PureOne/hsin-heart-exact84-two-intersection

Hsin-Heart Exact 84: Two-Intersection Families on 21 Points

Exact candidate theorem, complete extremal classification, sharp stability gap, prior-art audit, and replayable verification package.

Headline result

For a family

[ \mathcal F\subseteq {[21]\choose 13} ]

such that every two distinct sets satisfy

[ |A\cap B|\in{7,9}, ]

the release proves the candidate exact value

[ \boxed{|\mathcal F|\le 84}, ]

with equality attained. Equivalently, after complementation,

[ \boxed{A(J^{21,8},{4,6})=84}. ]

The strengthened release additionally gives:

• complete extremal rigidity: every 84-family lies in the unique self-orthogonal ([21,9,8]) code associated with (PG(2,4));
• explicit classification of every extremizer by one incident-line choice at each of the 21 projective points;
• sharp stability: every valid family of size at least 82 lies in the (PG(2,4)) core, while a non-(PG(2,4)) family of size 81 exists;
• a direct 38-core verification route, replacing the earlier 18,216-shortening route;
• a wide prior-art and novelty audit.

Scholarly status: this is a reproducible candidate theorem/research release. The included verifier passes, but the result has not yet undergone independent peer review. Classical coding-theory, design-theory, Golay-code, and finite-geometry ingredients are explicitly separated from the claimed new conclusions.

Why this repository exists

This repository is designed for four audiences at once:

1. Researchers — a compact statement of the theorem, proof architecture, historical context, and references.
2. Referees / verifiers — deterministic source data, certificates, hashes, and a standard-library verifier.
3. Search engines — explicit parameter notation and equivalent formulations.
4. AI agents — machine-readable manifest, claim inventory, file map, reproduction command, and epistemic-status labels.

Equivalent searchable formulations

The same problem may appear under different terminology. This release deliberately indexes all of them:

• two-intersection family of 13-subsets of a 21-set, intersections (7,9);
• 8-subsets of a 21-set with pair intersections (2,4);
• fixed-distance Johnson-space code (J^{21,8}) with distances (4,6);
• (A(J^{21,8},{4,6}));
• binary constant-weight-8 code with Hamming distances (8,12);
• doubly-even ([21,9]) binary-code closure;
• unique self-orthogonal ([21,9,8]) code;
• (PG(2,4)) line symmetric differences;
• Golay / octad / Petersen conflict-graph geometry;
• Delsarte upper bound 105 for the historical parameter set.

Core theorem package

File	Purpose
paper/FINAL_BREAKTHROUGH_Exact84_Rigidity_Stability.md	Main strengthened manuscript
verification/hsin84_final_verify.py	Standalone deterministic verifier
verification/core38_final_data.json	38 direct maximal doubly-even ([21,9]) representatives and certificates
verification/hsin84_final_certificate.json	Frozen theorem and witness certificate
verification/core38_invariants.csv	Human-readable 38-core invariant table
prior_art/Prior_Art_Audit_Exact84_21_13_7_9.md	Detailed prior-art / novelty audit
prior_art/Prior_Art_Source_Matrix_Exact84.csv	Structured source matrix
AGENTS.md	Agent-readable orientation and epistemic rules
CLAIMS.json	Machine-readable claim inventory
RELEASE.json	Release metadata and reproduction entry point
SHA256SUMS.txt	Integrity hashes
release/Hsin_Heart_FINAL_Breakthrough_Exact84_Rigidity_Stability.zip	Frozen all-in-one archive

Reproduce

Requires Python 3 only; the verifier uses the standard library.

python verification/hsin84_final_verify.py \
  verification/core38_final_data.json \
  verification/hsin84_final_certificate.json

Expected headline output:

PASS
38 direct maximal doubly-even [21,9] core representatives verified
Non-Golay sharp upper: 81
Golay/PG(2,4) exact upper: 84
Extremizers per fixed PG(2,4) core: 476837158203125
84-witness size: 84
81 non-Golay witness size: 81

Main proof architecture

1. Complement the 13-sets to weight-8 binary incidence vectors.
2. Pairwise intersections (2,4) imply pairwise binary orthogonality.
3. Their span is therefore a doubly-even binary code.
4. The even-weight quadratic space in length 21 has Witt index 9, forcing dimension at most 9.
5. Every candidate family lies in a maximal doubly-even ([21,9]) core.
6. A published classification gives exactly 38 permutation classes of such cores.
7. The release verifies upper certificates over one representative of every class.
8. Every non-(PG(2,4)) core has maximum at most 81.
9. The (PG(2,4)) core decomposes its conflict graph into 21 disjoint Petersen graphs, giving (21\cdot4=84).
10. Equality in every Petersen component forces a star, producing the complete extremizer classification.

New-claim boundary

The release does not claim discovery of the classical ingredients. In particular, the following are prior art and are cited in the manuscript/audit:

• the historical 105 Delsarte bound;
• Gosselin's earlier 105-block incidence formulation;
• Calderbank's quasi-symmetric-design obstruction;
• doubly-even / self-orthogonal binary-code theory;
• classification of doubly-even ([21,9]) codes;
• the unique self-orthogonal ([21,9,8]) code;
• its (PG(2,4)) realization and Golay-related geometry.

The candidate new conclusions are the exact value 84, the sharp 81/82 rigidity threshold, the problem-specific 21-Petersen decomposition as used here, and the complete selector classification of extremizers.

Citation

A machine-readable CITATION.cff is included. Until external review establishes final bibliographic status, cite this repository by its Hugging Face URL and release commit/revision.

Licensing

This repository uses a mixed license:

• research prose and tables: CC BY 4.0;
• verification/source code: MIT;
• third-party mathematical results remain subject to their original publications and are cited rather than relicensed.

See LICENSE_RESEARCH.txt and LICENSE_CODE.txt.