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This was referenced Nov 6, 2026
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Extend the auxiliary-projector obstruction in #6726 to full-bank ordinary bases, arbitrary coupled width and singular or central-dependent auxiliary prefixes. A new query-debt bound prices both forward and inverse restoration, including mixed L/SL channels. An exact two-channel escape attains the query-count floor but cannot pay for itself.
For redesigned histories, prove three-query header saturation, a coupled carrier bound, and an obstruction to arbitrary six-query interleavings of two modules. A separately scoped fourth-root character argument prices extra positive body cycles. These results narrow explicit routes to a stronger bound; they do not claim a global binary optimum.
Base: 7c49fe3.
#6726 source: 435bbe9.
SOURCES.json and ORIGINS.json bind all retained inputs and original paths.
Reproduce (CPython 3.12, stdlib):
Expected: five exact-checker PASS results and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed all five checkers normally and with -O, registered replay, committed-byte/mode checks and publication-pin replay. Controls include word/source mutations, 17,391 finite-field rank fixtures, exact interpolation, four-field header checks and explicit counterexamples to omitted assumptions. Separate proof reviews cover fixed prefixes, saturation, interleavings and finite characters. Universal claims rest on the written proofs.
No new κ. Controls are F-valued and S-independent; coupled width uses a common S. Changed-history results state their atomic-stage, positive-body and tariff hypotheses. Two equal-orientation fragments, larger interleavings and more general physical controls remain open. No native admission, framework assumption, selected cold replay or full-history preflight is discharged. The paying root geometry of #6726 is not contradicted.
By @CubeJerry with substantial OpenAI assistance and separate mathematical review. Credit Cinna Davis, Ali Romman (#6348/#6398), thu-nmrc (#6716), zxv (#6726) and the preserved lineage. Apache-2.0.
Proofs, pins, controls and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/1153341fb288bd4cd3c9eba490b6dec4e2e6d43c/research/binary-full-bank-retirement