# Sources and integration notes for recent.tex

The fragment covers native harmonics (Sept. 19), negative-density localization,
density-only Q2/Q3/Q4 and log assessment, final coupled-convergence controls,
compatible rotating n3 family, pressure/gravity consistency, physical slices,
and the resulting next priorities. It is approximately 7,700 words, with 14
small wrapped longtables and two figures. It deliberately does not duplicate
the preceding Sep. 18 reduced-surface study except to explain the transition
from an affine restricted diagnostic to a native prescription.

## Integration

- Required standard packages: `amsmath`, `longtable`, `graphicx`, `hyperref` (as
  already planned). No custom command definitions are required. Grouped tables
  set tab spacing to 3 pt and explicitly wrap columns; total prescribed column
  widths are <=147 mm before internal tab spacing.
- Figure references assume compilation and Pandoc conversion with `reports/`
  as the resource directory: `figures/n3_ldu_meridional.png` and
  `figures/n1p5_rotating_refinement.pdf`.
- The third planned figure `figures/frozen_linear_rates.pdf` can be added by the
  root in the frozen-comparison section or an appendix; its multiple-panel layout
  may merit a full page.
- Sections describe Sept. 19 state and producer commits, not the still-pending
  final main integration. Root must separately write the exact selected branch
  disposition and final main commit.
- All internal labels begin `recent-`, avoiding likely collision with the early
  report fragment.
- No fresh FEM runs or audits were performed for this report fragment; it is a
  synthesis of the immutable studied outputs and their existing audits.
- The old negative-density report's recommendation for derived density is
  explicitly identified as historical and superseded by the user's decision to
  keep density independent. The later preference to defer positivity is also
  explicit. Do not silently carry that old recommendation forward.
- Important no-overclaim rules are repeated where they matter: finite-budget
  endpoints are not floors; equal-update trajectories are not same-model
  speedups; quadrature agreement is not mesh convergence; sampled positivity
  is not global positivity; the n3 force/h virial alone passing does not meet
  the overall milestone; raw J or angular velocity is not a critical fraction.

## Primary repository reports read

Original source checkout root:
`/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/preconditioner-improvement`

1. `experiments/SPHERICAL_HARMONIC_SURFACE_2026-09-19.md`
   - Native all-m L4 prescription, typed contracts, fixed reference basis,
     Q4/Q3 geometry/displacement, scalar trace-mass normalization, single-rank
     scope, five six-update live controls, excluded-force and local diagnostics.
   - Producer: `7b0eb8d92d0f550f1fe3fb19cd8eecc52dc80c56`.
2. `experiments/NEGATIVE_DENSITY_INVESTIGATION_2026-09-19.md`
   - Four frozen states, independent closure projection identity, all 96 outer
     cells, root-sliced measures, positive nodal values, seed-vs-root mechanism,
     exact one-dimensional quadratic projections, interpretive scope.
   - Producers: `9c22934d52b369f96364e781d6a4843939ec9523` and
     `d1a594808f960bbaeeb6df8a8ce6a95458765b19`.
3. `experiments/DENSITY_ORDER_STUDY_2026-09-19.md`
   - Frozen common-q26 and error-q30 controls; all 12 live density-order cases;
     selected quadrature rules, block sizes, cost scopes, central forcing and
     initializer distinctions; 48 contract tests per order.
   - Producers: frozen `241621c`; coupled Q2 `c0e8551`; Q3 `cd71f37`;
     Q4 `e054c74`; final comparison generator `c0a496e`.
4. `experiments/COUPLED_CONVERGENCE_STUDY_2026-09-19.md`
   - Five initial nonlinear controls, matched gravity-path pair, all 11 frozen
     inverse pairs, weak coupled direction analysis, compatible n3 pair,
     display refinement, validation and next priorities.
   - Final integration/report commit: `d7acb0d399aac5ac4c36d834a954c1c959042b81`.
5. `experiments/N3_COMPATIBLE_FAMILY_2026-09-19.md`
   - Explicit K factor, independent Lane–Emden compatibility check, exact
     shared specifications and caveats for the live pair.
6. `experiments/FIELD_COUPLING_REVIEW_2026-09-19.md`
   - Actual material triangular structure contains the exact h-to-rho action;
     independent vs LDU feedback, possible exact DG mass and H1 mass improvements,
     optional future EOS-aware field Schur and closure affine-target trial path.
7. `experiments/MODEL_SLICES.md`
   - Physical-plane cut and source immutability; element-owned DG traces,
     independent outlines, field raster before colormap, display errors/limits,
     use/replay parameters and geometry/plotting checks.
8. `experiments/REDUCED_SURFACE_STUDY_2026-09-18.md`
   - Read for bridge/context only; preceding early-history fragment should
     contain its detailed restricted-vs-full system results.

Separate source checkout:
`/Users/tboudreaux/.codex/.chatgpt-projects/g-p-6a6b3f2194ec8191bebcd59075897f1d/worktrees/log-density-assessment`

9. `experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md`
   - Mathematical distinction between exponentiating coefficients and pointwise
     exponential, vacuum singularity, independent vanishing factor, 1D moment
     fits, weighted-mass response/conditioning, general-EOS and support contracts.
   - Separate assessment branch at `90580b2`; no coupled stellar log solve.

## Raw evidence and concise analysis read

Archive target supplied by root:
`working/research/diagnostics/<campaign>/` in the main repository.
Original workspace `diagnostics/` will remain a symlink to that archived path.

Campaign names:

- `spherical_harmonic_surface_2026-09-19/`
- `negative_density_2026-09-19/`
- `density_order_2026-09-19/`
- `log_density_2026-09-19/`
- `coupled_convergence_2026-09-19/`
- `n3_mass_compatibility_2026-09-19/`

Additional analyses read directly:

- `coupled_convergence_2026-09-19/N3_COMPATIBLE_RESULTS.md`
- `coupled_convergence_2026-09-19/PRESSURE_MOMENT_NOTE.md`

High-value existing evidence links for root's archive appendix:

- `coupled_convergence_2026-09-19/STUDY_INDEX.json`
- `coupled_convergence_2026-09-19/CONTROL_RESULTS.md` and `.json`
- `coupled_convergence_2026-09-19/linear_analysis/FINAL_INTERPRETATION.md`
- `coupled_convergence_2026-09-19/n3_direction_difference/README.md`
- `coupled_convergence_2026-09-19/SLICE_REFINEMENT_SUMMARY.json`
- `coupled_convergence_2026-09-19/validation/validation.json`
- `coupled_convergence_2026-09-19/figures_n3_v2/slice_tests.log`
- `density_order_2026-09-19/final_analysis/coupled_outcomes.csv`
- `density_order_2026-09-19/final_analysis/trial_stages.csv`
- `negative_density_2026-09-19/regions_campaign_audit_v1.json`
- `spherical_harmonic_surface_2026-09-19/campaign_audit.json`

## Figure sources

All below are relative to original workspace diagnostics, or to the archive's
`working/research/diagnostics/` after reorganization.

- Report `figures/n3_ldu_meridional.png`:
  `coupled_convergence_2026-09-19/figures_n3_v2/n3_J0p035_K0p99_ldu_xz.png`
- Report `figures/n1p5_rotating_refinement.pdf`:
  `coupled_convergence_2026-09-19/figures_slices_v2/n1p5_rotating_xz_s4_s8.pdf`
- Optional report `figures/frozen_linear_rates.pdf`:
  `coupled_convergence_2026-09-19/figures/linear_rates/frozen_linear_rates.pdf`
- Additional useful density-order figure:
  `density_order_2026-09-19/final_analysis/density_order_comparison.png`
- Additional sign-location figure:
  `negative_density_2026-09-19/figures/surface_density_traces.png`

## External primary-source context retained from the studied reports

The numeric results and equations in the fragment are primarily local evidence
and explicit derivations. If adding a bibliography, these are the primary
sources referenced by the underlying reports (no new online verification in
this report-writing subtask):

- Cheng, Cheng and Lin, *Expanding solutions near unstable Lane–Emden stars*,
  equations (1.14)–(1.15) and Theorem 3.1:
  https://arxiv.org/html/2305.10218#S3.SS1
  This supplies the sharp energy bound and Lane–Emden mass relation. The beta
  bound is our inference with the steady virial identity, not a quoted theorem.
- MESA `star/private/star_utils.f90`:
  https://github.com/MESAHub/mesa/blob/main/star/private/star_utils.f90
- MESA T(tau) atmosphere documentation:
  https://docs.mesastar.org/en/latest/atm/t-tau.html
- Vijaywargiya and Fu, *Two Finite Element Approaches for the Porous Medium
  Equation That Are Positivity Preserving and Energy Stable* (2024):
  https://link.springer.com/article/10.1007/s10915-024-02642-x
- Fu and Xu, *High-order space-time finite element methods for the
  Poisson–Nernst–Planck equations: Positivity and unconditional energy stability*:
  https://arxiv.org/abs/2105.01163
- Zhang and Shu review (historical alternative only, not currently adopted):
  https://www.math.purdue.edu/~zhan1966/research/paper/review.pdf

The condensed fragment mentions MESA only as context and does not import PDE
positivity theorems as guarantees for this steady stellar system.