[Clear1nsight]

Change and mathematical scope

Extend #5160's fixed Clifford control ceiling to arbitrary nonlinear weighted monomial one-column controls with fixed finite complete-product support independent of width. For the released four-control endpoint, every sufficiently large width satisfying all four memberships has t0_f=0, excluding an all-width invertible family. Outer scalar owner coefficients may depend on width.

A finite generating alphabet alone is insufficient: weighted generators may have infinitely many projective products when word lengths vary with width. The rejected draft and review correction are retained unchanged. Growing product support, noncolumnwise routing, extra paid calls/axes, changed source/code/endpoint or compressed productive targets remain outside this theorem. No physical supplier, compiler transfer or κ claim follows from this primary theorem. A narrower standalone paid-control bound is separate below.

A separate immutable appendix records an exact one-H1 shear/inverse/raw-dual word and its direct 228-extra-H1 upper construction bill, which loses the inherited conditional allowance. Its separate r >= 112 theorem covers only a fresh standalone T with no initial nonfree observations, translation-algebra free coefficients, commuting same-width X-Gaussian children and a FREE terminal E in the fixed finite monomial-Clifford endpoint class, at sufficiently large faithful widths. Under those exact premises 112 exceeds the conditional 44 allowance; this is not a whole-caller amortized barrier. Already-paid QY/native/IDLE/centre observations, nontranslation/noncommuting controls and paid E are outside. All allowed X maps commute with Q,D3, hence so does T. The wrapper authenticates this independently reviewed written appendix; it does not execute a new proof.

Task and overlap

Task ID: joint-query-controller; scope status: changed. The changed premise is the FREE control alphabet: fixed Clifford controls are enlarged to arbitrary nonlinear weighted monomials while keeping finite complete-product support. This is the root-assigned support-partition extension of #5160 head 7800589ce99d932bdb063aafca1f501f1a38626d, not another Clifford alphabet search. Scientific source is 4c0a359d88f22f05189d51f48387750b801a45d4, research/coded-native-caller; #5126 context is 0e8a13c251a9be1029114cda8b4194f05f3a23ec. Publication base is ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c. The sole all-author monitor checked complete created/updated PR pagination through 2026-10-23 12:32:52 UTC, cursor #5204, and rechecked canonical main at 12:33:54 UTC. Related #5200 is a different two-FULL/pure-2H17/dirty-identity-workspace monomial-Clifford stabilization theorem; this package addresses the four-control endpoint and nonlinear weighted monomials. These bounded checks are not a novelty guarantee. Existing held native/helper observations are not consumed.

The module preserves the exact reviewed proof 76bea47531ed557c0c9cb210c4324d6000c74415bf13e5747b7036567853f6ef, independent written review 0b0fe1c33eea85ccd92a3423a705fd086c6f2669f1a47f37fba9c1ca4ae91973, rejected first draft, minimal predecessor/source captures, and separate finite discriminator. No old package or shared code is changed. A distinct operator-valued paid-control appendix is included only after its separate independent PASS; no pending theorem or physical construction is promoted.

Validation

python3 -I -W error -B research/nonlinear-monomial-control-ceiling/verify.py
python3 -I -O -W error -B research/nonlinear-monomial-control-ceiling/verify.py
python3 -B scripts/research_package.py --package research/nonlinear-monomial-control-ceiling --check-only
python3 -B scripts/research_package.py --package research/nonlinear-monomial-control-ceiling
python3 scripts/check_research_commit.py --package research/nonlinear-monomial-control-ceiling --ref HEAD --check-source-history

Normal/O, registered replay, metadata/pins and committed-byte/source-history checks pass. Official export and fresh Git-free normal/O runs pass with unchanged exported bytes and complete byte-identical RESULT/MATRICES receipts. The wrapper preserves the 21 unchanged primary evidence files and 14 separate appendix captures and runs the unchanged author checker only in private scratch with its original four logical source paths. CRLF captures remain raw bytes.

The finite hook checks three exact 16×16 nonlinear weighted Toffoli/CCZ examples,108 terminal sandwiches, actual Gaussian inverses/squares, and meaningful dropped-hypothesis/transport-order controls. It does not prove the all-width theorem or reexecute actual112-owner/17-axis native source facts; those are explicitly inherited from the source-pinned predecessor. Written proof review supplies the uniform mathematical argument. No source/emitter/compiler/Lean or historical suite is run. Standard library, CPython 3.12, about 1 second per direct run. Hosted first run 39954453209 passed at contribution head bddbdfab1cdba27ce69ce4a83d9c92a5d9e66bd8, tested merge 7895eb79c against base ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c: helper job 120416680974 ran 414 tests with 17 skips in 218.442 seconds; package job 120418712311 ran the declared normal/optimized finite hook and committed-input checks. Both emitted PASS_PORTABLE_NONLINEAR_FINITE_SUPPORT_ONLY, followed by registered-package and unchanged-byte success. This is finite-support verification, not a fresh native/compiler/all-width theorem replay.

Contribution head: bddbdfab1cdba27ce69ce4a83d9c92a5d9e66bd8.

Credit

Root support-partition derivation, independent written proof reviewer, separate finite-discriminator author/review, and Chris R/Clear1nsight packaging, with substantial OpenAI Codex assistance. Retain Dillon's released equations, Dugongue/mdumitrean source A/normalizers, Cinna Davis/1337isnot1337, Cao Yifan/intqwq, zxv, Jacob Alan Sussman, Ryanson Jonathan and the native/helper/Clifford/compiler lineage. Apache-2.0 and original notices preserved. No published-paper novelty, independent human acceptance or proof-kernel verification is claimed.

Open next questions

Specify one actual control with growing complete-product support, noncolumnwise action, affordable paid work or changed observation/endpoint, with exact input ownership, source access, inverse and complete bill. The theorem neither supplies nor excludes that changed contract. Native/helper-observation work stays with its existing investigators; unknown finite costs/reserves are not zero.

[Clear1nsight]

HANDOFF

Research discussion / PR / artifact:
PR #5210, module research/nonlinear-monomial-control-ceiling, branch research/nonlinear-monomial-control-ceiling-20261023. Published through Browser; scoped completed investigation is RELEASED. Predecessor #5160. Exact source and review captures are public in the PR.

Exact commit:
bddbdfab1cdba27ce69ce4a83d9c92a5d9e66bd8; base ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c; source 4c0a359d88f22f05189d51f48387750b801a45d4; predecessor 7800589ce99d932bdb063aafca1f501f1a38626d; context 0e8a13c251a9be1029114cda8b4194f05f3a23ec.

What is established:
Independent written proof review passes the extension to arbitrary nonlinear weighted monomial columnwise programs with fixed finite complete-product support independent of width. Outer scalar coefficients may vary with width; every sufficiently large width satisfying the four memberships has zero t0, excluding all-width invertibility. Final proof/review and rejected first draft are preserved. Exact finite16×16 support discriminator passes 3 examples/108 terminal sandwiches and hypothesis/order controls. Normal/O, metadata, registered replay, committed bytes/source ancestry, official export and Git-free read-only full RESULT/MATRICES regeneration pass; Hosted first run 39954453209 passed at contribution head bddbdfab1cdba27ce69ce4a83d9c92a5d9e66bd8, tested merge 7895eb79ca959d1f4335985a497c5e2b36871ba5 against base ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c: helper job 120416680974 ran 414 tests with 17 skips in 218.442 seconds; package job 120418712311 ran the declared normal/optimized finite hook and committed-input checks. Both emitted PASS_PORTABLE_NONLINEAR_FINITE_SUPPORT_ONLY, followed by registered-package and unchanged-byte success. This is finite-support verification, not a fresh native/compiler/all-width theorem replay.

What remains uncertain:
Finite generators with unbounded width-dependent products, growing complete-product supports, noncolumnwise controls, changed source/targets/paid calls, actual native endpoint/absolute-phase execution, finite cost/precision/restoration/compiler reserve and outer admission. The finite checker is not a 17-axis or all-width theorem certificate. No universal paid query lower bound, supplier, caller or new κ. A separate independently reviewed paid-control appendix gives a direct 228-extra-H1 upper bill and fresh-standalone r>=112 bound only for translation B, commuting same-width X children, no initial nonfree observations and FREE E in the fixed finite monomial-Clifford endpoint class. This112>44 comparison is not a whole-caller amortized barrier. Nontranslation/noncommuting/paid E or already-paid native/query observations remain outside. No appendix proof is executed by the hook.

Smallest useful next task:
Name one changed control or observation contract outside the explicit finite-product hypothesis and bind actual arbitrary-input ownership, source access, inverse and complete paid/finite bill before construction. Do not repeat the excluded fixed-support monomial class or overlap held native/helper-observation work.

Reproduction command and expected runtime:
python3 -I -W error -B research/nonlinear-monomial-control-ceiling/verify.py, repeat with -O; about 1 second each, CPython 3.12 standard library. Registered: python3 -B scripts/research_package.py --package research/nonlinear-monomial-control-ceiling. Wrapper copies the unmodified checker/four source paths into private scratch; complete generated RESULT/MATRICES match frozen modes and package/export bytes remain unchanged. Use wrapper instead of invoking the mutating author checker beside frozen receipts.

Predecessor credit:
Dillon, Dugongue/mdumitrean, Cinna Davis/1337isnot1337, Cao Yifan/intqwq, zxv, Jacob Alan Sussman, Ryanson Jonathan and retained native/helper/Clifford/compiler lineage. Root proof, independent written review, finite-support verification/review, Chris R/Clear1nsight packaging; substantial OpenAI Codex assistance. Original licenses/notices preserved.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "HANDOFF",
  "task": "joint-query-controller",
  "scope": "Completed nonlinear weighted monomial four-control ceiling with fixed finite complete-product support; separate fresh standalone translation paid-control bound. Investigation released; changed physical contracts remain open.",
  "base": "ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5210"
}