[petalas]

Summary

Scoped proof with finite exact corroboration. Grant a labelled linear-operator call a free coordinatewise polynomial before or after it, with every polynomial fixing zero. For an exact linear endpoint over an infinite field, this extension has the same Pareto-minimal operator-call profiles as the linear model, including all required dirty and clean returns.

b += p(Ta)  or  b += T p(a)
              |
              | extract degree one at the common zero state
              v
b += p'(0) T a
zero derivative: delete call; otherwise keep operator, ports, label and order

Scalar multiplication by p'(0) must belong to the comparison's permitted linear class. The classical degree-one quotient/chain rule proves the extraction. The new contribution is the explicit helper-interface profile statement, its uniform-family corollary and its bounded executable extractor. Produced by Nick Petalas. No predecessor scientific replay is the deliverable, no literature-priority claim is made, and no new kappa or selected-frontier change is claimed.

Only research/polynomial-helper-profile-domination/ is added. Advance CLAIM. The delivered scope matches that claim.

Evidence

Before, a linear operator-call obstruction leaves zero-preserving nonlinear data-value processing unspecified. After, every exact linear endpoint in this polynomial class supplies a linear word whose count in each label class is no larger.

Local CPython 3.12.15 on macOS arm64 passes PASS_POLYNOMIAL_HELPER_PROFILE_DOMINATION:

• 145 complete sparse-polynomial endpoint and inverse fixtures, including clean and arbitrary dirty returns, singular/nonsymmetric rational operators, both placements and successive noncommuting blocks.
• 1,029 directions each for a separate dual-number interpreter and literal rational execution, with 14 invalid-input or essential-boundary controls. These finite checks corroborate the written theorem.
• Registered warning-strict normal and optimized replay, syntax/function-size inspection, metadata/source pins and exact committed-byte/source-history preflight pass. Each registered argv was timed locally, 0.219 and 0.216 seconds, below its 120-second timeout. These are replay-budget measurements, not a performance claim.
• Official committed export integrity/input-coverage checks and a read-only, Git-free optimized replay pass.
• The five-gate dirty echo's illustrative labelled profile goes from T:3, read:2 to T:2, read:2. These abstract labels are not measured Fourier ranks or selected-word jobs. The generated receipt binds all seven proof/code/metadata sources by SHA256.

From the worktree root, exact reproduction commands:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/polynomial-helper-profile-domination/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/polynomial-helper-profile-domination/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/polynomial-helper-profile-domination
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/polynomial-helper-profile-domination --check-source-history

Any CPython 3.12 interpreter can replace the local interpreter path. No third-party dependencies are needed. The theorem has no polynomial degree or finite-workspace-count bound across different candidate words; the rational implementation enforces its separately documented finite bounds.

Limitations and unrun obligations

This is an optimistic labelled operator-call model. Actual scalar, frame, native, tape, precision, stock and compiler costs are not equated or supplied. The extractor does not certify that its input word's endpoint is linear; the verifier separately establishes that premise on the fixtures. Nonzero constant trajectories, branching, finite-grid/rounded execution, nonlinear address permutations and nonlinear output contracts are outside the theorem.

No selected-word replay, physical supplier, full multiplication theorem, Lean or Docker check was run. Independent human and formal proof-kernel review are unclaimed. Jose Manuel / jmbarrancoml's #1757 nonlinear finite-alphabet cut theorem and Chris R / Clear1nsight's #3992 typed Toom/Jacobian obstruction are credited and pinned context, not audited here.

Related open work

• Exclude finite-menu adaptivity for exact helper endpoints (no new κ) #6015 studies a data-dependent finite menu of fixed linear endpoint maps. This PR instead fixes one nonlinear arithmetic word and extracts its linear endpoint; neither result supplies the other's premise or a new selected supplier.

• Prove a sharp ten-unit floor for the nonlinear six-frame seed #5944 prices a nonlinear finite-alphabet six-frame circuit through two cut partitions. This PR has an infinite-field polynomial gate contract and a different operator-call price; its conclusion does not extend that finite-alphabet exclusion.

• Classify fixed nonlinear role recodings under exact ADD compatibility (no new κ) #6025 classifies fixed independent nonlinear role recodings through exact ADD conjugacies, including nondifferentiable bijections. This PR instead extracts a fixed polynomial arithmetic word at its common zero state; its hypotheses and labelled-call comparison differ. No common numerical minimum is computed.

Open next questions

• Can a nonzero constant trajectory yield a cheaper complete helper word? First price the constants, coordinate-dependent derivatives, retained outputs and restoration. The translated-square boundary witness only shows where this proof stops.
• Can a finite-grid polynomial or bit-operation mechanism save calls under an exact selected-source precision contract? First specify the grid, nonlinear consumer and arbitrary dirty inputs. The F2 square control prevents transferring this derivative theorem to finite grids.

CI notes

Both hosted checks passed at 90d1c7271bf0479c137acc340644d2d6e2130532: plan and replay (research/polynomial-helper-profile-domination), in run 40591995051. The completed check list was read once, and run metadata confirms the head. Hosted replay certifies the registered finite checks; the universal written proof and missing physical obligations retain their stated scope.

Merge Danger

Door: two-way. Blast Radius: isolated.

The change adds one opt-in research package and changes no shared code, frozen predecessor, selected record or frontier.