[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. Integral Fourier squares exclude 317 antipodal pairs in the length-sixteen D16 spinor distance model.

- spinor lines <= 317; roots plus spinors <= 557
+ spinor lines <= 316; roots plus spinors <= 556

With 317 antipodal pairs, every even-weight character sum is twice an odd integer. The inherited dual identity forces its weighted sum of squares to be 116,656, while integrality forces at least 146,992. The exact contradiction gap is 30,336. This is a new upper-bound step; attainment of 316 is unclaimed.

Produced by Nick Petalas. Credit DreamingOfClouds / #2244 for the model and original cap, Nick Petalas / #5206 for the exact real relaxation, dual identity and open integrality question, and their retained #2228/#2125/#968 lattice/decoder lineage. Character orthogonality and parity refinements are classical. No general literature-priority claim is made. No predecessor scientific executable is imported or replayed.

Evidence

Before, the real two-point optimum and even-cardinality rounding left 317 spinor lines possible. After, the written parity proof excludes that endpoint and gives at most 632 codewords.

Local CPython 3.12.15/macOS arm64 checks pass:

• All 20,998 selected even-weight masks, seven literal character/kernel values, and 25,440 possible sign counts.
• Fifteen small antipodal codes, with 314,970 character evaluations and 320 ordered pairs. These fixtures overlap and are not a census of large codes.
• Seven rejecting controls and three semantic discriminators, with identical normal/optimized receipts. The registered checks take 0.184 and 0.146 seconds in single local runs; these are timeout checks, not a speedup benchmark.
• Registered metadata/replay, committed bytes/modes/source history, official export integrity and declared input coverage. Both standalone Git-free, empty-PATH replay modes pass and preserve all nine exported files and modes.

From the contribution checkout root, using CPython 3.12 standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/spinor-integral-cap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/spinor-integral-cap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/spinor-integral-cap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/spinor-integral-cap --check-source-history --check-modes

Another CPython 3.12 path may replace the local interpreter path. Expected markers include PASS_SPINOR_INTEGRAL_CAP spinor_lines=316 roots_plus_spinors=556 gap=30336 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Every direct check emits a generated receipt bound to all six package files.

Limitations and unrun obligations

The universal bound rests on the written character/parity proof. Finite checks corroborate its identities; no exhaustive 634-word-code search is claimed. The D16+ model identification is a cited predecessor premise. This is a geometric side result, not a native supplier or a complete multiplication theorem. Physical decoder/compensation, frames, routing, restoration, compiler, precision/tape, stock and complete paid invoices remain separate. The selected frontier and all five retained assumptions remain unchanged. Independent human and formal proof-kernel review are unclaimed. No selected cold replay, Lean or Docker ran. Every registered command ran locally.

Open next questions

Can 316 antipodal pairs be excluded or attained? The even pair count permits zero character sums, so this proof does not settle it. First test integral or triple-distance feasibility at exactly 632 words. No stronger certificate or attaining construction is supplied, and no continuation is reserved.

Related open work

No open PR studying this antipodal spinor-code size quantity was identified in the fully paginated 539-body snapshot and the fresh prepublication open-PR check. #2244 and #5206 are merged predecessors. The merged #606 studies full-shell spinor circuits with a different boundary; no new circuit cost is compared here. Sparse notes, arbitrary historical diffs and unpublished work remain coverage limits.

Completion refresh: thirteen account open PRs created since the CLAIM at 2026-11-03T11:01:07Z were inspected, including this PR. None of the other twelve studies this spinor-code size quantity or reports a competing value. PR6590 applies Fourier integrality to matched P112 phase-event lower bounds, a different endpoint and quantity. No additional computations audited other contributions.

CI notes

Both hosted checks, plan and replay (research/spinor-integral-cap), passed at exact head 612b6e7cf4106af561fc3cbe49f29df35eb403f3 in run 41625088953. The completed check list was read once; run metadata confirms the same head, completed status and success. Hosted replay executes both registered warning-strict normal and optimized checks. Local registered, committed-byte/source-history and exported standalone checks also passed. Hosted execution corroborates these bounded checks; independent human and formal proof-kernel acceptance of the written proof remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Six new files in one opt-in research package. No shared code, frozen predecessor, selected result or historical evidence changes.