MeanField/experiments/COUPLED_CONVERGENCE_STUDY_2026-09-19.md

# Coupled harmonic-surface convergence study

The clearest measured improvement is the existing gravity-Schur inverse:
at matched saved states and a checked full linear tolerance of .003, total
construction/solve/check cost falls from 53.31 to 31.17 seconds for n=3,J=0,
and from 118.62 to 82.31 seconds for n=1.5,J=.1. These are frozen linear
improvements, not demonstrated complete-model speedups. The new surface/scalar
Schur correction is algebraically effective but is not the best total-cost
candidate in these tests and can worsen finite-step behavior.

The live rotating n=3 comparison also gives a substantive improvement:
existing stellar LDU with gravity refinement reaches a 12.687-times smaller
residual than gravity refinement alone after ten updates, at 14.593% more
Newton time. The energy h-pressure virial improves from 7.184e-5 to 1.811e-5,
but the full virial milestone is still unmet and small-step stagnation returns.
Density remains independent and no positivity enforcement is introduced.

Nonlinear progress is limited by more than Krylov accuracy. The n=2 control
stalls with positive sampled density and both virials below 1e-5. Its gravity
trial-path correction buys an initial improvement but no durable recovery.
At n=3, measured weak coupled direction differences are dominated by changes
in the surface monopole compensated by interior fields. Better treatment of
this coupled response is a stronger next hypothesis than positivity repair.

Physical xz/xy slices are implemented, exported from unchanged saved fields,
checked at two display resolutions and visually inspected. The rotating
n=1.5 model is smoothly oblate but remains unconverged and misses the virial
milestone. Images, linear histories, raw results and audits are linked below.

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 evaluates 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 the specifications written before each adaptive extension are
recorded below. 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`.

The completed n=1.5,J=.1 frozen pair subsequently gives the first clear
combined-candidate total-cost improvement: 246 to 95 iterations and 117.925
to 101.760 setup/solve/check seconds, with both eta=.003 gates passing and
the same largest sampled acceptable alpha=1/16. Add its matched gravity-only
control to the same follow-up manifest before attributing this 13.71% time
reduction to the new surface block. This two-case attribution hypothesis is
isolated on `research/gravity-coupling-ablation-20260919`; the original n=3
specification branch is retained. These controls add no new nonlinear runs
or trial grids.

## Live n=3 field-feedback follow-up, specified before execution

At the completed frozen n=3 control capture, the existing
`stellar-ldu-gravity-schur` policy reduces 99 to 41 Krylov iterations and
53.190 to 35.200 setup/solve/check seconds. Both eta=.003 gates pass.
Its largest sampled acceptable alpha remains 1/4, with F_trial/F_base
.674223 versus baseline's .702686; its full-step ratio is 2.77979 versus
7.42157. This is promising frozen evidence, not a live speedup or proof
that the policy transfers to rotation or removal of the central border.

Before executing the rotating compatible-family control, pair it with the
same ten-update case using this field-feedback inverse. Only the inverse
changes: both retain n=3, J=.035, K factor=.99, no central phase, all-m L4,
the gravity-curvature trial path, ordinary live eta=.03 and target1e-12.
The forcing tolerance differs from the frozen screen, so this is a practical
generalization test rather than a replay of its direction. The case is added
to `n3_compatible_family_cases_2026-09-19.json` on isolated branch
`research/n3-live-field-feedback-20260919`. No surface-order change is made.

## 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 exact continuum 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 recorded numerical
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.

## Completed frozen linear comparisons

All eleven retained-artifact audits pass, including independent arithmetic
checks of saved corrections, fresh full-J actions and endpoint defects.
This does not independently reassemble J or certify every recurrence norm.

Each arrow is independent baseline → candidate at the same frozen state and requested tolerance within that case. The true relative defect is ‖Jp+F‖/‖F‖, from the fresh endpoint check. The ordinary relative tolerance is 0.003; the explicitly marked tight case requests 1e−7. A speedup is reported only when both endpoints pass the requested true gate. Times are seconds, with rounding only for display.

| Case | Iterations B→C | True relative defect B→C | True gate B/C | Total seconds B→C | Candidate construction | Full J calls B→C | Equal-gate speedup |
|---|---:|---:|:---:|---:|---:|---:|---:|
| `n2_late_gravity` | 105→83 | 2.62169821e-03→2.53227246e-03 | pass/pass | 51.1251→50.2194 | 1.7656 | 106→84 | 1.018× |
| `n2_late_surface` | 105→300 | 2.62169821e-03→5.67029124e-03 | pass/fail | 51.6185→244.6878 | 21.2548 | 106→657 | not applicable |
| `n2_late_combined` | 105→109 | 2.62169821e-03→2.60837250e-03 | pass/pass | 51.4773→117.3489 | 24.4227 | 106→273 | 0.439× |
| `n2_late_tight_combined` | 300→300 | 1.87757546e-04→1.21976551e-04 | fail/fail | 145.5541→283.3130 | 24.7723 | 303→657 | not applicable |
| `n3_late_combined` | 99→41 | 2.82583089e-03→2.77409778e-03 | pass/pass | 53.1688→65.6294 | 27.7166 | 100→139 | 0.810× |
| `n1_rotating_late_combined` | 63→28 | 2.97251190e-03→2.02497346e-03 | pass/pass | 29.9344→44.3827 | 22.3112 | 64→111 | 0.674× |
| `n1p5_rotating_late_combined` | 246→95 | 2.99226031e-03→2.99968978e-03 | pass/pass | 117.9250→101.7603 | 23.2647 | 248→245 | 1.159× |
| `n2_late_ldu_gravity` | 105→78 | 2.62169821e-03→2.80557346e-03 | pass/pass | 51.4910→60.0450 | 2.0900 | 106→79 | 0.858× |
| `n3_late_ldu_gravity` | 99→41 | 2.82583089e-03→2.65862364e-03 | pass/pass | 53.1904→35.2005 | 2.6098 | 100→42 | 1.511× |
| `n3_late_gravity` | 99→46 | 2.82583089e-03→2.59308753e-03 | pass/pass | 53.3147→31.1652 | 2.1394 | 100→47 | 1.711× |
| `n1p5_rotating_late_gravity` | 246→137 | 2.99226031e-03→1.99325392e-03 | pass/pass | 118.6171→82.3113 | 1.7997 | 248→138 | 1.441× |

`gravity` means gravity-Schur on the independent base; `surface` means surface/border Schur on the independent base; `combined` means surface/border Schur on the gravity-Schur base; `ldu_gravity` means stellar LDU plus gravity Schur. The rotating cases are identified by their input case names; this note does not reconstruct their physical metadata from external artifacts.

**Cost scope.** Total = preconditioner construction + normalization binding + Krylov setup + Krylov solve + one fresh true-residual check. Construction includes coarse-Schur setup. Internal Schur Jacobian work is already included in inverse/Krylov time and must not be added again. Total excludes initial model/context setup, separate diagnostic inverse/Jacobian action checks, nonlinear trial sampling, and output observers. These are cold one-right-hand-side measurements, so they do not establish a benefit from future cache reuse or amortization. The independent baselines were freshly timed for each pair; variation across those timings is visible, but there are no statistical repeats or confidence intervals.

Full J calls count outer Krylov actions + coarse-Schur setup actions + full J actions inside Schur inverse applications + the fresh endpoint check. They exclude separate diagnostic checks and lower-level component operator work. Therefore they are a useful work counter, not a replacement for elapsed time. Restart bookkeeping can make outer J actions exceed the iteration count.


Gravity-only saves 41.54% for n=3 and 30.61% for rotating n=1.5. The n=2
saving is only 1.77%, too small for a robust timing claim from one pair.
LDU improves n=3 over default but is slower than gravity-only. The new coarse
wrapper pays 20-26 seconds of setup and an additional full J per inverse
application. Fewer outer iterations can therefore cost substantially more.

The tight n=2 endpoints improve within the budget, but neither passes 1e-7.
All n=2 directions with raw-trial samples have the same largest sampled
passing alpha=1/256; approximately
90.18% of the small-step squared Taylor remainder is Poisson. Its measured
quadratic scaling supports finite curvature over that range. More accurate
linear solutions have not enlarged the sampled usable step range. These
finite-budget endpoints are not demonstrated residual floors.

The rotating n=1 candidate leaves the pressure-preparation restriction
unchanged. The rotating n=1.5 candidate leaves its largest sampled acceptable
alpha at 1/16. Gravity-only n=3/n=1.5 followups omit raw trials and establish
linear cost benefits only. Full cost, gate, setup and trial tables are in
[FINAL_INTERPRETATION.md](../../../diagnostics/coupled_convergence_2026-09-19/linear_analysis/FINAL_INTERPRETATION.md).
The [linear histories](../../../diagnostics/coupled_convergence_2026-09-19/figures/linear_rates/frozen_linear_rates.pdf)
distinguish reported recurrence histories from checked endpoint defects.

## Measured weak coupled response

At the common n=3 frozen state, baseline, combined and LDU directions all
pass eta=.003 and have merit slopes within 8e-6 relative of -||F||^2. Nevertheless,
combined increases the normalized surface-direction norm from .0681146 to
.113440 and reduces the largest sampled passing alpha from 1/4 to 1/16.
LDU reduces that norm to .0287899, retains alpha=1/4 and improves its trial
residual ratio from .702686 to .674223. Full-step ratios are 7.42157,
14.7443 and 2.77979, respectively.

Saved-vector analysis measures complete direction differences of 58.44% and
61.89% of baseline norm but action differences of only .3197% and .3364%
of ||F||. Response per unit difference norm is 183-184 times smaller than
along the baseline direction, in the same normalized metric. This measures
two weak directions, not a condition number or an exact nullspace.

Roughly 43% of each squared normalized difference is surface, 34% potential
and 14% density. Of the surface difference alone, 97.67% and 96.77% are l=0.
The signed monopole is -.057083 for baseline, -.104359 for combined and
-.00681011 for LDU. This is a coupled radial-size/field response, not mainly
translation or high-degree roughness. Freezing the monopole is premature:
radius is physical and this control already has a central phase constraint.

Combined lowers its final surface/scalar defect/||F|| from 4.64933e-4 to
1.07519e-6 while worsening trials. LDU permits 9.45535e-4 there but has the
better quarter-step result. Per-application coarse recovery alone does not
select a good finite Krylov direction. The n=3 combined coarse matrix's
reported reciprocal condition is 1.31320e-9, not the conditioning of full J.
These coarse validation scalars are producer FEM observations, separate from
the independently audited endpoint vectors.

For e_i=Jp_i+F, the common gate bounds ||J(p_c-p_b)|| by ||e_c||+||e_b||;
it does not bound ||p_c-p_b|| without inverse information. This motivates
coupled step control and a better approximate field response. The measured
weak directions come from the centrally constrained nonrotating control;
their magnitude and modal structure must be rechecked in the unbordered
rotating family before treating them as its demonstrated mechanism. Full identities,
modes and provenance are in the
[direction comparison](../../../diagnostics/coupled_convergence_2026-09-19/n3_direction_difference/README.md).

## Physical slices and display refinement

The optional diagnostic exports physical xz/xy cuts, independent density and
enthalpy, and the actual deformed material boundary. Other planes and offsets
are available. Element-owned triangles preserve DG traces; exterior points
remain masked. Deformation is not exaggerated and signed values are retained.
[MODEL_SLICES.md](MODEL_SLICES.md) documents use and geometric limitations.

Four replay cases export final n=2,J=0 and n=1.5,J=.1 states at four/eight
subdivisions. All four retained-artifact audits pass, including source-state
provenance and unchanged source fields. Both planes were rendered at common
axes/color ranges on a 512-by-512 scalar raster and inspected visually.

| Model / plane | Area change | Signed 2D density integral change | Signed 2D h integral change | Boundary length change |
|---|---:|---:|---:|---:|
| n=2 / xz | .12333% | .07667% | .07036% | .03065% |
| n=2 / xy | .12333% | .07667% | .07036% | .03065% |
| n=1.5, J=.1 / xz | .12495% | .00319% | .08797% | .03165% |
| n=1.5, J=.1 / xy | .12395% | .01249% | .07838% | .03088% |

These relative absolute changes concern displayed straight triangles, including
any overlaps, not exact curved-FEM integrals or 3D mass/virials. Sampled chord
errors decrease about fourfold; axial surface extents agree. Broad shape and
central concentration are stable. Small seams from independently triangulated
curved faces remain a documented display limitation.

The rotating meridional cut has equatorial extent approximately +/-1.12084
and polar extents -.988807 and +.988480. It is smoothly oblate; the equatorial
cut is approximately circular. The north/south difference is retained.
Its refined meridional sampled density minimum is -8.59e-4, without clipping.
The n=2 outline is near spherical at radius 1.0049. Neither image establishes
equilibrium; only n=2 meets the recorded virial milestone. Exact comparisons
are in [SLICE_REFINEMENT_SUMMARY.json](../../../diagnostics/coupled_convergence_2026-09-19/SLICE_REFINEMENT_SUMMARY.json).


## Completed rotating n=3 field-feedback comparison

Both compatible-family runs complete ten accepted updates and pass the
retained-artifact audit. They share n=3, J=.035, K factor=.99, no central
constraint, all-m L4/Q2, the gravity-curvature trial path and eta=.03. The
initial completed residual is .101333855231 and the nonlinear target is
1e-12. Only the inverse differs. Neither converges to that target.

| Quantity | Gravity-Schur | Stellar LDU + gravity-Schur |
|---|---:|---:|
| Final normalized residual | 1.64730591e-4 | 1.29842469e-5 |
| Newton seconds | 918.881 | 1052.972 |
| Linear seconds / iterations | 348.441 / 534 | 335.399 / 410 |
| Trial preparation seconds | 477.771 | 621.957 |
| Final accepted alpha | 1/16 | 1/64 |
| Energy h-pressure virial | 7.18429e-5 | 1.81141e-5 |
| Force h-pressure virial | 5.23757e-5 | 2.90028e-6 |
| Energy density-pressure virial | 6.96335e-5 | 3.51558e-5 |
| Force density-pressure virial | 5.01663e-5 | 1.41415e-5 |
| Absolute mass error | 5.45536e-5 | 3.37148e-6 |
| EOS density RMS / sampled density scale | 3.99773e-4 | 4.00529e-4 |
| Bernoulli RMS / sampled h scale | .00408345 | .00372207 |
| Hydrostatic-gradient RMS (model units) | .656446 | .588702 |
| T/abs(W) | .00497575 | .00497421 |

LDU produces a 12.687-fold lower residual after the same update budget, at
14.593% more Newton time. Total linear time decreases 3.743%, while trial
preparation rises 30.179%. These are evolving trajectories, not a speedup
to a common converged model. Every reported full linear gate passes. All
rejected trials fail sufficient decrease; neither run has a pressure or
geometry rejection. q18 sampled density remains positive in both, without
any positivity repair. This is not a global positivity certificate.

The LDU trajectory makes its major gain by update seven, reaching
1.30261014e-5 with a full step. Three further updates reduce that only .3213%,
at alpha=1/64. Its final squared residual is 36.500% Poisson, 34.432%
hydrostatic, 20.234% surface and 6.742% mass. The gravity-only endpoint is
80.254% hydrostatic and 10.967% mass. Field feedback is a useful live
improvement in this configuration, but does not remove the small-step stall.

The q14/q18 energy h-pressure virials differ by less than 8e-14 in both
runs; this is a quadrature check, not a mesh-convergence result.

The LDU force h-pressure virial passes 1e-5, but **the model does not meet the
virial milestone overall**: energy h-pressure and both density-pressure
virials fail. Its signed h-force result is -2.90028e-6; signed h-energy is
+1.81141e-5. Their gap is about 2.10144e-5, consistent with the separately
reported gravity-energy discrepancy. Changing from h-pressure to
rho-pressure shifts the signed virials by about +1.70418e-5. Reporting only
the passing force/h diagnostic would conceal these consistency errors.

The gravity-energy discrepancy remains about 2e-5 in both trajectories,
and EOS density RMS barely changes despite the large algebraic improvement.
This motivates separate checks of gravity/pressure consistency and spatial
discretization after improving the coupled solve. It does not prove a mesh
error floor: neither state solves the discrete equations, and this pair does
not vary mesh or field order. The finite-element closure tests a projected
EOS relation; a small algebraic closure norm alone is not pointwise EOS
accuracy. No exterior-only cause is inferred from these energy diagnostics.

At fixed density, gravity and rotation, any pressure-only change shifts both
signed virials equally. Their 2.10144e-5 separation therefore implies a best
possible maximum absolute error of at least 1.05072e-5 under such a shift.
Pressure repair alone cannot make both diagnostics pass in this frozen state.
This is not a bound on a newly solved or refined model.

A useful pressure-consistency identity for the present polytrope is
P(rho)-P(h)=(h/n)[rho-rho_EOS(h)]+D, with D>=0 for nonnegative fields.
Current DGQ2 closure tests do not generally contain the H1Q3 h field, so even
zero weak closure does not cancel the linear integrated pressure term.
Compatible test-space inclusion would cancel it under the same quadrature,
but the convex remainder, quadrature difference and finite residual remain;
near vacuum that remainder is not uniformly quadratic. This is an untested
consistency hypothesis, not evidence for a particular order change or a
reason to derive density from h. The derivation and scope are in the
[pressure-moment note](../../../diagnostics/coupled_convergence_2026-09-19/PRESSURE_MOMENT_NOTE.md).

Angular velocities are .401764 and .401388 in model units. Neither is a
calibrated fraction of critical rotation. The compatible physical inputs
make this a meaningful rotating test; changing K and removing the central
border are shared by both runs and are not counted as a preconditioner gain
against the earlier n=3,J=0 screen.

Final xz/xy slices were rendered and inspected for both runs. They show strong
central concentration and modest oblateness. These n=3 exports use four
subdivisions only; the separate n=2/n=1.5 display-refinement check must not be
misrepresented as a refinement test of these n=3 images. The complete results,
trial accounting, states and physical metrics are in
[N3_COMPATIBLE_RESULTS.md](../../../diagnostics/coupled_convergence_2026-09-19/N3_COMPATIBLE_RESULTS.md)
and its adjacent JSON.

## Architecture and interpretation for the next investigation

The library fields, orders, all-m L4 surface, normalization and physics-facing
API remain unchanged. No clipping, logarithmic density, derived-density
replacement or new positivity acceptance gate was introduced. The new coarse
inverse is an experiment-only frozen policy. It uses generated row/column
partitions and an explicitly unbordered base inverse; compile-time concepts
check structural contracts, while generation/dimension/rank checks guard facts
known only after preparation. Changed Jacobians require reconstruction.

Checked restart support is separated from read-only diagnostic observation.
The context's explicit diagnostic initializer validates compatible finite
physical coordinates, prepares and refreshes before commit, clears stale
warm starts and rolls back on failure. An unconverged snapshot requires an
explicit experimental override. None of this silently accepts an invalid
model or changes the physical equations.

The current evidence supports the following order of work:

1. Retain gravity-Schur as the frozen linear-cost comparator and stellar
   LDU plus gravity as the better live candidate in the tested rotating n=3
   configuration. That live result is conditional on its gravity-curvature
   trial path; it does not establish the best raw-Newton default. Candidate
   iteration counts alone are insufficient. Do not promote the new surface
   wrapper on this evidence.
2. Investigate the weak coupled radial-size/field response and finite-step
   curvature. A coupled trust region or regularized step with a physical
   displacement scale is a plausible swappable experiment. Keep radius free,
   keep surface/interior simultaneous and evaluate the full residual and both
   virials. The data do not yet select a regularization strength or prove
   this remedy works.
3. Improve the approximate material response in controlled stages: exact local
   DG density mass inversion and stronger enthalpy mass inversion are narrower
   candidates before a more expensive EOS-aware material/gravity Schur solve.
   The existing material inverse already includes the h-to-density derivative;
   claiming it is missing would be incorrect. This remains an independent
   density formulation. See [FIELD_COUPLING_REVIEW_2026-09-19.md](FIELD_COUPLING_REVIEW_2026-09-19.md).
4. Define a compatible n=3 rotating family and calibrate dimensionless rotation
   before targeting 50% critical. At fixed M and K, n=3 homology changes J with
   scale; changing input J alone is not a calibration of critical fraction.
   This physical parameter issue is separate from the n=2 numerical stall.

A projected surface-equipotential equation and native lower harmonic degree
were reviewed as possible future operator studies but were not implemented
or tested here. They change the discrete system. In particular, a surface
potential derivative must respect the actual ALE pullback/field transport;
substituting an effective-gravity derivative without that analysis would not
be sound. The present mechanical surface equation and pressure-query guards
are retained.


## Validation, artifacts and reproduction

Release builds of the diagnostic and targeted test executables pass. The
new coarse inverse passes 70 assertions in nine cases, including typed
normalization/partition behavior and algebraic recovery. The transactional
initializer passes 71 assertions in three real-FEM cases. Pure slice geometry
controls pass; all 14 slice plotting/comparison tests pass, including after
the final label/bounds fixes. The artifact auditor passes 11 controls, and
both convergence plotters pass 11 tests each with actual rendering enabled.
No unrelated full-suite claim is made.

All 24 campaign cases pass retained-artifact verification: five initial
nonlinear controls, two matched gravity-path restarts, eleven frozen inverse
comparisons, four unchanged-state slice replays and two live n=3 controls.
A passed artifact audit can describe a failed linear gate or an unconverged
model; it certifies the stated records and accounting, not scientific success.
The [study index](../../../diagnostics/coupled_convergence_2026-09-19/STUDY_INDEX.json)
records producer revisions and hashes of all six audit files.

Numerical campaigns and heavy builds/audits/rendering were serialized.
Each campaign retains exact arguments, clean producer commit, private copied
executable, build settings, mesh/input hashes, logs, saved states and output
hashes. Source-provenance commits differ because followups were declared and
committed as evidence accumulated. Later documentation/plotting edits do not
change the copied numerical executables. Timings have no statistical confidence
intervals and are not portable hardware benchmarks.

The integration branch is `research/coupled-convergence-study-20260919`.
Separate retained branches cover the linear coarse inverse, slice exporter,
n=3 compatible family, gravity-only attribution and live field feedback.
The original programming checkout is unchanged. The report and case manifests
are tracked; large artifacts are outside the source checkout in the evidence
folder. Production defaults, physical field orders and positivity handling
remain unchanged.

Campaign manifests in this directory are `convergence_screen_cases_2026-09-19.json`,
`convergence_gravity_path_cases_2026-09-19.json`,
`convergence_linear_cases_2026-09-19.json`,
`convergence_linear_followup_cases_2026-09-19.json`,
`convergence_slice_cases_2026-09-19.json` and
`n3_compatible_family_cases_2026-09-19.json`. Run `run_plateau_campaign.py`
from the matching clean source revision, supplying that manifest, source,
build, mesh and a new output directory. Predecessor paths in restart/replay
manifests must resolve to their recorded saved states. The runner deliberately
refuses to overwrite campaigns or use modified tracked source.

Review artifacts:

- [Rotating n=3 meridional slice](../../../diagnostics/coupled_convergence_2026-09-19/figures_n3_v2/n3_J0p035_K0p99_ldu_xz.png)
  and [equatorial slice](../../../diagnostics/coupled_convergence_2026-09-19/figures_n3_v2/n3_J0p035_K0p99_ldu_xy.png).
  These are preliminary four-subdivision images of the unconverged feedback
  model, with corresponding PDFs in the same folder.
- [Refined n=1.5 rotating slice](../../../diagnostics/coupled_convergence_2026-09-19/figures_slices_v2/n1p5_rotating_xz_s8.png)
  and [same-state display comparison](../../../diagnostics/coupled_convergence_2026-09-19/figures_slices_v2/n1p5_rotating_xz_s4_s8.pdf).
- [Eleven frozen linear histories](../../../diagnostics/coupled_convergence_2026-09-19/figures/linear_rates/frozen_linear_rates.pdf),
  [n=2 trial-path comparison](../../../diagnostics/coupled_convergence_2026-09-19/figures/convergence/n2_paired_convergence.pdf)
  and [n=3 tolerance comparison](../../../diagnostics/coupled_convergence_2026-09-19/figures/convergence/n3_paired_convergence.pdf).
- [Audited nonlinear controls](../../../diagnostics/coupled_convergence_2026-09-19/CONTROL_RESULTS.md),
  [live n=3 comparison](../../../diagnostics/coupled_convergence_2026-09-19/N3_COMPATIBLE_RESULTS.md),
  [implementation validation](../../../diagnostics/coupled_convergence_2026-09-19/validation/validation.json)
  and [final slice test log](../../../diagnostics/coupled_convergence_2026-09-19/figures_n3_v2/slice_tests.log).