readme: |
  # Finite large-field diagnostics and certificates

  The scripts in this directory support the accompanying mathematical notes.
  The small tracked `.log` files are mathematical certificate/diagnostic
  summaries. Raw arrays, job logs and submission tooling are not included.

  **Standard-library checks:** `bad_branch.py`, `chart_bounds.py`,
  `coverage_checks.py`, `rough_certificate.py`, `touching_checks.py` and
  `scales.py` run without numerical packages. Their current outputs were
  reproduced byte-for-byte during upstream integration. The rough-field
  certificate checks exact integer identities; scalar inequalities in
  `chart_bounds.py` use rational arithmetic. These checks do not substitute
  for the analytic arguments or extend their hypotheses.

  **Numerical diagnostics:** the other chart/domain scripts use NumPy and,
  where imported, SciPy; threshold/ledger scripts use mpmath. The stored
  threshold computations predate later domain and environment obstructions;
  they are conditional ledger calculations, not certified working island
  thresholds. See the latest section of `../ROADMAP.md` for current status.

  Three additional standalone scientific programs accompany the newer work:

  - `branch_controls.py --device cpu --reference --batch 64 --batches 8`
    uses NumPy/JAX to check constraint Jacobians against finite differences.
    A path-only singularity does not establish divergent integrated transport.
  - `frontier_diagnostics.py` uses NumPy/SciPy for the scalar transverse fold,
    separate hard island weights through a bounded scalar collar, and finite
    spectral-mixture checks. These are the diagnostics of
    `../TRANSVERSE_FOLD.md`, `../ISLAND_KERNEL.md` and
    `../POSITIVE_TIME_TIGHTNESS.md`.
  - `frontier_calibration.py` uses the standard library for the Bessel,
    oscillator, overlap, locality and normalisation calibrations of
    `../FRONTIER_LEMMAS.md`.

  These three programs each write `result.json` in their current working
  directory; use separate scratch directories to keep generated data out of
  the source checkout. They contain no scheduler or cluster transport code.
  Numerical diagnostics alone close neither the bad-side transport problem
  nor the environment recursion or multiscale estimate.

  **Island thresholds (mpmath, standard library for the enumerations):**
  `island_threshold.py` computes the block-scale threshold of the island
  theorem (the largest root of the chessboard exponent against the entropy
  of the collar graph); `composed_threshold.py` the thresholds at every
  scale of the composed cut-off average with the box sizes of the fine
  witnesses, and the scale of the defect propagation at the same couplings;
  `witness_geometry.py` enumerates the witness boxes and their overlaps;
  `abelian_growth.py` iterates the cut-off average on an abelian field of
  constant curvature and reproduces the factor four per step;
  `grid_check.py` compares the two scales on a grid of couplings. The
  tracked `.log` files are their outputs.

  **The two sides of the path-link law (mpmath interval arithmetic):**
  `fold_intervals.py` is the interval library (rigorous enclosures of the
  cut-off average and of its transverse derivative on boxes, built on
  `mpmath.iv` with outward rounding); `fold_certificate.py` verifies the two
  inequalities of the fold proposition that are not proved by hand: the
  transverse component of the cut-off average is positive off the torus on
  the strip, and the transverse stretching lies in the stated ranges at the
  branches; `fold_selftest.py` checks the enclosures against floating-point
  evaluation; `goodside_constants.py` prints the contraction constant and
  the response constants of the good side at radius 1/32. The tracked `.log`
  files are their outputs.

  **Two islands and one kernel:** `island_width_threshold.py` recomputes
  the island threshold with the joining distance 2s+7 that the
  factorisation theorem requires, and the radius at which the chart
  calmness hypothesis of the layer theorem holds, with the threshold at
  the joining distance 2s+8. `layer_constants.py` computes the constants of
  the Hessian of the layer law in the label coordinates (lower bound
  0.0185 lambda, weighted norm 2068.4 lambda, admissible rate 2.79e-4) and
  the corrected threshold. The tracked `.log` files are their outputs.

  **The witness radius for SU(2):** `witness_radius.py` encloses the
  critical radius kappa_c = 0.0837613, the constant 28.04 of the sharp form
  of the small-frames lemma, the constants 16.003 and 12.002 of the coarse
  witness, and the counterexample at L = 32, n = 10. `island_radius.py`
  recomputes the island thresholds at the radii 1e-8 and 1e-4 and the gap
  k_* - j_max over 2001 couplings. Both use interval arithmetic.

  **The prefactor of the layer bound:** `lambda2_constants.py` computes the
  constants c_W <= 99.0001 and c_psi <= 4.65e6 of the bound on Lambda_2, the
  plaquette and path counts behind them, and the width of the good layer at
  which the two-island cross-ratio falls below one at the threshold.