P429: certify dual-cycle blockers on the two real N425 trigger graphs

[LightChainr]

Result

Advances #429 / #334 and the theorem-facing question in #466. This is a result on the existing real N425 checkpoints, not another random graph-class census or generic synthetic framework.

The minimal black rank-two completion hypergraph is the blocker of the complementary white essential-cycle family, within the existing digital-Alexander domain. A short sufficient theorem follows: if two essential white cycles intersect only at singleton-trigger vertices, every minimal safe trigger pair joins their disjoint safe portions. Hence the trigger-pair graph is bipartite; Ferrers/nested neighborhoods do not follow.

Both real N425 second-geometry witnesses from 6147e22, subsequently analyzed at c827cd8/1b5a9de, admit vertex-disjoint cycle certificates:

checkpoint	cycle lengths	nonisolated sides	pair triggers	W2
43042514269	20, 25	12, 14	108	926
43042505280	19, 43	5, 29	108	1466

The cycle windings are (12,-19) and (-12,19) in [[425,268],[0,1]]. All ordered vertices, occupied masks and source anchors are retained. The original N16 singleton-overlap control is included in tests.

What changes scientifically

The observed 14x12 versus 5x29 bipartitions now have concrete positions on two complementary homology carriers. The next question can be an essential-cycle packing/bottleneck theorem, rather than another abstract graph-class guess. This does not turn the preceding 84% benchmark decomposition into a causal population percentage or eliminate genuine triple triggers.

The note also gives the exact safe-complex union-of-simplices description and safe-update/link identity. No finite-dimensional continuum closure is claimed.

Verification actually run

Discovery used a separately written integer-lift BFS. The committed verifier uses an independent potential-union-find, directly checks supplied cycles without running a discovery search, and rejects corrupted certificates.

python -m compileall -q scripts tests
python scripts/verify_p429_dual_cycle_blocker.py --full-pairs
python -m unittest discover -s tests -p 'test_p429_dual_cycle_blocker.py' -v

• 12/12 focused tests passed, CPython 3.13.5.
• 29,756 safe pair checks, each including black/white rank duality and every trigger edge's cycle-side assignment.
• Existing pair counts, two-star counts, side sizes and isolates reproduce.
• Source/result SHA-256 list, environment metadata and actual test log included.
• All eight uploaded blobs were checked against the locally tested bundle by rebuilding the same tree from locally calculated Git blob hashes: tree SHA remained 1586d10e87f0fa0cd05f8c9ef7e240ed5222a27d.

The repository-wide test suite was not run. No new Monte Carlo or server access occurred.

Provenance and integration

Source checkpoint commit: 6147e22f53902a94e5f133739f2c1d423691d0b8, results/local-20260831/P334-cooperative-closure/scalar_state_collisions.json.

Integration parent: 75fbfec706d5d6dc2210db16a0d41c01ffba6804. New commit: 0e52dbaeed53dfffa94592e53e38129c179c5078.

Only eight new files. No existing topology engine, frozen scorer, raw archive, navigation ledger or historical result is modified. No dependency on unmerged code. The imported two states retain their original dependency group; this is not a new independent statistical block.

Claim boundary / next task

The two-cycle theorem is a conditional finite combinatorial implication, and the two actual certificates are exact. Existence of such a cycle pair for every HNF/checkpoint is not proved, nor is necessity for bipartiteness. Search failure is not automatically an odd-cycle witness.

Next, request these geometric certificates on the already selected 22 checkpoints, then prove the appropriate packing statement or return its minimal obstruction. For H2>0, overlapping at forced singleton bottlenecks must be allowed. Matching diagonals/digital cell semantics must be retained in any annular proof.

No CFT field, physical Jordan block, new arm exponent, threshold bound or autonomous finite-state scaling limit is identified.