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Prove sharp five-thirds bounds for two binary helper classes - #5455
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This scoped negative result proves sharp 5/3 bounds for two different binary helper-controller classes. It does not improve the complete kappa calculation. The proof, integer certificate, and replay receipt separate the claims.
The global theorem assumes commuting rank-three stage projectors and query terms supported on both sides by their stages, each of Jordan type
J_2(0) direct-sum [1]. If their sum is an idempotent target that commutes with every stage projector, its rank is at most5*f/3forfFULL terms. The target can include D-only coordinates. A support-component proof establishes the bound without a call order; three FULL terms on five fields attain it.The finite theorem permits outside-support helper feedback and paid THIN queries, but retains at most three live coordinates and removes completed coordinates immediately. All 20,020 legal transitions in the reachable 94-state grammar satisfy an exact integer potential. The 10,035 dominant edges have maximum
(outputs - THIN calls)/FULL = 5/3, with an attaining word. This is not an unrestricted controller bound. A later local-only catalyst extension reports broader 188-state coverage with the same ratio and credits the released 94-state search; this PR does not claim that successor result.Publication base:
32890bc5de034ab8fc93ac88ead7279a6a416d04. There are no pending-PR runtime dependencies. Source pins retain the base record, task card, and earlier proof as provenance. Cinna Davis / @1337isnot1337 supplied the earlier commuting-projector reduction and support-pair obstruction. Dillon's PR #5388 motivated the early-helper observation interface. Their physical source work and retained contributor notices remain separate from this abstract certificate.Use Python 3.12 and its standard library:
The registered command ends with:
Local checks passed: registered replay, normal/optimized byte equality, the full binary polynomial endpoint of the sharp motif, all 28,224 two-term support cases, committed-byte preflight, export integrity, and direct replay from the export without Git. Controls reject a changed potential, missing state, unpaid THIN query, invalid native direction, and wrong endpoint coefficient. The package also records a counterexample when the Jordan-type hypothesis is removed.
No complex Laurent lift, native payload execution, full caller tariff, supplier admission, or outer multiplication replay was run. Hosted CI is pending publication. No selected frontier file changes. New proof, code, and certificate by @aliromman, with substantial OpenAI Codex assistance; parallel AI review is disclosed and is not independent human review. Apache-2.0.