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Prove the sharp 240-return minimum in the frozen p2 oracle model - #5618
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This scoped negative result proves that the frozen p2 caller needs exactly 240 new returns in the enlarged arbitrary-linear-oracle model, for every width f >= 72. It closes the previous 225–240 gap. The original paid inputs, all 39 observations, ordinary algebra, capacities and targets stay fixed; feedback between new returns is allowed. Complete kappa is unchanged. This settles the return-count optimization in this contract and directs productive caller work to a changed contract.
Proof, exact verifier, receipt, source pins. Base: canonical main 6cf7a68, priority 7. Consumed PR #5587 is pinned to its proof at 49811a9.
The new second-moment certificate is 27/343 for each of the two intersection losses. It gives a complete target deficit of at least (82266/343)|H| > 239|H|. Every return supplies at most |H| dimensions, while the credited 240-return oracle construction attains the integer bound. An invertible row-space rescaling is used only in the rank proof; it does not add a free physical division by 11.
Normal and optimized registered checks pass. They check the pinned involution, full exact projector identities, balanced row spaces, all 12432 ordered off-diagonal product classes and 213248 independent signed basis actions. Direct pair-group summation agrees with the closed second moment. Four defective controls reject. Committed-byte, export-integrity and Git-free optimized replay checks pass.
Python 3.12 and the standard library suffice. Expected markers:
PASS_P2_EXACT_ORACLE_REPAIR_240andPASS_REGISTERED_RESEARCH_PACKAGE.The source full-history deficit and oracle upper construction, and PR #5526 pointwise Clifford faithfulness, remain credited premises. Their full scientific replays were not run here. This is not a native 240-call supplier. No physical compiler, selected cold replay, complete kappa calculation, full source-history ancestry or independent human/formal review was run. The checkout is shallow; hosted CI is separate. The earlier source tariff obstruction already excluded this caller; this contribution settles its exact oracle repair count.
Credit Dugongue's PR #4948 native source; Chris R / Clear1nsight's PR #4978 D-left audit and clarifications; Ali Romman's linked history and faithfulness proofs and PR #5449 history formulation; and josecarlosgonzalezroque-design's PR #5570 bridge audit. Retain Cao Yifan, Cinna Davis, DLamdata, zxv and their native/controller/Clifford/interval lineage recorded in the pinned source proof. Projection moments and the product rank inequality are classical. Coordination. Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.