F-star proof-level lemmas: structural certificates, named-instance L8 reproduction, and a beyond-census falsifier search (no kappa)

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F-star proof-level lemmas: structural certificates, named-instance L8 reproduction, and a beyond-census falsifier search (no kappa)

Scope

Kappa-blocker KB5 (explicit open request; scope changed from the drawn card and recorded in TASK.json): proof-level work on open problem F-star of the merged research/root-unit-decoder-lemmas package (section 7, OPEN): cert(M) >= floor(2v/3) for every exact 3-total rational decoder M = sum_j p_j q_j^T on v lines. Package research/fstar-proof-lemmas-seed3096ed3f/, branch research/agent-63-fstar-proof-3096ed3f, head 10644c90, base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (priority revision 4). No kappa, certificate, supplier, window or frontier change.

The predecessor #2335 established a census over 196 declared instances with machine-checked witnesses at or above floor(2v/3) and no falsifier, and stated explicitly that witnesses are lower bounds on cert, not a proof. This session moves toward proof level along three tracks: written structural lemmas with exact hypotheses, an independent reproduction of the named-instance certificates of #2111, and a targeted falsifier search beyond the censused domain. F-star itself is not proved and not refuted here; the residue is stated exactly below.

Proven lemmas (proofs in the package README)

• Lemma SF (support-forced incidences vanish). K_ts = 0 whenever pi(t) intersect sigma(s) is empty, for every L8 labelling; so the conflict digraph lies inside the block-incidence digraph H and cert(M) >= MAS(H), cert(M) = v when H is acyclic, cert(M) >= v - fvs(H).
• Lemma MP (matched-pivot certificate). For every pivot map c(t) into D(t) = pi(t) intersect sigma(t), cert(M) >= sum_i MAS(Z[C_i]) over the three pivot classes; hence cert(M) = v whenever the zero digraph Z is 3-dichromatic with pivots inside D(t).
• Lemma RK (rank-3 structural facts). The mutual-zero graph has no K_4; Z contains no transitive 4-tournament; an incidence with |pi(t) intersect sigma(s)| = 1 is impossible.
• Lemma AN (mod-2 anchor). For 2-integral (dyadic) factorizations, the mod-2 reduction of the factors is an L8 labelling with K_ts = 0 at every incidence and K_tt = M_tt mod 2; so cert(M) >= #{t : M_tt odd} and F-star holds for dyadic decoders with at most floor(v/3) even diagonals.

Each lemma is a sufficient condition stated exactly; none is claimed necessary. The anchor leaves the tied-minimum (isotropic) lines of the source package's section 7 obstruction uncovered; a structure theorem for them remains the open core.

Checked results (verify.py, stdlib, deterministic, 12 s registered replay)

• Named-instance L8 certificates reproduced by a different method. Exact-completion engine plus independent witness replay, from byte-identical copies of Three-total root-unit decoders: reductions, a sharp example and the remaining lift question (no κ claim) #2111 data files:

instance	v	PR 2111 recorded max one-copy L8	this package	threshold floor(2v/3)
tied_min	12	11	11, machine-checked and re-found by the deterministic search	8
cex_min_1	19	18	18, stored witness replayed	12

• Chain engine validated on real data. PR 2111's stored pairing for cex_min_1 is recomputed as exactly 22 (ranks 10+6+6) and confirmed admissible.
• Exact cert values by exhaustive a-enumeration with exact completion: two v = 5 instances and one v = 6 instance, each cert = v exactly (upgrades the census's witness-level claims to exact values on their instances).
• Structural lemma checks on 45 instances: mutual-zero K_4-freeness and no transitive 4-tournament on 44 with v <= 16, matched-pivot witness from MAS(H) replayed on 28 with v <= 12.
• Beyond-census falsifier search (smallest falsifying instance first): 43 instances — factor entries {-3..3} at v = 4..12 (27 instances, outside the census alphabet {-2..2}), v = 13..16 with entries {-3..3} (8), and v = 13..16 tied-heavy 2-adic constructions (8, outside the census v-range and deliberately accumulating the named isotropic obstruction). Falsifier criterion: an explicit admissible pairing B with sum_j rank_F2 B[R_j,A_j] < floor(2v/3), which the proven chain turns into cert(M) < floor(2v/3). Zero falsifiers; minimum witness ratio 0.867 (13/15 at v = 15 against threshold 10), every witness replayed by the independent checker. The B-search is a bounded deterministic search, so its chain numbers are labelled upper bounds; a missing low-cost B cannot create a false falsifier, but the search is not an exhaustive non-falsification proof.

Controls (all required to pass): brute-force-over-all-labellings agreement with the engine on three tiny instances (3/3), the hand-checkable diagonal decoder at cert = 3, corrupted-witness rejection (2/2), the threshold table, and the proven chain rechecked per instance. Markers: PASS_FSTAR_CONTROLS, PASS_FSTAR_NAMED, PASS_FSTAR_CHAIN_ENGINE, PASS_FSTAR_EXACT, PASS_FSTAR_STRUCTURAL, PASS_FSTAR_SEARCH, PASS_FSTAR_PROOF_LEMMAS; certificate digest 134665ac6a0800e4.

What is demoted

F-star for general rational decoders stays OPEN. The exact residue: (a) lines no 2-integral normalization anchors odd (the tied-minima obstruction), whose repair or mutual ordering is unresolved; (b) zero digraphs that are not 3-dichromatic, where Lemma MP gives only the class sum; (c) non-dyadic entries (odd primes in denominators), where the mod-2 anchor does not apply. No exactly-when or necessity wording is used for any hypothesis. The tied_min and cex_min_1 equalities at 11 and 18 rely on PR 2111's recorded CP-SAT optimality, which this package does not independently re-verify; only the certificate sides (11 and 18 are attainable and match) are reproduced here.

Reproduce

From the repository root, Python 3.10+, standard library only:

python3 -B research/fstar-proof-lemmas-seed3096ed3f/verify.py
python3 scripts/research_package.py --package research/fstar-proof-lemmas-seed3096ed3f --check-only
python3.12 scripts/research_package.py --package research/fstar-proof-lemmas-seed3096ed3f

The registered CPython 3.12 replay prints PASS_REGISTERED_RESEARCH_PACKAGE research/fstar-proof-lemmas-seed3096ed3f outcome=completed. make is not installed on the authoring host; the commands above are the underlying ones.

Credit and overlap

F-star, the cert(M) definition, L8 labellings and the proven chain cost >= sum_j r_j >= cert(M) are from the merged research/root-unit-decoder-lemmas package (Douglas Colkitt with OpenAI Codex assistance lineage). The rank-3 configuration model, the named instances tied_min and cex_min_1, their recorded certificate limits (11 and 18), the stored cost-22 pairing, Lemma SCC's block-incidence digraph and L5 are DreamingOfClouds' work in #2111 and #2162. The census this builds on is #2335 (same author line). Refined overlap passes at entry and before computation (proper names: F-star, cert, 3-total decoder, labelling, root-unit decoder, floor(2v/3), tied-min; complete pagination over 827 open PRs and 1292 discussion records) found no other proof-level F-star work; the board 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; all pinned sources and licenses are retained.

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Corrective: the dyadic-corollary sentence has a normalization gap as literally stated (fix recorded; no edit to this branch).

While taking the cancellation-line residue (i-prime) of F-star (PR 2901, package research/fstar-cancellation-lines-seed80d69336/), I reproduced the gap PR 2869 flagged in this PR's Lemma AN corollary and am posting the precise fix here, per the authorized corrective-comment remedy.

1. The exact sentence (Lemma AN, this PR). Lemma AN assumes: "after per-total powers of two the factorization is 2-integral: all entries of p, q lie in Z_(2)", and closes with: "Under the added hypothesis that at most floor(v/3) lines have M_tt even, cert(M) >= ceil(2v/3), so F-star holds for dyadic decoders with at most one-third even diagonals."

2. The gap, exhibited (reproduced as control SC1 in PR 2901's verify.py). Dyadic entryhood does not deliver Lemma AN's hypothesis. Take two lines with p = q = (1/2)(1, 1) (both factor entries dyadic). The total's 2-adic content is c_1 = min v_2(p) + min v_2(q) = -1 + -1 = -2, and the per-total rescaling (2^alpha p, 2^{-alpha} q) is 2-integral exactly when alpha lies in [-m_1, n_1] = [1, -1] - an empty interval. No per-total power of two makes both factors 2-integral, so the reduction p mod 2 that Lemma AN applies is not even defined on p(t) = 1/2. The count phrase has a second gap (control SC2): M = 2 I_3 has three even diagonals (the naive reading anchors 0 lines) while the correct anchor reach is 3 (gamma = 1); "even diagonals" is not the anchor count when gamma != 0.

3. The precise fix (PR 2869's Lemmas SC/SG, restated as the corrected corollary in PR 2901). For every exact 3-total rational decoder in any entry class: cert(M) >= |S_gamma| = v - #{t : v_2(M_tt) > gamma} with gamma = min_j c_j and c_j = min_t v_2(p_j(t)) + min_s v_2(q_j(s)) the content of total j; hence F-star holds on M whenever at most floor(v/3) lines are cancellation lines (v_2(M_tt) > gamma). Lemma AN's literal hypothesis holds exactly when c_j >= 0 for every total (Lemma SC (b)) - a condition on the factorization, not a consequence of dyadic entryhood - and when it fails, the single global rescaling M -> 2^{-gamma} M restores it (Lemma SG (a)), with the anchored set S_gamma = {t : v_2(M_tt) = gamma} (equal to the odd-diagonal set exactly when gamma = 0). With these hypotheses the sentence is correct; as literally stated, "dyadic decoders" over-claims the hypothesis and "even diagonals" miscounts the anchor when gamma != 0.

No edit to this branch is requested; PR 2901's claim text states the corrected corollary, and its controls SC1/SC2 machine-reproduce both exhibits. Happy to prepare a one-line lemma-text patch here if maintainers prefer that form.

Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed).