17 KiB
One-hour reduced smooth surface study — 2026-09-18
Outcome: independent smooth surface parameters remove the immediate full-step obstruction and reduce cold linear solve times by 2–5 times in this screen. They do not remove the original shape-force imbalance. A three-step seed chain reaches P(h) virial errors below 7e-7 while retaining substantial excluded force and imperfect interior fields.
Authorized wall-clock window: 19:30:15–20:30:15 UTC. Branch:
research/reduced-smooth-surface-20260918, parent 6ceda61.
Keep Q3 displacement/surface, Q4 reference geometry, Q2 DG density/potential, Q3 enthalpy and RT2 gravity. No central-density constraint. n=1.5,J=0. No production API, operator or extension change. Single MPI rank.
Frozen affine embedding q=q0+B a. The base q0 is never projected. Construct real spherical harmonics through degree L at actual surface radial nodes; orthonormalize the Q3 interpolants using q14 prepared physical surface area. Thin QR of S B and R^-1 B supplies separate normalized state and dual test maps. Solve Qf^T Jnormalized Qx p=-Qf^T Fnormalized. The projected full inverse Qx^T Mnormalized^-1 Qf is an approximation, not an exact reduced inverse. All other unknowns/equations remain active. Cold FGMRES, eta=.03, max200, restart160, same gravity-Schur preconditioner (one sweep, relaxation.5).
Priority: fresh full-space direction at the prior Q3 six-update endpoint, then L4 there, then L2 and L8 (L6 optional); analytic-seed L4 only if time permits. Stop launching FEM cases by20:20; reserve remaining time for independent artifact audit and reporting. No convergence claim from frozen steps. Cases, builds, and full numerical audits run serially. A library-only Q3 warm rebuild may overlap editing experiment-only headers; no production source changes.
Raw independent trial alphas 1,1/4,1/16,1/1024. Preserve original and reduced residuals, separate linear gates, full/reduced slopes, excluded shape forces, and exact restoration checks. Observational merit criterion is norm(Ftrial)<=(1-1e-4 alpha)*norm(Fbase), evaluated separately in both spaces. No trial accepted, no SOC, no positivity repair. q14/q18 physical measurements at base and the first/largest prepared trial only; all trial residuals saved. Direction physical observer q10/q14 reused from the preceding audited study.
Acceptance of an observation requires algebra/transpose/full-rank control tests, true reduced gate classification, finite vectors, source/input/output hashes, and exact saved-state restoration. Reduced success is not full success. Virial1e-5, interior physical errors, density sign, and omitted force remain separate measurements. No selected norm reduction can certify a physical root.
Implementation
Experiment-only headers:
reduced_surface_basis.hpp: scalar radial-amplitude FE trace basis and quadrature-defined physical surface mass metric, explicit rank/orthogonality checks, and raw data for independent verification.reduced_surface_operators.hpp: two separate QR maps, reduced operator, projected flexible preconditioner and deterministic algebra checks.reduced_surface_diagnostics.hpp: frozen solve, both original/reduced gates, excluded-force decomposition, raw trials and restoration invariants.audit_reduced_surface.py: independent artifact algebra and corruption tests.
For physical state factors S and residual factors R, the shape maps obey
S B = Qx Tx, R^-1 B = Qf Tf,
Qf^T (R Fphysical) = Tf^-T B^T Fphysical.
The separate state/test QR maps therefore preserve physical dual pairing despite nonuniform normalization and distinct shape row/value offsets. The area whitening measures scalar radial amplitude, not the norm of the generated three-dimensional displacement. The latter is measured separately over the stellar volume. Changing the area whitening changes coordinates without changing the raw harmonic span.
The diagnostic executable adds experiment-only flags --reduced-surface-probe and --surface-degree (0 means full-space control); no production-facing API is modified. Degrees are runtime experimental choices while operator/inverse compatibility and the inherited preconditioner application contract remain compile-time constrained. Actual mesh dimensions and normalization factors are necessarily checked at runtime. Existing physical observers are reused.
Evidence root: ../../diagnostics/reduced_surface_2026-09-18 relative to this
checkout. The preceding study remains on its two committed research branches.
Results: late-state comparison
At the common six-update n=1.5, J=0 Q3 endpoint, the normalized original residual is 1.4971971e-5. The unrestricted control reproduces the previous fresh direction. Its unit trial is rejected at pressure preparation (thermodynamic domain); alpha=1/4 and 1/16 prepare but increase the original norm. Only alpha=1/1024 passes among the sampled values. Intermediate steps were not tested.
| Shape space | Independent parameters | Krylov iterations | Cold solve seconds | Original relative linear defect | Reduced relative linear defect | Original residual after unit trial |
|---|---|---|---|---|---|---|
| Full Q3 trace | 866 | 61 | 32.184 | 0.02767 | 0.02767 | Pressure preparation rejected |
| L2 harmonics | 9 | 12 | 6.447 | 0.15121 | 0.01719 | 2.263903e-6 |
| L4 harmonics | 25 | 21 | 11.249 | 0.16187 | 0.01608 | 2.432960e-6 |
| L8 harmonics | 81 | 29 | 15.745 | 0.19207 | 0.01409 | 2.964119e-6 |
All reduced directions pass their independent 3% linear gate, and all four trial alphas prepare and decrease both norms. None of the reduced directions passes the original full Newton gate. Original unit-step residual decreases are 84.88%, 83.75%, and 80.20%. Thus the benefit is measurable in the original residual, but it comes from solving a restricted set of shape equations. These directions do not meet the requested 3% accuracy for the original Newton system; they could be interpreted as much looser inexact full-system steps.
These are one-shot cold linear solve timings, not end-to-end Newton timings or repeated performance statistics. The remaining non-shape unknowns and equations stay active: full dimension 178073, L4 dimension 177232. The full operator and full inverse are still applied inside the reduced wrappers. The dimension change is less than 0.5%; fewer Krylov iterations provide most of the measured speedup.
Curvature and physical motion
| Shape space | Volume displacement direction L2 norm (q14) | Unit-trial original Taylor remainder |
|---|---|---|
| Full | 3.96028e-3 | Inadmissible; alpha=1/4 remainder1.85956e-4 |
| L2 | 4.86622e-7 | 1.84872e-10 |
| L4 | 9.14977e-5 | 2.21090e-7 |
| L8 | 1.46119e-4 | 7.19540e-7 |
L4 cuts physical displacement motion by 43.28 times relative to the full direction. L2 is nearly a field-only relaxation; admitting higher harmonics introduces substantially more compensated shape/field motion and nonlinear curvature. The degree dependence is not monotone improvement of a physical solution. This supports the approximation-space compatibility hypothesis without proving that every omitted mode is unphysical or identifying a continuum gauge.
After the L4 unit trial, excluded shape forces account for 98.38% of the remaining original squared residual. A single trial does not establish a new residual floor. Because q0 is retained, this late-state test also retains its existing roughly 1.1% nodal radius spread; restricting subsequent increments cannot remove the general fine-scale part of that inherited geometry.
Physical measurements
The table below compares the common base with unit trials, all using independent q18 evaluation. Pressure is evaluated through enthalpy, P(h). These energy- and force-based virial diagnostics must not be confused with the density-pressure diagnostic, which remains undefined where the interpolated density is negative.
| Measurement | Base | L2 unit | L4 unit | L8 unit |
|---|---|---|---|---|
| P(h) energy virial error | 4.22590e-6 | 8.66263e-7 | 8.11913e-7 | 8.15910e-7 |
| P(h) force virial error | 2.71367e-6 | 5.94930e-7 | 6.41398e-7 | 6.24187e-7 |
| Mass error relative to 1 | 3.41685e-6 | 8.81237e-10 | 2.39038e-8 | 5.67517e-8 |
| EOS density mismatch RMS | 1.82986e-4 | 1.82925e-4 | 1.82990e-4 | 1.83010e-4 |
| Hydrostatic gradient RMS | 1.06811e-2 | 1.06631e-2 | 1.07065e-2 | 1.06864e-2 |
| Minimum sampled density | -7.73185e-4 | -7.24527e-4 | -7.27628e-4 | -7.33640e-4 |
| Negative-density mass fraction | 2.89683e-5 | 2.89242e-5 | 2.89458e-5 | 2.89452e-5 |
Both P(h) virial balances are comfortably below 1e-5. Their q14-to-q18 differences are about 1.83e-9. Local EOS and hydrostatic errors remain essentially unchanged; the negative density has not been repaired. Negative-density mass estimates are quadrature-sensitive (about 2.23e-5 at q14 versus2.89e-5 at q18), so their magnitude is diagnostic rather than a precision estimate. All quoted gradient errors are the existing interior observers in code units, including the broken DG-potential gradient comparison where applicable.
The full-control physical.csv sample is alpha=1/4, its first prepared trial, which fails the norm screen. It is not a selected full-space update and must not be compared as an equivalent accepted unit step.
Seed continuation
The planned analytic-seed L4 probe was followed by two externally replayed physical unit candidates, an adaptive extension recorded before launching the chain. Each predecessor passed preparation, the reduced independent linear gate, and both conservative norm-decrease screens. All three observed unit trials satisfy the same conditions. This is a cold-solve experimental reduced-Newton chain, not the production accepted/warm trajectory. The first physical shape is retained, and each subsequent base is its preceding saved candidate.
| Unit steps from seed | Original norm | Retained norm | Excluded shape norm | P(h) energy virial q18 | Hydrostatic gradient RMS q18 | Nodal radius spread / mean |
|---|---|---|---|---|---|---|
| 0 | 2.736592e-4 | 2.590732e-4 | 8.815010e-5 | 2.410694e-5 | 1.528098e-3 | 0.003424% |
| 1 | 8.624827e-5 | 9.387845e-6 | 8.573583e-5 | 2.180785e-6 | 5.724618e-3 | 0.081116% |
| 2 | 8.473526e-5 | 3.147803e-6 | 8.467677e-5 | 8.660498e-7 | 7.177698e-3 | 0.152605% |
| 3 | 8.465191e-5 | 7.357283e-8 | 8.465188e-5 | 6.769653e-7 | 7.255981e-3 | 0.154699% |
The retained norm falls by about 3521 times. The original norm falls by only 3.23 times overall; after the first step, improvements are 1.75% and 0.0984%. At the final trial, excluded shape force accounts for 99.999924% of the original squared norm. These three updates show a pronounced separation between progress in the retained equations and the remaining force; they do not establish an asymptotic residual floor or convergence of the retained problem to arbitrary tolerance.
Linear solves take 18/20/18 iterations and 10.574/11.341/9.594 seconds. Reduced relative defects are 2.940%, 2.122%, 2.323%; original relative defects are 31.45%, 98.18%, 99.90%. All full-system 3% gates fail.
The final P(h) force virial is 5.769064e-7, mass error 3.79142e-9, sampled minimum density -4.07472e-4, and negative-density mass fraction 2.89249e-5. EOS density mismatch RMS scaled improves from 1.72399e-4 at seed to 1.25532e-4, but the hydrostatic-gradient error grows 4.75 times. Radius spread includes the reference geometry's initial interpolation error; this nodal statistic is not a full surface norm. The restricted chain produces much less shape distortion than the older unrestricted six-update state, but does not eliminate nonrotating drift or establish pointwise correctness.
This is the decisive limitation of a permanently truncated surface space: good virial balance and a small retained norm coexist with substantial original shape force and imperfect interior fields. The late-state trial benefits from fine-scale balance already developed by the unrestricted trajectory. Smooth parameters relieve the immediate curvature/step obstruction; they do not by themselves establish a compatible approximation of all original equations.
Interpretation and next design
The evidence supports separating independent surface parameters from the interpolation used to carry volume displacement. An internal typed discretization prescription can select a nested smooth surface space while preserving the physics-facing API. State injection, physical dual restriction, normalization, and excluded-force monitoring should be distinct interfaces. A reduced inverse can initially wrap the existing full inverse; later optimizations should preserve the same contracts.
A fixed small harmonic space is not yet justified for production. It changes which shape forces are enforced, and smaller reduced residuals do not certify the full equations. The next controlled study should test whether the excluded mechanical imbalance decreases under mesh or field enrichment at a spherical state, while measuring interior EOS/hydrostatic error. That can distinguish approximation error from an inconsistent force/operator coupling before a low-degree space is promoted to production. Degree and mesh sensitivity, the pressure/EOS representation, and compatibility of the volume extension with global radial dilation remain separate questions. This study makes no rotation or n>1.5 convergence claim and changes no production field order or volume lift.
Verification and reproducibility
Implementation commit 21cdbda224, on research/reduced-smooth-surface-20260918, parent 6ceda61. The diagnostic includes 19 C++ algebra checks (maximum error 2.22e-16) and 22 independent Python algebra/corruption tests. Actual-case audits independently reconstruct the raw nodal harmonics, quadrature-defined scalar trace Gram matrix, both QR maps, physical dual identity, full/reduced defects and norm decomposition, all trial remainder/norm classifications, and exact saved-state restoration. Individual artifact audits do not independently reimplement the FEM physics observers.
The observation flags use norm(Ftrial)<=(1-1e-4 alpha)norm(Fbase). This is a deliberately conservative norm-decrease screen, stricter than the production baseline squared-merit criterion, not an exact replay of production AcceptStep. Physical measurements cover the base and the first prepared trial only. No probe commits a production accepted state, and no SOC or positivity repair is enabled.
All builds, FEM cases and large artifact audits were serial. Raw commands, mesh/input/output and executable hashes, compiler settings, runtime and CPU details are in each campaign provenance and run JSON. The source checkout stays on the isolated branch; the original checkout and its pre-existing edits are untouched. Full regression tests were not rerun because production sources are unchanged; verification is scoped to the new diagnostic contracts, actual cases and snapshot chain.
All seven case audits pass: 2,098 saved-artifact checks in total. The snapshot-chain audit passes 107 checks with all three individual reviews bound. Both physical-state and residual handoffs are bitwise identical; normalized-state round trips differ by at most 1.39e-17 per entry. Layout, state/residual factors and raw harmonic basis hashes match exactly. Shape-base increments remain in the original raw span to relative errors below 9.4e-16. Twelve small chain-auditor tests include corruption and JSON-serialization regressions.
The chain auditor initially failed while serializing a NumPy boolean; it also emitted warnings during its small BLAS matrix product. The checker was corrected to emit native JSON scalar types and use an explicitly checked elementwise contraction with finite-value guards. The corrected audit passes without warnings. No FEM state, recorded observation, producer binary or numerical source changed in response to this checker issue.
Archived campaigns
Evidence directory (relative to the source checkout): ../../diagnostics/reduced_surface_2026-09-18/
| Campaign | Cases | Case audit |
|---|---|---|
| core_v1 | late_full_control, late_L4 | core_audit_v1.json |
| orders_v1 | late_L2, late_L8 | order_audit_v1.json |
| seed_v1 | seed_L4 | seed_audit_v1.json |
| seed_chain_v1 | seed_L4_step2 | seed_step2_audit_v1.json |
| seed_chain_v2 | seed_L4_step3 | seed_step3_audit_v1.json |
seed_chain_audit_v1.json binds the three seed stages. TEST_PLAN.md records the original scope and the conditional extension. late_summary.json and seed_summary.json collect the displayed values. plot_results.py regenerates the figure from the immutable case CSVs. COMPLETED_STUDY.json records final source and evidence hashes.
To reproduce, build plateau_diagnostic from producer commit 21cdbda using the compiler/MFEM configuration in core_v1/provenance.json. Replay each recorded run command, substituting a fresh output directory. All cases use the same saved sandbox.smesh and one MPI rank. Late cases bind the prior Q3 accepted_state.txt by hash. Replay the seed stages in order using physical_candidate_a0 from the previous stage; the case manifests and input-state hashes specify these links. The runner requires a clean tracked source checkout and records the executable and all case output hashes.
No order changes were made during this study. Optional L6, rotational models, higher indices, mesh/enrichment sweeps and production integration were outside this bounded experiment. All seven FEM probes finished by 20:09:58 UTC, before the 20:20 launch cutoff and 20:30:15 study deadline.