# Physical validation findings — 2026-09-09

## Outcome

The nonrotating n=1 model reaches a well-converged **discrete** equilibrium, and
its global virial balance passes the initial 1e-6 screening budget. It does **not**
yet pass the full analytic-accuracy screen. In particular, the exterior potential
has roughly 1.3–1.4% errors near the surface, and several interior field/shape
errors exceed the declared 1e-4 budgets. This is not yet a physical-accuracy
sign-off for performance work.

These results do not establish that the formulation is wrong: one discretization
cannot distinguish ordinary approximation error from a formulation bias or
implementation defect. They do establish that further Newton convergence alone
is not an adequate verification strategy.

The implemented experiment and commands are documented in
[POLYTROPE_VALIDATION.md](POLYTROPE_VALIDATION.md).

## Scope and reproducibility

- Added an opt-in `polytrope_validation_experiment` target and experiment-only
  reference, reconstruction, integration, profile, replay, and reporting code.
- No production physics, solver defaults, sandbox source/executable, or input
  mesh was changed by this implementation.
- Ran **one** production Newton solve, approximately 864 seconds including
  context setup and diagnostics. Three subsequent saved-field replays required
  no Newton context or linear solves. Timings are not an isolated performance
  benchmark; a diagnostic build overlapped part of the solve.
- Ran 140 independent analytic checks, mesh-based analytic controls, and angular
  sampling checks. Did not run the full test suite or ESTER.
- The input mesh and sandbox executable SHA-256 hashes were unchanged:
  `ce11ae99e21e6c3bbfbee402acd2e191c1da0d8261d2227b4203f73f2f337a74`
  and `a904b39935a5add938129e5c3edb72f94b416f0e09ba5b3b937a8004fa463057`,
  respectively. Every mesh-based run retains its own `input.smesh` copy.

The model has G=M=R=1, K=2/pi, central density pi/4, zero angular momentum,
and zero surface pressure. The current mesh has 1,216 elements and the state has
178,074 values. Density and potential use DG order 2, enthalpy and displacement
use H1 order 3. Gravity uses MFEM RT index 2 (reported element order 3).
The geometry's order 4 does not make all solution fields fourth order.

## Data and plots

| Artifact | Location |
|---|---|
| Original solve, seed baseline, coefficient/GF snapshots, residual blocks | [polytrope_solution_2026-09-09](../polytrope_solution_2026-09-09/) |
| Corrected analytic-mesh control | [polytrope_analytic_checked_2026-09-09](../polytrope_analytic_checked_2026-09-09/) |
| Default 6x12 angular replay | [polytrope_replay_2026-09-09](../polytrope_replay_2026-09-09/) |
| 12x24 angular replay, 64 exterior shells | [polytrope_replay_dense_2026-09-09](../polytrope_replay_dense_2026-09-09/) |
| 24x48 angular replay, 64 exterior shells | [polytrope_replay_angular24_2026-09-09](../polytrope_replay_angular24_2026-09-09/) |
| Main generated numerical report | [polytrope_summary.md](../polytrope_replay_angular24_2026-09-09/polytrope_summary.md) |

![Fixed-reference profiles, mean errors, angular scatter, and coverage](../polytrope_replay_angular24_2026-09-09/polytrope_profiles.svg)

The original `polytrope_analytic_mesh_2026-09-09` control is retained for
provenance, but its angular RMS statistic contained the roundoff artifact
described below. Use the **checked** control above for current interpretation.
The default-grid saved GF replay reproduced every pre-existing aggregate physical metric exactly;
new pressure/projection metrics were then added without rerunning Newton.

## Independent controls

All 140 closed-form checks pass, including non-unit scales, central/surface
limits, exterior potential, derivatives, and independent radial mass/energy
integrals. No production Lane–Emden integration or seed helper supplies the
reference values.

On the actual undeformed mesh, exact analytic fields give:

| Control | Result |
|---|---:|
| Relative mass error | 3.92e-10 |
| Relative binding-energy error | 2.62e-10 |
| Virial error | 2.61e-10 |
| Force-based virial error | 5.23e-10 |
| Change in virial between quadrature orders 14 and 18 | 1.38e-13 |
| Located profile points | 4,020 / 4,020 |
| Maximum scaled pointwise sampling error | 4.96e-11 |
| Corrected maximum scaled angular RMS | 2.12e-11 |

The first weighted variance update initially introduced a one-ulp contribution
to the second moment, creating spurious angular RMS values near 1e-9. Exact
first-sample initialization fixed this; a constant-field zero-variance check now
guards it. This did not change the numerical volume integrals or Newton solve.

Control limitations matter: volume field errors are zero by construction because
the same independent reference supplies the analytic fields and comparisons.
Those zeros are not tests of FE representability. The nonzero global integral
errors test integration/geometry, and requested-radius profile errors test the
inverse mapping. Quadrature agreement alone does not resolve the extremely thin
layer where the approximate surface crosses the exact analytic support.

The undeformed surface has RMS radius error **8.92e-6 R**, despite a much smaller
volume-equivalent radius bias of 9.06e-9 R. Signed surface errors cancel in the
volume; this is not 1e-8 local surface accuracy.

## Nonlinear and physical results

The experiment used nonlinear absolute tolerance 1e-8, relative tolerance 1e-8,
linear relative tolerance 0.03, and an 80-iteration linear limit. It stopped
before attempting another increasingly expensive near-floor correction.

| Accepted step | Nonlinear residual afterward | Linear iterations | Step length |
|---|---:|---:|---:|
| Initial seed | 1.81187e-4 | — | — |
| 1 | 7.41173e-6 | 24 | 1 |
| 2 | 2.20576e-7 | 25 | 1 |
| 3 | 6.60533e-9 | 28 | 1 |

All field L2 errors below use the full three-dimensional, physical-volume-weighted
stellar domain and fixed analytic scales/radii, not fitted spherical profiles.

| Diagnostic | Final value | Initial screening budget |
|---|---:|---:|
| Relative mass error | 2.36e-11 | 1e-4 |
| W = 0.5 integral rho Phi | -0.750000539807 | Exact -0.75 |
| Integral P(rho) | 0.250000102482 | Exact 0.25 |
| Virial ratio 3 integral P(rho) / abs(W) | 0.999999690184 | Exact 1 |
| Virial error using P(rho) | 3.10e-7 | 1e-6 |
| Force-based virial error using P(rho) | 3.73e-7 | 1e-6 |
| Virial error using production P(h) | 2.62e-7 | Same 1e-6 comparison |
| Force-based virial error using P(h) | 3.25e-7 | Same 1e-6 comparison |
| Gravity-energy consistency error | 6.30e-8 | 1e-6 |
| Density relative L2 error | 4.70e-4 | 1e-4 — fails |
| Enthalpy relative L2 error | 3.27e-4 | 1e-4 — fails |
| Potential relative L2 error, stellar interior only | 1.26e-4 | 1e-4 — fails |
| Gravity-gradient relative L2 error | 5.98e-4 | 1e-4 — fails |
| Surface radius RMS error / R | 2.75e-4 | 1e-4 — fails |
| Volume-equivalent radius error / R | 2.07e-4 | 1e-4 — fails |

The full screen also flags pointwise EOS mismatch and Bernoulli variation. All
original budgets remain visible and unchanged. No density/enthalpy negativity
was found at the volume quadrature samples. Higher-order integration changes
the numerical virial by only 3.14e-14: these discrepancies are not explained by
the diagnostic volume quadrature order.

Comparing seed and final state at the **same quadrature order**, density error
worsens 1.43x, enthalpy error 12.8x, and potential error 1.31x. Gravity error falls
to 0.694 of its initial value; mass, binding energy, moment of inertia, and virial
balance improve. Newton solves the discrete equations, not the continuum
reference-error minimization problem.

## What is and is not responsible

### Geometry remains healthy; spherical accuracy does not

All 19,968 stellar corner/inset samples are valid. The worst element condition
number is 3.466, versus 3.464 before the solve. The smallest sampled relative
mapping determinant is 0.99632 at the corner/inset samples. This is not the old
folding/step-collapse mechanism.

Nevertheless, surface RMS radius error grows approximately 31x. The final mean
radius is 0.999793481 R, and sampled radii range from 0.999381621 R to
1.000574426 R. There is both a mean contraction and an aspherical component
(approximately 1.82e-4 R RMS). Center-of-mass displacement is only 1.79e-6 R and
cannot explain that shape error.

### The central-density border is small

The central border coefficient is -4.64e-13. Its normalized residual action is
1.25e-9; removing it changes the full residual norm from 6.61e-9 to 6.72e-9.
The unbordered equations still satisfy the 1e-8 screening budget. This is not a
large artificial center force masking the observed 1e-4-level field errors.
The constrained central enthalpy is exactly 1; the independently sampled central
DG density is 0.7853956513 versus the prescribed pi/4 = 0.7853981634.

### The pointwise EOS mismatch has a verified projection component

The code enforces closure weakly in the density space, while the pressure force
uses P(h). Put delta = h - 2K rho. Then

    integral P(h) - integral P(rho)
        = integral delta^2/(4K) + integral rho delta.

For a converged density-space projection, the last term vanishes, but the
nonnegative squared-error term can remain because density and enthalpy use
different spaces. Numerically:

- Measured pressure-integral difference: 1.20602555e-8.
- Predicted squared projection contribution: 1.20605303e-8.
- Difference: -2.75e-13, or -1.10e-12 of the analytic pressure integral.
- Normalized closure-block residual: 6.71e-12.
- Pointwise EOS RMS scaled by central enthalpy: 8.57e-5.

Thus the remaining pointwise EOS RMS is not evidence that Newton failed to solve
the weak closure. This explanation does **not** remove the independent field,
shape, or exterior-potential discrepancies.

## Spatial discrepancies that global virial balance misses

### Exterior potential

At r=1.001 R, the 24x48 angular sample has mean potential error **+0.012866 GM/R**,
approximately 1.29% of the analytic potential magnitude there. The angular RMS
is only 8.76e-5 GM/R: the error is predominantly radial.

The worst saved directional sample has error 0.0139741 GM/R on the (+,-,+) body
diagonal, in exterior element 923; all eight body diagonals have almost the same
error. Its inverse-location error is only 1.73e-15 R. This is not a single-element
or locator accident.

Along that ray, samples from 1.001 R through 2 R all lie in the same exterior
element. The error changes sign with radius rather than behaving like a constant
potential offset. A coarse exterior potential representation is a plausible
cause, but an exact-space approximation comparison is needed to establish it.
The small stellar-interior potential L2 error in the earlier table excludes this
vacuum region.

### Cube-aligned gravity traces and angular sampling

At r=0.556875 R, fixed rays give gravity-gradient errors of approximately
+0.01167 GM/R^2 on body diagonals, -0.00355 on face diagonals, and +0.000674 on
axes. Each symmetry family is internally close. These are significant localized
trace errors, not deteriorated element conditioning. Because RT tangential
components can have one-sided traces on block seams, their solid-angle extent
has not yet been established.

Angular sampling sensitivity is measurable:

| Angular grid | Mean radial-gravity error at 0.556875 R | Mean potential error at 1.001 R |
|---|---:|---:|
| 6x12 | -6.0520e-4 | +1.27950e-2 |
| 12x24 | +5.3099e-4 | +1.29368e-2 |
| 24x48 | -1.8616e-5 | +1.28660e-2 |

Errors use fixed GM/R^2 and GM/R scales, respectively. The exact radial means
should not yet be treated as angularly converged. The roughly 1.3% exterior
potential discrepancy survives all three grids. The full 3D volume errors and
energy integrals are independent of this spherical sampling choice. The densest
replay located all 114,268 requested points and used 64 finite-exterior shells.

## Recommended next work

1. Use the saved fields for a cheap representation audit: compare the exterior
   potential with the best approximation of exact -GM/r in the identical
   potential space on the finite first exterior cells; densely sample both
   one-sided interface traces. Do not form a global physical L2 potential norm
   over the entire infinite exterior, where the exact 1/r potential is not
   square-integrable.
2. Perturb the axis/face/body-diagonal rays slightly off the block seams to
   determine whether the gravity extrema occupy finite angular regions or are
   mainly trace effects.
3. Perform a controlled mesh/order convergence study, retaining separate field,
   surface, virial, and border metrics. Do not replace this with a tighter Newton
   tolerance on the same discretization.
4. Only after the analytic case is satisfactory, use the separately planned
   ESTER comparison for rotating/nonanalytic models. ESTER was not run here.

The tools needed to separate nonlinear convergence from physical accuracy are
now in place. Global balance is encouraging; the field and exterior checks
show why it is too early to certify this model as physically verified.