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Source: petalas/integer-mult-bounds@e7b08ac. Original attribution and bytes preserved.
Resolve the irreducible-quadratic pair obligation in CrocSwap#6698. Preserve the scalar-idempotent model, exact predecessor provenance and all scope limits. Prepared for @CubeJerry with substantial OpenAI Codex assistance.
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Extend #6698's common-rational-flag theorem to a common flag over Qbar, retaining F_2 coefficients. Every pair of invertible rational m-by-m targets needs at least 2m paid calls when the rank-one frame span is at most m+1, with any number of exactly restored helpers. A quadratic address extension suffices; commuting relatives and invertible pencils satisfy the general mt bound.
This resolves #6698's irreducible-quadratic pair obligation. Sharp rational examples lie outside the old flag premise. A feasible nontriangularizable triple records the remaining boundary, not a counterexample.
Base: 7c49fe3.
#6698 source: e7b08ac.
Exact source bytes, Git/SHA256 pins and the retention ancestor are preserved.
Reproduce (CPython 3.12, stdlib):
Expected: PASS_EXACT_FINITE_CONTROLS and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed: normal/-O exact formal identities and seven rejection controls; separate AI proof and regular-representation checks; committed bytes/modes, export integrity and both Git-free replays with read-only permissions. Missing cleanup, rank-two frames, bad scalar idempotence and incorrect address arithmetic reject. Complete remote tree equals the tested tree.
No new κ. This is the exact whole-matrix scalar-idempotent model. Nontriangularizable triples, larger spans, nonidentity helper outputs and free address maps remain outside scope. Native/caller costs and all five selected assumptions remain. No human/formal acceptance, hosted CI or selected cold replay. Full-history preflight refused the shallow checkout.
By @CubeJerry with substantial OpenAI Codex assistance and separate proof review. Credit Nick Petalas (#6698/#5988), unpaidmoon (#6338), Ali Romman (#468), eumemic (#288) and retained lineage. Apache-2.0.
Proof, pins and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/a079d89c7c49d4ee418828f93e31f319e018c2c1/research/clean-helper-algebraic-flag-bound