✈️ Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim)

[DreamingOfClouds]

Characteristic-2 shadows of rank-3 decoders: the anti-Fano bound by first-order jets, with a Lean-checked core (no κ claim)

Package research/char2-jets. Model, cost formula and lemmas are those of the merged #1590 (exact decoders, Lemma A, Lemma B), of #2111 (three-total decoders, the sharp example tied_min) and of #2125 (where characteristic 2 starts to pay); the Lean project formalises the rank argument behind Lemma B and its characteristic-2 twin. This PR studies the incidence pattern of a configuration over a field of characteristic 2.

PROVEN (proofs in the README). Lemma CR2 (a characteristic-2 realisation of the incidence pattern gives cost ≥ v for every B) with its tropical / 2-adic corollary, and Lemma A (T ≤ 2) as a one-line corollary; Theorem B2 (the map φ with φ(−1) = 1: cost ≥ v − |H| − |F| + MAS(W[F]); no wild incidence and no harmonic line gives (A)); Lemma VCR2 (a valued realisation whose diagonal dominates every directed cycle gives cost ≥ v) and Corollary J (a B-independent linear-algebra jet test); Theorem T′: tied_min (the anti-Fano configuration of #2111) has cost ≥ 12 for every admissible B, by a first-order jet around its Fano collapse, with no SAT call and no enumeration; Lemma F (the first-order obstruction is a binary flow of even incidence count). The generic-rank / split bound is proved; its completeness (equivalence with (A)) rests on a standard matching min-max that is quoted, not proved, and checked on instances.

Lean (checked). lean/DecoderDuality.lean formalises the core lemma hadamard_diag_rank_sum, Lemma B and Lemma CR2; no sorry, axioms propext, Classical.choice, Quot.sound only. It does not formalise B2, VCR2, Corollary J, T′ or Lemma F. verify.sh does not need Lean (LEAN=1 runs the build if lake is present).

CHECKED (exact, verify.sh). T′: the explicit jet satisfies (J1) on all 72 incidences and (J2) on all 12 lines, and 518 stored matrices B (217 optima, the full B, 300 random) have rank 12; stored CR2 realisations of 21 configurations are verified (all realise over GF(2) except cex_min_1 over GF(4), so cost(cex_min_1) ≥ 19 for every B); tied_min is not CR2 over GF(2), GF(4), GF(8), GF(16); the 44 minimal non-CR2 sub-configurations (32 of six lines, tight); Corollary J on all 4095 sub-configurations of tied_min and on 1500 random all-harmonic configurations; Lemma A and B2 spot checks. Recorded, not re-run: the SAT enumeration of the 217 optima, 1889 of 1892 random configurations passing the jet test, the 400 B2 configurations, 300 wild-cycle configurations (all CR2), the split min-max on 300 instances.

Jet failures (CHECKED). On three random configurations jf0, jf1, jf2 a harmonic line cannot be moved at first order, and the order-2 diagonal vanishes identically on the first-order tangent space, so jets of order 1 and 2 give no B-independent certificate there. (A) still holds on them: exact minimum cost 12 (jf1, v = 9) and 14 (jf2, v = 9); cost ≥ 16 for jf0 (v = 11; "cost ≤ 15" is UNSAT; the exact value is unknown).

Not claimed / OPEN. No κ gain, no word, no window closed. (A) at T = 3 is neither proved nor refuted. Open: the "2-adic shadow" conjecture (every configuration over Q has a characteristic-2 valued realisation whose diagonal dominates every directed cycle); whether the Lemma F flow obstructs at every order on jf0..2 (orders ≥ 3 untested); a theorem for wild cycles (three-term incidences), which φ breaks; the min-max behind the split bound.

Verify. bash research/char2-jets/verify.sh /fresh/dir (python3 with python-sat and numpy, and cc; about one minute; step 0 checks MANIFEST.sha256 and stops hard on mismatch). Optional: LEAN=1 bash research/char2-jets/verify.sh /fresh/dir, or cd research/char2-jets/lean && lake build (Mathlib at the pinned revision).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.
🤖 Generated with Claude Code

[Mister-Meeseeks]

Maintainer review of the pinned head 113a718265c62aec301c5695f4bfb9c421bef7ab: the independent CR2/B2 audit #2293 and the counterexamples #2334/#2340 have now been replayed, reviewed within their stated scope, and merged.

Please make B2(iii)'s MAS definition explicitly the cardinality of a maximum induced acyclic vertex set. The triangular-submatrix argument supports that reading. The acyclic-edge reading is false: #2334 gives an admissible nine-line instance with cost 21 versus edge RHS 27 and an all-k family; #2340 independently gives five- and six-line counterexamples, with complete pairing minima. Repeated indexed lines are permitted by the stated model. Neither audit refutes the vertex version or establishes a supplier consequence.

Credit Aaron Wang and ROTl24 for these complementary audits. The remaining full proof/formalization review of this PR is separate; this comment records a specific statement clarification, not a rejection of the whole contribution.

[DreamingOfClouds]

Thank you, and apologies for the lack of clarity in the original wording.

Fixed at head 642bfaf4325f0c8294ab72e8911c3eab4fea354f. Theorem B2(iii) now states that MAS(W[F]) is the maximum size of an induced acyclic vertex set: a set Y ⊆ F of lines whose induced wild subgraph has no directed cycle. The bound is applied with X = ([v] \ (H ∪ F)) ∪ Y, which is the vertex reading the proof uses (N[X,X] is triangular in a topological order of W[X]). The text now also says explicitly that the acyclic-edge reading is false, citing #2334 and #2340, with credit to Aaron Wang and ROTl24. No other statement changed. MANIFEST.sha256 is updated, and verify.sh from a fresh export passes (58 PASS, 0 FAIL).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.