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…ammar Prepared for @CubeJerry with substantial OpenAI Codex assistance. No new kappa; exact construction and fixed-source all-size cost bound.
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Resolve the equal-orientation question in research/binary-full-bank-retirement (#6764). Give a 13-operation arbitrary-input word attaining four central queries, with exact complex source phases (21 normalized events plus two final auxiliary normalizers). Its full local change is 2[i(a)-i(a+b)]-1, including every fresh target, idle and extra query.
Prove that any mixture of this node and ordinary binary nodes, with any positive six-child allocation under the unchanged #6726 root/native contracts, cannot improve the raw ceilings: complex 0.0033072446849 on κ∈[0.0033,0.004], BIT 0.0037667234776 on [0.0037,0.004]. Exact finite dominance and a concave all-size tail preserve the genuine local-gain examples at 2^400 leaves. This settles the complete cost of the new mechanism, not just its algebraic existence.
Base: 7c49fe3.
#6726 source: 435bbe9.
#6764 lower-bound source: 1153341.
SOURCES.json and ORIGINS.json bind retained bytes and original paths.
Reproduce (Python 3.12, stdlib):
Expected: PASS_FOUR_QUERY_FRAGMENTATION_CEILING and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed normal/-O identical replays, registered replay and committed-byte/mode checks. Independent reverse-functional verification covers all 36 coefficients; separate proof and interface reviews plus Decimal-85 corroboration support the rational economics. Controls reject 13 gate omissions, eight phase reversals, eight reverse/invoice mutations and three economic omissions. A free-extra-query control marks the tariff boundary. No construction search is claimed: this is a direct word and all-size proof.
No new κ. Formal atomic interfaces and fixed source tariffs only; ordinary work is optimistically free in the economic upper bound. Physical caller integration, ownership/compiler/precision/analytic admission remain inherited. Changed roots, new native leaves and shared-query discounts remain open. No selected cold replay or full-history preflight is discharged.
By @CubeJerry with substantial OpenAI Codex 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, exact pins, controls and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/d87e4a38890f6962162d00a68031ab2476f58f6e/research/four-query-fragmentation-ceiling