[lm1909]

Change and mathematical scope

Audit, no new κ. This is an independent check of one hand-argued claim in Jacob Alan Sussman's open #4628 (head ee4fd87f064017c23e5a86fc1dcfe0a5237caed3): "m ≥ 4: at least 2.9447 calls per array (2.99172 at m = 5, 2.97290 at m = 6)", in the engine game.

Verdict: the numbers reproduce exactly and a sound route to the bound exists. Two inputs are uncertified, and the cited written proof is not public. No error found.

• Projection counts N_r(s). I derived the closed form myself and brute-forced it over all 15 / 135 / 2,295 Lagrangians at m = 2, 3, 4, for every nonzero p, plus a 60-Lagrangian sample at m = 5. All match.
• Exact recursion, own code. T_4 = 41/13, T_5 = 834241/278850 = 2.991720 and T_6 = 1188222/399685 = 2.972896. T_m decreases; the minimum over m = 5..60 is 2.944709234.
• All-m tail. The author's tail.py checks the coupling's transition inequalities for k ≤ 80. I proved them algebraically for every k and checked them exactly for k ≤ 300, giving c* = 2.944709234 > 2.9447.
• Soundness. An independent route, argued by hand in README §2: one copy of the block-(m−1) certificate per nonzero Pauli, Burnside–Brauer for programs right at every width, and Bruhat transitivity. It gives the recursion exactly, with the cap input as an upper bound and the demand inputs as lower bounds.

Gaps (none refutes the claim):

1. The cited written proof is not public. At commit 86e575e2 of the author's public repository only the programs and outputs are present. Its README lists the cited PROOF.md theorems under proofs/induction/ of a private archive.
2. Two block-4 inputs rest on one author program. T_5 uses a_3 ≥ 68,320 and T_6 also uses a_2 ≥ 48,576. Both are mod-p rank lower bounds from the author's hookG.py; no certificate is stored, and Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628's verify.py does not check them. Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 cites 48,576 as a certified upper bound, but the induction needs a lower bound. With only the triangle bounds (31,680 and 59,136) the route gives 2.832 at m = 5, 2.772 at m = 6 and 2.708 at m = 60.
3. The tail check stopped at k = 80. This package closes that with the all-k proof.

Open shapes: the smallest are 2 in 7 and 3 in 9 at m = 5, and 2 in 8 and 3 in 9 at m = 6, matching #4628. This route cannot exclude 3 in 9, because its limit 2.9447 is below 3 and falls with m.

Not covered: recomputing the block-4 hook ranks a_2, a_3, a_4 (dimension 102,960), the Lean proofs, the author's private proof, and #4628's other claims (W+m and 2W, one array, no-phase m = 3, widened-game keptness, section 7).

Task and overlap

• Card: independent-moment, scope changed (reason in TASK.json): an independent-verification check of a lower-bound claim.
• Base: fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, current main at submission.
• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 had no comments or reviews at submission, and no Handoffs and requests for independent checks #1166 post covers it.

Validation

python3.12 scripts/research_package.py --package research/projection-recursion-check (stdlib only, about 35 s) prints PASS_REGISTERED_RESEARCH_PACKAGE locally on Linux, and check_research_commit.py passes.

• Nothing is imported from Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 or the author's repository. Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628's README.md and PROOF.md are pinned as head: provenance.
• Ten failure controls are all rejected.
• These are exact finite checks plus a hand-argued route; there is no Lean.

Credit

Jacob Alan Sussman (#4628), eumemic (#288, Theorem R / FO lemma), Swapnil Jain (P1/P2 via #288), and OpenAI (engine letters). Prepared by lm1909 with Claude Code (Claude Opus 5.5) assistance; there has been no independent human review. Apache-2.0.

Open next questions

The smallest useful follow-up is to certify a_3 ≥ 68,320 and a_2 ≥ 48,576, for example with a stored rank certificate checked by an independent program; T_5 and T_6 hang on them. The 3-in-9 shapes need a different route.

🤖 Generated with Claude Code