.} +\nolinkurl{experiments/COUPLED_TRIAL_CORRECTION_2026-09-18.md} and +\nolinkurl{experiments/DAMPED_COUPLED_TRIAL_2026-09-18.md}. The latter covered +58 serial numerical/solver cases, 23 correction cases on two MPI ranks, +56 Python checker tests, twelve frozen cases including repeats, and the +matched live pair. Its tests covered fallback, exceptions, budgets, +rank-local rejection, preserved outer reports, and typed composition. + +\section{Shape freedom, field transport, and the displacement-order control} + +The next investigation targeted the weak coupled response itself, preserving +the equations and avoiding an assumed gauge interpretation. At independent +quadrature orders 10 and 14, scalar pullback changes were decomposed as +\begin{equation} + \delta f_{\mathrm{Eulerian}}=\delta\widehat f-\nabla f\cdot v, +\end{equation} +where $v$ was the generated physical displacement. Density and potential +gradients were broken within-element gradients; facet jumps and RT transport +were not included. Small Eulerian/pullback ratios therefore suggested +transport compensation but could not certify a pure relabeling nullspace. + +At the old twelve-update $Q_3$ endpoint, these ratios were 0.05769 for +density, 0.18870 for enthalpy, and 0.01288 for potential, versus seed values +0.3243, 0.3080, and 0.1993. A six-block sampled singular-response minimum +fell from 0.0580574 at seed to 0.000143543 late, with largest sampled response +near 22.1--22.6. The normalized block direction span itself changed with the +state, so this was not a full-Jacobian spectrum or a rank proof. + +Despite an enthalpy energy virial of $4.06909\,10^{-6}$, the late +nonrotating nodal surface radii spanned 0.997674--1.010093. Its direction was +99.9995107 percent cubic-invariant in sampled squared norm, while degree-eight +spherical harmonics left 0.893679 of its nodal norm unexplained. This was +fine angular structure consistent with mesh/discretization coupling, not +evidence that all omitted modes were unphysical. The bulk and gravity +discretizations also shared cubic symmetry. + +\subsection{The isolated $Q_4$ displacement experiment} + +The original $Q_3$ displacement could not exactly represent the order-four +reference coordinate map: the affine identity control gave relative value +error $9.01277\,10^{-5}$ and gradient error $1.23472\,10^{-3}$. +On an isolated branch, only +\nolinkurl{libmeanfield/interface/field/field_registry.cppm}, +\texttt{Displacement::vectorOrder}, changed from $3+$ uniform increment +to $4+$ uniform increment, with the increment fixed at zero. The rationale +and date were recorded in source. The shared scalar surface collection +therefore also changed from $Q_3$ to $Q_4$, increasing surface unknowns from +866 to 1538. Density/potential $Q_2$, enthalpy $Q_3$, RT2 gravity, geometry, +and the radius-squared lift were retained. + +This was not a pure interpolation intervention: it added shape modes, +changed surface normalization/preconditioning, increased derived pressure +quadrature by one and gravity-force quadrature by two, and raised the +geometry-inspection request from 18 to 24. Both variants started from analytic +seeds; incompatible coefficient snapshots were not transferred. + +$Q_4$ reduced affine value and gradient errors to +$2.11629\,10^{-16}$ and $3.24701\,10^{-15}$. Nevertheless, seed physical +displacement direction norm grew 31.63-fold, from 0.00121967 to 0.0385749, +and seed linear iterations grew from 26 to 67. Both initial linear gates +passed, and small trial remainders remained quadratic. The fixed six-attempt +live comparison was unfavorable: + +{\small +\begin{longtable}{@{}p{70mm}p{38mm}p{38mm}@{}} +\hline +Measurement & $Q_3$ displacement/surface & $Q_4$ displacement/surface \\ +\hline +Root unknowns & 178,073 & 178,745 \\ +Accepted updates & 6 & 3 \\ +First three accepted-step seconds & 269.31 & 505.65 \\ +Residual after three / own initial & 0.056755 & 0.906188 \\ +All attempted Newton seconds & 547.34 & 638.42 \\ +Krylov iterations, including failure & 354 & 492 \\ +$P(h)$ energy virial & $4.22590\,10^{-6}$ & $8.89666\,10^{-5}$ \\ +$P(h)$ force virial & $2.71367\,10^{-6}$ & $8.27467\,10^{-6}$ \\ +Hydrostatic-gradient RMS & 0.0106811 & 0.325413 \\ +Nodal radius span / nodal mean & 1.09849\% & 12.2559\% \\ +\hline +\end{longtable} +} + +The fourth $Q_4$ solve and an independent cold endpoint solve both failed +after 200 iterations at relative defect 0.0868564. Even its first three +gate-passing corrections had already produced poor progress. The differing +spaces and stopped endpoints prevented an equal-accuracy comparison; the +result nevertheless ruled out matching displacement order alone as a +sufficient cure in this configuration. The $Q_4$-only change was explicitly +not recommended for promotion. It did not rule out compatible enrichment +of multiple fields. + +\subsection{Two structural cautions} + +A naive replacement of mechanical shape rows by surface Bernoulli moments +was identified as rank-deficient at an ideal $B=0,J=0$ root for the current +trace spaces. On the actual closed surface, continuous $Q_p$ had +$96p^2+2$ degrees of freedom, versus $96p^2$ for broken $Q_{p-1}$ potential +traces. After enforcing the zero enthalpy trace, $Q_3$ surface supplied 866 +rows against 864 DG $Q_2$ trace coefficients; $C$ added no independent +constant trace dimension and the direct shape-measure derivative vanished +with $B$. This was an objection to that proposed replacement, not proof of +rank deficiency in the existing mechanical operator. Any alternative +boundary formulation needed compatible spaces and a stability analysis. + +Second, correcting ambient interpolation did not make the radius-squared +extension generate a physical dilation. If $b$ was the boundary trace of +reference position $X$, the existing lift $E_0(b)$ differed from $X$; at a +logical radius $1/8$ vertex it gave one eighth of the desired affine value. +A proposed but unimplemented rank-one correction was +\begin{equation} + E(z)=E_0(z)+\bigl[X-E_0(b)\bigr]\ell(z),\qquad\ell(b)=1. +\end{equation} +It would give $E(b)=X$ while retaining the supplied trace and admit an exact +transpose. This could place physical dilation in the mechanical test space, +but would not automatically enforce the potential-energy virial, which uses +the independent mixed potential. It was kept separate from surface-space +reduction and was not claimed as an implemented remedy. + +\paragraph{Evidence.} +\nolinkurl{experiments/SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md}. The $Q_4$ +production change was isolated at commit +\texttt{4be1a23c3ec2650af6a2a49528a8f5248cb001c3}; 22 targeted cases and +380,856 assertions passed. That correctness coverage did not convert its +negative numerical outcome into a recommendation. + +\section{The bounded smooth-surface study} + +The user authorized a one-hour study, recorded as 19:30:15--20:30:15 UTC on +18 September. It retained $Q_3$ displacement, $Q_4$ reference geometry, +$Q_2$ density/potential, $Q_3$ enthalpy, RT2 gravity, the same extension, +$n=1.5,J=0$, and no central constraint. It changed only independent surface +variation through a frozen affine embedding +\begin{equation} + q=q_0+Ba. +\end{equation} +The existing base $q_0$ was never projected. Real spherical harmonics through +degree $L$ were evaluated at actual radial surface nodes and their +interpolants orthonormalized in a quadrature-defined physical surface-area +metric. All azimuthal components in that harmonic span were included. + +Correct primal and dual normalized maps were essential. For physical state +scales $S$ and residual scales $R$, separate thin QR decompositions gave +\begin{equation} + SB=Q_xT_x,\qquad R^{-1}B=Q_fT_f, +\end{equation} +so that +\begin{equation} + Q_f^T(RF_{\mathrm{physical}})=T_f^{-T}B^TF_{\mathrm{physical}}. +\end{equation} +The reduced operator was $Q_f^T\widehat AQ_x$, not a blind transpose of a +normalized state basis. Other fields and scalar equations remained active. +The projected existing inverse was an approximation, not an exact reduced +inverse. Both retained and original full residuals, and excluded shape force, +were always exported. + +\subsection{Frozen late-state result} + +At the common six-update $Q_3$ endpoint, +$\lVert F\rVert=1.4971971\,10^{-5}$, cold FGMRES used restart 160, cap 200, +$\eta=0.03$, and one gravity-Schur sweep with relaxation 0.5. + +{\small +\begin{longtable}{@{}p{40mm}p{22mm}p{22mm}p{22mm}p{30mm}@{}} +\hline +Shape space & Parameters & Iterations & Solve seconds & Unit-trial full norm \\ +\hline +Full nodal trace & 866 & 61 & 32.184 & Pressure rejection \\ +$L=2$ & 9 & 12 & 6.447 & $2.263903\,10^{-6}$ \\ +$L=4$ & 25 & 21 & 11.249 & $2.432960\,10^{-6}$ \\ +$L=8$ & 81 & 29 & 15.745 & $2.964119\,10^{-6}$ \\ +\hline +\end{longtable} +} + +All reduced directions passed their retained linear gates, with relative +defects 0.01719, 0.01608, and 0.01409. Their original full linear defects +were 0.15121, 0.16187, and 0.19207: none passed the original full Newton +gate. Nevertheless, all four sampled trial alphas prepared and decreased +both norms, and full residual decreases at unit step were 84.88, 83.75, +and 80.20 percent. This was a real improvement in the original residual, +obtained by restricting the shape equations, not a more accurate solution +of the original Newton system. + +The $L=4$ root had 177,232 unknowns, only 841 fewer than the full root; +its less than 0.5 percent dimension reduction did not explain a factor 2--5 +drop in cold solve time by itself. Fewer Krylov iterations supplied most +of the observed gain. Physical displacement norm fell from +$3.96028\,10^{-3}$ for the full direction to $9.14977\,10^{-5}$ at +$L=4$, 43.28-fold. The $L=2$ direction was nearly field-only, while admitting +higher modes increased coupled displacement and curvature. + +After the $L=4$ unit trial, excluded shape force accounted for 98.38 percent +of the remaining original squared norm. The inherited fine-scale geometry +also remained, since $q_0$ had not been projected. Enthalpy-based energy and +force virials improved to $8.11913\,10^{-7}$ and +$6.41398\,10^{-7}$, but local EOS/hydrostatic errors were almost unchanged +and minimum sampled density remained $-7.27628\,10^{-4}$. + +\subsection{Three cold reduced updates from the analytic seed} + +A conditional extension, recorded before launching, chained three saved +unit candidates from the analytic seed. Each stage passed preparation, +the retained independent linear gate, and both recorded norm-decrease +screens. This was an externally replayed cold reduced-Newton chain, not a +production warm-start trajectory or a fully converged model. + +{\small +\begin{longtable}{@{}p{14mm}p{30mm}p{30mm}p{30mm}p{30mm}@{}} +\hline +Updates & Full norm & Retained norm & Excluded shape norm & Energy virial $P(h)$ \\ +\hline +0 & $2.736592\,10^{-4}$ & $2.590732\,10^{-4}$ & $8.815010\,10^{-5}$ & $2.410694\,10^{-5}$ \\ +1 & $8.624827\,10^{-5}$ & $9.387845\,10^{-6}$ & $8.573583\,10^{-5}$ & $2.180785\,10^{-6}$ \\ +2 & $8.473526\,10^{-5}$ & $3.147803\,10^{-6}$ & $8.467677\,10^{-5}$ & $8.660498\,10^{-7}$ \\ +3 & $8.465191\,10^{-5}$ & $7.357283\,10^{-8}$ & $8.465188\,10^{-5}$ & $6.769653\,10^{-7}$ \\ +\hline +\end{longtable} +} + +The retained norm fell about 3521-fold but the original norm only 3.23-fold. +At the final trial, excluded shape force represented 99.999924 percent of +the original squared norm. Cold solves took 18, 20, and 18 iterations and +10.574, 11.341, and 9.594 seconds. Full relative linear defects were 31.45, +98.18, and 99.90 percent; these were not full-system Newton solutions. + +The final force virial was $5.769064\,10^{-7}$ and mass error +$3.79142\,10^{-9}$. Yet hydrostatic-gradient RMS rose from 0.001528098 +at seed to 0.007255981, about 4.75-fold, while density remained negative at +some samples. Nodal radius spread grew to 0.154699 percent, substantially +less than unrestricted late distortion but not an exact spherical result. +Small retained residual and excellent virial could therefore coexist with +significant omitted mechanical force and imperfect interior fields. + +The result supported decoupling smooth independent surface parameters from +the ambient displacement carrier, with nested configurable spaces and +consistent dual restriction. It did not establish that a fixed degree-four +space was sufficient at all rotation rates, that excluded force was +irrelevant, or that the complete original mechanical system had converged. +The subsequent native spherical-harmonic implementation and higher-index +studies should be read as later developments, not retroactively attributed +to this bounded experiment. + +All seven case audits passed 2,098 saved-artifact checks; a separate chain +audit passed 107 checks with bitwise physical-state and residual handoffs. +Actual builds, FEM cases, and expensive audits were serialized, and all seven +FEM probes completed by 20:09:58 UTC. No production order, operator, or lift +changed in this study. Optional $L=6$, rotating cases, higher indices, +mesh/enrichment sweeps, and production integration were outside its scope. + +\paragraph{Evidence.} +\nolinkurl{experiments/REDUCED_SURFACE_STUDY_2026-09-18.md}, producer commit +\texttt{21cdbda22486e408adc52b65b25413f9380c7c30}, and the archived +\nolinkurl{working/research/diagnostics/reduced_surface_2026-09-18} campaigns. Together with the +failed $Q_4$ experiment, this was the main reason to prefer a controlled +smooth surface span over indiscriminate addition of nodal surface freedom. + +\section{Native spherical-harmonic surface prescription} +\label{sec:recent-harmonics} + +The reduced-surface experiments motivated making spherical harmonics a native, +selectable surface prescription. The implementation on +\nolinkurl{research/spherical-harmonic-surface-20260919} preserved the simultaneous +Newton solve: surface coefficients, material fields, gravity fields and global +multipliers remain coordinates of one coupled nonlinear system. This was not a +separate surface-relaxation iteration. The preferred research candidate became +maximum degree $L=4$ with all real azimuthal orders, hence 25 independent +coefficients. The unrestricted nodal radial prescription remained available as +a control. The choice of $L=4$ was an experimentally useful starting point, not +a demonstrated resolution requirement for 50\% critical rotation. + +At reference boundary node $i$, with fixed radial direction $\widehat r_i$, the +radial displacement is +\[ + q_i=\sum_{\ell=0}^{L}\sum_{m=-\min(\ell,M)}^{\min(\ell,M)} + a_{\ell m}Y_{\ell m}(\widehat r_i). +\] +Omitting the optional azimuthal cutoff retains $M\ge L$. The real harmonics are +normalized to unit squared integral over solid angle, include the +Condon--Shortley phase, and use negative $m$ for sine and positive $m$ for cosine. +The catalog is ordered by increasing $\ell$ and then signed $m$. Coefficients +have dimensions of length. Zero coefficients preserve the reference mesh; +$\ell=0$ is a genuine monopole rather than a removed gauge. + +\begingroup +\small +\setlength{\tabcolsep}{3pt} +\begin{longtable}{@{}p{115mm}p{30mm}@{}} +Configuration & Independent coefficients \\ +\hline +\endfirsthead +Configuration & Independent coefficients \\ +\hline +\endhead +$L=4$, all $m$ & 25 \\ +$L=4$, $\lvert m\rvert\le2$ & 19 \\ +$L=4$, $m=0$ & 5 \\ +$L=8$, all $m$ & 81 \\ +\end{longtable} +\endgroup + +No odd-degree or nonaxisymmetric modes are silently excluded. Choosing $m=0$ +constrains the shape about the fixed global $z$ axis; it does not impose +axisymmetry on the three-dimensional interior fields. The reference center is +configurable, but the frame does not move as another Newton unknown. + +\subsection{Architecture and normalization} + +The prescription implements the existing forward, transpose and pullback +contracts, with a cached reference basis and zero derivative of that fixed +pullback. It composes with the existing interior and fixed-infinity vacuum +extensions. The numerical discretization now owns typed surface, interior and +vacuum prescriptions, instead of the modern equilibrium compiler hardcoding a +nodal surface. Existing two-argument discretization calls preserve their nodal +behavior. The physics-facing EOS, pressure boundary, mass and angular-momentum +specification is unchanged. + +Prescription/extension compatibility and coupled operator structure remain +compile-time constrained. Requested degree, sampled rank and actual mesh +resolution are runtime facts. Sampled aliasing is rejected rather than silently +reducing the mode set. Modal normalization and the default material/surface +inverse use a diagonal of the pulled-back reference vector trace mass matrix; +state and residual factors retain their primal/dual relationship. Metric +assembly checks FEM identity and a canonical trace-layout capability. The older +nodal $H^1$ surface surrogate explicitly rejects harmonic coordinates. + +Geometry remained Q4 and displacement remained Q3. The harmonic samples feed +the existing finite-element trace, so the resulting surface between nodes is +not an exact analytic spherical-harmonic graph. Reference-mesh interpolation +error remains. The native implementation also differs from the earlier frozen +reduced-space experiment, which retained an arbitrary nodal base shape and used +area whitening and QR maps. Physical residual and Jacobian equivalence is the +appropriate comparison; identical normalized Krylov trajectories are not +expected from those different coordinate metrics. + +The implementation was single-rank, consistent with the radial volume +extension. Successful two-rank rejection tests do not establish distributed +harmonic-solver support. + +\subsection{Native coupled results} + +Five live cases used the same mesh, cold Lane--Emden initialization, no central +constraint, a six-update budget, restart 160, at most 200 linear iterations and +a true relative linear gate $\eta=0.03$. Nonlinear convergence required +$\lVert F\rVert/\lVert F_0\rVert\le10^{-8}$, with zero absolute tolerance. All 29 accepted updates +were full steps, with one trial each and passing linear gates. + +\begingroup +\small +\setlength{\tabcolsep}{3pt} +\begin{longtable}{@{}p{47mm}p{15mm}p{30mm}p{31mm}p{18mm}@{}} +Case & Updates & Final $\lVert F\rVert$ & $\lVert F\rVert/\lVert F_0\rVert$ & Newton s \\ +\hline +\endfirsthead +Case & Updates & Final $\lVert F\rVert$ & $\lVert F\rVert/\lVert F_0\rVert$ & Newton s \\ +\hline +\endhead +$n=1,J=0$, all $m$ & 5 & $1.52715\,10^{-12}$ & $9.64952\,10^{-9}$ & 146.00 \\ +$n=1.5,J=0$, all $m$ & 6 & $1.29965\,10^{-12}$ & $5.01751\,10^{-9}$ & 221.40 \\ +$n=1,J=.01$, all $m$ & 6 & $3.86299\,10^{-13}$ & $4.29274\,10^{-10}$ & 197.23 \\ +$n=1,J=.01$, $m=0$ & 6 & $3.08442\,10^{-13}$ & $3.42757\,10^{-10}$ & 164.90 \\ +$n=1.5,J=0$, gravity-Schur & 6 & $2.19823\,10^{-11}$ & $8.48662\,10^{-8}$ & 189.54 \\ +\end{longtable} +\endgroup + +The first four cases converged by their stated criterion. The gravity-Schur +case exhausted its budget, although its last update still reduced the residual +by approximately a factor of 40; calling that endpoint a plateau would be +incorrect. Process wall times, including setup and observations, were 210.55, +290.70, 265.89, 232.54 and 257.49 seconds, respectively. Timings are single-run +observations on the workstation, not replicated benchmark statistics. + +At q18, absolute energy/force virial errors formed with $P(h)$ were +$3.87036\,10^{-7}$ / $2.65938\,10^{-7}$ for the nonrotating $n=1$ case, +$6.55153\,10^{-7}$ / $6.00565\,10^{-7}$ for $n=1.5$, and approximately +$3.87\,10^{-7}$ / $2.66\,10^{-7}$ for both low-rotation $n=1$ cases. They also passed +$10^{-5}$ at q14. The corresponding density-pressure virials passed for the +sampled $n=1$ cases, but were undefined for $n=1.5$, whose independent density +was negative at some integration points. The later sign-region investigation +also detected a thin $n=1$ negative layer missed by ordinary volume quadrature. + +Global balance did not establish pointwise accuracy. For nonrotating $n=1.5$, +the normalized EOS-density RMS was $1.25551\,10^{-4}$ and the hydrostatic-gradient +RMS increased from $1.52810\,10^{-3}$ in the seed to $7.27475\,10^{-3}$ at the +root, a factor of 4.76. This gradient diagnostic has acceleration units in the +model normalization; it is not the normalized weak residual. The four converged +cases had mass errors below $5.3\,10^{-14}$ and angular-momentum errors below +$3.6\,10^{-16}$. + +The full nodal mechanical force remained almost entirely outside the harmonic +span: Euclidean nodal-dual norms were approximately $9.22\,10^{-6}$ for +$n=1,J=0$ and $8.99\,10^{-6}$ for $n=1.5,J=0$. In the latter case the retained +modal force norm was only $3.61\,10^{-14}$. This is convergence of the selected +Galerkin shape problem, not convergence of the former unrestricted nodal shape +problem. The excluded dimensional Euclidean norm cannot be directly compared +with a normalized nonlinear norm or the dimensionless virial tolerance. + +Nonaxisymmetric coefficients persisted in nominally spherical models. The +$(\ell,m)=(4,4)$ coefficients were $8.97263\,10^{-5}$ for $n=1$ and +$9.23867\,10^{-4}$ for $n=1.5$, compared with $a_{40}=1.09313\,10^{-3}$ in the +latter. These are evidence to examine mesh/discretization imprint, not evidence +of a physical nonaxisymmetric branch. + +Total Krylov counts were 212, 335, 296, 219 and 206 in the table's order. The +$m=0$ rotating control used 26.0\% fewer Krylov iterations and 16.4\% less Newton +time than the all-$m$ control, but changes the admissible shape and is not a pure +preconditioner speedup. All $m$ remained the preferred three-dimensional +candidate. Ten new harmonic tests and 64 selected existing regression cases +passed; independent audits checked saved basis pullbacks, excluded force and +physical/nodal equivalence. Those implementation checks do not validate the +higher-rotation target. + +\section{Negative density: a located discrete representation defect} +\label{sec:recent-negative} + +The subsequent investigation was observational: it changed no equilibrium +equation, field order, shape prescription, normalization, preconditioner or +Newton policy, and solved no new equilibrium. It examined four frozen states: +converged $n=1$ and $n=1.5$ nonrotating harmonic models, the original $n=1.5$ +seed, and the $n=3$ nonrotating seed. Only the $n=3$ case had a central condition; +it was not a converged $n=3$ model. + +\subsection{What the weak EOS equation determines} + +Density and enthalpy are independent unknowns. Baseline density is DGQ2 and +enthalpy is continuous Q3. On a stellar element the closure is +\[ + R_i=\sum_q w_q\varphi_i(q)\left[\rho_h(q)-\rho_*(h_h(q))\right], + \qquad \rho_*(h)=\left[\frac{h}{(n+1)K}\right]^n, +\] +with physical mapped weights. At zero closure residual, +\[ + M\boldsymbol\rho=\boldsymbol b(h),\quad + M_{ij}=\sum_qw_q\varphi_i\varphi_j,\quad + b_i=\sum_qw_q\varphi_i\rho_*(h_h). +\] +Thus the closure selects a quadrature-defined weighted $L^2$ projection, not +pointwise EOS equality. Its unique polynomial can have negative regions even +when the sampled EOS target is nonnegative. + +Independently assembled element mass matrices and Cholesky solutions reproduced +the converged $n=1.5$ density coefficients within $1.40\,10^{-14}$ and the field +within $2.25\,10^{-15}$ in physical $L^2$, while its negative density was of order +$10^{-4}$. Independent/production closure-residual differences were approximately +$1.2\,10^{-17}$ to $2.1\,10^{-17}$ across the four states. Raising projection +quadrature to q26 changed coefficients by at most $2.36\,10^{-7}$ and did not +remove the undershoot. These comparisons identify the defect at the measured +discrete root; it is not explained by incomplete closure convergence or the +small measured quadrature difference. + +The implemented dependency also corrects an earlier intuitive explanation: +negative density does not directly generate the represented enthalpy. Pressure +is evaluated from independently represented enthalpy; negative sampled enthalpy +is rejected during preparation. All negative-density q18/q26 volume samples in +this investigation had positive enthalpy. Some surface witnesses had exactly +zero enthalpy. Density influences enthalpy indirectly through the coupled +stellar equations. + +\subsection{Spatial location and root-based integration} + +All 96 surface-adjacent stellar cells contained measured negative regions; no +negative intervals were found on the sampled lines through the other 736 +stellar cells. All stored density nodal values nevertheless remained positive. +The nodes are interior Gauss--Legendre points, and positivity of their values +does not imply positivity between them. + +Ordinary quadrature can miss thin sign regions: q18 and q26 missed the $n=1$ +volume layer, while its sampled boundary trace was negative. For the same +unchanged $n=1.5$ field, negative-mass estimates changed from +$2.89\,10^{-5}$ at q18 to $2.18\,10^{-5}$ at q26 and the apparent negative volume +nearly halved. The investigation therefore isolated zero crossings of the +polynomial along reference-cell lines and integrated negative intervals. +Define $M=\int\rho_h\,dV$ and $M_-=-\int\min(\rho_h,0)\,dV$. The mass denominator +below is $\lvert M\rvert$, not $\int\lvert \rho_h\rvert\,dV$. + +\begingroup +\small +\setlength{\tabcolsep}{3pt} +\begin{longtable}{@{}p{41mm}p{32mm}p{30mm}p{42mm}@{}} +Frozen state & $M_-/\lvert M\rvert$ & $V_-/V$ & Measured physical radius range \\ +\hline +\endfirsthead +Frozen state & $M_-/\lvert M\rvert$ & $V_-/V$ & Measured physical radius range \\ +\hline +\endhead +$n=1$ equilibrium & $6.09585\,10^{-8}$ & $0.0327346\%$ & $.999643$--$.999886$ \\ +$n=1.5$ seed & $3.66771\,10^{-5}$ & $1.83795\%$ & $.993813$--$1.000019$ \\ +$n=1.5$ equilibrium & $2.24535\,10^{-5}$ & $1.49252\%$ & $.994917$--$1.001222$ \\ +$n=3$ seed & $2.88372\,10^{-5}$ & $12.0835\%$ & $.934409$--$.978469$ \\ +\end{longtable} +\endgroup + +Radii are in reference-radius units and are not distances from the local, +slightly deformed surface. The first three cases have intervals reaching the +surface; the $n=3$ seed has an interior pocket in the same outer cell layer. Its +surface trace is positive. Minimum-density witnesses were approximately +$-1.14756\,10^{-4}$, $-9.26332\,10^{-4}$, $-7.21171\,10^{-4}$ and +$-8.54870\,10^{-5}$, respectively; the $n=3$ witness had enthalpy $0.0455581$. +These are sampled witnesses, not certified global minima. + +Tangential rules with 8, 12 and 16 points per direction agreed on the negative +integrals to roundoff in these cases. The finest two negative masses differed +by at most $10^{-14}$ relative, and whole mass/volume agreed with separate +observations within $1.77\,10^{-13}$ relative. Eight points integrated each +root-partitioned interval. Analytic controls, decimal-arithmetic audits and +state/coefficient hash checks support the measurements. They are not rigorous +three-dimensional sign certificates: unsampled angular pockets or roots below +reconstruction uncertainty remain possible. + +\subsection{Mechanism controls and their limits} + +The initial seed independently interpolates tabulated Lane--Emden density and +enthalpy into their respective spaces. The converged closure instead projects +the EOS image of represented enthalpy. Both operations can undershoot; neither +Newton nor rotation is necessary. With inward surface-cell coordinate +$t\in[0,1]$, a linear enthalpy gives target density proportional to $t^n$. +Exact quadratic $L^2$ projections in a flat one-dimensional cell are +\[ + \Pi_2t^{3/2}=-\frac{2}{105}+\frac{16}{35}t+\frac47t^2, + \qquad + \Pi_2t^3=\frac1{20}-\frac35t+\frac32t^2. +\] +The fractional profile is negative for $0\le t<0.0396969$. The cubic profile +has a negative pocket $0.1183500$. An extra central-density +condition cannot repair incompatible mass/EOS inputs; artificial forcing could +instead conceal the mismatch. The energy inequality and its Lane--Emden mass +relation are documented in equations (1.14)--(1.15) and Theorem 3.1 of Cheng, +Cheng and Lin, \emph{Expanding solutions near unstable Lane--Emden stars}; the +rotation bound above is our consequence of that inequality and the virial +identity. + +The numerical calibration is only approximately critical. Independent DOP853 +Lane--Emden integrations using five tolerance/start/step controls found +$M_{\rm LE}=.9999999918534539$ for recorded $K=.36393905995216302$, permitting +only $\beta\lesssim2.71552\,10^{-9}$. The control spread in mass was +$1.95\,10^{-14}$, a numerical sensitivity observation rather than a rigorous +interval enclosure. This near-critical input cannot represent the intended +moderate rotational support under the continuum assumptions. + +The discrete equations need not satisfy every assumption: independent +$\rho/P(h)$, weak EOS closure, negative density, approximate gravity and central +forcing all require separate scrutiny. The argument chooses a sensible target; +it is not a density limiter or a proof that a discrete residual cannot vanish. + +A separate experiment reduced $K$ by 1\%, kept target $M=1$, prescribed +$J=.035$ and omitted the central condition. Its necessary bound became +$\beta\le.00500000269$. The seed's central density was multiplied by +$.99^{3/2}$ to retain its reference radius, making its seed mass approximately +$.99^{3/2}$ while target mass remained one. This is an explicitly off-constraint +initial guess, not density rescaling during Newton. The seed Bernoulli scalar +still used target mass; that mismatch remained in the initial residual. + +Nonzero prescribed $J$ already breaks the nonrotating homology, so an additional +central condition was not added. At fixed $M,K$ an $n=3$ homology can change size +and $J$ without specifying a new dimensionless shape/rotation fraction. Neither +input $J$ nor an uncalibrated angular velocity should be called a fraction of +critical rotation. + +\subsection{Matched live field-feedback result} + +Two cases shared $n=3$, $J=.035$, $K$ factor $.99$, no central condition, +all-$m$ L4/Q2, the gravity-curvature trial path, $\eta=.03$, ten updates and +absolute target $10^{-12}$. Their initial completed norm was $.101333855231$. +Only the inverse differed. Both exhausted the budget. + +\begingroup +\small +\setlength{\tabcolsep}{3pt} +\begin{longtable}{@{}p{73mm}p{35mm}p{37mm}@{}} +Quantity & Gravity-Schur & Stellar LDU + gravity-Schur \\ +\hline +\endfirsthead +Quantity & Gravity-Schur & Stellar LDU + gravity-Schur \\ +\hline +\endhead +Final normalized residual & $1.64730591\,10^{-4}$ & $1.29842469\,10^{-5}$ \\ +Newton seconds & 918.881 & 1052.972 \\ +Linear seconds & 348.441 & 335.399 \\ +Krylov iterations & 534 & 410 \\ +Trial preparation seconds & 477.771 & 621.957 \\ +Final accepted $\alpha$ & $1/16$ & $1/64$ \\ +Energy virial, $P(h)$ & $7.18429\,10^{-5}$ & $1.81141\,10^{-5}$ \\ +Force virial, $P(h)$ & $5.23757\,10^{-5}$ & $2.90028\,10^{-6}$ \\ +Energy virial, $P(\rho)$ & $6.96335\,10^{-5}$ & $3.51558\,10^{-5}$ \\ +Force virial, $P(\rho)$ & $5.01663\,10^{-5}$ & $1.41415\,10^{-5}$ \\ +Absolute mass error & $5.45536\,10^{-5}$ & $3.37148\,10^{-6}$ \\ +EOS RMS / density scale & $3.99773\,10^{-4}$ & $4.00529\,10^{-4}$ \\ +Bernoulli RMS / enthalpy scale & $.00408345$ & $.00372207$ \\ +Hydrostatic-gradient RMS & $.656446$ & $.588702$ \\ +$T/\lvert W\rvert$ & $.00497575$ & $.00497421$ \\ +Angular velocity, model units & $.401764$ & $.401388$ \\ +\end{longtable} +\endgroup + +LDU reached a 12.687-times lower residual after the same update count, at +14.593\% more Newton time. Linear time fell 3.743\%, while preparation rose +30.179\%. These are evolving trajectories, not a speedup to the same converged +solution. All full linear gates passed, and every rejected trial failed +sufficient decrease; neither trajectory had pressure or geometry rejection. +q18 density samples were positive without repair, not a global positivity +certificate. Thus forcing positivity is not supported as the immediate remedy +for this measured live obstruction. + +The LDU trajectory reached $1.30261014\,10^{-5}$ after update seven with a full +step; three subsequent $1/64$ steps improved it only .3213\%. Its final squared +residual was 36.500\% Poisson, 34.432\% hydrostatic, 20.234\% surface and 6.742\% +mass. Gravity-only ended 80.254\% hydrostatic and 10.967\% mass. Field feedback +is a substantive improvement on this trial path but does not remove small-step +stagnation. Its merit over raw-Newton policies has not been established by this +pair, since both shared the gravity-curvature strategy. + +\section{Virial consistency: pressure and gravity must be distinguished} +\label{sec:recent-virial} + +\begin{figure}[htbp] +\centering +\includegraphics[width=.95\linewidth]{figures/n3_ldu_meridional.png} +\caption{Physical meridional density and enthalpy slices of the final rotating +$n=3$, $J=.035$, $K$-factor $.99$ LDU plus gravity-Schur endpoint. The image +shows the actual surface deformation without magnification. The model remains +unconverged and fails the overall virial milestone; only four display +subdivisions were used for this image.} +\label{fig:recent-n3-slice} +\end{figure} + +The final rotating LDU model does not meet the overall $10^{-5}$ milestone. +Only force virial formed with $P(h)$ passes. Its signed enthalpy virials are +approximately +\[ + v_E^h=+1.811408\,10^{-5},\qquad v_F^h=-2.900278\,10^{-6}, +\] +where both use $\lvert W_E\rvert$ as denominator. Their difference, approximately +$2.10144\,10^{-5}$, agrees with the separately measured gravity-energy +consistency discrepancy. Replacing $P(h)$ by $P(\rho)$ shifts both signed +virials by approximately $+1.70418\,10^{-5}$, yielding +$+3.515584\,10^{-5}$ and $+1.414148\,10^{-5}$. + +At one fixed state, pressure changes both signed virials equally: +\[ + v_E^\rho-v_E^h=v_F^\rho-v_F^h + =\frac{3(\Pi_\rho-\Pi_h)}{\lvert W_E\rvert},\qquad + v_E-v_F=\frac{W_E-W_F}{\lvert W_E\rvert}. +\] +Consequently, even an arbitrary common pressure-only shift leaves at least one +absolute virial at or above half their separation, approximately +$1.05072\,10^{-5}$. Pressure adjustment alone cannot make both pass in this +frozen state. This is a fixed-state arithmetic observation, not a bound on a +newly solved or refined model. + +q14/q18 energy virials agreed to below $8\,10^{-14}$ in both rotating runs. +Quadrature stability must not be mistaken for mesh accuracy. The gravity gap +remained about $2\,10^{-5}$, while EOS RMS stayed near $4\,10^{-4}$ despite the +large algebraic gain. Since neither state solves the discrete equations and +neither mesh nor order was varied, these observations do not establish a +spatial error floor or identify an exterior-only cause. + +\subsection{Pressure-moment identity and the density test space} + +For nonnegative $h,\rho$, set $\rho_*=[h/((n+1)K)]^n$ and +$\delta\rho=\rho-\rho_*$. Convexity of $P(\rho)=K\rho^{1+1/n}$ gives +\[ + P(\rho)-P(h)=\frac hn\delta\rho+D_P(\rho,\rho_*),\qquad D_P\ge0, +\] +because $P'(\rho_*)=h/n$. For a represented density test +$v=\sum_i c_i\varphi_i$ and the same mapped quadrature $Q$ used in closure, +\[ + Q\left[\frac hn\delta\rho\right] + =\frac1nQ[(h-v)\delta\rho]+\frac1n\sum_i c_iR_i. +\] +Current DGQ2 tests do not generally contain Q3 enthalpy. The first term can +therefore remain even at zero weak closure. Compatible DGQ3 tests would permit +$v=h$ elementwise, canceling the linear moment at exact closure under that same +quadrature. Density would remain an independent field. At finite residual the +remaining term is $c_h^TR/n$, which is not directly measured by a normalized +Newton norm without undoing residual scaling. + +The convex remainder persists, and is not uniformly quadratic near vacuum: +for $n>1$, $P''(\rho)$ diverges as density approaches zero; at $\rho_*=0$, +$D_P=K\rho^{1+1/n}$, with exponent $4/3$ for $n=3$. Separate diagnostic quadrature +can also retain a difference after assembled moment cancellation. The identity +is conditional on nonnegative fields where evaluated and is specific to a +constant-$K$ polytrope; a general EOS uses its own thermodynamic derivatives. +The contribution of each term has not yet been measured in the final models. +This is a consistency hypothesis and a diagnostic to implement, not proof that +raising order, clipping density or changing pressure will solve convergence. + +\section{Physical two-dimensional model slices} +\label{sec:recent-slices} + +The added optional diagnostic exports physical $xz$ and $xy$ cuts, with $yz$ and +offsets available, showing independent density and enthalpy and the actual +material boundary. It changes no equations or field representation. The +important geometric contract is a cut by a physical plane, not a reference +plane deformed afterward. + +Each stellar hexahedron is subdivided in reference coordinates into a lattice +and tetrahedra. Crossed edges solve the actual mapped plane equation using a +safeguarded bracketed root calculation. Fields are evaluated at those reference +coordinates on the owning element. Density traces are not averaged between DG +elements; supplied connectivity prevents triangulation across vacuum. A plane +coincident with an internal face uses the positive-normal-side trace. Boundary +curves are separately evaluated from stellar/nonstellar faces, and the +undeformed outline is independently cut by the same physical plane. + +The plotter uses equal physical axis scales, no deformation magnification, and +signed field values without clipping, logarithms or EOS replacement. Scalar +values are rasterized on a 512-by-512 grid before color mapping, avoiding +color-interpolation artifacts under triangle refinement. Exterior pixels stay +masked. Images distinguish solid deformed and optional dashed reference +outlines and mark known unconverged endpoints. + +Straight triangles and boundary chords approximate the curved cut. Neighboring +parent elements may choose different face diagonals; small gaps/overlaps can +therefore occur on curved interfaces. A covered raster pixel uses the first +supplied triangle, without cross-element averaging. Tangencies or multiple +crossings between subdivision vertices can be missed. These limits are +explicitly documented and prevent using the image as a pointwise positivity +certificate or interpreting every seam as a physical feature. + +\subsection{Display-refinement measurements} + +Four unchanged-state replays compared four/eight subdivisions for final $n=2$ +and rotating $n=1.5,J=.1$ states, in both planes. Source buffers remained +bitwise unchanged, and both resolutions used common axes and color ranges. + +\begingroup +\small +\setlength{\tabcolsep}{3pt} +\begin{longtable}{@{}p{40mm}p{24mm}p{28mm}p{28mm}p{25mm}@{}} +Model/plane & Area change & $\int\rho\,dA$ change & $\int h\,dA$ change & Perimeter change \\ +\hline +\endfirsthead +Model/plane & Area change & $\int\rho\,dA$ change & $\int h\,dA$ change & Perimeter change \\ +\hline +\endhead +$n=2$, $xz$ & $.12333\%$ & $.07667\%$ & $.07036\%$ & $.03065\%$ \\ +$n=2$, $xy$ & $.12333\%$ & $.07667\%$ & $.07036\%$ & $.03065\%$ \\ +$n=1.5,J=.1$, $xz$ & $.12495\%$ & $.00319\%$ & $.08797\%$ & $.03165\%$ \\ +$n=1.5,J=.1$, $xy$ & $.12395\%$ & $.01249\%$ & $.07838\%$ & $.03088\%$ \\ +\end{longtable} +\endgroup + +These are relative absolute changes of displayed straight-triangle measures, +including overlaps; they are not curved-FEM integrals, three-dimensional mass, +virial errors or a mesh convergence study. Sampled chord errors decreased about +fourfold and axial extents agreed. The rotating meridional model was smoothly +oblate, with equatorial extent approximately $\pm1.12084$ and polar extents +$-.988807,+.988480$; the equatorial cut was approximately circular. Its sampled +negative density minimum $-8.59\,10^{-4}$ was retained. The $n=2$ outline was +nearly spherical at radius 1.0049. Only the latter met its recorded virial +milestone, and both remained unconverged algebraically. + +\begin{figure}[htbp] +\centering +\includegraphics[width=.98\linewidth]{figures/n1p5_rotating_refinement.png} +\caption{Same-state physical slice comparison for the unconverged rotating +$n=1.5$ endpoint at four and eight display subdivisions. Axes and color ranges +are shared. This checks the displayed geometry and fields; it is not a change +of the FEM mesh, a three-dimensional integral check or evidence of equilibrium.} +\label{fig:recent-display-refinement} +\end{figure} + +Final $n=3$ meridional/equatorial plots show strong central concentration and +modest oblateness. Those exports used four subdivisions only. The separate +$n=2/n=1.5$ display study must not be described as refinement verification of +the $n=3$ images. Visual plausibility supplies an additional diagnostic, not +physical validation. + +\section{What to carry forward and what remains unresolved} +\label{sec:recent-next} + +The recent work establishes a usable native harmonic prescription, diagnoses a +specific independent-density projection defect, measures order effects without +changing density's physical status, adds reproducible physical slices, and +shows real frozen-linear and live-nonlinear gains. It does not establish a +converged moderately rotating $n=3$ model, a calibrated 50\%-critical sequence, +or the ultimate 99\%-critical ambition. No ESTER comparison was performed. +Global virial balance at $10^{-5}$ remains a primary acceptance milestone, +accompanied by internal EOS/hydrostatic checks; an attractive image or a small +subset of residuals cannot replace it. + +The evidence supports the following staged continuation. +\begin{enumerate} +\item Retain gravity-Schur as the strongest measured frozen linear-cost +comparator and stellar LDU plus gravity as the better live candidate in the +tested rotating $n=3$ configuration. The latter is conditional on the shared +gravity-curvature trial path. Compare complete time to a common convergence +and physical target before selecting a production default. Do not promote the +new coarse surface/scalar wrapper merely because it lowers iterations. +\item Remeasure coupled direction differences in the unbordered compatible +rotating family, including true defects, field/modal composition and finite +trial remainders. The centrally constrained nonrotating weak-monopole evidence +is a motivated hypothesis, not a transferred diagnosis. A coupled trust-region +or regularized step with a physical displacement scale is a reasonable swappable +experiment. Radius must remain free, and surface/interior remain simultaneous. +\item Strengthen the material response in controlled stages if its frozen block +defects warrant it. Existing material preconditioning already contains the +exact prepared enthalpy-to-density derivative; it is incorrect to claim that +cross derivative is missing. Exact local mapped DG density mass inversion is a +narrow candidate (27 Q2 coefficients per hexahedron). A stronger stationary +enthalpy mass inverse must respect H1 connectivity and replaced boundary rows. +Only then consider a more costly EOS-aware material/gravity Schur approximation. +Such algebraic eliminations belong inside the inverse and need not make density +a derived nonlinear field. +\item Measure pressure-moment and gravity-energy consistency separately. Export +the linear pressure moment, convex remainder, quadrature differences and both +signed virials on identical states. Resolve the gravity relation/source and +energy/force discrepancy with interior accuracy as the goal; the exterior need +only be accurate insofar as it supports correct interior fields. The current +observations do not justify allocating effort solely to exterior resolution. +\item Define a physically compatible $n=3$ sequence in mass/EOS/rotation and +calibrate a dimensionless rotation measure and critical reference. Input $J$ +alone is not that calibration. Keep central homology handling optional and +separate from incompatible physical specifications. +\item After useful convergence is obtained, perform mesh and harmonic-degree +studies, measure excluded surface force in a meaningful norm, and compare local +profiles with analytic/nonrotating controls and eventually ESTER. Density-order +results remain available as consistency controls; positivity intervention and +logarithmic refactoring are deferred per the user's direction. +\end{enumerate} + +A closure-consistent seed and an affine-target closure correction are additional +untested hypotheses, not completed remedies. At fixed admissible enthalpy and +geometry, exact local density mass solves can populate independent density or +correct its weak moments. Any trial correction must preserve the original +Newton tangent and be freshly verified. If combined with gravity correction, +density must be corrected first and gravity rebuilt against its original affine +target afterward. That operation can change mass, inertia and mechanical +balance, cannot repair unchanged inadmissible enthalpy, and must pass the full +residual acceptance test. For fractional $1` have equivalent patches in the integration history. “Semantically integrated” covers build-registration and combined cherry-picks whose exact patch IDs differ but final source was inspected. This is more discriminating than treating every plus-sign from `git cherry` as missing code. + +| Branch | Tip | Status and action | +|---|---|---| +| `integration/research-consolidation-20260920` | `b1fe359` | New consolidation descendant, created during this audit: adds preserved user formatting (`6950dc6`) and six log-assessment documentation/reference commits (`549b1a7` through `b1fe359`) after `d7acb0d`. These were not represented in the base snapshot. | +| `main` | `75cc638` | Ancestor; already represented. No additional merge. | +| `research/baseline-20260916` | `80cc3d8` | Ancestor; already represented. No additional merge. | +| `research/central-constraint-20260916` | `2e19c2c` | All 4 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/combined-20260916` | `95d9fba` | Ancestor; already represented. No additional merge. | +| `research/coupled-convergence-study-20260919` | `d7acb0d` | Ancestor; already represented. No additional merge. | +| `research/coupled-shape-coarse-20260918` | `4a6c1b5` | Semantically integrated: two equivalent commits; isolated test-registration 4a6c1b5 is already in CMakeLists. Keep diagnostic-only coarse experiment; no additional merge. | +| `research/coupled-trial-correction-20260918` | `46249d0` | Ancestor; already represented. No additional merge. | +| `research/damped-coupled-trial-20260918` | `db854c0` | Ancestor; already represented. No additional merge. | +| `research/density-order-study-20260919` | `ef76e0c` | Ancestor; already represented. No additional merge. | +| `research/diagnostic-campaign-20260917` | `7a07cae` | Ancestor; already represented. No additional merge. | +| `research/experiment-harness-20260916` | `899cb18` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-compact-application-20260917` | `ff1b889` | Semantically integrated: seven equivalent patches; e599900 and 2c64a92 module/test registration already present. No additional merge. | +| `research/gravity-consistent-schur-20260918` | `c10d019` | All 2 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-coupling-20260916` | `db0ddd3` | Not integrated: failed one-way completion. Archive README/recipe/manifest and retain branch. Do not adopt old driver or policy. | +| `research/gravity-coupling-ablation-20260919` | `38c7184` | Ancestor; already represented. No additional merge. | +| `research/gravity-curvature-correction-20260917` | `3762c74` | Semantically integrated: four equivalent patches; e599900 registration already present. Keep opt-in; no additional merge. | +| `research/gravity-direction-diagnostics-20260917` | `47775a9` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-prepared-roundoff-20260917` | `782827c` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-reduced-newton-20260917` | `d25d141` | Semantically integrated: seven equivalent patches; 5952691 registration already present. Keep opt-in; no additional merge. | +| `research/gravity-trial-diagnostics-20260917` | `7f4b068` | All 2 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/krylov-20260916` | `7127383` | All 4 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/krylov-window-20260916` | `1f9184b` | Semantically integrated: documentation identical, sandbox restart is 160. 937d2f2 was integrated alongside other sandbox changes; no additional merge. | +| `research/linear-coupling-study-20260919` | `0e6688f` | Semantically integrated: core wrapper/tests identical; one patch equivalent and 1d95602 integrated as 2398118 with adjusted surrounding driver/build context. Frozen experiment only. | +| `research/log-density-assessment-20260919` | `90580b2` | Not integrated: six documentation/reference-only commits. Import final assessment and reference script as deferred research; no positivity/runtime change. | +| `research/model-slices-20260919` | `5da3f42` | Semantically integrated: seven equivalent patches; 966a4d4 integrated as 9bac57a. Slice implementation/plotter content identical. No additional merge. | +| `research/n3-compatible-family-20260919` | `1d746b0` | Ancestor; already represented. No additional merge. | +| `research/n3-linear-ablation-20260919` | `e78af98` | Ancestor; already represented. No additional merge. | +| `research/n3-live-field-feedback-20260919` | `1ed03f5` | Ancestor; already represented. No additional merge. | +| `research/negative-density-diagnosis-20260919` | `7ed4d23` | Ancestor; already represented. No additional merge. | +| `research/nonlinear-step-obstruction-20260918` | `2944dd5` | Ancestor; already represented. No additional merge. | +| `research/numerical-strategies-20260917` | `10adf7e` | Ancestor; already represented. No additional merge. | +| `research/polynomial-endpoint-certification-20260917` | `cc6f4b0` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/polynomial-positive-newton-20260917` | `9aefb36` | Semantically integrated: eight equivalent patches; a773c9a registration already present. Available explicit strategy only, not a default; do not enable. | +| `research/preconditioner-benchmarks-20260918` | `dc1da62` | Ancestor; already represented. No additional merge. | +| `research/preconditioner-cached-cross-actions-20260918` | `07769e7` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/pressure-obstruction-20260917` | `c6572d3` | All 3 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/reduced-smooth-surface-20260918` | `2171b8b` | Ancestor; already represented. No additional merge. | +| `research/shape-curvature-20260917` | `b24e6e0` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/shape-field-displacement-q4-20260918` | `785a7ae` | Implementation intentionally excluded: retain Q3. Final report/comparison scripts already identical in integration, preserving negative result. | +| `research/shape-field-weak-response-20260918` | `6ceda61` | Ancestor; already represented. No additional merge. | +| `research/spherical-harmonic-surface-20260919` | `1148c09` | Ancestor; already represented. No additional merge. | + +## Verified false-positive unmerged commits + +The following unique patch IDs do not indicate missing capabilities: + +- `4a6c1b5`: `tests/solver/plateau_coupled_coarse.cpp` is already registered at CMakeLists line 391. +- `e599900`, `2c64a92`, `5952691`, `a773c9a`: gravity subsystem/reduction, compact gravity, polynomial admissibility modules and tests are already in CMakeLists and the exported mean_field module. Reapplying isolated registration commits would only duplicate source entries or conflict. +- `937d2f2`: sandbox explicitly uses FGMRES restart 160. The final Krylov-window report is byte-identical to the topic branch. +- `1d95602`: `plateau_surface_border_schur.hpp` and its regression tests are byte-identical to the final topic branch. Integration commits `2398118` and `522a2cf` carry the implementation and followup test changes. +- `966a4d4`: physical-plane slice headers and current plotter are byte-identical to the final topic branch. Integration starts at `9bac57a`, with final label changes at `07799bc` and `2c6523b`. +- Q4 topic `785a7ae`: `SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md`, `compare_shape_order_runs.py` and `plot_shape_surface_comparison.py` are already byte-identical to the topic tip. The source-order change from `4be1a23` is deliberately absent. + +## What is active versus selectable at d7acb0d + +It is essential that the consolidated report distinguish library defaults, example configuration, diagnostic-driver defaults, and recommended research candidates. + +| Layer | Current behavior | Consolidation recommendation | +|---|---|---| +| Independent physical fields | Density remains independent DG Q2; enthalpy independent H1 Q3; displacement H1 Q3; gravity RT2/Q2; reference geometry Q4 | Retain. Do not import the Q4 trial or log-density runtime. | +| Density approximation configuration | `MEAN_FIELD_DENSITY_POLYNOMIAL_ORDER=2` compile-time CMake/registry option independent of other fields; uniform increment remains available | Retain configurable order and corrected closure quadrature. Default stays Q2. | +| Surface factory default | Two-argument `makeStellarDiscretization` remains `NodalRadialSurface`, with power-law radial interior and fixed-infinity radial vacuum extensions | Preserve compatibility; avoid silently changing all clients. | +| Sandbox preferred surface | Explicit `SphericalHarmonicSurface` maximum degree 4, all real m, 25 coefficients, coupled with all fields | Retain explicit preferred candidate; nodal remains selectable. | +| Harmonic surface limit | Single MPI rank; sampled basis rank and trace contracts checked. H1 nodal surface surrogate explicitly rejects harmonic coordinates | Report this limitation; do not imply general distributed harmonic support. | +| Normalization | Existing `PhysicalRieszDiagonal` in sandbox; harmonic reference trace metric adds correct primal/dual coefficient scaling | Retain. This is coordinate scaling, not an exact surface response inverse. | +| Sandbox physical model | n=1, J=0, fixed total mass/angular momentum; central density is Lane–Emden seed data only | Retain. Central constraint remains explicitly specifiable for n=3 studies, not a hidden universal constraint. | +| Library FGMRES option default | `FGMRESOptions.restartLength=50` | Unchanged by research. Do not claim it is 160 globally. | +| Sandbox FGMRES configuration | Restart 160; Newton maximum 30; relative nonlinear target 1e-8, linear target .03/cap 200 | Retain known baseline. | +| Diagnostic no-argument controls | Nodal surface, restart 40, independent preconditioner, baseline strategy; runs require output. Campaign manifests explicitly select harmonic/restart160 variants | Keep manifests authoritative; do not confuse convenient CLI defaults with the latest experimental configuration. | +| Warm-start guard | Reused linear correction is reset if its fresh full-J true residual is nonfinite or worse than the cold RHS norm | Retain correctness/performance safeguard; endpoint true-residual check remains authoritative. | +| `makePreconditioner()` | Default stellar structure: independent material/surface and gravity inverse, with generated scalar-border completion | Preserve default until matched full convergence evidence warrants promotion. | +| Default material inverse | SurfaceThenMaterialTriangular with diagonal density/enthalpy/surface solves; exact prepared h-to-density EOS cross derivative and surface-to-material actions included | Retain. Do not describe it as missing h-to-density coupling. | +| Default gravity inverse | GravityApproximateLDU; matrix-free Chebyshev order5/powerIterations20 mass inverse, three fixed AMG cycles | Preserve base comparator; refined consistent Schur remains available. | +| Consistent gravity Schur | Typed `GravityConsistentSchurLDU`, fixed number of sweeps; library policy defaults 1 sweep/relaxation1, diagnostic default relaxation .5 | Retain useful optional implementation and measured manifests; do not conflate the option defaults. | +| Live experimental recipes | `gravity-schur`, `stellar-ldu`, `stellar-ldu-gravity-schur` live in experiments owning recipe layer | Retain. LDU+gravity is currently best live candidate for the tested compatible rotating n3/gravity-curvature configuration, not established universal default. | +| Newton numerical defaults | `SampledEnthalpy`, `MonolithicGravity`, `UncorrectedTrial` | Preserve. No added positivity forcing or auxiliary corrections execute by default. | +| Optional Newton paths | Polynomial enthalpy certificate/repair, gravity-curvature, reduced gravity, fully coupled and damped second-order corrections | Preserve explicit selection and tests as experimental tools; no general convergence claim. | +| Coarse surface/scalar Schur | Experiment-only frozen wrapper; reconstruct after changed Jacobian | Retain diagnostic infrastructure. Do not promote; lower iteration counts did not generally justify setup/application expense. | +| Restart initializer | Explicit diagnostic API validates state/layout/coordinate provenance, prepares/refreshes transactionally, clears warm starts, rolls back failures | Retain. Unconverged snapshots require explicit experimental override. | +| Model slices | Optional experiment export/plot of actual mapped stellar domain; xz/xy/yz/offset options; signed fields retained | Retain. Display approximation/refinement is distinct from FEM convergence or physical validation. | + +Source anchors: `sandbox.cpp`; `libmeanfield/interface/equilibrium/stellar_discretization.cppm`; `libmeanfield/interface/field/field_registry.cppm`; `libmeanfield/interface/normalization/stellar_equilibrium.cppm`; `libmeanfield/interface/preconditioning/{stellar_recipe,stellar_structure,material_surface,gravity_field}.cppm`; `libmeanfield/interface/solver/{linear_backend,newton_numerics,stellar_context}.cppm`; `experiments/plateau_diagnostic.cpp`; `experiments/plateau_newton_preconditioner.hpp`. + +## Why the failed one-way branch stays historical + +`db0ddd3` documents a same-state n3,J0 cold-seed control with original material/gravity subcomponents unchanged, only `MaterialThenGravityTriangular` replacing independence. At restart160/cap200/target .03, true defect improves .0402590 to .0381781 but both fail; solve cost rises 98.915 to117.712s (19%). At restart40, defect worsens .0563080 to .0715826. Direction norm doubles .757644 to1.540825; sampled admissible step tightens .152435 to .0913461. This is not an improvement sufficient for default promotion. It does not rule out the later two-way LDU with better gravity inverse. + +The branch's shared harness commit `ad86e2e` includes some source changes also represented later in the main cumulative harness; it is not all useful missing code simply because patch-id differs. Importing its old `plateau_diagnostic.cpp` would erase subsequent functionality. + +## Useful cumulative integration already present + +- Baseline `80cc3d8` snapshots the user's optimized prepared-operator/reference-table/mapping/normalization path and diagnostics that were not in old main. The main-to-d7 diff includes this earlier user performance work; do not attribute every changed file to the subsequent AI-led numerical experiments. +- Optional central-density example cleanup, restart160 example choice and safeguarded warm starts are cumulative by `95d9fba`. +- Pressure/shape diagnostic instrumentation and typed Newton numerical strategies are integrated by `10adf7e`; prepared gravity arithmetic and compact application fixes are included. +- Consistent Schur, prepared cross-action reuse and honest full-cost frozen benchmarking are integrated by `dc1da62`. +- Coupled/damped trial tools and their bounded negative/conditional results are integrated by `db854c0`. +- Q3-restored weak-response/Q4 comparison report, reduced surface experiments and native all-m harmonic surface are integrated by `1148c09`. +- Negative-density localization, independent density-order configuration, closure quadrature repair and audited order studies are integrated by `ef76e0c`. +- Checked restart, frozen coupled coarse trials, physically compatible n3 controls, slices and final convergence/cost/virial evidence are integrated by `d7acb0d`. + +## Consolidation and validation risks + +1. **Do not promote experimental settings merely because they enter main.** Main can contain tested optional research modules. The final report must explicitly state which settings are defaults, preferred research candidates and failed historical alternatives. +2. **Preserve source provenance.** Campaign numerical executables were copied from specific producer commits. Main integration changes do not retroactively change the meaning of old runs. Keep original commands, metadata, hashes and branch tips, and include relocation manifests when moving results. +3. **Preserve old paths carefully.** Many tracked research reports and JSON manifests refer to evidence under workspace `diagnostics/` using worktree-relative or absolute paths. Relocation requires either stable links or a documented path mapping and adjusted browse links. Do not rewrite original run metadata/hashes simply to make them look current. +4. **Untracked build trees have absolute cache paths.** Moving them may invalidate rebuilds. Preserve them as historical artifacts or use explicit compatible links; configure a fresh consolidation build instead of treating a moved cache as portable. +5. **Do not rerun a full campaign unnecessarily.** A source-equivalent integration needs targeted build/smoke/regression checks around added files and resolved conflicts, not new scientific claims. The previous 24-case audit is a provenance check, not proof that every model converged. +6. **Harmonic support is serial.** A two-rank rejection test is not MPI solver validation. Report this accurately. +7. **No end-to-end target claim.** The latest compatible rotating n3 LDU endpoint remains unconverged; energy/h virial1.81141e-5 and both rho-pressure virials fail1e-5. No 50%-critical calibration exists. Lower finite-budget residuals are not measured mathematical floors. +8. **Documentation import is not new policy.** The log-density reference model is historical isolated analysis. Keep its label explicit and do not present it as an implemented solver option. + +## Audit limits + +This audit checks ancestry, patch equivalence, source content, defaults and recorded evidence. It does not constitute a new exhaustive correctness review of all 57k lines added since old main, nor a numerical rerun. Exact source comparisons established the particular equivalences above; registration line numbers refer to d7acb0d and can shift during consolidation. The parent integrator should record the final consolidation SHA, any actual conflicts/resolutions, validation results, and the eventual status of the preserved user formatting edit. + +## Consolidation update + +After the base-snapshot audit, the integrator preserved the user formatting change as `6950dc6` and imported all six log-density documentation/reference commits, ending at `b1fe359` on `integration/research-consolidation-20260920`. Consequently the earlier base-snapshot recommendation to import the log assessment is fulfilled. No numerical behavior was added by those seven commits. The original branch table entries otherwise describe their relationship to `d7acb0d`, not their eventual status after consolidation. diff --git a/reports/archive/branches_before.txt b/reports/archive/branches_before.txt new file mode 100644 index 0000000..7b51722 --- /dev/null +++ b/reports/archive/branches_before.txt @@ -0,0 +1,41 @@ + main 75cc638 perf(allocations): reduced overall allocations by 95%, increaseed jacobian applicatin by 2x + research/baseline-20260916 80cc3d8 Snapshot optimized solver and audit diagnostics before numerical experiments ++ research/central-constraint-20260916 2e19c2c (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/central-constraint) docs: record combined optional-density validation + research/combined-20260916 95d9fba docs: finalize controlled solver campaign and transfer limits ++ research/coupled-convergence-study-20260919 d7acb0d (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement) Record coupled convergence results, virial limitations and validated model slices ++ research/coupled-shape-coarse-20260918 4a6c1b5 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/coupled-shape-coarse) Register isolated coupled shape coarse regression tests + research/coupled-trial-correction-20260918 46249d0 Record coupled correction benefit cost and nonlinear curvature limits + research/damped-coupled-trial-20260918 db854c0 Record audited damping and eligibility results through six Newton updates + research/density-order-study-20260919 ef76e0c Record density-order study, convergence limits, and log-density implications + research/diagnostic-campaign-20260917 7a07cae experiment: record pressure positivity and shape curvature evidence ++ research/experiment-harness-20260916 899cb18 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/measurements) docs: distinguish conservative screening from virial milestone + research/gravity-compact-application-20260917 ff1b889 Validate compact gravity application probe records ++ research/gravity-consistent-schur-20260918 c10d019 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/gravity-consistent-schur) Aggregate gravity baseline equivalence assertions ++ research/gravity-coupling-20260916 db0ddd3 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/gravity-coupling) Document controlled gravity-coupling counterexample + research/gravity-coupling-ablation-20260919 38c7184 Add gravity-only attribution for rotating higher-index gain + research/gravity-curvature-correction-20260917 3762c74 Handle null reduced actions with an absolute derivative floor ++ research/gravity-direction-diagnostics-20260917 47775a9 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/gravity-reduction) Record gravity relift work before descent rejection + research/gravity-prepared-roundoff-20260917 782827c Scale prepared gravity consistency by uncancelled field contributions + research/gravity-reduced-newton-20260917 d25d141 Resolve reduced derivative trend with a recorded four-step diagnostic grid ++ research/gravity-trial-diagnostics-20260917 7f4b068 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/gravity-correction) Update numerical strategy documentation with verified evidence ++ research/krylov-20260916 7127383 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/krylov) Document matched Krylov safeguard results and limitations ++ research/krylov-window-20260916 1f9184b (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/krylov-window) docs: record nonrotating larger-window validation ++ research/linear-coupling-study-20260919 0e6688f (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/linear-coupling-study) Test normalized unbordered adapter and report its storage scope ++ research/log-density-assessment-20260919 90580b2 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/log-density-assessment) Document stellar support contract for log density ++ research/model-slices-20260919 5da3f42 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/model-slices) Preserve slice titles and colorbar labels in saved figure bounds + research/n3-compatible-family-20260919 1d746b0 Add explicit mass-compatible rotating n3 physical-family control + research/n3-linear-ablation-20260919 e78af98 Specify matched n3 gravity-only ablation before attribution + research/n3-live-field-feedback-20260919 1ed03f5 Specify matched live field-feedback trial for rotating n3 family + research/negative-density-diagnosis-20260919 7ed4d23 Record negative density mechanism, interval audits, and resolution options + research/nonlinear-step-obstruction-20260918 2944dd5 Record audited nonlinear obstruction findings and next experiment +* research/numerical-strategies-20260917 10adf7e Record numerical strategy validation and remaining convergence limits ++ research/polynomial-endpoint-certification-20260917 cc6f4b0 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/positivity-certification) Add read-only polynomial endpoint certification mode ++ research/polynomial-positive-newton-20260917 9aefb36 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/positivity-strategy) Reject unsupported pressure traces before polynomial state certification + research/preconditioner-benchmarks-20260918 dc1da62 Finalize preconditioner measurements, physical limits, and reproduction record ++ research/preconditioner-cached-cross-actions-20260918 07769e7 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-cached-cross-actions) Reuse prepared gravity force for stellar cross preconditioning ++ research/pressure-obstruction-20260917 c6572d3 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/pressure-obstruction) Export enthalpy nodal values on limiting diagnostic elements + research/reduced-smooth-surface-20260918 2171b8b Record reduced surface study and audit frozen seed chains ++ research/shape-curvature-20260917 b24e6e0 (/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/shape-curvature) experiment: split frozen operator curvature into shape and field contributions + research/shape-field-displacement-q4-20260918 785a7ae Record audited negative displacement-order trial and physical comparison + research/shape-field-weak-response-20260918 6ceda61 Preserve completed weak-response study while retaining Q3 baseline + research/spherical-harmonic-surface-20260919 1148c09 Record coupled harmonic surface validation and preconditioner baseline diff --git a/reports/archive/early_sources.md b/reports/archive/early_sources.md new file mode 100644 index 0000000..7b0eebb --- /dev/null +++ b/reports/archive/early_sources.md @@ -0,0 +1,148 @@ +# Early-history fragment evidence index + +Authored fragment: `early.tex`. Scope: original 16 September audit through the bounded reduced-surface study on 18 September. The 19 September native-harmonic, negative-density, density-order, and coupled-convergence material belongs to the companion later fragment. + +The fragment cites the report files below; those reports identify exact producer commits, immutable commands, manifests, states, binaries, raw vectors/CSVs, timing exclusions, and audit results. No new FEM solve or numerical result was generated for this section. Archived initial audits are referenced at their proposed final repository paths. Raw campaigns are referenced at the consolidation archive path. + +## Sources and hashes + +- Final reference: `libmeanfield/interface` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/libmeanfield/interface` + - Exists at consolidation review: `True` +- Final reference: `libmeanfield/impl` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/libmeanfield/impl` + - Exists at consolidation review: `True` +- Final reference: `reports/sources/original_audits/MeanField_audit_2026-09-16.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/MeanField_audit_2026-09-16.md` + - Exists at consolidation review: `True` + - SHA-256: `1249447f7957747575324d2598750f6f819a3023df2ab6fd00c64de39596759d` +- Final reference: `reports/sources/original_audits/MeanField_solver_diagnosis_2026-09-16.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/MeanField_solver_diagnosis_2026-09-16.md` + - Exists at consolidation review: `True` + - SHA-256: `d00befb23099256f8d38e57f3f9f2d5ed4192c2984afb935ec94e36e64675c0c` +- Final reference: `reports/sources/original_audits/formulation_audit_notes.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/formulation_audit_notes.md` + - Exists at consolidation review: `True` + - SHA-256: `e49441556f34295e140e4841fc95e6c9d6b675c8f2b1ff6a113d81d8f3006311` +- Final reference: `experiments/POLYTROPE_VALIDATION_FINDINGS_2026-09-09.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/POLYTROPE_VALIDATION_FINDINGS_2026-09-09.md` + - Exists at consolidation review: `True` + - SHA-256: `53c40e6c519bf1e5bf945b7cae2d4f8de0897aa3180cda4bc3876900b533078b` +- Final reference: `experiments/POLYTROPE_H_CONVERGENCE_2026-09-09.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/POLYTROPE_H_CONVERGENCE_2026-09-09.md` + - Exists at consolidation review: `True` + - SHA-256: `bdcc74edcd7b0caf48592d630dffac1fa758f576a4a2f8835f0bf3ca2fcc8bbc` +- Final reference: `experiments/NUMERICAL_CAMPAIGN_2026-09-16.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/NUMERICAL_CAMPAIGN_2026-09-16.md` + - Exists at consolidation review: `True` + - SHA-256: `ace2bc10d645e53fdbc6a6c61e2bd26ba9ab8bdffd37e918ba633a7dd3ce7b11` +- Final reference: `experiments/CENTRAL_CONSTRAINT_HYPOTHESIS_2026-09-16.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/CENTRAL_CONSTRAINT_HYPOTHESIS_2026-09-16.md` + - Exists at consolidation review: `True` + - SHA-256: `cde87179c80a71c133005ca9e28ca8152ff6d7d95cb8a12c4689407540dd1f74` +- Final reference: `experiments/KRYLOV_WINDOW_HYPOTHESIS_2026-09-16.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/KRYLOV_WINDOW_HYPOTHESIS_2026-09-16.md` + - Exists at consolidation review: `True` + - SHA-256: `0c45627584da5c6fcf6f1c2bdbe1350634467bc44214e83a8bb2c9ad4e0d647a` +- Final reference: `experiments/KRYLOV_HYPOTHESIS_2026-09-16.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/KRYLOV_HYPOTHESIS_2026-09-16.md` + - Exists at consolidation review: `True` + - SHA-256: `cf98c38da0a60f289f6ef9050f4fc6dccb805d51350eeb6ebe7f31f22cff284a` +- Final reference: `experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/PRESSURE_AND_SHAPE_CAMPAIGN_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `a605a69ed95a227515738be264f87f9e82110309c159df23909affbf2c78a1d0` +- Final reference: `experiments/PRESSURE_OBSTRUCTION_METHODOLOGY_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/PRESSURE_OBSTRUCTION_METHODOLOGY_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `b44afc0095ecc19497788ba02f71313b59c362e378d652df8a7becaa2d7bb180` +- Final reference: `experiments/POLYNOMIAL_ENTHALPY_ADMISSIBILITY_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/POLYNOMIAL_ENTHALPY_ADMISSIBILITY_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `232dd20d2ba3eba6139d0d80762b49b4f6d532caaeca7a3294923904a168e639` +- Final reference: `experiments/NUMERICAL_STRATEGIES_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/NUMERICAL_STRATEGIES_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `f06b7ca146aa77e4cb1f2a4f345463c0f6678a1965a6741944f68e3a6b88cf92` +- Final reference: `experiments/GRAVITY_SUBSYSTEM_IMPLEMENTATION_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/GRAVITY_SUBSYSTEM_IMPLEMENTATION_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `221660080749ac28505786df4a7c9f501e441512304116d869ec1ef660b1136f` +- Final reference: `experiments/GRAVITY_REDUCTION_IMPLEMENTATION_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/GRAVITY_REDUCTION_IMPLEMENTATION_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `193d33f5e219918e62dc266aeac00a59a2c44e7b917a461062f7820593763474` +- Final reference: `experiments/GRAVITY_COMPACT_APPLICATIONS_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/GRAVITY_COMPACT_APPLICATIONS_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `5949c6e62eb78f34af5a5a2651a8b28af6ad997cfc34d2d8b0491105917e3109` +- Final reference: `experiments/GRAVITY_PREPARED_ROUNDOFF_2026-09-17.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/GRAVITY_PREPARED_ROUNDOFF_2026-09-17.md` + - Exists at consolidation review: `True` + - SHA-256: `2383781c80b7c251859456df7a66acc323bed3a07816fe94d6d410b083effa77` +- Final reference: `experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/PRECONDITIONER_IMPROVEMENT_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `6aecb98da933a9aaa00f071f1723b2848b0c33e36e3cd98e52b198dba3fe8353` +- Final reference: `experiments/GRAVITY_CONSISTENT_SCHUR_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/GRAVITY_CONSISTENT_SCHUR_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `8f54dac778a276f78fc37b724ab616b96e5197f8ace7018f0d1b9527f23e970d` +- Final reference: `experiments/PRECONDITIONER_CACHED_CROSS_ACTION_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/PRECONDITIONER_CACHED_CROSS_ACTION_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `79f1b1a2be1de88cd3f1aefd9905fabc86eee7370a67e160dbecba43f0963fe9` +- Final reference: `experiments/NONLINEAR_STEP_INVESTIGATION_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/NONLINEAR_STEP_INVESTIGATION_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `241dd723f310892761b4b11074530b70e216f39097cf7a9c4f6ffe2d54befc5d` +- Final reference: `experiments/COUPLED_TRIAL_CORRECTION_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/COUPLED_TRIAL_CORRECTION_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `7995e496f90e634127a30c22f0a2499b7b18b15882acec3ea2a6bf1ec7cee5c7` +- Final reference: `experiments/DAMPED_COUPLED_TRIAL_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/DAMPED_COUPLED_TRIAL_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `cc4d5e29b71695796fe7c2bed58346a017895daf21abb217b23bd26494bc3f22` +- Final reference: `libmeanfield/interface/field/field_registry.cppm` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/libmeanfield/interface/field/field_registry.cppm` + - Exists at consolidation review: `True` + - SHA-256: `45b54aaf7f56c11f74d6fc0f4785612b3f2aca3922e10feaedb6db5048881084` +- Final reference: `experiments/SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `971f1dfd08a83625cb267c4be0d660dd63c4b5c894845ad69d11f7e93843696b` +- Final reference: `experiments/REDUCED_SURFACE_STUDY_2026-09-18.md` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/experiments/REDUCED_SURFACE_STUDY_2026-09-18.md` + - Exists at consolidation review: `True` + - SHA-256: `353474b9a7bc3eee0677a068cd906f1f136fc7ad73ed2fdbf710a7901210b5b6` +- Final reference: `working/research/diagnostics/reduced_surface_2026-09-18` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/diagnostics/reduced_surface_2026-09-18` + - Exists at consolidation review: `True` + +## Numerical interpretation controls + +- Original audit HEAD alone was not the source under test; incoming optimized changes were preserved at baseline `80cc3d8`. +- All time-to-convergence comparisons require both frozen endpoints to pass the same independently measured full-J true gate. Failed controls have no finite measured time-to-convergence. +- Nonlinear matched pairs follow evolving trajectories; fixed-update endpoint improvements are not speedups to the same solution. +- Sampled and certified positivity are distinguished; enthalpy certification never certifies independent density. +- All early J and Omega values are reference-unit inputs/results, never fractions of critical rotation. +- Q4-only displacement enrichment is a retained negative experiment, not a recommended main-branch order change. +- The L2/L4/L8 frozen reduced-surface study changes test/approximation space; reduced gates do not imply original full gates. +- Original coarse shape residuals/excluded-force results remain visible even when the reduced norm and virial improve. +- Original 40/160-restart controls, n=3 central border, initial beta/alpha grids, timer exclusions and unavailable timeout endpoint are described explicitly. + +## Source limitations + +This history is synthesized from the saved primary project reports and their recorded artifact measurements. It does not independently rerun every FEM measurement. Later corrections and clarifications within those reports take precedence over earlier provisional summaries. The consolidated report should retain the reports and raw campaign directories so future investigators can reproduce individual measurements. + +## Consolidation precision checks + +The normalization summary was checked directly against the separate value/residual registrations in `physical_riesz.cppm` (lines 408–463). The seed description was checked directly against its two `ProjectCoefficient` calls in `stellar_equilibrium_projection.cpp`; it now distinguishes coefficient initialization from solving the weak closure weighted L2 projection. Mathematical Jacobians are denoted A, while J continues to denote angular momentum. + +- Final source reference: `libmeanfield/interface/normalization/physical_riesz.cppm` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/libmeanfield/interface/normalization/physical_riesz.cppm` + - SHA-256: `5f47f07687843705bc29d908dd8b96f35b3a1138c833e18d0d28b56c77d034be` +- Final source reference: `libmeanfield/impl/seed/stellar_equilibrium_projection.cpp` + - Read source: `/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement/libmeanfield/impl/seed/stellar_equilibrium_projection.cpp` + - SHA-256: `c7e6c795bebccd723fea25f02c6b49fe192031d6dc58d7acf1e8a88770425152` diff --git a/reports/archive/figure_sources.json b/reports/archive/figure_sources.json new file mode 100644 index 0000000..f04054f --- /dev/null +++ b/reports/archive/figure_sources.json @@ -0,0 +1,37 @@ +[ + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures_n3_v2/n3_J0p035_K0p99_ldu_xz.png", + "report_figure": "reports/figures/n3_ldu_meridional.png", + "sha256": "3c83ed97704b81668880a63c246fb911bb42be03835342748118525dfb872f64" + }, + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures_n3_v2/n3_J0p035_K0p99_ldu_xy.png", + "report_figure": "reports/figures/n3_ldu_equatorial.png", + "sha256": "3d690de1b53bce5b20fceb8b070b43f8c66782ccdebff84f280a8094329b6b2b" + }, + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures_slices_v2/n1p5_rotating_xz_s4_s8.pdf", + "report_figure": "reports/figures/n1p5_rotating_refinement.pdf", + "sha256": "ba0fe7bc9d104119f155aeae46d9e1e61d71948cadcce24e088a6ccf370e5dd0" + }, + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures/linear_rates/frozen_linear_rates.pdf", + "report_figure": "reports/figures/frozen_linear_rates.pdf", + "sha256": "7d3849e7e65af6c2f0cb37607e5741b80a1c75772e8ca09488b8265494097dc3" + }, + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures/convergence/n2_paired_convergence.pdf", + "report_figure": "reports/figures/n2_paired_convergence.pdf", + "sha256": "aac7c7f935d3d1d46000fb08569093c2bbdf9f2c09e14b10929fdc549a9a45a2" + }, + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures/convergence/n3_paired_convergence.pdf", + "report_figure": "reports/figures/n3_initial_paired_convergence.pdf", + "sha256": "314d51d6848fb5369b4e89940b84f1ae9cc25deafb1980df544f65203de1498f" + }, + { + "source": "working/research/diagnostics/coupled_convergence_2026-09-19/figures_slices_v2/n1p5_rotating_xz_s4_s8.png", + "report_figure": "reports/figures/n1p5_rotating_refinement.png", + "sha256": "4561b09a64ab3492bc2ed85b3ce042079435cb70ee297bdc0e838dde9673f952" + } +] diff --git a/reports/archive/history_before.txt b/reports/archive/history_before.txt new file mode 100644 index 0000000..4a2a9d3 --- /dev/null +++ b/reports/archive/history_before.txt @@ -0,0 +1,239 @@ +d7acb0d399aac5ac4c36d834a954c1c959042b81 2026-09-19T17:09:32-04:00 Record coupled convergence results, virial limitations and validated model slices +2c6523b1ce8f080c073a818cfee58f01b737045b 2026-09-19T17:05:46-04:00 Preserve slice titles and colorbar labels in saved figure bounds +5da3f42d397355964d5fb53c5395aa04a2f2cf6d 2026-09-19T17:05:46-04:00 Preserve slice titles and colorbar labels in saved figure bounds +07799bca34ee191baeb9b56d16d6c9e7a15b1492 2026-09-19T16:25:45-04:00 Space slice colorbar labels in normalized color coordinates +91e3813838f9511f89d858ef3fc52b6de129443f 2026-09-19T16:25:45-04:00 Space slice colorbar labels in normalized color coordinates +1ed03f56833817114bc99bec4e6f12b0dfd8add6 2026-09-19T16:10:38-04:00 Specify matched live field-feedback trial for rotating n3 family +38c7184b5c32d47c4e98cdaaf7f2f782297ee49b 2026-09-19T16:01:29-04:00 Add gravity-only attribution for rotating higher-index gain +2aabfc4d35aabbda58990b6dec86dfbfb3c13e9e 2026-09-19T15:52:11-04:00 Plot audited frozen linear histories and checked endpoint gates +76ba3535913c572f8bb687e73af180763fb364a9 2026-09-19T15:52:11-04:00 Plot audited frozen linear histories and checked endpoint gates +e78af98e2b62efc84c0042dd78a357ca7a37822e 2026-09-19T15:44:25-04:00 Specify matched n3 gravity-only ablation before attribution +df86e84b76ba77282444c843e2d84257afdc2bf3 2026-09-19T15:29:16-04:00 Record audited nonlinear controls and interior diagnostics +476118cfaa87f735a7b9abd7a69fe5ebab6e417f 2026-09-19T15:11:09-04:00 Show local restart progress alongside full convergence threshold +8fc43379d5641c5ed93d1f9e94774684a5f3f37c 2026-09-19T15:11:09-04:00 Show local restart progress alongside full convergence threshold +f2c1a369a6f63fc1d105e52c17d0224d2dd1ac02 2026-09-19T15:06:17-04:00 Specify tight frozen-direction trials after gravity-path stall +a458af0a7a14b03368b80a27ae208928e2e07250 2026-09-19T14:57:43-04:00 Plot audited paired Newton convergence with actual thresholds +bb1ca336819ba7637792eed4b76349a7c3e9c17d 2026-09-19T14:57:43-04:00 Plot audited paired Newton convergence with actual thresholds +d62487f2d10789308848cebe5d1e97b9b471e195 2026-09-19T14:48:26-04:00 Add field-feedback controls and document the actual material factorization +cc22383562e9d6c41a150e74e7058cb2ce5e131a 2026-09-19T14:27:04-04:00 Target the combined rotation stall and its physical slice refinement +99eb86fedc72196b1e7bcb1e7e369b4693216f58 2026-09-19T14:19:59-04:00 Audit frozen slice replays and specify same-state refinement controls +a1df4d9f84f57f2ad4d5d6a5041b7bd3fbb64df3 2026-09-19T14:16:35-04:00 Record control outcomes and specify a matched rotating-state benchmark +6614a5bdf65aa2faee6c50991e6e910af256280b 2026-09-19T14:15:35-04:00 Compare physical slice refinement with shared scalar rendering +4b3da551abb7ab1a51af792f4d6d1fdd8aed306d 2026-09-19T14:15:35-04:00 Compare physical slice refinement with shared scalar rendering +5f8458c9cc416da8f354ebc540723cdf58c6c331 2026-09-19T14:07:40-04:00 Audit retained convergence evidence and distinguish compatible physical families +f9355bdc8f28eb5d7b5e6c059b1c970ce2763b6b 2026-09-19T14:03:31-04:00 Document curved slice conformity and refinement limits +f2d8f21d98e362c496d0b93429a8ae9d4537c56c 2026-09-19T14:03:31-04:00 Document curved slice conformity and refinement limits +1d746b028ef001782158a0a82d81b40eb933b68a 2026-09-19T13:58:08-04:00 Add explicit mass-compatible rotating n3 physical-family control +6d423cba46c6e7d2e4b3020902fd8e21f79ed4a6 2026-09-19T13:56:36-04:00 Specify gravity-path and matched Schur experiments from the measured n2 stall +9bac57a1df7bb9c6cca949b62eb40c5639d88ded 2026-09-19T13:46:11-04:00 Add physical-plane FEM model slice diagnostics and plots +7793fd91118cab950abbc0b2cfcac8023593fb31 2026-09-19T13:46:33-04:00 Keep restart input provenance separate from destination output records +522a2cfc021c8ba0b803b4d732c8e56461f10f70 2026-09-19T13:45:57-04:00 Test normalized unbordered adapter and report its storage scope +966a4d4e0583832ee0e6df9b0b4f298b71300f55 2026-09-19T13:46:11-04:00 Add physical-plane FEM model slice diagnostics and plots +0e6688f500873fb5cb3ecae783c90faa6d952159 2026-09-19T13:45:57-04:00 Test normalized unbordered adapter and report its storage scope +2398118b347b2b47b8004be8334b21cf3d11a615 2026-09-19T13:43:21-04:00 Add frozen surface and scalar Schur preconditioner experiment +53faac1cd202971bb7fc2a29898c6467a4e5b0b3 2026-09-19T13:43:49-04:00 Add checked harmonic snapshot continuation and matched linear trial diagnostics +1d956023446217802ea943a2af8bad0b25786db7 2026-09-19T13:43:21-04:00 Add frozen surface and scalar Schur preconditioner experiment +77130fa6033512e954b59f6d396d5c3eb1f5f406 2026-09-19T13:31:11-04:00 Define coupled harmonic convergence screen and evidence controls +ef76e0c7860d3972f4993e406defe9dd24dcc05b 2026-09-19T13:12:41-04:00 Record density-order study, convergence limits, and log-density implications +90580b23f9b170db7fffd3e5bb86a9a1f9927f9c 2026-09-19T13:05:53-04:00 Document stellar support contract for log density +c0a496e9d6666403a50df79869c07256380c30c3 2026-09-19T12:46:04-04:00 Preserve all trial preparation and disposition counts in reports +e054c7464c6b2942c714315eb55d4fc4092345dd 2026-09-19T12:16:56-04:00 Check non-density layouts across separately audited orders +4f7eaff5be0ccf28a805a2511a9995252df40b08 2026-09-19T12:14:07-04:00 Document MESA log density and atmosphere analogy +903615f10f45356a58fd14d5c914639baf7ecde0 2026-09-19T12:06:37-04:00 Separate uncertain density observations from plotted trends +ecde703c83badefab6a78da677d34f61633d7579 2026-09-19T12:04:27-04:00 Clarify finite precision positivity of log density +66debb6aae1dc048813204fbc44ff9994d4febbf 2026-09-19T11:22:04-04:00 Clarify local log-matrix scope and independent pressure admissibility +cd71f374f0b11db74680530bd03e273371708fca 2026-09-19T11:21:26-04:00 Audit trial rejection stages and expose line-search obstruction counts +54102b4f3c8d61360b963b7d7288d8b4c461b4d5 2026-09-19T11:15:28-04:00 Retain unbordered residual and central-border diagnostics in summaries +d34ae09fc4d790b594e88b57f70b2854941fb59a 2026-09-19T11:09:06-04:00 Audit actual stellar closure quadrature rules and support +c0e8551cbd8d54a141d3ad06ddd75a93037be146 2026-09-19T11:04:49-04:00 Summarize audited density-order results and scientific comparisons +241621ce8667647fc7fa0dfe252479139225f660 2026-09-19T10:53:34-04:00 Report sliced moment quadrature differences without identity gates +4ff1b10693255eaef5ec0f5fa6c4d81929a4ce14 2026-09-19T10:52:33-04:00 Separate surface and interval-witness enthalpy observations +e21f723fdfd43fcea834d317819488e10124d048 2026-09-19T10:51:44-04:00 Exercise rectangular gravity coupling and normalization in order-study checks +8c1cc3187409df764afa646f198b71f576e59058 2026-09-19T10:49:37-04:00 Audit density-order evidence and distinguish observation from certification +8d5ed607dc94540cb95f5b5b919a6d968e840e9a 2026-09-19T10:49:29-04:00 Clarify scope of coefficient positivity in log assessment +2ca6890a187721afff4bddb0568908302303c4e8 2026-09-19T10:48:14-04:00 Add controlled density-order projection and coupled-solve observations +dd955abca5fbe680d3769ae16718c8e0c30315b4 2026-09-19T10:47:58-04:00 Cover both density mass and EOS source in closure quadrature +47f27b2f6e6a4c7c5a511564dd39a577460aa99c 2026-09-19T10:35:48-04:00 Assess independent log density coordinates and exact vacuum +d6554cae6bd5b04673643f92fe0821aad3eeeeb6 2026-09-19T10:26:54-04:00 Reuse field and closure validation in a focused density-order test target +c7383e12ffb25892e9353ec323d63fa7d4c3c603 2026-09-19T10:25:55-04:00 Make density approximation order an independent compile-time choice +7ed4d232d0209dc209082c983f18f2e256a32742 2026-09-19T10:16:03-04:00 Record negative density mechanism, interval audits, and resolution options +d1a594808f960bbaeeb6df8a8ce6a95458765b19 2026-09-19T10:06:10-04:00 Integrate frozen negative density over polynomial root intervals +9c22934d52b369f96364e781d6a4843939ec9523 2026-09-19T09:46:52-04:00 Add frozen negative density localization and independent EOS projection diagnostics +1148c099e3c5839a63880ae13d2ad6602402b13f 2026-09-19T09:22:36-04:00 Record coupled harmonic surface validation and preconditioner baseline +7b0eb8d92d0f550f1fe3fb19cd8eecc52dc80c56 2026-09-19T08:51:42-04:00 Add native spherical harmonic surface coordinates to coupled equilibrium +2171b8b76ad78856323fdbcf3d2c537d9039201f 2026-09-18T16:14:34-04:00 Record reduced surface study and audit frozen seed chains +21cdbda22486e408adc52b65b25413f9380c7c30 2026-09-18T15:45:05-04:00 Add frozen smooth-surface restriction study with physical dual maps +6ceda61bddc0d4881d9f7ac98d170d07980a963d 2026-09-18T14:27:15-04:00 Preserve completed weak-response study while retaining Q3 baseline +785a7aedac72c810bb624c2b7be6b4f9a05ef6e8 2026-09-18T14:26:36-04:00 Record audited negative displacement-order trial and physical comparison +4be1a23c3ec2650af6a2a49528a8f5248cb001c3 2026-09-18T13:46:11-04:00 Trial Q4 displacement and surface representation with all material and gravity orders fixed +211be17190fd71ee979805b067b8c31d4bd91550 2026-09-18T13:45:27-04:00 Record audited shape-field compensation and cubic surface structure before order trial +2c4a5e05a669eadcdc02c29d08e0dc4d56010320 2026-09-18T13:31:14-04:00 Audit physical compensation artifacts and verify quartic affine displacement representation +d79db9445bfd41dc204927c91784a8bd3cddb6f3 2026-09-18T13:20:03-04:00 Add frozen physical shape-field compensation diagnostics and controlled order study plan +db854c0f7cdf5137e377273499f0ac5b6053b752 2026-09-18T12:13:53-04:00 Record audited damping and eligibility results through six Newton updates +18762298c1c4f51abfde4ada1e98374694ba7ae7 2026-09-18T11:40:36-04:00 Record independent campaign clocks and matched Newton follow-up cases +25816d4328c2da88cd476877ad682fde4676a886 2026-09-18T10:49:44-04:00 Match frozen trial metric validation to production backtracking +422c46bcbf7e19f7b52dcbb98c7e6c150249c696 2026-09-18T10:45:45-04:00 Add bounded coupled damping and prediction-based attempt selection +46249d0758e1c30f7585d35821f78cd558197cec 2026-09-18T10:21:19-04:00 Record coupled correction benefit cost and nonlinear curvature limits +0c926f40df41402812997cd67cfba2b0e4e1e116 2026-09-18T10:14:09-04:00 Generate matched scalar controls for coupled trial attribution +5aa48dcf8047cbd53dafb0c377e260629675ae9e 2026-09-18T10:07:46-04:00 Keep coupled evidence audit linear in state size and repeat raw timing +2d0d32b886407eee4b3f7f654a4507da9c0fa243 2026-09-18T09:52:30-04:00 Audit coupled trial artifacts and physical replay independently +0d87b8b919a9eecac73e23aa006f336d94194786 2026-09-18T09:41:00-04:00 Add bounded fully coupled second-order trial correction and frozen screen +2944dd5019ab8aaca60e147db010d7fded7fd916 2026-09-18T09:10:28-04:00 Record audited nonlinear obstruction findings and next experiment +db93bf73b629b7a05638245d6fb0e617c37ce6db 2026-09-18T08:59:52-04:00 Record constrained two-direction algebraic control +0b0ab9cd1836da75553673149fd15da4eb33d36f 2026-09-18T08:47:19-04:00 Plot audited nonlinear paths and matched direction controls +b8d1c6a5abc2bcd634175d45059b2da60248e00c 2026-09-18T08:37:54-04:00 Independently audit frozen nonlinear path vectors and provenance +ba057b3983ff6006e460a1fa80e3e6523ceeb4a5 2026-09-18T08:26:51-04:00 Add frozen nonlinear path and mixed curvature investigation +690c8ddc9ac339ce9ee2cc4d9ef3366121050f4a 2026-09-18T08:19:26-04:00 Clarify frozen endpoint scope and gravity target interpretation +1d98fff171e30fca5aeac8f0e3657caacd69003c 2026-09-18T08:16:27-04:00 Record frozen late-state probe and matched direction controls +d82540a13c4245cc2fc02233a13154da1b112b8b 2026-09-18T08:14:07-04:00 Record preconditioner conclusions and define next nonlinear obstruction investigation +dc1da62b2c2856112f49239c42b9020dc151cd78 2026-09-17T18:17:54-04:00 Finalize preconditioner measurements, physical limits, and reproduction record +83afd824ef49925550fe00926fc567c3b46443fc 2026-09-17T18:09:41-04:00 Record rotating linear gate success and bounded nonlinear follow-through +5298d88ed0ce98830bd900553de6a2912ff0fb95 2026-09-17T18:03:56-04:00 Document higher-index nonlinear limits after Schur refinement +031b02f3da6b9deee233ecc3b4f186d7cc95de75 2026-09-17T17:53:07-04:00 docs: clarify preconditioner experiment results and timing +30a99435a805abc9b592e1c169fae06eb0bd621f 2026-09-17T17:51:12-04:00 Document equal-accuracy gain and preserved virial regression +7b4f674ed5327d47e67854164ede181a5fd0b5b8 2026-09-17T17:36:48-04:00 Enable combined stellar LDU gravity Schur Newton experiment +107c66b192baff9a0b88b2774b9cd272d4c6ef9c 2026-09-17T17:35:21-04:00 Record complementary rotating coupling confirmation +bed1893cfbadb2946403cea26c8c8c66e0c960cf 2026-09-17T17:18:42-04:00 Record measured preconditioner screen and validation limits +4a6c1b5c26a352a8afb30feb0a2f3821aac9ee19 2026-09-17T17:18:18-04:00 Register isolated coupled shape coarse regression tests +15464da5951092f74e440b97dba6a44a04ead141 2026-09-17T16:55:20-04:00 Compile and verify radial coarse basis through existing FEM surface map +87569c03dd47d4ce3435c3410d41b0c94a94ccfe 2026-09-17T17:11:29-04:00 Record Newton and virial regressions for consistent gravity preconditioning +74d94bd275ae53679c707f1302f9a411e160be7b 2026-09-17T17:10:15-04:00 Record equal-accuracy gravity Schur confirmation cases +c2bd8fc9c9c4e79854683044d0f9aa2131cb4d16 2026-09-17T17:06:26-04:00 Document verified Schur coverage and MPI fixture limitation +fcb7833844e1636572f03c001e7fe51243789349 2026-09-17T17:05:44-04:00 Wire typed experimental preconditioners into Newton diagnostics +dc8d4b0501e2775b0d1938005e142411319d78b1 2026-09-17T16:55:20-04:00 Compile and verify radial coarse basis through existing FEM surface map +b3bd2440246ebabb2b470284c4dbd3785967a7c9 2026-09-17T16:48:14-04:00 Record matched preconditioner controls and candidate rationale +52474587ddee8bd6684668d52935b9729be946d5 2026-09-17T16:45:55-04:00 Guard coarse exactness diagnostics and report complete setup work +48cb8c527fb25e3fb82866e80d578cdb43a8c2ef 2026-09-17T16:42:22-04:00 Validate coarse preconditioner evidence and fallback comparisons +370dbbf73bd976f11cb80893ec8d6a4842e1faa3 2026-09-17T16:37:32-04:00 Compare cached coupling and checked coarse preconditioner variants +995d49b81347a1c68609e248a4d21c485bf2d9d5 2026-09-17T16:35:10-04:00 Prototype deterministic coupled shape coarse inverse for frozen benchmarks +47f76968a819a25c57997222375579edbb42d983 2026-09-17T16:35:10-04:00 Prototype deterministic coupled shape coarse inverse for frozen benchmarks +fac147774aeb965e0166d9c8629c13b5f17e7b62 2026-09-17T16:28:56-04:00 Reuse prepared gravity force for stellar cross preconditioning +07769e7c5d572964a9b159c8dbe89871a067346a 2026-09-17T16:28:56-04:00 Reuse prepared gravity force for stellar cross preconditioning +7f7d7b1cfafb74a2adafa31e09ac115f17765f9b 2026-09-17T16:28:42-04:00 Recognize zero-sweep gravity benchmark controls +30b94b75b3d90eff1d0d7fcdad5cf0e42ac41895 2026-09-17T16:24:47-04:00 Aggregate gravity baseline equivalence assertions +c10d019411fbde9dde45c4c7cadc80e0e6334dd7 2026-09-17T16:24:47-04:00 Aggregate gravity baseline equivalence assertions +61a44fc853de826a101f3f16c3af556b01808cd3 2026-09-17T16:24:42-04:00 Validate preconditioner evidence before claiming speedups +bc3c39dcfdc80a94960ddf0861296fe3ed0d1526 2026-09-17T16:23:13-04:00 Benchmark isolated and coupled gravity Schur refinements +b18f41f88f54cebc408f40b0650ce59456611a24 2026-09-17T16:19:22-04:00 Add fixed-work mass-consistent gravity Schur refinement +389a4fd7d76b9011eda28e91c5446b34b89655d1 2026-09-17T16:19:22-04:00 Add fixed-work mass-consistent gravity Schur refinement +018fc1c9924b73640074b823a9d16270de945148 2026-09-17T16:17:21-04:00 Record reproducible hard-state preconditioner screen cases +e557e6b481a49609d4f4da816f3bb4c6db5a7ce1 2026-09-17T16:16:46-04:00 Add matched frozen-state preconditioner benchmarks +10adf7ec3783b27976cb9cf52728f8269819a685 2026-09-17T12:40:48-04:00 Record numerical strategy validation and remaining convergence limits +c82e68dd7e73bdb31fa8ce9d8f80652e087203c4 2026-09-17T12:17:10-04:00 Record gravity relift work before descent rejection +47775a9aaa71874837be17b0c61a3fe1d1a491f1 2026-09-17T12:17:10-04:00 Record gravity relift work before descent rejection +b9e679499988bc7124699abe9e613dd6e1a07868 2026-09-17T12:09:57-04:00 Update numerical strategy documentation with verified evidence +7f4b068a39e45f58ba85acf84c488697aa5a8055 2026-09-17T12:09:57-04:00 Update numerical strategy documentation with verified evidence +8b307b43d37381415f466be78dcf1f5ddd5b4e66 2026-09-17T11:53:12-04:00 Retain iteration snapshots in numerical strategy campaigns +a2871e11e21527d8c7b8d8d436cb0f543bf93cd6 2026-09-17T11:01:13-04:00 Clear unavailable gravity measurements before each attempt +82568639f73efe3b7f04c10f7692f6d7d176b8a7 2026-09-17T11:01:13-04:00 Clear unavailable gravity measurements before each attempt +7e1ab69fcd191717bdf88c918c35ae6ddc725f23 2026-09-17T10:51:07-04:00 Add read-only polynomial endpoint certification mode +cc6f4b08190f1c5e4b63e13f0643e592dfae7a90 2026-09-17T10:51:07-04:00 Add read-only polynomial endpoint certification mode +b21fa4024963678cf8e0a214f62cd7767ef0f259 2026-09-17T10:29:23-04:00 Exercise combined polynomial and reduced gravity model steps +01169861c620d939e58f92b66631ccd77c6b43a2 2026-09-17T10:23:43-04:00 Validate compact gravity application probe records +ff1b889e2c316f1adc2cc4aac5aa169e665994bd 2026-09-17T10:23:43-04:00 Validate compact gravity application probe records +0bf260f04d2594f3a9063761f85579c2127f0607 2026-09-17T10:22:25-04:00 Record gravity arithmetic diagnostics and bounded model campaign +6a91cfaf067e1c4fb8c6cd05ca4c06108dfc5d7f 2026-09-17T10:21:05-04:00 Scale prepared gravity consistency by uncancelled field contributions +782827c25890d849c58acd3a27840ddea3e5fb09 2026-09-17T10:21:05-04:00 Scale prepared gravity consistency by uncancelled field contributions +f77c7d2b292882bce362206d5154225d046d21c0 2026-09-17T10:19:17-04:00 Compare compact gravity applications against full FEM blocks +3307191f2df60d1e60b411516447ed3d61087985 2026-09-17T10:17:19-04:00 Keep unrecognized physical runtimes on gravity fallback +497de707f59ceba2cf6e76ef4c0a14c2fcd86e21 2026-09-17T10:17:19-04:00 Keep unrecognized physical runtimes on gravity fallback +071417f0c11fe87bd1cd189f2951b3763961ad59 2026-09-17T10:15:57-04:00 Reject unsupported pressure traces before polynomial state certification +6e812aeca29e7af69efbd6ec515ac2df5b09f6b3 2026-09-17T10:15:35-04:00 Apply canonical gravity blocks without full root work +2c64a9259838a494effd32aeb08ed33539da5fa4 2026-09-17T10:16:14-04:00 Register compact gravity tests on isolated branch +9aefb364a42440ddb38d68df9e64c61761131113 2026-09-17T10:15:57-04:00 Reject unsupported pressure traces before polynomial state certification +842bd9bf808e19292edabdc6905938e1ec29ca48 2026-09-17T10:15:35-04:00 Apply canonical gravity blocks without full root work +b7ea1bdcf8e75e6e663ee7fbc20c08323c1424db 2026-09-17T10:09:24-04:00 Resolve reduced derivative trend with a recorded four-step diagnostic grid +d25d14147110fb4aa0485c5a0920f8d88390a1ad 2026-09-17T10:09:24-04:00 Resolve reduced derivative trend with a recorded four-step diagnostic grid +2e319172693f5a29d0e371bf260852b29b70c6e4 2026-09-17T10:09:22-04:00 Distinguish strategy failures and expose the actual trial tangent +5f3eb0b18cc0e0173851fe17681913632ebda225 2026-09-17T10:00:46-04:00 Mark unvisited enthalpy repair checks as unresolved +6b10e68ed3889996bfb896028089844185b97ac7 2026-09-17T09:54:34-04:00 Use nondeprecated subnormal prerequisite for C++23 certificates +485e80245374843b184fbeb13d7ea9cc0bfcffd8 2026-09-17T09:50:21-04:00 Make zero-correction strategy fixture independent of equality stopping +cc738b0b906edd76ff4e74500d9ea9efeaaca9ad 2026-09-17T10:00:46-04:00 Mark unvisited enthalpy repair checks as unresolved +6485da95c042a52be3848b039a7634eb93a97be3 2026-09-17T09:54:34-04:00 Use nondeprecated subnormal prerequisite for C++23 certificates +7f8b6fb72d30aae9918cfe64b7552bb8f563279c 2026-09-17T09:50:21-04:00 Make zero-correction strategy fixture independent of equality stopping +4ae2f3f13494f95f3589d7ad21ef2c730f0a07f2 2026-09-17T09:45:56-04:00 Handle null reduced actions with an absolute derivative floor +3762c74f033941e8e97f4dee1748ee8f3fb49c36 2026-09-17T09:45:56-04:00 Handle null reduced actions with an absolute derivative floor +c393847c6f6d7730d6964632b9cd506384ed0bf6 2026-09-17T09:45:05-04:00 Integrate typed Newton strategies with checked trial gravity and feasibility +a7d646d04979692f780cdaf00d6c1ec7310323fc 2026-09-17T09:44:04-04:00 Check frozen numerical strategies without conflating incomplete evidence with success +e5999007f5b75b9d579d92e741cb871fc87f7b03 2026-09-17T09:44:38-04:00 Register isolated gravity-correction module and tests +b32ec078d4b09e7b0dd6fe31ff515035595e3ca0 2026-09-17T09:44:04-04:00 Check frozen numerical strategies without conflating incomplete evidence with success +237485de5102d70ea72e82720456c7decfabb553 2026-09-17T09:39:39-04:00 Expose cached gravity target for prepared trial verification +be9da74dc98c59b5a1ea9ab97aced798a4a12520 2026-09-17T09:39:39-04:00 Expose cached gravity target for prepared trial verification +5e03188816a13247fb480e3b1f36f1b0425ac9cb 2026-09-17T09:37:35-04:00 Test reduced Newton equivalence within the intended root basin +a773c9a0943677b12f884976851c0895501eb321 2026-09-17T09:38:42-04:00 Register isolated polynomial positivity modules and tests +595269117366d2ac7750111babf204303340285d 2026-09-17T09:38:33-04:00 Register gravity reduction modules and tests on isolated branch +23f8a3c8f92b0446b1f751ebbee1a32a2d4a3430 2026-09-17T09:37:35-04:00 Test reduced Newton equivalence within the intended root basin +fc944c1bfeea06b4720af0bfeed2f7488eb2d31b 2026-09-17T09:35:59-04:00 Enforce floating arithmetic prerequisites for polynomial certificates +4418b25d4535b07a5c22ac08a1654a7dd839b923 2026-09-17T09:32:41-04:00 Preserve nonfinite input rejection at feasible-repair application boundary +97204a3253c36edfdbd51cbcb344c1239d772da6 2026-09-17T09:31:31-04:00 Add frozen polynomial positivity probes and tighten certificate diagnostics +dfba40b1f5938ae5f8cfc3a818c6f0c1f7ead63f 2026-09-17T09:35:24-04:00 Exercise typed stellar numerical strategies and rollback contracts +a7f080f2655fbd3d11f4bcde49877ac4a83db6a6 2026-09-17T09:30:12-04:00 Probe FEM gravity strategies and make reduced contracts collective +f8037238f930d9d827918a2f41a561b74516a342 2026-09-17T09:35:59-04:00 Enforce floating arithmetic prerequisites for polynomial certificates +f9af1d9fabc4f027034d0c1d29c66d48babe37a0 2026-09-17T09:35:24-04:00 Exercise typed stellar numerical strategies and rollback contracts +ee5b2f3f0333d49790756b22659dfdb99662e6a5 2026-09-17T09:32:41-04:00 Preserve nonfinite input rejection at feasible-repair application boundary +870427fb379a73cb53911c732da495b9022071bd 2026-09-17T09:31:31-04:00 Add frozen polynomial positivity probes and tighten certificate diagnostics +9831031564ddc888d320fd403b610d2451d1e17d 2026-09-17T09:30:12-04:00 Probe FEM gravity strategies and make reduced contracts collective +9e0e1bc42af6cf11986db113dfbeed694fdf4132 2026-09-17T09:28:01-04:00 Integrate polynomial feasibility with typed stellar enthalpy strategy +3f844061849034917b76ab0048800eb6ed440681 2026-09-17T09:28:01-04:00 Integrate polynomial feasibility with typed stellar enthalpy strategy +8e0ed800d707c77f77c97e2e5db1e03e2b8a279a 2026-09-17T09:27:12-04:00 Add prepared gravity subsystem and second-order trial correction +6628e237d3c4d837b4931e8f952b49cd9c5a66bf 2026-09-17T09:23:19-04:00 Add prepared Schur gravity reduction and contract tests +2fa83db8ab8fadb394cd96063a12236ebcde5c38 2026-09-17T09:27:12-04:00 Add prepared gravity subsystem and second-order trial correction +882e8feef76d848a027c4c8dfd85614ba2656e2b 2026-09-17T09:27:12-04:00 Add prepared gravity subsystem and second-order trial correction +2c084b1c621912680e999b5c4b2faa6b90f6ad2d 2026-09-17T09:23:19-04:00 Add certified polynomial enthalpy bounds and feasible direction repair +cc6e31d95595155a5579ff4ce7475a18977564a6 2026-09-17T09:23:19-04:00 Add certified polynomial enthalpy bounds and feasible direction repair +4ebd50f2bef4883f4bddb318269373847e760270 2026-09-17T09:23:19-04:00 Add prepared Schur gravity reduction and contract tests +7a07caeee33e2addb897b89682678b4cc41b1f3e 2026-09-17T08:55:55-04:00 experiment: record pressure positivity and shape curvature evidence +4280c532e3620f587a346269fc549384e04e2ba0 2026-09-17T08:40:24-04:00 Export enthalpy nodal values on limiting diagnostic elements +c6572d34c33f59085b62e3eb420b4687572f6a38 2026-09-17T08:40:24-04:00 Export enthalpy nodal values on limiting diagnostic elements +9c975422c1033092c790430066e09252b0a62e83 2026-09-17T08:35:49-04:00 experiment: inspect surface carrier and isolate closure bound crossing +b1dc2acf35c4a2b339f61a3dd7d6ebc84318d22f 2026-09-17T08:34:49-04:00 Add preparation-free surface carrier trace diagnostics +ae93c21d844b014cc61be1cda8513034c9a139ec 2026-09-17T08:34:49-04:00 Add preparation-free surface carrier trace diagnostics +b24e6e0c859d98e90648dfe38e467ec4b5d62561 2026-09-17T08:18:09-04:00 experiment: split frozen operator curvature into shape and field contributions +9e1b65dbde2c609b60c2ff8c2637df51646004b8 2026-09-17T08:18:09-04:00 experiment: replay captured candidates and run focused frozen probes +c10725beceff7787828102ff5e9d61f03a566820 2026-09-17T08:18:09-04:00 experiment: split frozen operator curvature into shape and field contributions +6d24c2399f40e80b3086547d1a9a7996eb9edb78 2026-09-17T08:16:37-04:00 experiment: localize pressure preparation admissibility boundaries +6753cc5ddc79933ee7f7b8f0d36951a60e4ae934 2026-09-17T08:16:37-04:00 experiment: localize pressure preparation admissibility boundaries +fd08f7b0e935883a77e2c85c88d26efd31f8bb04 2026-09-17T08:06:37-04:00 experiment: capture frozen live Newton trials without changing solver policy +95d9fbafe893257e85685ccaecde3ab4254eba71 2026-09-16T16:04:49-04:00 docs: finalize controlled solver campaign and transfer limits +5714a9b22271e93c266b476a0e507246834dd105 2026-09-16T15:59:28-04:00 docs: distinguish conservative screening from virial milestone +9e0ffaf1feedd33059881ed635afab53847e81d3 2026-09-16T15:59:28-04:00 docs: record nonrotating larger-window validation +084859037724c5d3de88f0509051cffc2cd624a8 2026-09-16T15:59:27-04:00 docs: record combined optional-density validation +899cb18f2c77171a5cde4d173471199d54affd26 2026-09-16T15:59:28-04:00 docs: distinguish conservative screening from virial milestone +1f9184b7cc217a0c31573675d5d425c0f56a2c84 2026-09-16T15:59:28-04:00 docs: record nonrotating larger-window validation +2e19c2c48c37408a08932268aeb5a164ca8a258b 2026-09-16T15:59:27-04:00 docs: record combined optional-density validation +c88baa19c30a33e837e353c6fc405e018cb86ef8 2026-09-16T15:47:23-04:00 Add a stronger-rotation transfer check for the combined solver +2eab110bef5ae915f75e2322d90ca697af0dcc4d 2026-09-16T15:46:09-04:00 Document matched Krylov safeguard results and limitations +71273836ed20c884c5f3872f85e646cbb06122ec 2026-09-16T15:46:09-04:00 Document matched Krylov safeguard results and limitations +b3a752b16b781d735c24ee57d37dab615e06e639 2026-09-16T15:43:38-04:00 Record controlled numerical outcomes and remaining operator questions +da38801b31307366771fb091ab9b8b822db1d3c2 2026-09-16T14:49:47-04:00 Define matched live controls for the Krylov safeguard +61174cf19225a6b81e29bd83b939392399288069 2026-09-16T14:49:47-04:00 Safeguard reused FGMRES guesses against the current true residual +21b2c2b38e224a6a9bcb3800bd9e749edd50e5ce 2026-09-16T15:20:01-04:00 Define the nonrotating combined-example validation case +db0ddd3454a2c33cfe3ee9edb90177d6b68b655e 2026-09-16T15:19:36-04:00 Document controlled gravity-coupling counterexample +5f83bcba749636aed92f7e5c3639cf6809819cd4 2026-09-16T15:04:21-04:00 Use the validated wider Krylov window in the stellar example +4ec1db230f793be118ebd49452512765b87250e4 2026-09-16T15:04:33-04:00 Record matched central-constraint control results +a1d7d4e67948d47d9ddd7a38320bf57ede306664 2026-09-16T14:39:55-04:00 Keep central density as seed data in the n1 sandbox +0470e3b2a16836ebc37d6d349ab36e10b48bb5dd 2026-09-16T15:17:38-04:00 Gate saved models on solver convergence and the virial milestone +5167744b7a2dc9f40eebcb23389d0fdea8e3e6a8 2026-09-16T15:04:33-04:00 Record matched central-constraint control results +937d2f2d56467995c231c5c7ddebf440aa0baadf 2026-09-16T15:04:21-04:00 Use the validated wider Krylov window in the stellar example +8c1f823da22eac200de101e0393155b7bfcb90aa 2026-09-16T14:51:36-04:00 Define a matched n3 seed test for gravity completion +1911a26a9a283eed7e53b03fc3481ee894e61bfb 2026-09-16T14:49:47-04:00 Define matched live controls for the Krylov safeguard +80d5dfed47d8dcfc92794726350fc1fb8dc67190 2026-09-16T14:42:41-04:00 Record reproducible plateau campaigns and selected-direction probes +ed758307c0da15fa25c26b7c84435b38e5e93879 2026-09-16T14:49:47-04:00 Safeguard reused FGMRES guesses against the current true residual +f9c74aae1054c4001361b96b08721ac410c8091a 2026-09-16T14:42:41-04:00 Record reproducible plateau campaigns and selected-direction probes +ad86e2e21e923fa078c167ccc6569942547af2d2 2026-09-16T14:42:41-04:00 Record reproducible plateau campaigns and selected-direction probes +9bd4af699650e2a6bbe795c618e6ef3fbcb48e12 2026-09-16T14:42:41-04:00 Record reproducible plateau campaigns and selected-direction probes +d34a9c3dffe147331933e548bd1f923d09d9a813 2026-09-16T14:39:55-04:00 Keep central density as seed data in the n1 sandbox +a8e0abf4610f217909037d54afce6e510ec1d0e2 2026-09-16T14:39:15-04:00 Experiment with default-block material-to-gravity completion +80cc3d83d752bd43b7be23cee3d7be6e67cd985f 2026-09-16T14:34:06-04:00 Snapshot optimized solver and audit diagnostics before numerical experiments +75cc63873992984966ea20011cc04d9984f54be4 2026-09-10T06:50:56-04:00 perf(allocations): reduced overall allocations by 95%, increaseed jacobian applicatin by 2x +b3c04d507a1d7bc321ba6edd9dde33f6713345c1 2026-09-08T06:36:39-04:00 feat(newton): first newton solver implementation +76818f2f828ca12606a495a0b84f5d6ce4b78dc6 2026-09-06T10:15:00-04:00 feat(libmeanfield): variadic refactor +71423d543f1775d3b943d9c0361f49edd56b59c1 2026-09-04T07:54:10-04:00 feat(preconditioner): major work on preconditioner system +25510008ddf9810eaf9821149734d526c76b087a 2026-09-02T17:01:50-04:00 perf(jacobian-action): major updates to jacobian action application by removing redudant quadrature work. ~5x increase in speed +85500fef3bbd38ae2342ca4c942c6a454f8308fd 2026-09-01T11:50:13-04:00 feat(surface): surface deformation prescriptions +0a7f18c5c72aa562c020feeab6845e853bfe5217 2026-08-30T16:41:14-04:00 feat(surface): major work on implementing surface constraints in a presciption agnostic manner +36adfa11745cf3eb97d89f836609f1d791a9f8d8 2026-08-29T08:56:36-04:00 feat(FieldDofMap): Completed FieldDofMap migration +177ae8b38aa34d88c07203376961145c5403ccaa 2026-08-24T15:44:24-04:00 feat(support): continued migration to new field dof support system +0f3ca8050b78f0c5c1ef98d485b9fbc2ba06183c 2026-08-23T10:13:53-04:00 feat(field-support): added field support system, mid migration +dc912fd15e51038c999625d7fb0ebfb909f7079a 2026-08-04T14:24:55-04:00 feat(libmeanfield): centrifugal + pressure +9bc4f2758a3325962148d8fbf36719ba3a693fc1 2026-07-15T09:44:43-04:00 feat(mean_field): added initial implementation diff 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These files document a negative experiment. They are excluded from the active build; the old driver and default-policy change were not adopted. The later stellar LDU experiments are different algorithms. The branch is retained for complete reconstruction. diff --git a/reports/archive/one_way_gravity_completion/plateau_gravity_cases.json b/reports/archive/one_way_gravity_completion/plateau_gravity_cases.json new file mode 100644 index 0000000..985faa6 --- /dev/null +++ b/reports/archive/one_way_gravity_completion/plateau_gravity_cases.json @@ -0,0 +1,17 @@ +[ + { + "id": "n3_J0_seed_probe160", + "arguments": [ + "--index", + "3", + "--angular-momentum", + "0", + "--max-newton", + "0", + "--probes", + "--probe-restart", + "160" + ], + "notes": "Fresh matched n3 seed, central homology phase retained. Compare both cold windows and test the restart160 correction without committing trials." + } +] diff --git a/reports/archive/one_way_gravity_completion/plateau_material_gravity_README.md b/reports/archive/one_way_gravity_completion/plateau_material_gravity_README.md new file mode 100644 index 0000000..2f7a0d0 --- /dev/null +++ b/reports/archive/one_way_gravity_completion/plateau_material_gravity_README.md @@ -0,0 +1,115 @@ +# Material–gravity coupling candidate + +This branch changes only the plateau experiment's preconditioner prescription. +The numerical baseline is commit `80cc3d83d752bd43b7be23cee3d7be6e67cd985f` with +independent material/surface and gravity subsystems; the matched run below uses +the shared diagnostic harness on top of it. The candidate uses the existing +`MaterialThenGravityTriangular` policy and copies the control's default material +and gravity descriptors verbatim: surface-then-material with its uncalibrated +diagonals, gravity approximate LDU, Chebyshev order 5, and three fixed AMG cycles. +The scalar border is prepared anew around the candidate. Its physical model, +normalization, Jacobian, and Newton acceptance rules are unchanged. + +For structure blocks `[A B; C D]`, the candidate computes + +``` +m = Atilde^-1 b_material +g = Dtilde^-1 (b_gravity - C m) +``` + +The hypothesis is that completing the material correction's gravitational source +will reduce the Poisson-dominated defect seen on actual frozen Newton right-hand +sides. `C` includes both density and shape contributions, so success would support +their combined completion, not establish a density-only cause. Reverse gravity +feedback into hydrostatic and surface balance remains approximate; the added cross +action can also cost more than it saves. + +## Controlled outcome: retain as a counterexample, do not promote + +The Release executable compiled successfully with Homebrew Clang 22.1.8. On +16 September 2026 the candidate at +`8c1f823da22eac200de101e0393155b7bfcb90aa` was compared with the independent control +at `9bd4af699650e2a6bbe795c618e6ef3fbcb48e12`. Both used a fresh n=3, J=0 seed, +the central constraint, M=R_reference=G=1, K=0.36393905995216302, +rho_c=12.935114545826922, and: + +```sh +plateau_diagnostic --index 3 --angular-momentum 0 --max-newton 0 \ + --probes --probe-restart 160 --output /absolute/path/to/new-results +``` + +Each run performed cold window-40 and window-160 solves, with cap 200 and true +relative target 0.03. Each policy's own window-160 correction was then tested +without committing trials. This was a **frozen probe, not live Newton convergence**. +The seed-only driver exit code was 3; it did not indicate a build failure. + +Build settings, mesh, root layout, seed physical measurements, initial residual +block norms, and accepted physical snapshot were identical. Mesh SHA-256 was +`ce11ae99e21e6c3bbfbee402acd2e191c1da0d8261d2227b4203f73f2f337a74`; snapshot SHA-256 +was `a620600e2a8c7c5d54b628b7919749db47ef7c90822fa940925ae6cd99a68630`. + +| Cold solve | Control true defect / RHS | Candidate true defect / RHS | Control / candidate seconds | +|---|---:|---:|---:| +| Window 40, 200 iterations | 0.0563080093 | 0.0715826233 | 98.873 / 117.956 | +| Window 160, 200 iterations | 0.0402590213 | 0.0381780551 | 98.915 / 117.712 | + +Neither candidate solve passed. Window 160 improves the defect only 5.17% for +19.0% more solve time; window 40 worsens it by 27.1%. These are single-run solve +times, excluding setup and nonlinear diagnostics. At window 160, the candidate +reduces the gradient-relation defect substantially, but Poisson remains 87.96% +of its squared defect; hydrostatic and surface defects increase. + +The selected correction norm grows from 0.757644 to 1.540825. Its sampled +thermodynamic alpha limit tightens from 0.152435 to 0.0913461, making the +candidate's alpha=0.1 trial inadmissible. The control's alpha=0.1 trial reduces F +from 0.0031005934 to 0.0029951212. The candidate's largest sampled admissible step, +alpha=0.01, gives F=0.0030693165. The sampling does not identify either direction's +optimal step. On admissible segments, the measured Taylor remainder follows +approximately 0.111780 alpha² for the control and 0.267383 alpha² for the +candidate; the larger candidate direction therefore has more finite-step +curvature. This is not evidence of a derivative inconsistency. + +The candidate and control R160 corrections have normalized cosine 0.996157. +Their difference has norm 0.788889. For defects `r=Jp+F` on this identical system, + +``` +J(p_candidate-p_control) = r_candidate-r_control +||J(p_candidate-p_control)|| / ||p_candidate-p_control|| + <= (||r_candidate||+||r_control||) / ||p_candidate-p_control|| + <= 3.083e-4. +``` + +This identifies a weak direction in the current normalized coordinates, not a +particular symmetry, a physical error, or the full Jacobian condition number. +The old control R40 direction points nearly opposite to both R160 directions; +its comparison with the candidate gives cosine -0.987233 and a response upper +bound 1.571e-4. These bounds use existing true defects, not additional Jv probes. +No gauge or centering change follows from this evidence alone. + +The scalar Schur is unchanged, as expected because this one-way completion adds +only gravity coordinates while the present scalar constraints depend on material. +Completed border columns show mixed density-source response: the central-column +Poisson defect falls from 315020.6 to 65499.5, while the Bernoulli-column defect +grows from 47.3016 to 56.3085. Both columns have q=0 and unchanged rho/h, isolating +density completion in those probes. Improving one column does not establish +improved full-system performance. + +No state was advanced. The shared seed's P(h) energy/force virial errors remain +5.32642e-5 and 8.34070e-5, above the 1e-5 milestone. Trial virials and interior +errors were not measured. This result rejects promotion of this particular +one-way composition; it does not rule out two-way coupling, a better gravity +inverse, or a more accurate coupled Schur approximation. + +Raw outputs remain outside the repository in the audit workspace under +`diagnostics/numerical_campaign_2026-09-16/{baseline_n3,gravity}/n3_J0_seed_probe160`. +The companion `coupling_comparison.md/.json`, `coupling_baseline_analysis.md`, +and `coupling_direction_comparisons.json` retain detailed analysis. + +The existing diagnostic flags and output rules are described in +[the plateau README](plateau_diagnostic_README.md). `metadata.txt` identifies this +candidate as `material_then_gravity_triangular_default_subcomponents`. + +The prescription is local to this experiment and uses the existing owning-value +and ADL preparation API. Production defaults and the physics-facing public API +are unchanged. No source-completion probes or additional policy variants are +included in this first candidate. diff --git a/reports/archive/one_way_gravity_completion/plateau_material_gravity_prescription.hpp b/reports/archive/one_way_gravity_completion/plateau_material_gravity_prescription.hpp new file mode 100644 index 0000000..cb84e39 --- /dev/null +++ b/reports/archive/one_way_gravity_completion/plateau_material_gravity_prescription.hpp @@ -0,0 +1,28 @@ +#pragma once + +// Experiment-only prescription. Include after `import mean_field;`. +#include +#include + +namespace experiment::material_gravity_candidate { +namespace pc = mean_field::preconditioning; + +inline constexpr char Name[] = "material_then_gravity_triangular_default_subcomponents"; + +struct Prescription final : pc::StellarPreconditionerPrescriptionTag {}; + +// ADL binds this owning prescription to a prepared problem. Copy the default +// descriptors so that only the structure coupling policy changes. Preparing a +// new border also rebuilds its cached structure responses and scalar Schur. +template +[[nodiscard]] auto prepareStellarPreconditioner(Prescription, const Problem &problem) { + auto defaults = pc::stellarStructureBlock(problem); + static_assert(std::same_as, + "This experiment expects an independent-subsystems baseline."); + auto structure = pc::stellarStructureBlock( + problem, defaults.materialSurfaceComponent(), defaults.gravityComponent(), + pc::MaterialThenGravityTriangular{}); + auto block = pc::specificationBorderBlock(problem, std::move(structure), pc::backend::DenseDirect{}); + return pc::prepare(problem, std::move(block)); +} +} // namespace experiment::material_gravity_candidate diff --git a/reports/archive/organize_artifacts.py b/reports/archive/organize_artifacts.py new file mode 100644 index 0000000..25b977c --- /dev/null +++ b/reports/archive/organize_artifacts.py @@ -0,0 +1,67 @@ +"""Execute the reviewed move plan without deleting or rewriting evidence.""" +from pathlib import Path +import hashlib +import json +import os +import subprocess +import time + +HERE = Path(__file__).resolve().parent +REPO = Path('/Users/tboudreaux/Programming/4DSTAR/MeanField') + + +def inventory(path): + entries = [] + paths = [path] if not path.is_dir() else sorted(path.rglob('*')) + for item in paths: + relative = '.' if item == path else str(item.relative_to(path)) + if item.is_symlink(): + entries.append({'path': relative, 'kind': 'symlink', 'target': os.readlink(item)}) + elif item.is_file(): + digest = hashlib.sha256() + with item.open('rb') as stream: + for block in iter(lambda: stream.read(1024 * 1024), b''): + digest.update(block) + entries.append({'path': relative, 'kind': 'file', 'bytes': item.stat().st_size, + 'sha256': digest.hexdigest()}) + elif item.is_dir(): + entries.append({'path': relative, 'kind': 'directory'}) + else: + raise RuntimeError(f'Unsupported evidence type: {item}') + return entries + + +plan = json.loads((HERE / 'move_plan.json').read_text()) +tracked = [REPO / x for x in subprocess.check_output( + ['git', 'ls-files', '-z'], cwd=REPO).decode().split('\0') if x] +for move in plan: + source, destination = Path(move['source']), Path(move['destination']) + if not source.exists() or source.is_symlink(): + raise RuntimeError(f'Missing/already-moved source: {source}') + if destination.exists() or destination.is_symlink(): + raise RuntimeError(f'Destination already exists: {destination}') + if any(x == source or source in x.parents for x in tracked): + raise RuntimeError(f'Plan would move tracked source: {source}') + +manifest = {'started_at_unix': time.time(), 'moves': []} +record = HERE / 'move_manifest.json' +for move in plan: + source, destination = Path(move['source']), Path(move['destination']) + before = inventory(source) + entry = {**move, 'entries': before, 'verified': False} + manifest['moves'].append(entry) + record.write_text(json.dumps(manifest, indent=2) + '\n') + destination.parent.mkdir(parents=True, exist_ok=True) + # Atomic rename preserves contents, permissions and directory structure. + source.rename(destination) + if move['compatibility_symlink']: + source.symlink_to(destination, target_is_directory=destination.is_dir()) + if inventory(destination) != before: + raise RuntimeError(f'Post-move evidence mismatch: {destination}') + entry['verified'] = True + record.write_text(json.dumps(manifest, indent=2) + '\n') + print(f'Preserved and verified: {source.name} ({len(before)} entries)', flush=True) +manifest['completed_at_unix'] = time.time() +manifest['all_verified'] = all(x['verified'] for x in manifest['moves']) +record.write_text(json.dumps(manifest, indent=2) + '\n') +print('All moves verified; no artifacts deleted.', flush=True) diff --git a/reports/archive/original_status.txt b/reports/archive/original_status.txt new file mode 100644 index 0000000..63b6d7c --- /dev/null +++ b/reports/archive/original_status.txt @@ -0,0 +1,75 @@ + M libmeanfield/interface/quadrature/mfem.cppm +?? .cex/ +?? .codex/ +?? accuracy_budget.csv +?? application_optimizations_01/ +?? central_density_homology_coupling.csv +?? cmake-build-profile-homebrew-llvm/ +?? cmake-build-profile-homebrew/ +?? cmake-build-profile-ninja/ +?? cmake-build-release-homebrew-p1/ +?? coupled_gauge_modes.csv +?? docs/optimization_and_mpi_report.aux +?? docs/optimization_and_mpi_report.log +?? docs/optimization_and_mpi_report.out +?? docs/optimization_and_mpi_report.pdf +?? docs/optimization_and_mpi_report.tex +?? docs/optimization_and_mpi_report.toc +?? final_optimizations_01/ +?? geom_tests_2/ +?? geometry_quality_results_2026-09-08_baseline/ +?? geometry_quality_results_2026-09-08_profile/ +?? geometry_quality_results_2026-09-08_vertices/ +?? gravity_completed_null_space.csv +?? gravity_completed_null_space_restricted.csv +?? hdiv_isolation_01/ +?? homology_mass_h0_p0.csv +?? homology_mass_h0_p1.csv +?? homology_mass_h1_p0.csv +?? libmeanfield.zip +?? libmeanfield/interface/field.zip +?? libmeanfield/interface/quadrature.zip +?? linear_derivatives_01/ +?? new.patch +?? package-lock.json +?? package.json +?? performance_optimization_2026-09-09/ +?? performance_profile_2026-09-09/ +?? phase_h_reduced_surface_modes.csv +?? polytrope_analytic_checked_2026-09-09/ +?? polytrope_analytic_mesh_2026-09-09/ +?? polytrope_h1_analytic_2026-09-09/ +?? polytrope_h1_mesh_2026-09-09/ +?? polytrope_h1_solution_2026-09-09/ +?? polytrope_replay_2026-09-09/ +?? polytrope_replay_angular24_2026-09-09/ +?? polytrope_replay_dense_2026-09-09/ +?? polytrope_self_checks_2026-09-09/ +?? polytrope_solution_2026-09-09/ +?? preconditioning_p0_identity.csv +?? preconditioning_p0_identity_extended.csv +?? preconditioning_p0_identity_extended_v2.csv +?? preconditioning_p10_full_stellar.csv +?? preconditioning_p10_gravity_backend_finalists.csv +?? preconditioning_p10_gravity_backend_finalists_v2.csv +?? preconditioning_p10_physical_newton.csv +?? preconditioning_p4_followup.csv +?? preconditioning_p4_gravity.csv +?? preconditioning_p4_gravity_chebyshev_sweep.csv +?? preconditioning_p4_gravity_chebyshev_sweep_v2.csv +?? preconditioning_p9_material_surface_baseline.csv +?? preconditioning_p9_material_surface_h1.csv +?? preconditioning_p9_material_surface_refinement.csv +?? preconditioning_p9_surface_calibration_objectives.csv +?? preconditioning_p9_surface_h1_tuning.csv +?? preparation_optimizations_01/ +?? rebuilt_homology_results.csv +?? results.txt +?? runtime_optimizations_01/ +?? scripts/ +?? stellar_derivatives_01/ +?? stellar_null_space.csv +?? surface_frequency_limits.csv +?? test_summary.html +?? testing.sh +?? tests.zip diff --git a/reports/archive/preserved_quadrature.patch b/reports/archive/preserved_quadrature.patch new file mode 100644 index 0000000..ec7ae5c --- /dev/null +++ b/reports/archive/preserved_quadrature.patch @@ -0,0 +1,27 @@ +diff --git a/libmeanfield/interface/quadrature/mfem.cppm b/libmeanfield/interface/quadrature/mfem.cppm +index 21560a3..37003ae 100644 +--- a/libmeanfield/interface/quadrature/mfem.cppm ++++ b/libmeanfield/interface/quadrature/mfem.cppm +@@ -16,16 +16,16 @@ export namespace mean_field::quadrature { + class RuleFactory { + public: + explicit RuleFactory(Policy policy); +- MfemRule +- get(const Query &query, +- mfem::Geometry::Type geometry) const; +- MfemRule +- get(Term term, ++ MfemRule get(const Query &query, mfem::Geometry::Type geometry) const; ++ ++ MfemRule get( ++ Term term, + QuadratureRole role, + mfem::Geometry::Type geometry, + int base_order, + utils::DOMAINS domain = utils::DOMAINS::ALL, +- MappingKind mapping = MappingKind::none) const; ++ MappingKind mapping = MappingKind::none ++ ) const; + + Resolution configure_gravity_hdiv_mass( + mfem::VectorFEMassIntegrator &integrator, diff --git a/reports/archive/recent_sources.md b/reports/archive/recent_sources.md new file mode 100644 index 0000000..a63f5cf --- /dev/null +++ b/reports/archive/recent_sources.md @@ -0,0 +1,167 @@ +# Sources and integration notes for recent.tex + +The fragment covers native harmonics (Sept. 19), negative-density localization, +density-only Q2/Q3/Q4 and log assessment, final coupled-convergence controls, +compatible rotating n3 family, pressure/gravity consistency, physical slices, +and the resulting next priorities. It is approximately 7,700 words, with 14 +small wrapped longtables and two figures. It deliberately does not duplicate +the preceding Sep. 18 reduced-surface study except to explain the transition +from an affine restricted diagnostic to a native prescription. + +## Integration + +- Required standard packages: `amsmath`, `longtable`, `graphicx`, `hyperref` (as + already planned). No custom command definitions are required. Grouped tables + set tab spacing to 3 pt and explicitly wrap columns; total prescribed column + widths are <=147 mm before internal tab spacing. +- Figure references assume compilation and Pandoc conversion with `reports/` + as the resource directory: `figures/n3_ldu_meridional.png` and + `figures/n1p5_rotating_refinement.pdf`. +- The third planned figure `figures/frozen_linear_rates.pdf` can be added by the + root in the frozen-comparison section or an appendix; its multiple-panel layout + may merit a full page. +- Sections describe Sept. 19 state and producer commits, not the still-pending + final main integration. Root must separately write the exact selected branch + disposition and final main commit. +- All internal labels begin `recent-`, avoiding likely collision with the early + report fragment. +- No fresh FEM runs or audits were performed for this report fragment; it is a + synthesis of the immutable studied outputs and their existing audits. +- The old negative-density report's recommendation for derived density is + explicitly identified as historical and superseded by the user's decision to + keep density independent. The later preference to defer positivity is also + explicit. Do not silently carry that old recommendation forward. +- Important no-overclaim rules are repeated where they matter: finite-budget + endpoints are not floors; equal-update trajectories are not same-model + speedups; quadrature agreement is not mesh convergence; sampled positivity + is not global positivity; the n3 force/h virial alone passing does not meet + the overall milestone; raw J or angular velocity is not a critical fraction. + +## Primary repository reports read + +Original source checkout root: +`/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement` + +1. `experiments/SPHERICAL_HARMONIC_SURFACE_2026-09-19.md` + - Native all-m L4 prescription, typed contracts, fixed reference basis, + Q4/Q3 geometry/displacement, scalar trace-mass normalization, single-rank + scope, five six-update live controls, excluded-force and local diagnostics. + - Producer: `7b0eb8d92d0f550f1fe3fb19cd8eecc52dc80c56`. +2. `experiments/NEGATIVE_DENSITY_INVESTIGATION_2026-09-19.md` + - Four frozen states, independent closure projection identity, all 96 outer + cells, root-sliced measures, positive nodal values, seed-vs-root mechanism, + exact one-dimensional quadratic projections, interpretive scope. + - Producers: `9c22934d52b369f96364e781d6a4843939ec9523` and + `d1a594808f960bbaeeb6df8a8ce6a95458765b19`. +3. `experiments/DENSITY_ORDER_STUDY_2026-09-19.md` + - Frozen common-q26 and error-q30 controls; all 12 live density-order cases; + selected quadrature rules, block sizes, cost scopes, central forcing and + initializer distinctions; 48 contract tests per order. + - Producers: frozen `241621c`; coupled Q2 `c0e8551`; Q3 `cd71f37`; + Q4 `e054c74`; final comparison generator `c0a496e`. +4. `experiments/COUPLED_CONVERGENCE_STUDY_2026-09-19.md` + - Five initial nonlinear controls, matched gravity-path pair, all 11 frozen + inverse pairs, weak coupled direction analysis, compatible n3 pair, + display refinement, validation and next priorities. + - Final integration/report commit: `d7acb0d399aac5ac4c36d834a954c1c959042b81`. +5. `experiments/N3_COMPATIBLE_FAMILY_2026-09-19.md` + - Explicit K factor, independent Lane–Emden compatibility check, exact + shared specifications and caveats for the live pair. +6. `experiments/FIELD_COUPLING_REVIEW_2026-09-19.md` + - Actual material triangular structure contains the exact h-to-rho action; + independent vs LDU feedback, possible exact DG mass and H1 mass improvements, + optional future EOS-aware field Schur and closure affine-target trial path. +7. `experiments/MODEL_SLICES.md` + - Physical-plane cut and source immutability; element-owned DG traces, + independent outlines, field raster before colormap, display errors/limits, + use/replay parameters and geometry/plotting checks. +8. `experiments/REDUCED_SURFACE_STUDY_2026-09-18.md` + - Read for bridge/context only; preceding early-history fragment should + contain its detailed restricted-vs-full system results. + +Separate source checkout: +`/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/log-density-assessment` + +9. `experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md` + - Mathematical distinction between exponentiating coefficients and pointwise + exponential, vacuum singularity, independent vanishing factor, 1D moment + fits, weighted-mass response/conditioning, general-EOS and support contracts. + - Separate assessment branch at `90580b2`; no coupled stellar log solve. + +## Raw evidence and concise analysis read + +Archive target supplied by root: +`working/research/diagnostics//` in the main repository. +Original workspace `diagnostics/` will remain a symlink to that archived path. + +Campaign names: + +- `spherical_harmonic_surface_2026-09-19/` +- `negative_density_2026-09-19/` +- `density_order_2026-09-19/` +- `log_density_2026-09-19/` +- `coupled_convergence_2026-09-19/` +- `n3_mass_compatibility_2026-09-19/` + +Additional analyses read directly: + +- `coupled_convergence_2026-09-19/N3_COMPATIBLE_RESULTS.md` +- `coupled_convergence_2026-09-19/PRESSURE_MOMENT_NOTE.md` + +High-value existing evidence links for root's archive appendix: + +- `coupled_convergence_2026-09-19/STUDY_INDEX.json` +- `coupled_convergence_2026-09-19/CONTROL_RESULTS.md` and `.json` +- `coupled_convergence_2026-09-19/linear_analysis/FINAL_INTERPRETATION.md` +- `coupled_convergence_2026-09-19/n3_direction_difference/README.md` +- `coupled_convergence_2026-09-19/SLICE_REFINEMENT_SUMMARY.json` +- `coupled_convergence_2026-09-19/validation/validation.json` +- `coupled_convergence_2026-09-19/figures_n3_v2/slice_tests.log` +- `density_order_2026-09-19/final_analysis/coupled_outcomes.csv` +- `density_order_2026-09-19/final_analysis/trial_stages.csv` +- `negative_density_2026-09-19/regions_campaign_audit_v1.json` +- `spherical_harmonic_surface_2026-09-19/campaign_audit.json` + +## Figure sources + +All below are relative to original workspace diagnostics, or to the archive's +`working/research/diagnostics/` after reorganization. + +- Report `figures/n3_ldu_meridional.png`: + `coupled_convergence_2026-09-19/figures_n3_v2/n3_J0p035_K0p99_ldu_xz.png` +- Report `figures/n1p5_rotating_refinement.pdf`: + `coupled_convergence_2026-09-19/figures_slices_v2/n1p5_rotating_xz_s4_s8.pdf` +- Optional report `figures/frozen_linear_rates.pdf`: + `coupled_convergence_2026-09-19/figures/linear_rates/frozen_linear_rates.pdf` +- Additional useful density-order figure: + `density_order_2026-09-19/final_analysis/density_order_comparison.png` +- Additional sign-location figure: + `negative_density_2026-09-19/figures/surface_density_traces.png` + +## External primary-source context retained from the studied reports + +The numeric results and equations in the fragment are primarily local evidence +and explicit derivations. If adding a bibliography, these are the primary +sources referenced by the underlying reports (no new online verification in +this report-writing subtask): + +- Cheng, Cheng and Lin, *Expanding solutions near unstable Lane–Emden stars*, + equations (1.14)–(1.15) and Theorem 3.1: + https://arxiv.org/html/2305.10218#S3.SS1 + This supplies the sharp energy bound and Lane–Emden mass relation. The beta + bound is our inference with the steady virial identity, not a quoted theorem. +- MESA `star/private/star_utils.f90`: + https://github.com/MESAHub/mesa/blob/main/star/private/star_utils.f90 +- MESA T(tau) atmosphere documentation: + https://docs.mesastar.org/en/latest/atm/t-tau.html +- Vijaywargiya and Fu, *Two Finite Element Approaches for the Porous Medium + Equation That Are Positivity Preserving and Energy Stable* (2024): + https://link.springer.com/article/10.1007/s10915-024-02642-x +- Fu and Xu, *High-order space-time finite element methods for the + Poisson–Nernst–Planck equations: Positivity and unconditional energy stability*: + https://arxiv.org/abs/2105.01163 +- Zhang and Shu review (historical alternative only, not currently adopted): + https://www.math.purdue.edu/~zhan1966/research/paper/review.pdf + +The condensed fragment mentions MESA only as context and does not import PDE +positivity theorems as guarantees for this steady stellar system. diff --git a/reports/archive/refs_before.txt b/reports/archive/refs_before.txt new file mode 100644 index 0000000..cd0159d --- /dev/null +++ b/reports/archive/refs_before.txt @@ -0,0 +1,45 @@ +75cc63873992984966ea20011cc04d9984f54be4 refs/heads/main +80cc3d83d752bd43b7be23cee3d7be6e67cd985f refs/heads/research/baseline-20260916 +2e19c2c48c37408a08932268aeb5a164ca8a258b refs/heads/research/central-constraint-20260916 +95d9fbafe893257e85685ccaecde3ab4254eba71 refs/heads/research/combined-20260916 +d7acb0d399aac5ac4c36d834a954c1c959042b81 refs/heads/research/coupled-convergence-study-20260919 +4a6c1b5c26a352a8afb30feb0a2f3821aac9ee19 refs/heads/research/coupled-shape-coarse-20260918 +46249d0758e1c30f7585d35821f78cd558197cec refs/heads/research/coupled-trial-correction-20260918 +db854c0f7cdf5137e377273499f0ac5b6053b752 refs/heads/research/damped-coupled-trial-20260918 +ef76e0c7860d3972f4993e406defe9dd24dcc05b refs/heads/research/density-order-study-20260919 +7a07caeee33e2addb897b89682678b4cc41b1f3e 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No remote branch was pushed and +no research branch or worktree was deleted. + +The useful changes from the hypothesis branches were already largely combined +in the cumulative research history, often through cherry-picks. Consolidation +verified their final contents instead of forcing every historical tip into the +ancestry or reintroducing old driver versions. + +- Retained the optimized baseline, safeguards, optional central constraint, + typed numerical policies, gravity-Schur refinement, cached cross actions, + native all-m harmonic surface, density-order controls, checked restarts, + physical diagnostics, model slices, tests and experiment recipes. +- Preserved the existing user quadrature formatting edit in `6950dc6`. +- Imported the missing log-density assessment and standalone reference script + through `b1fe359`. They remain deferred research; density is still an + independent Q2 field and no log-density runtime or new positivity policy was + enabled. +- Kept Q3 displacement. The failed Q4 change remains isolated; its findings are + included in the report. +- Archived the unsuccessful one-way material-to-gravity recipe under + `archive/one_way_gravity_completion/`; its old driver/default change was not + adopted. Experimental corrections/coarse solvers remain explicit options. +- Preserved existing library defaults. All-m L4/restart160 are explicit sandbox + choices; stellar LDU plus gravity-Schur is a measured research candidate. + +The report's consolidation-validation section and `validation/` records give +this delivery's actual checks. The exact original audit is preserved in +`archive/branch_audit_initial.md`. The table and source audit below describe the +**pre-consolidation d7acb0d comparison**; in particular, old main=75cc638 and the +then-missing log assessment are historical states, not the completed result. + +--- + +# MeanField branch consolidation audit — 20 September 2026 + +This is a read-only source/history audit. No source files, branch references, worktrees or artifacts were changed by the auditor; this document is its only output. No numerical test was run. Conclusions refer to integration commit `d7acb0d399aac5ac4c36d834a954c1c959042b81` unless a different revision is named. + +## Recommendation + +Use `research/coupled-convergence-study-20260919` (`d7acb0d`) as the integration base and advance `main` to a tested consolidation descendant. It is a descendant of current `main` (`75cc638`) and contains all successful cumulative implementation work, all current experiment infrastructure, and most unsuccessful-study records. It already incorporates most topic branches by cherry-pick or equivalent integration; blindly merging every branch would reintroduce obsolete driver versions or failed defaults and create unnecessary conflicts. + +Preserve historical branches. Record their disposition in the consolidated report. A branch need not become an ancestor of main to have its useful contents incorporated; the patch/content mapping below supplies that provenance. Do not use an `ours` merge to falsely suggest discarded behavior was adopted. + +Two divergent topics merit explicit treatment: + +1. **Log-density assessment:** import the two final files from `research/log-density-assessment-20260919` (`90580b2`) as deferred research documentation/reference: `experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md` and `experiments/log_density_reference_model.py`. The six unique commits change only these files. There is no FEM log-density implementation to merge, and adding the historical assessment does not enable positivity enforcement or contradict the current decision to defer it. +2. **One-way material-to-gravity completion:** preserve the final README, recipe and case manifest from `research/gravity-coupling-20260916` (`db0ddd3`) in a clearly historical/archived experiment location, or retain the branch with a report link and extracted evidence. Do **not** transplant its old driver or make its prescription the default. This was a measured counterexample, not the later successful two-way LDU policy. + +The failed Q4 displacement implementation remains isolated on `research/shape-field-displacement-q4-20260918`; its final report and comparison scripts are already present in the integration branch with Q3 restored. There is no scientific reason to activate Q4 during consolidation. + +## Main checkout preservation + +At audit start, `/Users/tboudreaux/Programming/4DSTAR/MeanField` was on `research/numerical-strategies-20260917` at `10adf7e`, not on main. The only tracked local edit was formatting in `libmeanfield/interface/quadrature/mfem.cppm`: method declaration wrapping, no signature/semantic change. Preserve it explicitly by backup/commit or an exactly reapplied patch; it is the user's work. Numerous untracked result folders, CSVs, archives, build trees, `.cex`, `.codex`, scripts and optimization report files were also present. This audit does not classify those as deletable. + +`main`, `origin/main` and `local/main` all pointed to `75cc638` at audit start. Do not push remote main without further authorization; local integration is the requested action. No AGENTS.md exists in the programming checkout or its checked ancestors. The project mirror's AGENTS.md says every file under `sources/` is read-only; do not relocate or change synced sources. + +## Branch-by-branch disposition + +“Ancestor” means reachable from `d7acb0d`. “Patch-equivalent” means all topic commits reported by `git cherry d7acb0d ` have equivalent patches in the integration history. “Semantically integrated” covers build-registration and combined cherry-picks whose exact patch IDs differ but final source was inspected. This is more discriminating than treating every plus-sign from `git cherry` as missing code. + +| Branch | Tip | Status and action | +|---|---|---| +| `integration/research-consolidation-20260920` | `b1fe359` | New consolidation descendant, created during this audit: adds preserved user formatting (`6950dc6`) and six log-assessment documentation/reference commits (`549b1a7` through `b1fe359`) after `d7acb0d`. These were not represented in the base snapshot. | +| `main` | `75cc638` | Ancestor; already represented. No additional merge. | +| `research/baseline-20260916` | `80cc3d8` | Ancestor; already represented. No additional merge. | +| `research/central-constraint-20260916` | `2e19c2c` | All 4 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/combined-20260916` | `95d9fba` | Ancestor; already represented. No additional merge. | +| `research/coupled-convergence-study-20260919` | `d7acb0d` | Ancestor; already represented. No additional merge. | +| `research/coupled-shape-coarse-20260918` | `4a6c1b5` | Semantically integrated: two equivalent commits; isolated test-registration 4a6c1b5 is already in CMakeLists. Keep diagnostic-only coarse experiment; no additional merge. | +| `research/coupled-trial-correction-20260918` | `46249d0` | Ancestor; already represented. No additional merge. | +| `research/damped-coupled-trial-20260918` | `db854c0` | Ancestor; already represented. No additional merge. | +| `research/density-order-study-20260919` | `ef76e0c` | Ancestor; already represented. No additional merge. | +| `research/diagnostic-campaign-20260917` | `7a07cae` | Ancestor; already represented. No additional merge. | +| `research/experiment-harness-20260916` | `899cb18` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-compact-application-20260917` | `ff1b889` | Semantically integrated: seven equivalent patches; e599900 and 2c64a92 module/test registration already present. No additional merge. | +| `research/gravity-consistent-schur-20260918` | `c10d019` | All 2 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-coupling-20260916` | `db0ddd3` | Not integrated: failed one-way completion. Archive README/recipe/manifest and retain branch. Do not adopt old driver or policy. | +| `research/gravity-coupling-ablation-20260919` | `38c7184` | Ancestor; already represented. No additional merge. | +| `research/gravity-curvature-correction-20260917` | `3762c74` | Semantically integrated: four equivalent patches; e599900 registration already present. Keep opt-in; no additional merge. | +| `research/gravity-direction-diagnostics-20260917` | `47775a9` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-prepared-roundoff-20260917` | `782827c` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/gravity-reduced-newton-20260917` | `d25d141` | Semantically integrated: seven equivalent patches; 5952691 registration already present. Keep opt-in; no additional merge. | +| `research/gravity-trial-diagnostics-20260917` | `7f4b068` | All 2 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/krylov-20260916` | `7127383` | All 4 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/krylov-window-20260916` | `1f9184b` | Semantically integrated: documentation identical, sandbox restart is 160. 937d2f2 was integrated alongside other sandbox changes; no additional merge. | +| `research/linear-coupling-study-20260919` | `0e6688f` | Semantically integrated: core wrapper/tests identical; one patch equivalent and 1d95602 integrated as 2398118 with adjusted surrounding driver/build context. Frozen experiment only. | +| `research/log-density-assessment-20260919` | `90580b2` | Not integrated: six documentation/reference-only commits. Import final assessment and reference script as deferred research; no positivity/runtime change. | +| `research/model-slices-20260919` | `5da3f42` | Semantically integrated: seven equivalent patches; 966a4d4 integrated as 9bac57a. Slice implementation/plotter content identical. No additional merge. | +| `research/n3-compatible-family-20260919` | `1d746b0` | Ancestor; already represented. No additional merge. | +| `research/n3-linear-ablation-20260919` | `e78af98` | Ancestor; already represented. No additional merge. | +| `research/n3-live-field-feedback-20260919` | `1ed03f5` | Ancestor; already represented. No additional merge. | +| `research/negative-density-diagnosis-20260919` | `7ed4d23` | Ancestor; already represented. No additional merge. | +| `research/nonlinear-step-obstruction-20260918` | `2944dd5` | Ancestor; already represented. No additional merge. | +| `research/numerical-strategies-20260917` | `10adf7e` | Ancestor; already represented. No additional merge. | +| `research/polynomial-endpoint-certification-20260917` | `cc6f4b0` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/polynomial-positive-newton-20260917` | `9aefb36` | Semantically integrated: eight equivalent patches; a773c9a registration already present. Available explicit strategy only, not a default; do not enable. | +| `research/preconditioner-benchmarks-20260918` | `dc1da62` | Ancestor; already represented. No additional merge. | +| `research/preconditioner-cached-cross-actions-20260918` | `07769e7` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/pressure-obstruction-20260917` | `c6572d3` | All 3 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/reduced-smooth-surface-20260918` | `2171b8b` | Ancestor; already represented. No additional merge. | +| `research/shape-curvature-20260917` | `b24e6e0` | All 1 unique topic commits patch-equivalent; already represented. No additional merge. | +| `research/shape-field-displacement-q4-20260918` | `785a7ae` | Implementation intentionally excluded: retain Q3. Final report/comparison scripts already identical in integration, preserving negative result. | +| `research/shape-field-weak-response-20260918` | `6ceda61` | Ancestor; already represented. No additional merge. | +| `research/spherical-harmonic-surface-20260919` | `1148c09` | Ancestor; already represented. No additional merge. | + +## Verified false-positive unmerged commits + +The following unique patch IDs do not indicate missing capabilities: + +- `4a6c1b5`: `tests/solver/plateau_coupled_coarse.cpp` is already registered at CMakeLists line 391. +- `e599900`, `2c64a92`, `5952691`, `a773c9a`: gravity subsystem/reduction, compact gravity, polynomial admissibility modules and tests are already in CMakeLists and the exported mean_field module. Reapplying isolated registration commits would only duplicate source entries or conflict. +- `937d2f2`: sandbox explicitly uses FGMRES restart 160. The final Krylov-window report is byte-identical to the topic branch. +- `1d95602`: `plateau_surface_border_schur.hpp` and its regression tests are byte-identical to the final topic branch. Integration commits `2398118` and `522a2cf` carry the implementation and followup test changes. +- `966a4d4`: physical-plane slice headers and current plotter are byte-identical to the final topic branch. Integration starts at `9bac57a`, with final label changes at `07799bc` and `2c6523b`. +- Q4 topic `785a7ae`: `SHAPE_FIELD_WEAK_RESPONSE_2026-09-18.md`, `compare_shape_order_runs.py` and `plot_shape_surface_comparison.py` are already byte-identical to the topic tip. The source-order change from `4be1a23` is deliberately absent. + +## What is active versus selectable at d7acb0d + +It is essential that the consolidated report distinguish library defaults, example configuration, diagnostic-driver defaults, and recommended research candidates. + +| Layer | Current behavior | Consolidation recommendation | +|---|---|---| +| Independent physical fields | Density remains independent DG Q2; enthalpy independent H1 Q3; displacement H1 Q3; gravity RT2/Q2; reference geometry Q4 | Retain. Do not import the Q4 trial or log-density runtime. | +| Density approximation configuration | `MEAN_FIELD_DENSITY_POLYNOMIAL_ORDER=2` compile-time CMake/registry option independent of other fields; uniform increment remains available | Retain configurable order and corrected closure quadrature. Default stays Q2. | +| Surface factory default | Two-argument `makeStellarDiscretization` remains `NodalRadialSurface`, with power-law radial interior and fixed-infinity radial vacuum extensions | Preserve compatibility; avoid silently changing all clients. | +| Sandbox preferred surface | Explicit `SphericalHarmonicSurface` maximum degree 4, all real m, 25 coefficients, coupled with all fields | Retain explicit preferred candidate; nodal remains selectable. | +| Harmonic surface limit | Single MPI rank; sampled basis rank and trace contracts checked. H1 nodal surface surrogate explicitly rejects harmonic coordinates | Report this limitation; do not imply general distributed harmonic support. | +| Normalization | Existing `PhysicalRieszDiagonal` in sandbox; harmonic reference trace metric adds correct primal/dual coefficient scaling | Retain. This is coordinate scaling, not an exact surface response inverse. | +| Sandbox physical model | n=1, J=0, fixed total mass/angular momentum; central density is Lane–Emden seed data only | Retain. Central constraint remains explicitly specifiable for n=3 studies, not a hidden universal constraint. | +| Library FGMRES option default | `FGMRESOptions.restartLength=50` | Unchanged by research. Do not claim it is 160 globally. | +| Sandbox FGMRES configuration | Restart 160; Newton maximum 30; relative nonlinear target 1e-8, linear target .03/cap 200 | Retain known baseline. | +| Diagnostic no-argument controls | Nodal surface, restart 40, independent preconditioner, baseline strategy; runs require output. Campaign manifests explicitly select harmonic/restart160 variants | Keep manifests authoritative; do not confuse convenient CLI defaults with the latest experimental configuration. | +| Warm-start guard | Reused linear correction is reset if its fresh full-J true residual is nonfinite or worse than the cold RHS norm | Retain correctness/performance safeguard; endpoint true-residual check remains authoritative. | +| `makePreconditioner()` | Default stellar structure: independent material/surface and gravity inverse, with generated scalar-border completion | Preserve default until matched full convergence evidence warrants promotion. | +| Default material inverse | SurfaceThenMaterialTriangular with diagonal density/enthalpy/surface solves; exact prepared h-to-density EOS cross derivative and surface-to-material actions included | Retain. Do not describe it as missing h-to-density coupling. | +| Default gravity inverse | GravityApproximateLDU; matrix-free Chebyshev order5/powerIterations20 mass inverse, three fixed AMG cycles | Preserve base comparator; refined consistent Schur remains available. | +| Consistent gravity Schur | Typed `GravityConsistentSchurLDU`, fixed number of sweeps; library policy defaults 1 sweep/relaxation1, diagnostic default relaxation .5 | Retain useful optional implementation and measured manifests; do not conflate the option defaults. | +| Live experimental recipes | `gravity-schur`, `stellar-ldu`, `stellar-ldu-gravity-schur` live in experiments owning recipe layer | Retain. LDU+gravity is currently best live candidate for the tested compatible rotating n3/gravity-curvature configuration, not established universal default. | +| Newton numerical defaults | `SampledEnthalpy`, `MonolithicGravity`, `UncorrectedTrial` | Preserve. No added positivity forcing or auxiliary corrections execute by default. | +| Optional Newton paths | Polynomial enthalpy certificate/repair, gravity-curvature, reduced gravity, fully coupled and damped second-order corrections | Preserve explicit selection and tests as experimental tools; no general convergence claim. | +| Coarse surface/scalar Schur | Experiment-only frozen wrapper; reconstruct after changed Jacobian | Retain diagnostic infrastructure. Do not promote; lower iteration counts did not generally justify setup/application expense. | +| Restart initializer | Explicit diagnostic API validates state/layout/coordinate provenance, prepares/refreshes transactionally, clears warm starts, rolls back failures | Retain. Unconverged snapshots require explicit experimental override. | +| Model slices | Optional experiment export/plot of actual mapped stellar domain; xz/xy/yz/offset options; signed fields retained | Retain. Display approximation/refinement is distinct from FEM convergence or physical validation. | + +Source anchors: `sandbox.cpp`; `libmeanfield/interface/equilibrium/stellar_discretization.cppm`; `libmeanfield/interface/field/field_registry.cppm`; `libmeanfield/interface/normalization/stellar_equilibrium.cppm`; `libmeanfield/interface/preconditioning/{stellar_recipe,stellar_structure,material_surface,gravity_field}.cppm`; `libmeanfield/interface/solver/{linear_backend,newton_numerics,stellar_context}.cppm`; `experiments/plateau_diagnostic.cpp`; `experiments/plateau_newton_preconditioner.hpp`. + +## Why the failed one-way branch stays historical + +`db0ddd3` documents a same-state n3,J0 cold-seed control with original material/gravity subcomponents unchanged, only `MaterialThenGravityTriangular` replacing independence. At restart160/cap200/target .03, true defect improves .0402590 to .0381781 but both fail; solve cost rises 98.915 to117.712s (19%). At restart40, defect worsens .0563080 to .0715826. Direction norm doubles .757644 to1.540825; sampled admissible step tightens .152435 to .0913461. This is not an improvement sufficient for default promotion. It does not rule out the later two-way LDU with better gravity inverse. + +The branch's shared harness commit `ad86e2e` includes some source changes also represented later in the main cumulative harness; it is not all useful missing code simply because patch-id differs. Importing its old `plateau_diagnostic.cpp` would erase subsequent functionality. + +## Useful cumulative integration already present + +- Baseline `80cc3d8` snapshots the user's optimized prepared-operator/reference-table/mapping/normalization path and diagnostics that were not in old main. The main-to-d7 diff includes this earlier user performance work; do not attribute every changed file to the subsequent AI-led numerical experiments. +- Optional central-density example cleanup, restart160 example choice and safeguarded warm starts are cumulative by `95d9fba`. +- Pressure/shape diagnostic instrumentation and typed Newton numerical strategies are integrated by `10adf7e`; prepared gravity arithmetic and compact application fixes are included. +- Consistent Schur, prepared cross-action reuse and honest full-cost frozen benchmarking are integrated by `dc1da62`. +- Coupled/damped trial tools and their bounded negative/conditional results are integrated by `db854c0`. +- Q3-restored weak-response/Q4 comparison report, reduced surface experiments and native all-m harmonic surface are integrated by `1148c09`. +- Negative-density localization, independent density-order configuration, closure quadrature repair and audited order studies are integrated by `ef76e0c`. +- Checked restart, frozen coupled coarse trials, physically compatible n3 controls, slices and final convergence/cost/virial evidence are integrated by `d7acb0d`. + +## Consolidation and validation risks + +1. **Do not promote experimental settings merely because they enter main.** Main can contain tested optional research modules. The final report must explicitly state which settings are defaults, preferred research candidates and failed historical alternatives. +2. **Preserve source provenance.** Campaign numerical executables were copied from specific producer commits. Main integration changes do not retroactively change the meaning of old runs. Keep original commands, metadata, hashes and branch tips, and include relocation manifests when moving results. +3. **Preserve old paths carefully.** Many tracked research reports and JSON manifests refer to evidence under workspace `diagnostics/` using worktree-relative or absolute paths. Relocation requires either stable links or a documented path mapping and adjusted browse links. Do not rewrite original run metadata/hashes simply to make them look current. +4. **Untracked build trees have absolute cache paths.** Moving them may invalidate rebuilds. Preserve them as historical artifacts or use explicit compatible links; configure a fresh consolidation build instead of treating a moved cache as portable. +5. **Do not rerun a full campaign unnecessarily.** A source-equivalent integration needs targeted build/smoke/regression checks around added files and resolved conflicts, not new scientific claims. The previous 24-case audit is a provenance check, not proof that every model converged. +6. **Harmonic support is serial.** A two-rank rejection test is not MPI solver validation. Report this accurately. +7. **No end-to-end target claim.** The latest compatible rotating n3 LDU endpoint remains unconverged; energy/h virial1.81141e-5 and both rho-pressure virials fail1e-5. No 50%-critical calibration exists. Lower finite-budget residuals are not measured mathematical floors. +8. **Documentation import is not new policy.** The log-density reference model is historical isolated analysis. Keep its label explicit and do not present it as an implemented solver option. + +## Audit limits + +This audit checks ancestry, patch equivalence, source content, defaults and recorded evidence. It does not constitute a new exhaustive correctness review of all 57k lines added since old main, nor a numerical rerun. Exact source comparisons established the particular equivalences above; registration line numbers refer to d7acb0d and can shift during consolidation. The parent integrator should record the final consolidation SHA, any actual conflicts/resolutions, validation results, and the eventual status of the preserved user formatting edit. + +## Consolidation update + +After the base-snapshot audit, the integrator preserved the user formatting change as `6950dc6` and imported all six log-density documentation/reference commits, ending at `b1fe359` on `integration/research-consolidation-20260920`. Consequently the earlier base-snapshot recommendation to import the log assessment is fulfilled. No numerical behavior was added by those seven commits. The original branch table entries otherwise describe their relationship to `d7acb0d`, not their eventual status after consolidation. diff --git a/reports/build_report.py b/reports/build_report.py new file mode 100644 index 0000000..4758db1 --- /dev/null +++ b/reports/build_report.py @@ -0,0 +1,67 @@ +#!/usr/bin/env python3 +"""Build the handoff's Markdown with Pandoc and its PDF with LaTeX. + +The LaTeX is authoritative. Generated Markdown is deliberately not post-edited, +so a reader can regenerate both delivery formats from the same research record. +""" +from pathlib import Path +import argparse +import hashlib +import json +import shutil +import subprocess +import sys +import time + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--markdown-only', action='store_true') + args = parser.parse_args() + reports = Path(__file__).resolve().parent + stem = 'MeanField_research_handoff_2026-09-20' + source = reports / (stem + '.tex') + build = reports.parent / 'working/report_builds/research_handoff' + build.mkdir(parents=True, exist_ok=True) + commands = [[ + 'pandoc', '--from=latex', '--to=gfm-tex_math_gfm+tex_math_dollars', '--standalone', + '--toc', '--toc-depth=2', '--lua-filter=pandoc_tables.lua', + '--wrap=none', '--output=' + stem + '.md', source.name, + ], [sys.executable, 'check_report_conversion.py', + '--output', str(build / 'conversion_audit.json')]] + if not args.markdown_only: + commands.append([ + 'latexmk', '-pdf', '-interaction=nonstopmode', '-halt-on-error', + '-outdir=' + str(build), source.name, + ]) + record = {'source_sha256': hashlib.sha256(source.read_bytes()).hexdigest(), + 'commands': [], 'outputs': {}} + for index, command in enumerate(commands): + executable = shutil.which(command[0]) + if executable is None: + raise SystemExit(f'Required program is unavailable: {command[0]}') + command[0] = executable + start = time.monotonic() + log = build / f'build_{index}.log' + with log.open('w') as stream: + result = subprocess.run(command, cwd=reports, stdout=stream, + stderr=subprocess.STDOUT) + record['commands'].append({'command': command, 'returncode': result.returncode, + 'seconds': time.monotonic() - start}) + (build / 'build_manifest.json').write_text(json.dumps(record, indent=2) + '\n') + if result.returncode: + raise SystemExit(f'Report build failed; inspect {log}') + if not args.markdown_only: + shutil.copy2(build / (stem + '.pdf'), reports / (stem + '.pdf')) + for suffix in ['.tex', '.md'] + ([] if args.markdown_only else ['.pdf']): + path = reports / (stem + suffix) + record['outputs'][path.name] = { + 'sha256': hashlib.sha256(path.read_bytes()).hexdigest(), + 'bytes': path.stat().st_size, + } + (build / 'build_manifest.json').write_text(json.dumps(record, indent=2) + '\n') + print('Generated ' + ', '.join(record['outputs'])) + + +if __name__ == '__main__': + main() diff --git a/reports/check_report_conversion.py b/reports/check_report_conversion.py new file mode 100644 index 0000000..5dfb5f1 --- /dev/null +++ b/reports/check_report_conversion.py @@ -0,0 +1,305 @@ +#!/usr/bin/env python3 +"""Check the handoff's LaTeX-to-Markdown conversion without modifying inputs. + +Pandoc's parsed tables preserve cell boundaries, which catches both lost values +and accidental columns caused by vertical bars in mathematical expressions. +Figure captions need a separate comparison because GFM stores them as raw HTML. +Only the explicitly requested JSON record is written; no report is rebuilt. +""" +from pathlib import Path +from collections import Counter +from html.parser import HTMLParser +import argparse +import copy +import datetime +import hashlib +import json +import re +import shutil +import subprocess +import sys + + +STEM = 'MeanField_research_handoff_2026-09-20' +MARKDOWN_FORMAT = 'gfm-tex_math_gfm+tex_math_dollars' + + +def nodes(value, skip_figures=False): + if isinstance(value, dict): + if skip_figures and value.get('t') == 'Figure': + return + yield value + for child in value.values(): + yield from nodes(child, skip_figures) + elif isinstance(value, list): + for child in value: + yield from nodes(child, skip_figures) + + +def semantic_tokens(value): + """Keep values and math distinct; disregard layout-only whitespace.""" + if isinstance(value, dict): + kind = value.get('t') + if kind == 'Str': + return [('text', value['c'])] + if kind == 'Code': + return [('code', value['c'][1])] + if kind == 'Math': + return [('math', value['c'][0]['t'], + re.sub(r'\s+', ' ', value['c'][1]).strip())] + return [token for child in value.values() + for token in semantic_tokens(child)] + if isinstance(value, list): + return [token for child in value for token in semantic_tokens(child)] + return [] + + +def row_values(row): + # Retaining spans and cell boundaries rejects values shifted into a neighbor. + return [(cell[2], cell[3], semantic_tokens(cell[4])) for cell in row[1]] + + +def table_values(table, omit_empty_head=False): + content = table['c'] + head = [row_values(row) for row in content[3][1]] + if omit_empty_head and all(not cell[2] for row in head for cell in row): + head = [] + body = [row_values(row) for group in content[4] + for row in group[2] + group[3]] + foot = [row_values(row) for row in content[5][1]] + return {'columns': len(content[2]), 'caption': semantic_tokens(content[1]), + 'head': head, 'body': body, 'foot': foot} + + +def expected_filtered_table(table): + result = copy.deepcopy(table) + head, bodies = result['c'][3][1], result['c'][4] + removed = False + if len(head) == 1 and bodies and bodies[0][3]: + head_contents = [cell[4] for cell in head[0][1]] + first_contents = [cell[4] for cell in bodies[0][3][0][1]] + if head_contents == first_contents: + bodies[0][3].pop(0) + removed = True + return result, removed + + +class HTMLInventory(HTMLParser): + """Compare rendered captions, including their inline mathematical markup.""" + def __init__(self): + super().__init__(convert_charrefs=True) + self.ids = [] + self.figures = [] + self.figure = None + self.in_caption = False + + def handle_starttag(self, tag, attrs): + attrs = dict(attrs) + if 'id' in attrs: + self.ids.append(attrs['id']) + if tag == 'figure': + self.figure = {'id': attrs.get('id', ''), 'images': [], + 'caption_count': 0, 'caption': []} + if self.figure is None: + return + if tag == 'img': + self.figure['images'].append( + {'src': attrs.get('src', ''), 'alt': attrs.get('alt', '')}) + if tag == 'figcaption': + self.in_caption = True + self.figure['caption_count'] += 1 + elif self.in_caption: + self.figure['caption'].append( + ('start', tag, sorted(attrs.items()))) + + def handle_startendtag(self, tag, attrs): + self.handle_starttag(tag, attrs) + if tag not in {'img', 'br', 'hr', 'input', 'meta', 'link'}: + self.handle_endtag(tag) + + def handle_endtag(self, tag): + if self.figure is None: + return + if tag == 'figcaption': + self.in_caption = False + elif tag == 'figure': + self.figures.append(self.figure) + self.figure = None + elif self.in_caption: + self.figure['caption'].append(('end', tag)) + + def handle_data(self, data): + if self.in_caption: + self.figure['caption'].extend(('text', word) for word in data.split()) + + +def math_counts(blocks): + return Counter((node['c'][0]['t'], re.sub(r'\s+', ' ', node['c'][1]).strip()) + for node in nodes(blocks, skip_figures=True) + if node.get('t') == 'Math') + + +def counter_entries(counter): + return [{'value': value, 'count': count} for value, count in sorted(counter.items())] + + +def file_record(path): + data = path.read_bytes() + return {'path': str(path), 'sha256': hashlib.sha256(data).hexdigest(), + 'bytes': len(data)} + + +def check(args, record): + pandoc = shutil.which('pandoc') + if pandoc is None: + raise RuntimeError('Pandoc is required but was not found on PATH.') + record['pandoc_version'] = subprocess.check_output( + [pandoc, '--version'], text=True).splitlines()[0] + + def convert(text, source_format, target_format, filtered=False): + command = [pandoc, '--from=' + source_format, '--to=' + target_format] + if filtered: + command.append('--lua-filter=' + str(args.lua_filter)) + return subprocess.run(command, input=text, text=True, capture_output=True, + check=True, cwd=args.tex.parent).stdout + + inputs = [args.tex, args.markdown, args.lua_filter] + record['inputs'] = [file_record(path) for path in inputs] + tex_text, markdown_text = args.tex.read_text(), args.markdown.read_text() + raw = json.loads(convert(tex_text, 'latex', 'json')) + source = json.loads(convert(tex_text, 'latex', 'json', filtered=True)) + markdown = json.loads(convert(markdown_text, MARKDOWN_FORMAT, 'json')) + errors = record['errors'] + + # The generated TOC is not document content. Verify its membership before + # excluding it from the independent body-math comparison. + expected_toc = [node['c'][1][0] for node in nodes(source['blocks']) + if node.get('t') == 'Header' and node['c'][0] <= args.toc_depth] + toc_blocks = [] + if markdown['blocks'] and markdown['blocks'][0].get('t') == 'BulletList': + toc_blocks = [markdown['blocks'].pop(0)] + toc_targets = [node['c'][2][0][1:] for node in nodes(toc_blocks) + if node.get('t') == 'Link' and node['c'][2][0].startswith('#')] + if toc_targets != expected_toc: + errors.append('Generated TOC entries differ from the source headings.') + + tables = lambda ast: [node for node in nodes(ast['blocks']) + if node.get('t') == 'Table'] + raw_tables, source_tables, markdown_tables = map(tables, (raw, source, markdown)) + record['table_counts'] = {'source': len(raw_tables), 'filtered': len(source_tables), + 'markdown': len(markdown_tables)} + if len({len(raw_tables), len(source_tables), len(markdown_tables)}) != 1: + errors.append('Table counts differ between source, filtered source and Markdown.') + record['tables'] = [] + for index, (original, filtered, actual) in enumerate( + zip(raw_tables, source_tables, markdown_tables), start=1): + expected, removed = expected_filtered_table(original) + # GFM requires a header separator even when LaTeX has no header. Its + # resulting empty row carries no data and is the sole allowed addition. + expected_values = table_values(filtered) + actual_values = table_values(actual, omit_empty_head=not expected_values['head']) + unchanged = expected == filtered + equivalent = expected_values == actual_values + record['tables'].append({ + 'index': index, 'removed_duplicate_header': removed, + 'filter_changes_exactly_expected': unchanged, + 'all_cell_values_and_math_preserved': equivalent, + 'body_rows': len(expected_values['body']), + 'numeric_tokens': sum(any(char.isdigit() for char in token[-1]) + for token in semantic_tokens(filtered)), + }) + if not unchanged: + errors.append(f'Table {index}: unexpected Lua-filter mutation.') + if not equivalent: + errors.append(f'Table {index}: cell values, boundaries, math or caption changed.') + + source_math, markdown_math = map(math_counts, + (source['blocks'], markdown['blocks'])) + record['body_math'] = { + 'source_count': sum(source_math.values()), + 'markdown_count': sum(markdown_math.values()), + 'source_only': counter_entries(source_math - markdown_math), + 'markdown_only': counter_entries(markdown_math - source_math), + } + if source_math != markdown_math: + errors.append('Body mathematics differs; figure-caption math is checked separately.') + + paths = re.findall(r'\\(?:nolinkurl|path)\{([^}]*)\}', tex_text) + code_counts = lambda ast: Counter(node['c'][1] for node in nodes(ast['blocks']) + if node.get('t') == 'Code') + source_code, markdown_code = map(code_counts, (source, markdown)) + path_differences = [{'path': path, 'source_count': source_code[path], + 'markdown_count': markdown_code[path]} + for path in sorted(set(paths)) + if path not in markdown_text or source_code[path] != markdown_code[path]] + record['paths'] = {'source_occurrences': len(paths), 'distinct': len(set(paths)), + 'differences': path_differences} + if path_differences: + errors.append('Repository path text or occurrence counts changed.') + + inventories = [] + for ast in (source, markdown): + inventory = HTMLInventory() + inventory.feed(convert(json.dumps(ast), 'json', 'html5')) + inventory.close() + inventories.append(inventory) + expected_html, actual_html = inventories + record['figures'] = { + 'source_count': len(expected_html.figures), + 'markdown_count': len(actual_html.figures), + 'images_ids_and_captions_preserved': expected_html.figures == actual_html.figures, + 'source': expected_html.figures, + 'markdown': actual_html.figures, + } + if expected_html.figures != actual_html.figures: + errors.append('Figure identifiers, image paths or rendered captions differ.') + + missing_targets = [target for target in toc_targets if target not in actual_html.ids] + record['toc'] = {'expected_entries': len(expected_toc), 'entries': len(toc_targets), + 'matches_source_headings': toc_targets == expected_toc, + 'missing_targets': missing_targets} + if missing_targets: + errors.append('Some generated TOC links have no Markdown destination.') + + record['inputs_unchanged'] = record['inputs'] == [file_record(path) for path in inputs] + if not record['inputs_unchanged']: + errors.append('An input changed during the audit; rerun against a stable report.') + + +def main(): + reports = Path(__file__).resolve().parent + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--tex', type=Path, default=reports / (STEM + '.tex')) + parser.add_argument('--markdown', type=Path, default=reports / (STEM + '.md')) + parser.add_argument('--lua-filter', type=Path, default=reports / 'pandoc_tables.lua') + parser.add_argument('--toc-depth', type=int, default=2) + parser.add_argument('--output', type=Path, required=True, + help='Explicit destination for the JSON audit record.') + args = parser.parse_args() + for name in ['tex', 'markdown', 'lua_filter', 'output']: + setattr(args, name, getattr(args, name).resolve()) + if args.output.suffix.lower() != '.json': + parser.error('--output must name a JSON file; source documents are read-only.') + if args.output in {args.tex, args.markdown, args.lua_filter, Path(__file__).resolve()}: + parser.error('--output cannot overwrite an input or the checker.') + record = {'checked_utc': datetime.datetime.now(datetime.timezone.utc).isoformat(), + 'checker': file_record(Path(__file__).resolve()), 'errors': []} + try: + check(args, record) + except (OSError, RuntimeError, ValueError, KeyError, IndexError, + subprocess.CalledProcessError) as error: + record['errors'].append(str(error)) + if isinstance(error, subprocess.CalledProcessError): + record['pandoc_error'] = error.stderr + record['passed'] = not record['errors'] + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(record, indent=2) + '\n') + status = 'PASS' if record['passed'] else 'FAIL' + print(f'{status}: report conversion audit saved to {args.output}') + for error in record['errors']: + print(error, file=sys.stderr) + return 0 if record['passed'] else 1 + + +if __name__ == 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