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.

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 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.

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.