Independent audit of Lemma CR2 and Theorem B2 (item 2 of the 1166 independent-check request)

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Summary

Independent audit of Lemma CR2 and Theorem B2 of #2162 (research/char2-jets, section 2.1-2.2), answering item 2 of the open independent-check request comment 18934590. New package research/cr2-b2-audit-seed4f435ab7/. Scoped independent verification, no kappa claim; certificates/selected-result.json, both suppliers and upstream/ are unchanged.

Coordination claim (posted before computation, released in the DONE note below): #1166 (comment)

Prepared by Aaron Wang, with help from opencode (substantial AI assistance). The audited statements, the model and the configuration families are DreamingOfClouds's work in #2162 (head 113a718265c62aec301c5695f4bfb9c421bef7ab), #2111 (head 98da76c2839ba38538b510420dec54546ccabf0b) and #2125. No predecessor implementation is imported; the model is re-implemented from the pinned statements.

Task card

• Drawn assignment: track bit-mechanisms, chooser OS nonce 4f435ab70c339026e70839638467699a, card bit-helper-contract, base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4, catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7, selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Scope status: changed, recorded in TASK.json with a reason. The drawn bit-helper-contract object (a helper-mixing word with an H2/K replacement interface) is covered by existing packages (research/r439-copyshare-h2sem-admission, research/bit-copy-live-stabilizer, research/bit-live-copy-operator-intertwiner and others found by scripts/find_overlap.py at entry); Independently audit anti-Fano rank-three completions #2193 records the same coverage finding for its own bit-helper-contract draw. I therefore take the named open request from the handoffs thread instead.
• Request items 1 and 4 are covered by open Independently audit anti-Fano rank-three completions #2193 and Prove jf0 pairing cost exactly 19 with fixed minors #2176 (jf0 cost exactly 19); item 3 was closed by Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative) #2272; item 5 needs a Lean/Mathlib toolchain that is not installed on this host. Item 2 was live and unclaimed at the 03:54 UTC overlap refresh (512 open PRs, 1002 discussion records, complete pagination). Neighbouring open Research note: Desargues 10_3 / Pappus 9_3 as T=3 decoder atoms — frame-universal char-2 (CR2) certificates and the one uncertified Pappus frame (conditional on #2162, no κ claim) #2274 uses Lemma CR2 conditionally (Desargues/Pappus certificates) but does not audit it.

Result 1: proof audit (written re-derivation, each implication checked)

1. Lemma GR is sound, and its rank comparison holds with equality. For a 0/1 matrix M and any field K of characteristic 2, rank_K(M) = rank_F2(M): the entries lie in the prime field of K, so Gaussian elimination over F2 is valid in K (equivalently a maximal nonzero minor over F2 is odd, hence nonzero in K). The printed chain in PR 2162 section 1 is correct with its last step an equality.
2. Lemma CR2 is sound. N = B ∘ P~Q~^T is diagonal with nonzero diagonal under (ii)-(iii), so rank N = v and GR gives cost(B) >= v.
3. Theorem B2's structural steps check out: binomial incidences are preserved by phi; the phi-diagonal vanishes exactly on the harmonic lines H (the hypothesis p(t) . q(t) != 0 excludes the opposite-sign case); every directed wild cycle lies inside F; and N[X, X] is triangular with nonzero diagonal for X a subset of the non-harmonic lines with no wild cycle.
4. One statement/proof gap: the MAS term. The printed argument proves cost(B) >= |X| and hence cost(B) >= v - |H| - |F| + masV, where masV is the largest A ⊆ F with W[A] acyclic (a vertex count). The printed statement glosses MAS(W[F]) as "maximum acyclic subset" of the wild-edge set, which read literally is an edge count masE, strictly stronger whenever masE > masV; the printed proof does not imply it. The verifier machine-checks on every instance that the vertex form equals the best bound obtainable from the printed X-form argument.

Result 2: exact falsification search (smallest instance first)

Declared deterministic domain: v in 4..7, entries in {-2,-1,1,2}, per-bucket quotas for no-wild, wild-acyclic and wild-cycle configurations (admissible-pair count free <= 14), plus targeted large-free and dense-wild instances (free <= 16..17) where masE > masV (built from a cross-product wild core). Every admissible pairing is enumerated exactly, so every minimum below is exact.

Bucket	Instances	Bound (i)/(ii)/(iii-vertex)/(iii-edge) violations
no-wild	24	0
wild-acyclic	24	0
wild-cycle	10	0

Machine-checked on all 58 instances: phi preserves every non-wild incidence and no wild one; the phi-diagonal vanishes exactly on H; every wild cycle lies inside F; the Xbest identity above. Tightness: the vertex form is attained once, the edge form twice; masE > masV on 4 instances (so the two readings genuinely differ in-domain). Controls (all pass): a two-line fixture with analytically known minimum 2 (positive fixture and sensitivity of the strengthened +1 bound), a dense three-line fixture with all six off-diagonal pairs wild (masE = 3 > masV = 1, exact minimum 3, tight for the edge form), a full-collapse phi0 disagreement control (48 domain disagreements), a second independent rank implementation cross-checked on sampled block ranks, and a frozen RECEIPT.json summary comparison.

Not checked: Lemma VCR2, Corollary J, Theorem T-prime, Lemma F, the split min-max of PR 2162 section 3, configurations with v > 7 or larger entries, and target (A) itself. This is a scoped verification of two lemmas, not a proof of (A) and not a theorem audit of PR 2162 as a whole.

Reproduce

Standard library only; from the repository root:

python3 -B research/cr2-b2-audit-seed4f435ab7/verify.py
python3 scripts/research_package.py --package research/cr2-b2-audit-seed4f435ab7 --check-only

Reference run (CPython 3.12, this Windows host): verify.py prints PASS_CR2_B2_AUDIT instances=58 buckets={"no-wild": 24, "wild-acyclic": 24, "wild-cycle": 10} and ends RECEIPT_MATCH; research_package.py --check-only prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check); the registered replay python3.12 scripts/research_package.py --package research/cr2-b2-audit-seed4f435ab7 ends PASS_REGISTERED_RESEARCH_PACKAGE research/cr2-b2-audit-seed4f435ab7 outcome=completed. make is not installed on this host, so no make-based target was run.

Open next questions (also in TASK.json)

1. Does the edge form cost(B) >= v - |H| - |F| + masE hold in general? It is strictly stronger than the printed proof and survived this domain with two tight instances. First step: extend the dense cross-product cores to v = 5, 6 and branch-and-bound over pairings. Blocked by exact enumeration past free ~ 18.
2. Is there a wild-cycle configuration violating the vertex form? A violation would falsify Theorem B2 itself; none appeared in 10 wild-cycle instances. First step: targeted structured v = 8 cores. Blocked by the same enumeration cost.

Remaining obligations

• The registered CPython 3.12 replay passes locally; hosted CI checks were queued at submission time and no hosted pass is claimed.
• Findings are provisional until maintainer review; claims on the discussion board are advisory.

Credit

Aaron Wang, with help from opencode. Statements audited, the model and configuration families: DreamingOfClouds (#2162, #2111, #2125). Items 1 and 4 of the same request were closed by #2193 and #2176. New work here is Apache-2.0.