MeanField/experiments/COUPLED_CONVERGENCE_STUDY_2026-09-19.md

Coupled harmonic-surface convergence study

The user has deferred positivity enforcement. This study keeps density an independent Q2 field and retains the simultaneous all-m degree-four harmonic surface/interior system. The primary milestones are useful convergence at higher index and angular momentum, and both enthalpy-pressure virials below 1e-5. Negative-density measurements remain diagnostic, not acceptance repairs.

Base revision: ef76e0c. Work is isolated on research/coupled-convergence-study-20260919, with separate research/linear-coupling-study-20260919 and research/model-slices-20260919 branches for independent implementations. The original programming checkout is unchanged.

First screen: n=2,J=0; n=3,J=0 at forcing tolerances 0.03 and 0.003; n=1,J=0.1; n=1.5,J=0.1. All use the same Q2 density, Q3 enthalpy and displacement, RT2/Q2 gravity, Q4 geometry, reference mesh, normalization, default inverse, 12-update budget, restart 160, and 300-iteration linear budget. Only n=3 includes the existing central phase condition. This tests whether a tenfold more accurate direction changes the n=3 trajectory, but it is not a same-state preconditioner comparison. Capture updates 0,3,7 for subsequent controlled probes. No input J is identified with a calibrated fraction of critical rotation.

Follow-up decisions depend on this screen and source review: frozen-state preconditioner/forcing comparisons; small coupled surface/border corrections; accepted-state continuation with checked field/coordinate transfer. Linear claims require independently applied full-J residuals; nonlinear claims use common stopping thresholds plus physical diagnostics, not residual alone. Report scalar-border forcing separately from the unbordered equations.

The slice feature will evaluate physical geometry and independent fields, show the material surface at true scale, retain signed density, and separate outside-domain points from valid interior samples. It must not silently turn a reference-coordinate cut into a physical planar cut.

Builds and expensive numerical cases run serially. The existing campaign runner records exact source, binary, mesh, arguments, saved states and output hashes. Initial evidence directory: ../../diagnostics/coupled_convergence_2026-09-19/screen_v1.

Results and any adaptive extensions will be recorded below before finalizing the study. Defaults and the physics-facing API are unchanged.

Completed nonlinear controls

The retained-artifact audit passes all five screen cases: provenance, complete output hashes, block accounting, reported full linear gates and independently recomputed virial accounting. Every case exhausts twelve updates; none meets its nonlinear stopping threshold. The audit checks the production linear reports, not a fresh FEM application of every live Newton Jacobian.

Case	Final normalized residual	Newton seconds	Total Krylov iterations	Energy / force h-pressure virials, q18
n=2, J=0, eta=.03	8.85263e-6	742.74	1009	3.25447e-7 / 2.24560e-6
n=3, J=0, eta=.03	8.24392e-4	499.89	429	9.46671e-5 / 6.20929e-5
n=3, J=0, eta=.003	5.50731e-4	734.09	821	2.23846e-4 / 1.98912e-4
n=1, J=.1, eta=.03	5.83293e-3	310.43	226	3.23282e-4 / 3.36216e-4
n=1.5, J=.1, eta=.03	7.22567e-5	550.70	717	5.10711e-5 / 5.20697e-5

The nonrotating initial norms are 3.81795e-4 and 3.09931e-3 respectively; actual stopping thresholds are 3.81795e-12 and 3.09931e-11. The rotating initial norms are .0857344 and .139488, with thresholds 8.57344e-10 and 1.39488e-9. q14/q18 differences in all six nonrotating h-pressure virial errors are below 8e-14 absolute. This quadrature agreement does not establish a physical equilibrium or pointwise accuracy.

After n=2's second update, ten more updates reduce its residual only 2.112%. Its last alpha is 1/256. Every rejection is insufficient residual decrease; the final squared norm is 94.770% Poisson and 4.076% gravity-gradient error. This is a useful globally balanced intermediate model, but a stalled nonlinear solve and not an established discrete residual floor.

For n=3, the tighter tolerance lowers the endpoint residual by 33.20%, but requires 46.85% more Newton time and 91.38% more Krylov iterations. It follows a different nonlinear trajectory, so these are equal-update-budget outcomes, not a matched-state preconditioner comparison. The control's final squared norm is 79.927% Poisson, 13.854% gravity gradient, and 6.144% closure. The tighter case instead has 71.329% closure and 25.876% Poisson. Its smaller residual coexists with worse virials and a smaller final alpha (1/32 versus 1/4). Linear accuracy alone has not resolved the nonlinear obstruction.

Both n=3 runs continue below their earlier eight-update endpoints; those endpoints were not demonstrated floors. Their final unbordered norms exceed bordered norms by only .180% and .229%, although the artificial central force still needs to be reported and may matter locally.

When the full Jacobian is the derivative of the prepared residual, a direction with ||J p + F|| <= eta ||F|| obeys

F dot J p <= -(1-eta) ||F||^2.

Thus both eta=.03 and .003 give strict infinitesimal descent. Small accepted steps despite a passing linear gate motivate measurements of finite-step curvature, preparation validity and tangent consistency; a faster inner solve alone need not remove that limitation.

The rotating n=1 case ends at alpha=1/65536, with 26 pressure-preparation rejections and no metric rejections. The n=1.5 case has seven of each and ends at alpha=1/64. Its first eight updates reach 7.60346e-5; four further updates give only a 4.97% reduction. These are distinct obstructions, not evidence that every difficult model is failing its linear solve.

Interior checks and residual location

The n=2 endpoint has positive sampled density (minimum 2.93907e-4), and its density-pressure energy/force virials also pass: 2.49026e-6/5.70105e-7. It supplies a stalled control independent of a sampled negative-density obstruction. This is not a pointwise positivity certificate.

Case	EOS density RMS / sampled density scale	Bernoulli RMS / sampled h scale	Hydrostatic-gradient RMS	Minimum sampled map determinant
n=2, J=0	1.87717e-4	2.80766e-4	.0264341	.982421
n=3, eta=.03	3.52140e-4	2.30017e-3	.398054	.999838
n=3, eta=.003	3.69501e-4	2.65289e-3	.479759	.943956
n=1, J=.1	1.26849e-3	2.21774e-3	.104394	.938419
n=1.5, J=.1	1.69365e-4	6.32866e-4	.0366157	.955286

These q18 volume observations are sampled error measures, not mesh-convergence results. Gradient RMS is in the diagnostic's G=M_reference=R_reference=1 units; it is not a relative percentage. Good virials do not remove these interior defects. Density-pressure virials are undefined for the two rotating screens because sampled density is negative; their h-pressure virials remain defined and are reported without modifying either independent field.

A separate source-derived row map locates the saved normalized Poisson residual: 84.437% of its squared norm is stellar at n=2 capture seven, 99.9768% for n=3 control, 99.9995% for n=3 tight, and over 99% for both rotating captures. The n=2 envelope alone contributes 70.602%. The stalled norm cannot be explained solely by exterior rows. These are algebraic residual-row shares, not pointwise div(g)-4pirho errors. The mapping and dependency/provenance caveats are recorded in the evidence directory's poisson_regions/README.md.

Saved nonrotating surface directions also do not support translation as their dominant component: at capture seven, l=1 contributes only .0086253%, .0000077557% and .000103632% of squared normalized surface-direction norm for n=2, n=3 control and n=3 tight. Degrees zero and four dominate. Physical scale recovery and the small COM drift are documented in translation_check/README.md; this does not prove that the Jacobian has no weak translational response.

Adaptive follow-up, specified before execution

The completed n=2 control has energy/force enthalpy virials 3.25e-7/2.25e-6 but a residual of 8.85e-6 after twelve updates. At captured update seven, 94.42% of squared residual is Poisson balance and 4.45% gravity-gradient relation. All observed rejections there are insufficient decrease; pressure preparation has not failed. This motivates a gravity-path comparison in the current harmonic space, independently of earlier nodal-surface conclusions.

Restart both raw and gravity-curvature Newton from the same update-three physical capture, with six updates, eta=.03, and a shared absolute residual target 1e-12. Initial guesses are installed through an explicit transactional experiment hook, separate from the read-only diagnostic callback. Coordinate transfer checks names, sizes, discretization metadata, mesh and saved hashes; captured/unconverged iterates require an explicit experimental override. Gravity-curvature correction retains the full coupled equations and Newton direction and changes only the prepared trial path. Gravity elimination is deferred until per-iteration reporting distinguishes its reduced and lifted full linear defects and its completed initial state.

Frozen tests at that same n=2 capture compare gravity refinement, a new surface-plus-scalar approximate Schur inverse, and their combination at eta=.003. Repeat the combined candidate at eta=1e-7 to measure attained accuracy within a 300-iteration budget, not assert an arithmetic floor. Test the combined candidate at n=3 update seven separately. Each benchmark also runs a fresh default inverse on the identical RHS and physical Jacobian. Setup costs and internal Jacobian applications belong to the comparison.

The rotating n=1,J=.1 control initially takes a full update, then its accepted alpha falls below .001 by update six, with pressure-preparation rejections. Add a combined inverse benchmark at its update-seven capture and eta=.003, including the same raw trial grid. This separates the rotation obstruction from n=2's gravity-dominated merit rejection without changing admissibility. The n=1.5,J=.1 control reaches 7.60e-5 after eight updates, then needs alpha 1/512 with both pressure and metric rejections. Also compare the combined inverse at its final saved state, once its original twelve-update budget is complete. This is the combined higher-index/rotation obstruction; it is not assumed to be a failed linear solve, since its update-eight linear gate passes.

Source review also identifies the existing stellar-ldu-gravity-schur policy as a distinct control: it introduces material/gravity feedback that the independent subsystem policy omits. Add frozen n=2 and n=3 comparisons at the same captures and eta=.003, with n=3 raw trial observations. The existing material inverse already includes the exact prepared h-to-density closure derivative, so a large nonlinear closure residual must not be described as proof of a missing EOS cross derivative. Frozen block defects and finite-step closure remainders distinguish those possibilities.

The live gravity-path comparison initially accepts an eight-times-larger step but returns to smaller steps by its third update. Before the frozen campaign, extend raw trial observations to the eta=1e-7 n=2 benchmark as well: if that linear accuracy is attained, its trial data can test whether a more accurate direction relieves curvature. A failed linear gate will remain labeled as such, regardless of whether a sampled nonlinear trial happens to decrease.

The small Schur block contains all 25 surface parameters and every generated scalar. Its field inverse is the unbordered structure inverse; restricting an already scalar-bordered inverse could spuriously constrain the field solve. Singular/ill-conditioned coarse blocks are reported, never silently repaired. The implementation is frozen-state only and must be reconstructed for a new Jacobian. No changed live default is implied by a successful frozen benchmark.

Saved linear directions receive optional raw trial observations on a fixed alpha grid. The squared-merit screen matches production's sufficient-decrease formula, but the sparse grid is not a replay of its adaptive line search. Remainder scalars are direct FEM observations; saved Jp, RHS and correction allow independent verification of the linear defect and merit slope.

Additional n=3 linear control, specified before execution

The frozen n=3 combined gravity/surface-scalar candidate passes eta=.003 in 41 iterations versus 99 for baseline, with 37.609 versus 51.077 Krylov seconds. Its construction raises total setup/solve/check cost to 65.629 versus 53.169 seconds. This favorable iteration count cannot yet be assigned to the surface correction: the same-state n=3 gravity-only ablation is missing. Before recommending coarse-setup optimization, add exactly that control with the same snapshot, RHS, target and 300-iteration cap, and a fresh baseline. No additional raw trial grid is needed for this attribution. The isolated branch is research/n3-linear-ablation-20260919, with case manifest convergence_linear_followup_cases_2026-09-19.json.

Completed matched gravity-path test

The retained-artifact audit passes both six-update restarts. Both begin at F=9.0134584624e-6 and every reported full linear gate passes. Raw Newton ends at 8.8762959137e-6 in 449.196 Newton seconds; gravity-path Newton ends at 8.6749533568e-6 in 887.543 seconds. The latter endpoint is only 2.2683% lower at 1.97585 times solver cost. Neither reaches the shared 1e-12 target.

The first gravity-path update accepts alpha=1/16 versus raw's 1/128, giving 90.1683% of its eventual six-step improvement immediately. Its final alpha is 1/2048, eight times smaller than raw's final 1/256. All rejections are metric failures, not pressure failures. Preparation time rises 3.72673-fold; 22 auxiliary gravity solves use 866 iterations. Their final prepared defect passes its own threshold, so the stalled full solve must not be described as a failed auxiliary gravity correction.

The final gravity-path squared norm remains 85.0361% Poisson, with larger hydrostatic (6.4051%) and mass (1.8661%) shares than raw. Its q18 h-pressure energy/force virials are 7.29743e-7/1.22487e-6; raw's are 3.27166e-7/2.25835e-6. Both meet the global milestone while local nonlinear progress remains poor. There is no demonstrated durable cure or speedup, and no default strategy change is warranted by this pair.

Full control tables, timing scopes and audit hashes are retained in the evidence directory's CONTROL_RESULTS.md and CONTROL_RESULTS.json.

Compatibility of the rotating n=3 target

This is a separate physical-model issue, not an explanation of the measured n=2 stall. The driver derives K from a nonrotating Lane–Emden model with M=R=1 and retains M=1 when J changes. At n=3 that mass/K combination is critical: the nonrotating mass M_LE(K) is independent of central density.

For nonnegative finite-energy density with the exact EOS P=K rho^(4/3), the sharp gravitational-energy bound gives

|W| <= 3 integral(P) (M/M_LE(K))^(2/3).

The bound holds for nonradial densities as well. Its relation to Lane–Emden mass is given in equations (1.14)–(1.15) and Theorem 3.1 of Cheng, Cheng and Lin, Expanding solutions near unstable Lane-Emden stars. The paper uses G=1; restoring G in W leaves the displayed mass-ratio form.

Our inference from this bound and the isolated steady virial identity is: at M=M_LE(K), W+3 integral(P)>=0, while nonzero angular momentum at finite moment of inertia requires T>=J^2/(2I)>0. Thus 2T+W+3 integral(P)=0 is impossible. A rotating n=3 family needs M>M_LE(K); at fixed M=1 this means K below the driver's nonrotating critical value. This necessary condition does not itself construct a rotating equilibrium or choose its rotation rate. Fixing an additional central density cannot repair incompatible physical specifications; a central artificial force could instead mask them.

The exact continuum statement must not be applied uncritically to the present discrete equations: independent rho and P(h), weak EOS consistency, negative density, gravity approximation and central forcing can bypass its assumptions. Numerical Lane–Emden integration also makes the driver's K only approximately critical. Independent Lane–Emden integration gives M_LE=0.9999999918534539 for the recorded K, allowing only T/|W| <= 2.7155e-9 under those continuum assumptions. Thus the actual rounded parameters permit negligible, rather than literally zero, rotational support. A small discrete virial in such a case would require careful interpretation.

The initial screen keeps its original mass/K parameters. A separate physical family control is specified in N3_COMPATIBLE_FAMILY_2026-09-19.md and isolated on research/n3-compatible-family-20260919: reduce K by one percent, leave M=1, omit the central border and prescribe J=.035. This tests a necessary physical compatibility condition, not a preconditioner speedup or a guarantee of equilibrium or a particular fraction of critical rotation.