# 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.

## Initial completed controls

The following scalar observations precede the final retained-artifact audit.
All three cases exhaust twelve updates and every reported full linear solve
passes its requested relative true-residual gate.

| 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 |

Initial norms are 3.81795e-4 and 3.09931e-3 respectively; actual stopping
thresholds are 3.81795e-12 and 3.09931e-11. q14/q18 differences in all six
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.

## 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 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. 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.