Certify cooperative homology continuation beyond one-step H2

[LightChainr]

Purpose and linked work

Exact finite-volume contribution to #401, with a frontier review linking #408 and the specialization control on #398. These issues remain open for their larger mechanism/production questions.

Commit: d09f9252dd18de497ab88bd30e92101813432a81. Base snapshot: 6f711e524d9b33693703109521e592ca893e9cb0.

What changes scientifically

The current-k0 pilot 625d602 already measures h2, the count of one-site additions that create rank two. This PR does not duplicate that pilot. It proves exact two- and three-step continuation identities and exhibits actual-history dependence after controlling progressively richer current-state statistics:

• N10, fixed (k,rank,line,h2): two-step cohort gap 1/44;
• N13, fixed (k,rank,line,h2): two-step gap 1/154;
• N13, fixed (k,rank,line,h2,cooperative-pair count): three-step gap 2/315.

The third-step identity isolates cooperative-pair overlaps, triangles and minimal triples. A two-independent-futures acquisition identity directly measures unresolved predictive geometry; its exact N10 residual variance is 1/450, falling to zero at horizon two once the cooperative-pair count is included.

Files and provenance

• New continuation scorer, 13 focused tests, integer/rational result certificate and local environment/hash metadata.
• One research note with repository chronology, proposed discriminators and 15 primary arXiv anchors, including checked 2026 revisions/publications.
• The existing topology oracle from fee33287 is included unchanged, using its original Git blob 62e06795fdfa91a956aedd62b7344e84aa5efc5c. It was absent from this main snapshot. No new topology implementation or unrelated branch ancestry is imported.

All six paths are additive relative to the base. No production engine, RNG domain, frozen prediction or historical result is modified.

Validation actually run

python scripts/p401_cooperative_continuation.py --output results/p401-cooperative-continuation/exact.json
python -m unittest discover -s tests -p 'test_p401_*.py' -v

13/13 focused tests pass locally. Four quotients contain 9,760 subset states; 2,822 two-step and 2,812 three-step identities pass. The independent N10 check enumerates all 30,240 length-five prefixes. The independent N13 check enumerates all 37,440 prefixes within the declared controlled stratum, not all 13P6. Tests also verify the inherited 1/57 witness, future-order survival probabilities, a full Bernstein expansion, unchanged topology hash, result regeneration and rejection of corrupted rank/line inputs.

This was run in a focused local source bundle, not a complete repository checkout. Repository-wide tests/CI, production simulations and remote compute were not run.

Claim level and boundary

Exact finite-volume identities and independently checkable counts; retrospective discovery, not a prospectively frozen production result. No continuum memory law, square-site universality theorem, typed-arm probability lower bound, physical Q4/Jordan identification or Markov closure of the reliability vector is asserted. The conditional six-/eight-arm proposals and formal-jet specialization control remain clearly separated from the tested continuation certificate.

Integration

The research branch is self-contained. It does not require merging the old fee33287 branch or the current geometry-pilot branch. No migration is required, and #401 should not be closed by this PR.