REPRODUCING THE ODD WEIL-FORM CERTIFICATE

Requirements
------------
Python 3.12, NumPy 2.3.5, and python-flint 0.9.0. Install into your chosen
environment with: python -m pip install -r requirements.txt
The programs use Arb real/complex balls through python-flint. The main matrix
checks use 192-bit working precision. Allow at least 16 GB of available memory;
24 GB RAM is recommended for the 4085 by 4085 matrices. Peak memory depends on
the allocator and thread count. NumPy proposes centers and a dyadic witness;
all accepted primitive conversions, source sums and matrix inequalities use Arb.

To rebuild the paper PDF, use the parent directory's build.py with a locally
installed Tectonic executable:

  python ../build.py --tectonic /path/to/tectonic

Tectonic is only a PDF-build dependency; the arithmetic programs do not use it.

Run from this directory, choosing an EMPTY output directory outside this source:

  python run_reproduction.py --mode scalar --output-dir ../../scalar-results
  python run_reproduction.py --mode replay --output-dir ../../replay-results --threads 8
  python run_reproduction.py --mode fresh --output-dir ../../fresh-results --threads 8

The output paths above are examples: choose paths outside the extracted package
if that package directory is to remain untouched. On systems using an external
BLAS, setting OPENBLAS_NUM_THREADS=4 and OMP_NUM_THREADS=4 can avoid oversubscription.
The modes have distinct scopes:

scalar: Regenerate the complete M8 arithmetic-tail bound, whole exterior floor
cover, low-band and Fourier scalar budgets, whole positive-exterior trace, exact
moment controls, and the actual M11 ambient odd exterior norm lower bound.

replay: Run scalar controls, then reconstruct the physical signed coefficient,
padded receiving covariance and dyadic witness congruence in a separate program.
All 4085 full off-diagonal Gershgorin row sums must pass. This mode authenticates
and reuses the supplied source-enclosure data; it does not regenerate primitive
quadratures or their full source sums.

fresh: Run scalar controls, regenerate every primitive, center, coefficient-block
and physical-Gram source enclosure, outer Fourier feature and final matrix cap,
then run the separate reconstruction. This is the complete arithmetic replay.
The producer can also be invoked separately with --stop-after primitives,
--stop-after centers, or --stop-after enclose. For interruption recovery it offers
--resume; its source/input hashes must match the completed stages.

On the reference eight-thread run, full producer arithmetic took about 20 minutes
and the separate finite reconstruction about 8 minutes. These are indicative
performance observations, not acceptance conditions. Every acceptance gate raises
an exception on failure and remains active under Python -O.

Data and normalization
----------------------
data/M8 arithmetic certificate.json contains the mathematical source-enclosure
data, exact parameters, per-block row and column error sums, outward numerical
radii and final cap.  Fresh
mode regenerates them from the formulae in the producer. The three bundled NPZ files hold
only finite numeric arrays (no Python objects): dyadic center
matrices, complete exterior Fourier features and the dyadic congruence witness.
The larger primitive arrays are regenerated in fresh mode and are not bundled.
The saved source hashes identify which data were checked. The cover JSON is a
whole-interval partition of [2400,4800], with an analytic all-ray continuation.

The signed physical coefficient is folded using D=diag(2 I_4080,I_5). The five
odd global rows comprise the original polar row and four exact derivative-null
rows, with the fixed invertible dyadic mixing matrix in the producer. The raw
same-state exterior covariance includes all phase/row and intercolumn entries.
The positive-frequency features carry the literal full-line factors 2 and 1/pi.

The programs certify the finite and scalar arithmetic used in the paper. The
paper supplies the analytic quadrature/tail lemmas, exact original-domain/null
return, chord inequality and full-form relaxation. These mathematical arguments are supplied in the manuscript.

Files
-----
verify_m8_odd_chord_certificate.py: complete M8 primitive/source/outer/matrix
generator and enclosure program.
verify_finite_witness.py: separate reconstruction of the physical matrix cap.
Check M8 complete arithmetic tail.py: complete paired arithmetic debit.
Certify M8 exterior floor.py: complete whole-interval symbol cover.
Verify M8 positive outer trace.py: whole positive exterior trace bound.
Verify M8 low band analytic budgets.py: exact low-band approximation budgets.
Verify M8 full Fourier analytic controls.py: full physical/frequency budgets.
Verify M8 complex profile and global row tails.py: changed-alpha complex
denominator, derivative-profile and complete global-row Gram-tail budgets.
Verify M8 rigorous pilot analytic interface.py: matching row and trace interface.
Verify outer moment receiver controls.py: exact rational moment controls.
Verify M11 positive exterior odd obstruction.py: actual M11 lower enclosure.

All program outputs are generated in the selected output directory. The source
contains no dependency on a private filesystem or service. No output directory,
runtime log, assessment or review record is part of the publication supplement.
