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](https://arxiv.org/html/2305.10218#S3.SS1).
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. A small discrete virial in such a case would require
careful interpretation. No mass/K change has been made in this campaign.