[petalas]

Summary

Scoped proof with exact finite corroboration. Arbitrary polynomial constant terms cannot improve a labelled linear-operator call minimum when every allowed operator has constant row sum. Dirty inputs, required workspace returns and chronological labels are preserved.

T e = lambda_T e
zero-input trajectory: register v = c_v e
b += p(Ta)  gives  delta_b += p'(lambda_T c_a) T delta_a
b += T p(a) gives  delta_b += p'(c_a) T delta_a
exact linear endpoint gives the same linear word with no larger label counts

Merged PR #6017 establishes the zero-preserving case and asks about nonzero trajectories. This answers that question for the constant-row-sum alphabet. Width-independent row sums also yield one uniform extracted word. The chain rule is classical; the new step is the homogeneous orbit and its explicit profile and uniform-family consequences. The contribution was produced by Nick Petalas. Predecessor and contextual credit are retained in the package.

Only research/homogeneous-polynomial-profile/ is added. No new kappa or selected-frontier change is claimed.

Evidence

Before, the cited theorem requires every polynomial to fix zero. After, the proof permits arbitrary constant terms under the stated operator hypothesis. It does not audit or replay the predecessor.

Local CPython 3.12.15 on macOS arm64 passes PASS_HOMOGENEOUS_POLYNOMIAL_PROFILE:

• 348 complete rational polynomial endpoints, all with nonzero constant trajectories, including singular and noncommuting operators and active clean or dirty workspace returned exactly.
• 1,392 literal rational input checks and complete inverse round trips; 24 nonlinear-prefix Jacobian checks.
• Twelve named boundary and corruption controls, including the failure of differentiating every gate at local zero.
• Warning-strict normal and optimized runs pass with identical generated, source-hash-bound receipts. The registered commands take 0.620 s and 0.280 s locally, each below its 120-second limit. These are replay-budget measurements.
• Registered metadata/pins and both commands, syntax/JSON/function bounds, whitespace, committed-byte/source-history preflight, and official committed-export integrity/input coverage pass.

From the worktree root:

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

Any absolute CPython 3.12 interpreter path can replace this local path. Expected final markers are PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. No third-party packages or runtime predecessor inputs are required. Scientific base and refreshed canonical main are 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision eight. The source-history preflight passes at head 6d38d4c55bf81afec80cc422c98d9b588432acf1.

Advance claim. Live overlap checks cover open PR metadata, the handoff discussion, a paginated cache of 734 open PR bodies and 6,836 discussion records, main-tree README searches and targeted merged searches. Unpublished work and arbitrary historical code diffs remain coverage limits.

Limitations and unrun obligations

The theorem requires an infinite field, coordinatewise scalar polynomials, exact linear endpoints on independent inputs, fixed topology and a linear comparison permitting scalar multiples at the original charge. Arbitrary vector constants, coordinate-dependent polynomials, nonconstant row sums, restricted grids, branching and rounded operations are outside it. Width-dependent row sums retain pointwise domination but do not automatically give uniform coefficients.

No native realization, physical supplier, scalar/tape/precision/storage/stock bill, complete signed invoice, selected cold replay, Lean or Docker run is supplied. The five selected assumptions remain. Independent human and formal proof-kernel review are unclaimed. The written proof is the universal claim; finite checks only corroborate it.

Open next questions

• Can an alphabet that does not preserve the constant-vector line yield a strictly cheaper complete polynomial helper profile? Supply its exact endpoint, every constant-preparation and restoration gate, and a linear-model lower bound. No such separation or physical native contract is supplied here.
• Can a named width-dependent-row-sum family still admit a uniform extracted word? Follow its orbit symbolically and specify the permitted coefficient ring. A source-bound native family is missing.

Related open work

No open PR studying this homogeneous polynomial call-profile question was found in the stated overlap coverage. The direct predecessor #6017 is merged. The merged #6025 fixed nonlinear role-recoding theorem and #1757 finite-alphabet cut theorem use different contracts; neither is rechecked here. No common numerical minimum is compared.

• Exclude dyadic Borel savings in a common integral polynomial algebra (no κ) #6792 studies a common integral polynomial algebra of linear Borel blocks and transfers a paid-rank floor. This PR studies nonlinear coordinatewise polynomial shears along homogeneous constant trajectories. Their contracts and claimed quantities differ.

The closeout check read this account's seven other open PRs created since the advance claim: #6785, #6787, #6788, #6789, #6790, #6791 and #6792. None computes a competing homogeneous-polynomial call-profile minimum. No additional computations were run to compare them.

CI notes

Both hosted checks passed at exact head 6d38d4c55bf81afec80cc422c98d9b588432acf1 in run 42223022774: plan (4m52s) and replay (research/homogeneous-polynomial-profile) (1m24s). The completed checks were read once and both report pass. Hosted replay certifies execution of the registered finite checks and metadata at this commit; the universal written proof and physical obligations retain their stated scope. All listed local checks also passed.

Merge Danger

Door: two-way

Blast Radius: isolated

Six new package files. Shared scripts, frozen evidence, existing credits and selected records are unchanged.