[petalas]

Summary

Scoped proof with exact finite corroboration, produced by Nick Petalas. For a known invertible binary affine permutation and an exact phase oracle promised to be quadratic over Z/4 with even mixed coefficients, the optimal deterministic adaptive membership test for the normalized affine tag family uses

1 + n + binom(n,2) queries
n = 9: exactly 46 phase evaluations; the cube has 512 addresses

The test queries zero, unit vectors and pair sums. A matching adversary constructs a nonmember with the same transcript for every shorter query set. The lower bound allows an unknown constant phase and unknown even-linear tag. This is an oracle query count, not a measured speedup, paid physical saving, new supplier or new kappa.

Only research/affine-tag-query-minimum/ is added. The proof defines its full problem locally and supplies a reusable checker and adversary. It builds on JoyboyBrian's #6106 affine-tag convention and stated coordinate-only limitation without auditing or replaying that contribution's warp, inverse or census. Credit ToikY #4002, Chris R / Clear1nsight #3987, Dugongue #3945/#3929, dataisfire #3734, Cinna Davis #2226 and the retained native lineage. Quadratic interpolation and binary elimination are classical; the sharp decision cost with nuisance tags is this contribution.

Advance CLAIM. Base: canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8, unchanged at the final fetch.

Evidence

Before, coordinate-only testing leaves quadratic residuals undetected. After, the complete test and matching lower bound settle the exact black-box cost in the stated grammar. CPython 3.12.15 on macOS arm64 returns PASS_AFFINE_TAG_QUERY_MINIMUM with:

• 2,192 complete quadratic-phase cases through three axes, checked against literal cube values and every possible tag.
• All 58,916 insufficient distinct query subsets through four axes. Each gets an explicit zero-transcript impostor whose nonmembership is checked literally.
• 36 affine constructions through 64 axes, with complete cube comparisons through five axes; 14 rejecting controls and one cubic-promise limitation control.
• Identical generated file-hash receipts in warning-strict normal and optimized modes. The two registered commands took 6.956 and 6.963 seconds locally, below their 180-second limits. These times establish replay budget only.
• PASS_REGISTERED_RESEARCH_PACKAGE, metadata/pin validation, exact committed-byte/source-history preflight, official export integrity and an unchanged read-only optimized Git-free export replay with empty PATH.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/affine-tag-query-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/affine-tag-query-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/affine-tag-query-minimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/affine-tag-query-minimum --check-source-history

Any CPython 3.12 path can replace the local interpreter path. Standard library only. Each direct run prints a generated receipt bound to all six package files. Publication head: 5677768cc9602055c21d4fecf2a101357bd07087.

Limitations and unrun obligations

The written theorem covers every positive n; the implementation supports 1 through 64 axes. Exact phase access, a known affine map and the structural quadratic promise are hypotheses. A cubic can fool these samples, as the limitation control demonstrates. The lower bound does not apply when coefficients are directly available, to bounded-error randomized tests, or to quantum queries.

No serializer integration, arbitrary native-circuit membership, complete tensor identity, physical caller, dirty restoration, invoice, stock/precision/tape or multiplication theorem is supplied. No selected cold replay, Lean, Docker, independent human review or formal proof-kernel acceptance is claimed. The selected frontier is unchanged.

Related open work

• Preserve translation-induced scalar phases in affine native tags #6106, JoyboyBrian, head 9d0126c134720258cfe685b661ce288da01008fa: supplies scalar-aware affine tag composition and a concrete warp. This contribution instead proves optimal membership-query complexity under the promised phase grammar; no predecessor number is audited or compared.

The other candidate returned by the broad tag/affine keyword check, #6146, concerns rounded Walsh stages with an interleaved integer shear. Its body was read; it does not study this phase-membership decision problem. The live pre-commit open-PR and local-claim checks found no competing target for this question.

The completion refresh read the seven other account PRs created since the CLAIM: #6149 and #6151 through #6156. They concern a P112 Walsh basis orbit, padding tapes, RootPlan backing, hidden invoice windows, publication byte checks and source-validation memory. None studies affine native phase-tag membership or reports a competing value for this decision problem. No additional mathematical computation was run for comparison.

Open next questions

• Bind the test to a serializer whose affine permutation and quadratic phase grammar are structurally authenticated. First expose its exact phase oracle and induction, then price the complete caller.
• Determine the optimal bounded-error randomized query cost for this promise problem, after specifying the error probability and one-sided versus two-sided rejection.

CI notes

Uncertified by CI. At head 5677768cc9602055c21d4fecf2a101357bd07087, run 40724809789 failed plan in the onboarding helper tests. replay (research/affine-tag-query-minimum) was skipped and did not run in CI. The local scientific and publication checks above passed; this hosted run certified none of the package's scientific checks. A maintainer re-run is needed after main is fixed.

The exact cause is the selected-record/documentation mismatch inherited from canonical main 721c03822c421b7f7fcbb2598993b947c4d88419. That PR6030 kappa-admission commit was pushed with [skip ci] and changed the selected record without updating the onboarding documents to match:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md. The document instead bolds 0.003125804 = 781451/250000000 as one expression, so the exact substring assertion fails.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

Both exact failures also occur in PR6120's run 40691776027, confirmed from its failed-job log. The completed PR checks were read once. Open PR6050 targets these documentation mismatches. This contribution changes neither failing document nor the selected record; no new commit or push was made for this unrelated blocker.

Merge Danger

Door: two-way. Blast Radius: isolated.

One opt-in research package is added. Shared code, frozen predecessors and selected-frontier records are unchanged.