[petalas]

Summary

Scoped proof and exact certificate tools for a finite-template helper interface:

finite fixed linear menu A_1,...,A_N, exact full endpoint T, infinite field K
[for every x, some A_j x = T x]  implies  [some fixed A_j = T]

Arbitrary data-dependent selection cannot repair a menu whose every fixed entry misses the full arbitrary-input endpoint. All dirty inputs and required restoration outputs belong to that same contract. The cost corollary preserves any simultaneous budget satisfied by every globally legal menu entry.

The package also gives a rational simultaneous counterexample constructor and the Cartesian-grid obstruction sum_j 1/W_j < 1, where W_j is the maximum product of coordinate sizes over an independent column basis of A_j-T. Equal coordinate sizes s give sum_j s^(-rank(A_j-T)) < 1. An invertible-shear fixture shows the strict threshold is necessary.

This application and implementation were produced by Nick Petalas. The finite-union, root-counting, fiber-counting and column-matroid methods are classical. Sagar Thakrar / thakrarsagar's #1133 already uses the same N(d-1)+1 polynomial-avoidance bound for support descent; that bound is credited and reused, not claimed as new. Its proof is pinned as bibliographic provenance. No predecessor scientific replay is the deliverable.

Evidence

Before, a fixed-menu exclusion alone leaves a data-dependent selector unspecified. The new exact-interface corollary excludes every selector for that same finite globally legal menu over an infinite field. It does not establish a selected P104 menu or a new fixed-program lower bound.

Local CPython 3.12 normal and optimized checks pass PASS_FINITE_MENU_ADAPTIVITY:

• 729 exact small matrices, 2,187 weighted grids and 11,139 enumerated output fibers. Independent determinant ranks and exhaustive column bases confirm elimination and maximum-product greedy bases.
• All 2,600 three-element menus of nonzero ternary rows, simultaneous counterexamples and success unions on 166,400 grid points. The four-map curve fixture requires nine prescribed tests and returns (1,8,64).
• Restricted-grid strictness, complete finite-field domains totaling 503 points, 81 infinite-menu fixtures, dirty-output and rational/dimension-one controls, and ten domain rejections.
• Three disposable optimized-code mutations reject a zero witness, reversed weighted greedy order and a false coverage bound.

Each registered command took under one second locally. Metadata and source pins, registered normal/optimized replay, in-memory syntax/function-size inspection, staged whitespace, committed bytes/modes and source-history preflight pass. The official six-file export passes its input-coverage check. Read-only Git-free normal/optimized replay with empty PATH passes without changing exported bytes or modes. Generated stdout binds the evidence to package SHA256 hashes; no receipt is handwritten.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/finite-menu-adaptivity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/finite-menu-adaptivity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/finite-menu-adaptivity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/finite-menu-adaptivity --check-source-history --check-modes

Another installed CPython 3.12 can replace this executable. Only the standard library is required.

Limitations

The whole-space statement requires an infinite field, finitely many fixed linear maps, a shared full endpoint and global legality of each program. A paid-call bound alone does not guarantee finiteness. Finite fields, restricted reachable inputs, infinitely many coefficient choices, approximate or rounded execution, nonlinear leaf arithmetic and extra retained branch outputs require separate treatment. Explicit counterexamples delimit several of these hypotheses.

The universal statements have written proofs with finite corroboration. No physical P104 menu, native implementation, compiler/stock/precision closure, complete supplier, new kappa or selected-frontier change is claimed. The original construction card has explicitly changed scope. Lean, Docker, selected scientific replay, independent human review and formal proof-kernel review are unrun.

Open next questions

• Export one actual source-bound complex helper menu, its full arbitrary-input/restoration endpoints and complete invoices, then apply the witness constructor. The missing globally legal physical menu blocks a source-specific exclusion.
• Study approximate endpoint selection after fixing a physical norm, magnitude domain and restoration tolerance. No such priced error contract is supplied here.

Related open work

• #5997 bounds adaptive cross-port observation capacity. This package selects among full endpoint maps and asserts no observation-capacity floor.
• #5961 counts adaptive finite-field minor choices. This package's whole-space theorem assumes an infinite field and does not recompute its probabilities.
• #5994 excludes fixed helper splits on the PR5907 word. This package supplies an abstract finite-menu selection criterion, does not instantiate that menu, and does not validate or strengthen that source-specific census.

Refreshed this account's fifteen other open PRs created since the advance CLAIM at 2026-10-27T21:55:16Z. Additional related helper-interface work:

• #6017 extracts a linear word from a fixed zero-preserving polynomial helper word. It excludes branching; this package allows arbitrary selection among finitely many fixed linear endpoint maps.
• #6025 classifies fixed nonlinear role recodings compatible with exact ADDs. This package assumes each menu endpoint is linear and imposes no regularity on its selector.
• #6019 computes chronological leakage and repair ranks for a prescribed helper-replacement word. This package gives a finite-menu endpoint criterion and does not instantiate that replacement model.
• #6016 optimizes two-role dyadic helper-basis conditioning. This package does not price basis conditioning or compute its optimum.

These contributions do not compute the same numerical quantity as this package; no additional comparison computation was run.

CI notes

Both hosted checks passed on exact head a96885d1a2061ad1c6f3c2b937591ccb818006dc: plan and replay (research/finite-menu-adaptivity), in run 40591663720. The completed check list was read once, and the completed successful run's recorded head matches the PR head. All local checks listed above also pass. Hosted replay certifies the registered finite checks; it does not independently certify the universal written theorem or physical obligations.

Merge danger

Door: two-way. Blast radius: package. Six isolated new research files; shared code, frozen predecessors and the selected result are unchanged.