This document is written for the extracted submission supplement, not for a Git checkout. It covers both Proposition 4.3 Newton configurations and the separate, external-preprint-dependent degree consequence.
All proof-bearing computer algebra commands use the immutable image
sagemath/sagemath@sha256:e068670ae5863b54b2550e72437ec637b0283acb0dc712c8584c124dbf44e667
It contains SageMath 10.9 and Singular 4.4.1. No floating-point value, prime enumeration, or unbounded certificate-degree search is a theorem premise.
From the extracted supplement root:
python tools/verify_supplement_manifest.py
This must report zero missing, mismatched, and unlisted files. Then run:
docker run --rm --memory 30g --memory-swap 30g `
-v "${PWD}:/repo" `
sagemath/sagemath@sha256:e068670ae5863b54b2550e72437ec637b0283acb0dc712c8584c124dbf44e667 `
bash -lc "export SAGE_DOT_SAGE=/tmp/sage; cd /repo; sage --version; Singular --version"All remaining commands are shown without the Docker wrapper. Run them inside
the same container with working directory /repo.
The integrity check is a pre-replay check. Sage creates .sage.py,
__pycache__, and pytest cache files, while several regenerated JSON reports
record nondeterministic elapsed_seconds. Consequently, rerunning the
top-level manifest verifier in the same mutated directory is expected to
report those new files and timing-only output differences. Re-extract the ZIP
to a fresh directory to repeat the archive-integrity check. The mathematical
statuses, fixed hashes, ranks, factors, and exact identities checked below are
deterministic.
The exact recurrence generator checks that the seven tail equations are
precisely d_11=...=d_17=0. The two Sage verifiers then consume the frozen
six-element lex basis, independently reconstruct the degree-35 field and all
lift operators, and decide the terminal compatibility by different methods.
python experiments/breakthrough_program/belyi_edge_solver.py
sage experiments/breakthrough_program/belyi_lift_over_number_field.sage
sage experiments/breakthrough_program/independent_belyi_lift_verifier.sage
The primary output must contain BREAKTHROUGH_GATE_5=True; the independent
output must end with a direct compatibility Groebner basis ["1"]. The
full characteristic-zero Singular input and FGLM scripts are included as
results/belyi_edge_Q.sing, results/belyi_edge_Q_fglm.sing, and
results/belyi_edge_Q_fglm_export.sing. They are the from-scratch basis
construction path; the two Sage commands above are the fast exact replay of
the serialized basis and all theorem-bearing lift equations.
These commands regenerate the 61/125 support, all 302 Laurent-Jacobian coefficient equations, layers 1--5, the layer-1--4 state, the principal generators, and the exceptional-chart generators. Thus the later certificate verifiers are bound to the original Newton coefficient system rather than to unexplained JSON files.
sage -python experiments/larger_polygon_program/primary_verifier.py
sage -python experiments/larger_polygon_program/principal_exact_generators.py
sage -python experiments/larger_polygon_program/exceptional_chart.py
sage experiments/larger_polygon_program/verify_logical_principal_source_chain.sage
The last command must print
LOGICAL_PRINCIPAL_SOURCE_CHAIN_VERIFIED
It reconstructs the degree-five descent from the degree-35 principal
generators, checks the field round trip and invertible coordinate change,
replays the exact 2 x 5 quotient matrix and its constant right inverse, and
reconstructs all six row-split generators.
sage experiments/larger_polygon_program/verify_exceptional_unit_certificate.py
sage experiments/larger_polygon_program/independent_verify_exceptional_certificate.py
Both commands read the regenerated exceptional_chart_layers_5_7.json. They
recompute every generator and multiplier hash and expand the five-multiplier
characteristic-zero identity to exactly one. The second implementation uses
a different reconstruction path and includes corruption controls.
The finite case split is driven by
b2*p0-a2*r7 = D*Y+E.
Every point therefore lies in D!=0 or D=0,E=0.
sage experiments/larger_polygon_program/verify_generic_short_subbranches.sage
sage experiments/larger_polygon_program/verify_degree18_denominator_linear_factor.sage
sage experiments/larger_polygon_program/verify_degree18_good_reduction_witness.sage
sage experiments/larger_polygon_program/verify_degree18_denominator_component_independent.sage
These commands close the a2=0, degree-one, and degree-18 strata. The last
stratum proves L | D,E, L^2 | Q0,P1, and nonvanishing of one fixed
11 x 11 Sylvester determinant. The finite-field calculation is only a
guarded homomorphic image of that fixed characteristic-zero determinant.
sage experiments/larger_polygon_program/construct_d0_projection_checkpoints.sage
sage experiments/larger_polygon_program/prove_d0_factorwise_evaluation.sage
sage experiments/larger_polygon_program/verify_d0_factorwise_evaluation_independent.sage
Required statuses are
D0_EXCEPTIONAL_BRANCH_CLOSED_BY_FACTORIZED_EVALUATION
D0_EXCEPTIONAL_BRANCH_INDEPENDENTLY_VERIFIED
The exact resultant Res_X(D,E) is squarefree of degree 19 with irreducible
factor degrees (1,18). Both factors are closed by exact fraction-free
evaluations in characteristic zero.
Still inside the container:
python experiments/larger_polygon_program/logical_coverage_audit.py
python experiments/larger_polygon_program/global_degree_bound_audit.py
python -m pytest -q tests/test_larger_polygon_logical_closure.py
The first status is
LARGE_POLYGON_EXCLUDED_BY_LOGICAL_BRANCH_CERTIFICATES
The global audit reruns both smaller-configuration Sage verifiers by default,
checks their numerical and unit-ideal invariants, verifies the five actual
distributed arXiv source archives byte-for-byte, and then emits
GLOBAL_DEGREE_BOUND_125. It does not require Git metadata and does not use
substring tests on project-authored Markdown as evidence. This final status
is explicitly conditional on the pinned external preprint reduction.
The three pytest tests are regression checks for the generated JSON objects; they are not described as an independent mathematical verifier.
The theorem replay above verifies serialized exact checkpoints. A stronger but much slower provenance audit can recompute the large resultants and all projection checkpoints from their displayed generators:
sage experiments/larger_polygon_program/sage_exact_resultant_r1.sage
sage experiments/larger_polygon_program/sage_exact_resultant_r4.sage
JACOBIAN_FORCE_RECOMPUTE=1 sage experiments/larger_polygon_program/verify_degree18_denominator_linear_factor.sage
JACOBIAN_FORCE_RECOMPUTE=1 sage experiments/larger_polygon_program/construct_d0_projection_checkpoints.sage
Each command compares the newly derived object with the distributed checkpoint
before replacing it. The R4, H, A, and B checks require exact equality and
print a RECOMPUTED_MATCHED marker. The historical R1 checkpoint differs
from the freshly computed resultant by a nonzero rational scalar. Because the
consumer immediately applies .monic() to R1 and R4, the R1 hook checks exact
equality up to that scalar, verifies equality of the monic representatives,
and prints R1_RECOMPUTED_MATCHED_UP_TO_SCALAR scale=.... This normalization
does not change the gcd, radical, factorization, or any theorem branch.
This path can take hours and is not needed to check the finite identities, but it prevents a cold-provenance audit from silently short-circuiting at H, A, B, R1, or R4.
The theorem proved in the manuscript excludes both exact Proposition 4.3
Newton configurations. Combining it with the explicitly pinned external
preprint reduction gives max(deg(P),deg(Q)) >= 125 for a hypothetical
characteristic-zero plane Jacobian counterexample. The work does not prove
the plane Jacobian conjecture and does not exclude degree 125 or larger.