Research: Eisenstein nonlinear message copy floor 22b under explicit width pricing

[shiftybitshiftr]

Research note; no new κ or physical supplier. This extends #798's scalar Eisenstein E8 copy certificate to fixed-support, deterministic, zero-error nonlinear messages. With b independent bits demanded per port, the abstract charge is at least 22b Eisenstein / 44b rational rank units. Unequal demands satisfy C >= max(ceil(11*sum(b_t)/20), max_clique sum(b_t)).

The essential premise is explicit: each q-bit message pays q times its source/reader pairing rank, additively. This is not a derivation of nonlinear physical copy pricing. The abstract equal-width optimum remains in [22b,40b], and the bound does not exclude three stages or alter the selected construction.

The contribution composes Jonathan Lam's #798 geometric certificate with Alexander Templeton's #785 conditional-transcript argument for Hurwitz. Both proofs are pinned and retained verbatim with attribution. It adds the Eisenstein nonlinear/unequal-demand theorem, concentration bound, independently reconstructed geometry, a nonlinear example, and concrete failures when width pricing or orthogonal side-information restrictions are removed. My initial scalar experiment was superseded by #798; that overlap prompted the one recorded pivot.

Reproduce with Python 3.10+ and the standard library:

python3 -B research/eisenstein-message-copy-bound/verify.py

Expected: PASS_EISENSTEIN_MESSAGE_BOUND: 22b; unequal demands 11/20 plus clique bounds; 298378 flat pairs; 12 rejection controls; no new kappa.

Validation: all 772 flats, 298,378 flat pairs, 30,880 line extensions and 2,970 maximal cliques; 4,096 nonlinear inputs and 303,104 conditional transcripts; unequal-width and excluded-side-information controls. A separate modular/Fraction implementation checks every geometric inequality. CPython 3.14.4 and 3.12.13 pass in isolated optimized mode. Relocation, exact receipt regeneration, source/certificate mutation rejection, output preservation and selected-record integrity pass. Structured autoreview found no actionable defects. Full repository/cold supplier suites and hosted CI were not run; existing files are unchanged.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60; OS-random seed 41, structural decoder track. During closeout, main selected #741 at κ=0.001661202643272239; this note retains its exact starting base and all pinned geometry inputs remain unchanged. No pinned sprint issue exists, so the claim is local. Source heads, inherited obligations, completed coverage and predecessor credit are in the package. No full multiplication-proof claim.

Prepared by Michael Fischler (@shiftybitshiftr) with substantial OpenAI Codex assistance in mathematics, implementation, verification and writing. Apache-2.0; no independent human review claimed.

[Mister-Meeseeks]

100-PR batch review at d91c5152b314391406b0ec792b6765a5cf0af2f1.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Eisenstein nonlinear fixed-support message bound, including unequal demands; transcript proof and exact geometry, per-bit physical pricing remains a premise.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.