Exact: full survival laws do not close the topology state (#429)

[LightChainr]

Scientific result

Advances #429, downstream of #401/#403 and PR #415. This is not another c2/c3 statistic extension.

Two N16 square-torus configurations have exactly the same complete remaining rank-one survival vector (1,7/8,9/14,5/14,4/35,0,0,0,0). Every unbranched future rank trace therefore has the same law. Nevertheless, one shared insertion followed by two independently continued clones gives joint survival probabilities 95/196 and 93/196, an exact 1/98 gap.

A selected-prefix calculation also proves non-Markovianity of the recomputed survival-signature process under the actual uniform-permutation law: at the same current signature, the probability that the next state is safe with three one-site exits is 1/6 versus 2/11 for first-birth cohorts K1=4 and K1=8, a 1/66 gap.

The six-quotient census separates full-survival equivalence from the coarsest strong Markov partition: N16 210 -> 214, N17 346 -> 390 rank-one classes. These are finite quotient classes, not physical field counts or an all-HNF minimality theorem.

Contents

• Reuse p334_birth_age_collision_review_20260830.py byte-identically from original blob 62e06795fdfa91a956aedd62b7344e84aa5efc5c / commit fee33287; no production topology changes.
• Exact integer survival recurrence, backward Markov partition refinement, branching witness and actual-law prefix weights.
• Independent C++ potential-union-find and direct selected-prefix enumeration, with physical-coordinate rather than opaque-mask witnesses.
• Exact JSON, environment/source hashes, test log and SHA-256 manifest.
• Research note reviewing the updated rooted-module, spatial-gauge and typed-arm frontiers, with 12 checked arXiv sources and a proposed branching-aware acquisition.

The pointwise observable-algebra criterion is an application of finite-state lumpability, not claimed as a new abstract theorem. A filtered belief predictor for rank-only traces remains valid; the counterexample concerns the signature recomputed on the actual microscopic successor.

Validation actually run

python scripts/p429_branching_continuation.py --output results/p429-branching-continuation/exact.json
python -m unittest discover -s tests -p 'test_p429*.py' -v
python -m compileall -q scripts tests

13 focused tests passed, zero skipped. Includes:

• independent BFS/potential-UF rank and primitive-line agreement on all 65,536 N16 configurations;
• exhaustive enumeration of 7,741,440 prefixes in the selected 192-state stratum, not all 16P8 prefixes;
• direct enumeration of both witnesses' 80,640 future suffixes and all delayed-fork choices;
• exact strong-partition transitions, translation invariance and deliberately corrupted certificates.

C++17, g++ 14.2.0, -O2 -Wall -Wextra -Wpedantic, clean compiler output. One development serialization test exposed tuple/list differences; normalized to JSON lists without altering scientific counts. Repository-wide CI and production simulation were not run.

Compatibility and scope

Based on f854f823a90be3395178fdfce04f1bae21435d46; all changes are additive. Existing frozen predictions, raw blocks, pilot scores and result chronology are unchanged. The legacy oracle is the same content as PR #415's inherited source, not a competing implementation.

No Monte Carlo, remote production, continuum-memory exponent, universal state dimension, Q4/Jordan identification or permission-policy change. The next scientific discriminator is a delayed-fork experiment at declared scaled horizons, with nested checkpoint/shared-prefix/clone covariance. Parent #429 stays open.