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Prove optimal 52-bit refined recovery over mode-preserving frames (no new κ) - #6327
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Mister-Meeseeks merged 1 commit intoOct 31, 2026
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Includes an explicit sparse construction, independent all-output verification, and exact caller-compatibility and cost limitations. Prepared for @CubeJerry with substantial OpenAI Codex assistance. No new kappa.
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Resolve #5444's two-step mixed-refinement question for every fixed integer unimodular frame U preserving the actual eight mode projections. For H_U(x)=K(U K(2x)), the minimum worst-case recovery alphabet is 2,601,832,488,601,921 states (52 packed bits), versus 56 bits in the parity-I subclass. Optimality holds exactly when the binary images of M and U^-1 M are disjoint. Supplies an explicit sparse frame, full-output codec and a complementary finite-depth parity-I theorem.
Base: a7638a4.
#5444 source head: 9004eb1.
Exact fixture and predecessor Git/SHA256 pins: SOURCES.json.
Reproduce (Python 3.12, stdlib):
Expected:
PASS_REFINED_NATIVE_RECOVERY,PASS_INDEPENDENT_OPTIMAL_FRAMEandPASS_REGISTERED_RESEARCH_PACKAGE.Executed: 400 optimal-codec and 384 chain roundtrips; independent 20,001 scalar cases, 65,536 binary pairs, 256 cosets, 224 inverse checks and 1,152 trajectories. Actual code/source mutations reject omitted parity, invalid frames and altered fixtures. Exact three-stage counterexample refutes one proposed proof extension, not the longer-chain optimum. Normal/-O, committed-byte/mode, export and read-only Git-free replays pass. Fixed domains; universal proofs are separate. Full-history preflight was not run in the shallow checkout.
No new κ. The output function changes. A linear consumer is compatible iff OV=0, but recovery still requires the full output. Restoring the old output without extra side information restores its 56-bit lower bound. Relative refinement, routing, codec workspace and cleanup remain payable; no selected caller or improved invoice is supplied. All five assumptions remain.
By @CubeJerry with substantial OpenAI Codex assistance in construction, independent proof/checking and caller-cost audit. Credit zxv (#5444), Ali Romman (#5323), Brian (#5035) and retained lineage. No human/proof-kernel acceptance.
Proof, pins, controls and limitations: https://github.com/CubeJerry/integer-mult-bounds/tree/f184d2a0257c1cb56a820c5c123ec6d7ba878741/research/refined-native-recovery
Complete remote Git tree matches the tested contribution.