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)-4*pi*rho 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](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. 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](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.