Research: F-star cert(M) falsifier census on exact 3-total decoders (scoped negative, no new kappa)

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Scope

Scoped negative result for the named open problem F-star of research/root-unit-decoder-lemmas (section 7, OPEN): an exact falsifier census for cert(M) >= floor(2v/3) on exact 3-total rational decoders. Package research/fstar-cert-census-seed298df375/, branch research/agent-51-fstar-cert-298df375, head 856cd760, base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (priority revision 4). No kappa, certificate, supplier, window or frontier change.

For a rational 3-total decoder M = sum_j p_j q_j^T on v lines with M_tt != 0, an L8 labelling chooses F2 vectors a(t), b(s) inside the factor supports and sets K_ts = a(t) dot b(s) mod 2; cert(M) is the largest ordered subset S with K_tt = 1 on S and K_ts = 0 at every incidence M_ts = 0 with t before s. The source package proves cost >= sum_j r_j >= cert(M) for every admissible pairing B and shows that F-star would close every open T = 3 window at once. It records F-star as OPEN with about 90 sampled configurations at cert = v.

Assignment and scope change. Chooser OS nonce 298df3754c1488b77620ddda3eb48b27 drew track adversarial-audit, card uniform-transfer-audit (scope key ce7411fb7cf6f3460bfc). All three drawn cards match recent claimed or completed objects, so per the pair-collision rule this session takes the explicit open-problem route instead (recorded in TASK.json): uniform-transfer-audit was completed today on its hosting-implication refinement (#2273), supplier-boundary-audit was completed today on its owner-map binding refinement (#2301), and selected-stock-envelope is covered by a live handoffs-thread claim (#1166 (comment)) and the first-stage coefficient work of #2084 and #2120. The adversarial-audit method is kept: an exact counterexample search at a claimed implication.

Method (exact, stdlib only, deterministic)

• Side B, lower bound on cert(M). Deterministic local search over L8 labellings; each labelling is scored by the exact maximum acyclic subset of the incidence digraph (subset DP); the maximizing subset and sink-peeling order are stored and replayed by an independent witness checker.
• Side A, falsifier hunt and upper bound on cert(M). The proven chain cert(M) <= sum_j rank_F2 B[R_j, A_j] for every admissible B (0/1, unit diagonal, B_ts = 1 only where M_ts = 0; the chain's proof uses only those properties, so asymmetric B are included). The cost is minimized exhaustively when the admissible pair count is at most 10 (always at v <= 5) and by deterministic single-bit local search otherwise. An instance with minimum cost below floor(2v/3) would falsify F-star.

Domain: v in 4..12, exactly 3 totals, factor entries in {-2,-1,0,1,2} with nonzero diagonal, fixed seeds 20261015 + 1000*family_index + 17*v + i. Three families: dense random; configuration-style M = P Q^T with P,Q over {-1,0,1} (the rank-3 configuration model of #2111); and tied-minima 2-adic constructions retained only when tied minimal valuations cover at least half the lines (the obstruction named in the source section 7).

Result

Scoped negative: no falsifier in the declared domain, and every instance is resolved by a machine-checked witness. Census digest 3237519deb9ec377f5a8a86b7b4b1e8197074ee8d0dd0e653cfcea1104c9e6e8.

family	instances	resolved with witness	falsifiers	min witness ratio
dense	70	70	0	8/9 = 0.889
config	70	70	0	6/7 = 0.857
tied	56	56	0	7/8 = 0.875

The worst instance is the config instance at v = 7, seed 20262138: witness 6 against threshold 4, so cert(M) lies in [6, 12] there (12 is its exact admissible-B cost minimum). The tied family reached 8 tied-minimum lines at v = 12 and stayed above threshold throughout. Because witnesses are lower bounds on cert(M), every recorded instance satisfies F-star under either reading of the quantifier (universal or unit-attached).

Controls (all required to pass): three tiny-domain instances at v = 3 are certified by exhaustive enumeration of every labelling and the search engine reproduces those exact values; a corrupted witness (one K_tt broken) and a wrong order at a binding pair are both rejected; the floor(2v/3) table is checked against hand values for v = 4..12; a hand-checkable diagonal decoder gives cert = 3; and the proven chain cert(M) <= cost(B) is rechecked on every instance, with one exact-chain control pinning brute_cert = 4 <= min_cost = 7.

What is claimed and what is not

Claimed: the declared-domain census, its witnesses and dual bounds, and the reusable engine. Not claimed: F-star itself (witnesses are lower bounds and only 196 instances are covered), exact cert(M) values above v = 3, any separation of the universal versus unit-attached readings, and any kappa, window or supplier consequence. The T = 3 window closure stays conditional on F-star.

Reproduce

From the repository root (Python 3.10+ standard library only; make is not installed on this host, so the underlying scripts are given):

python3 -B research/fstar-cert-census-seed298df375/verify.py
python3 scripts/research_package.py --package research/fstar-cert-census-seed298df375 --check-only
python3.12 scripts/research_package.py --package research/fstar-cert-census-seed298df375

Expected markers:

PASS_FSTAR_CERT_CENSUS instances=196 resolved=196 falsifiers=0 unresolved=0 min_ratio=0.8571 digest=3237519deb9ec377
PASS_FSTAR_CERT_CONTROLS bruteforce=3 diagonal_cert=3 chain_cert=4
PASS_REGISTERED_RESEARCH_PACKAGE research/fstar-cert-census-seed298df375 outcome=completed

All three commands were run on this host (CPython 3.13 for the first two, the registered CPython 3.12 replay for the third) and produced exactly those lines; PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check) is the metadata output. The repository suite python3 -B -m unittest discover -s tests -p 'test_research_*.py' also passes (65 tests, OK, 5 skipped).

Open next questions

1. Exact cert(M) for the worst instance (config, v = 7, seed 20262138, cert in [6, 7]) by branch-and-bound over labellings; it is the natural place to look for sub-v certificates. No structural blocker, just a longer exact run.
2. Universal versus unit-attached readings of the quantifier: wire A3/A4/D4 Gram tables into the pairing-cost minimizer and compare admissible-B minima on the same decoders. Blocked on unit tables not being in this package.
3. The tied-minima family at v = 13..18, where the named obstruction accumulates.

Credit and provenance

F-star, the cert(M) definition, L8 labellings and the proven chain are from the merged package research/root-unit-decoder-lemmas (root units, Lemmas A and B, the L8 lemma; PR 1590 by DreamingOfClouds, with Claude Opus 5.5 (Anthropic) via Claude Code assistance). The rank-3 configuration model and the tied_min / cex_min_1 examples with their certificate limits are DreamingOfClouds' work in #2111 and #2162; this package neither restates those examples nor supersedes those results. The adjacent independent CR2/B2 audit is #2293 (Aaron Wang). The source README is pinned as an input in SOURCES.json. The coordination claim is #1166 (comment).

Prepared by Aaron Wang with help from opencode (substantial AI assistance; no independent human review claimed). New work is Apache-2.0.