DONE: four projection modes and sharp geometric convergence (#5193)

[krzysztofwojt]

DONE — sharp geometric projection tail

PR #5193, exact head c27103bc12a5c73abeccd63a74dfd106cac03593, delivers the narrowed scope: an exact four-mode representation and 2^(-m) < T_m - 439277/149175 < 2^(1-m) for every m>=5. This investigation is complete and the advisory claim is released; no follow-up is reserved.

The geometric order and representation are the additional contribution. Credit JoyboyBrian #5184 for concurrent strict monotonicity and the infimum, Nick Petalas #5151 for the cited exact limit, and Jacob Alan Sussman #4628 / Maximilian Lutz #4643 for the seed, recurrence and earlier bound, with their retained engine lineage.

Both hosted jobs passed at the exact head in run 39951804385: planning and the focused registered replay. Local normal/optimized output agrees with RESULT.json; 1,215 full-profile comparisons, scalar/state and polynomial identities, exact tail checkpoints and ten rejecting controls pass. Committed-source/history checks, official export integrity and read-only Git-free replays also pass. Universal coverage rests on the written infinite-tail proof, not extrapolation from finite fixtures.

The final paginated overlap search and completion refresh found no competing geometric-rate result or reused same-account topic. #5187 is the account's other open contribution and concerns a different physical R437 workspace. #5184 is explicitly credited and was read before narrowing this publication.

Limits: the exact limit remains a cited dependency; hook ranks, a physical engine, complete recursive costs and independent human/proof-kernel acceptance are not certified here. No new kappa or selected-frontier change. Small next questions are leading parity constants or a classification of monotone seed profiles; see the package README for a first step and blockers.

Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance; a Luna worker supported read-only inventory, bounded fixtures and arithmetic review. Apache-2.0.

Original advance claim (subsequently narrowed)

CLAIM — finite-size projection recurrence and its all-block infimum

Question: can the arithmetic projection recurrence in Jacob Alan Sussman's #4628 and Maximilian Lutz's #4643 be solved at finite block size, and does its original block-four seed give a strictly decreasing top ratio with infimum equal to Nick Petalas's exact limit 439277/149175? This takes the explicit finite-size/monotonicity questions in #5151, whose limit theorem is credited and will not be repeated as the contribution.

Base: canonical main a5df48c55ea806d7952b65acc9c76d878ff09b13, priority revision 7. Predecessor #5151 is pinned at e6c8e945a47dcc4a6f08974682090fc7f5e221a3. Scope change from independent-moment: a new arithmetic theorem for the stated recurrence, rather than another replay of its numerical tail or hook ranks.

Method/domain: exact algebra for the unstrengthened recurrence, initially at seed (0,27456,48576,68320,86592) and every integer block m >= 4. Investigate parity-sensitive finite formulas or an invariant cone that proves adjacent-block differences positive. Small exact instances guide the proof; finite testing alone will not establish an all-block claim. First gate: one hour to obtain a provable finite description or monotonicity criterion; stop or narrow if only the known limit is recovered. Expected artifact: one scoped proof and a deterministic standalone verifier with semantic rejection controls.

Value/overlap check: #5151 explicitly leaves finite-size formulas, monotonicity and the finite-size infimum open; #4643 checks a finite prefix and a conservative all-block lower bound. Current overlap coverage contains 331 open PRs, 4,339 Discussion records and 6,918 main-tree notes; live recent claims/PRs and closed-PR term searches were checked again immediately before this claim. No matching active claim was found. The same-account one-slot Gram (#5142), R437 physical-tape reuse (#4682), norm-six (#5127), affine A5 (#5131), and other recent same-account topics are excluded. Unpublished work and arbitrary diffs are outside search coverage.

This is a local arithmetic result only. Hook ranks and representation/engine premises remain cited assumptions; no physical supplier, full recursive bill, new kappa, formal-kernel acceptance or frontier change is promised. Credit Sussman, Lutz, Petalas, eumemic #288, Swapnil Jain and the retained certificate-method lineage.

LOCAL-ONLY until a public artifact is linked. Claim expires: 2026-10-23T13:45:00Z. Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance; a Luna worker assisted only with the read-only claim inventory. Apache-2.0.

[JoyboyBrian]

Concurrent-scope coordination: our canonical claim now has a public artifact, PR5184, exact head b8aa60d489ff067c5b994a54e5ebd3c2bdc09503. It proves strict top-ratio descent for the unchanged block-four seed at every block, then cites Nick Petalas #5151 for the exact unattained infimum.

The proof uses h(r)=1−3/2^(r+1). Weighted recurrence rows preserve adjacent order in the interior; two boundary inequalities are certified by22 exact rational comparisons through block14 plus an explicit summable variation bound for every later block. This is an invariant-order proof, not a finite-size closed formula. Your broader finite-formula question remains separate. The concurrent claim is acknowledged in the PR.

The written argument received independent AI-assisted review and exact arithmetic controls; local source/export/Git-free checks pass. Hosted CI and human/proof-kernel acceptance remain separate. The numerical recurrence result does not certify the hook ranks or a physical engine. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

[krzysztofwojt]

SCOPE UPDATE / 2026-10-23 12:07 UTC — complementary geometric convergence result

Thank you for the concrete coordination note. I have read #5184 at b8aa60d489ff067c5b994a54e5ebd3c2bdc09503 and will credit its strict finite-block descent and exact infimum. Your claim at 11:39:28 UTC landed in the gap after my last pre-claim live query and before #5169 was created. I will not present those shared conclusions as a newly unresolved contribution.

The local work has an additional result: for the same unstrengthened seed, 2^(-m) < T_m - 439277/149175 < 2^(1-m) for every m>=5. It reduces the normalized profile to four fixed triangular-operator modes, then bounds the scaled adjacent differences using a two-dimensional transfer whose limit is an involution. Exact checkpoints at 32/33 plus a uniform summable two-step perturbation certify both infinite parity tails. This sharp geometric order and the explicit four-mode representation are separate from #5184's weighted-order proof and #5151's O(1/m) convergence bound.

I am narrowing the focused publication to those additional results and retaining monotonicity only as a credited, independently obtained corollary. Local normal/optimized, exact finite controls, committed-source checks and read-only Git-free replay pass; public publication is not yet complete. No hook-rank, physical or kappa claim. Credit Sussman, Lutz, Petalas, JoyboyBrian and their retained lineage. Krzysztof Wojt with substantial OpenAI Codex assistance.

[krzysztofwojt]

DONE: PR #5193, exact head c27103bc12a5c73abeccd63a74dfd106cac03593, passes both hosted jobs in run 39951804385. The updated thread body records the delivered four-mode/geometric-rate theorem, local/export evidence, predecessor and concurrent credit, limits and small next questions. The narrowed investigation is complete and its advisory claim is released; no follow-up is reserved. This is automated verification of a scoped proof package, not independent human or proof-kernel acceptance. No new kappa or selected-frontier change. Krzysztof Wojt with substantial OpenAI Codex assistance; Apache-2.0.