[petalas]

Summary

Scoped proof with finite corroboration. In the fixed-support nonlinear Eisenstein message model, fixing h individual input bits to constants that make every message affine forces C >= N-h, where N is total demand and C is additive per-bit pairing charge. Optimizing the face gives C >= N-h_*. The exact minimum zero-fixing number equals the hitting-set number tau of all nonlinear ANF monomial supports, so C >= N-tau also follows.

fix h coordinates -> independent remaining demands
                 -> affine messages -> credited linear floor N-h

This is a restriction corollary of Jose Manuel's #2017 block-linear theorem in Michael Fischler's #814 nonlinear model. Jonathan Lam and Joel Pulikkan supplied the geometry/model; ywz65's #2607 treats the distinct adaptive public affine-query boundary. The new contribution is the model-preserving coordinate restriction and its dependence-sensitive nonlinear obstruction. No predecessor verifier was replayed.

Evidence

• Before this corollary, the cited unrestricted nonlinear floor is 22b for equal demands. The new bound is max(22b,40b-h_*) for the same priced model. The new genuinely nonlinear forty-port fixture has h_*=1, so the necessary floor is 39; its exhibited charge is 40. No cost-39 construction is claimed.
• Warning-strict ordinary and optimized local checks pass 366 Boolean maps, 18,086 coordinate faces and 164,114 point/output substitution comparisons. The strict nonzero-face example has h_*=1 versus zero-fixing number two.
• The new code fixture passes sixteen local frame inputs, 64 receiver observations, 72 direct-bit component cases and forty rational message-price computations. Nine rejection controls and three semantic controls pass. The full forty-input statement uses the written product decomposition, without claiming a 2^40 census.
• Registered replay, source pins, exact committed bytes/source history, official export/integrity, and Git-free ordinary/optimized exported replay pass at 6595ed483ae6d3030d327b7855594d83980c0384. Exported source corruption, a cancellation mutation and output overwrite are rejected; all eleven export files including its manifest are restored unchanged.

Reproduce from the repository root. Another Python 3.12 executable may replace the local path.

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/eisenstein-nonlinear-restriction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/eisenstein-nonlinear-restriction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/eisenstein-nonlinear-restriction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/eisenstein-nonlinear-restriction --check-source-history

Local environment is CPython 3.12.15 on macOS, standard library only. The registered commands each run in well under one second locally, with a 120-second limit. This timing only establishes replay budget suitability, not a performance comparison.

Limitations

The all-width proof invokes the credited linear floor. Exact independent product inputs, fixed supports and widths, orthogonal side information and additive per-bit prices are required. Cross-port coupled restrictions, errors, adaptive/private routing and shared physical pricing are excluded. No unrestricted nonlinear optimum, attaining nonlinear saving, physical supplier, new kappa or selected-frontier change is established. Independent human and formal proof-kernel review, full supplier replay, compiler, hosting and precision obligations remain unrun or outside scope.

Open next questions

• Can an exact genuinely nonlinear code achieve C<N under unchanged supports and prices? Start with a small complete code and enumerate receiver observations, then compute a linearizing coordinate face.
• Can affine changes within each port expose a larger product restriction? Preserve independence between ports; the required invariant and bounded search remain open.

Related open work

• No directly overlapping open Eisenstein nonlinear-message PR was found in the live Eisenstein search immediately before publication. The cited Research: Eisenstein nonlinear message copy floor 22b under explicit width pricing #814, Prove sharp cost 40 for Eisenstein binary decoders #2017 and Research: sharp Eisenstein floor for adaptive affine public broadcasts #2607 are merged predecessors.

• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 studies real-E8 masked characteristic-zero decoder costs with a dense binary pairing table. Its geometry and encoder model differ from this nonlinear Eisenstein message corollary; no numerical comparison is made.

• Prove a strict Gaussian localized public-header surcharge (no new κ) #4897 proves a source-sensitive expected-cost surcharge for localized public Gaussian headers. It studies a different geometry and observation contract from the fixed-support zero-error Eisenstein coordinate restriction; no shared numerical quantity is compared.

• Prove the sharp Hurwitz budget versus decoding-success law (no new κ) #4900 proves a Hurwitz budget-versus-simultaneous-decoding-success law. Its geometry and permitted decoding errors differ from this Eisenstein zero-error obstruction; no shared numerical quantity is compared.

The pre-DONE refresh covered all 71 open PR records on this account. Ten were created since the advance CLAIM, including this PR and nine other submissions. The two adjacent nonlinear-message results are listed above; the other seven concern different objects. No competing value for this restriction parameter or fixture was found in those titles/bodies, and no additional scientific computation was run for the comparison.

CI notes

Both hosted checks passed at head 6595ed483ae6d3030d327b7855594d83980c0384 in Registered research packages run 39870231459: planning in 3m07s and registered package replay in 1m22s. Run metadata confirms this exact head, completed status and success. The hosted log contains both ordinary and optimized PASS_EISENSTEIN_NONLINEAR_RESTRICTION markers and PASS_REGISTERED_RESEARCH_PACKAGE. Local registered/direct replay, committed bytes/source history, official export integrity and Git-free ordinary/optimized replay also passed. These checks certify execution of the scoped automated checks; independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast Radius: isolated research package. Seven new files and no shared-code or selected-result edits.