Prove sharp cost 40 for Eisenstein binary decoders

[jmbarrancoml]

For the 40 Eisenstein lines, I prove that every binary scalar rectangle decoder has abstract pairing cost at least 40, matching Jonathan Lam's row-star upper bound. I allow arbitrary asymmetric supports, any finite number of terms and XOR cancellations. This closes the interval left by the floor 22 in #798, the floor 23 in #1977 and the non-clean screen in #1794.

I normalize the Hermitian Gram matrix over Q(omega), prove that its entries lie in Z[omega,1/3], and reduce it to F4. For every fitting B, the entrywise product Hbar o B is I40. Vanishing minors and the trace-pairing identity bound each summand's F4 rank by its existing abstract pairing charge. Rank subadditivity then gives C >= 40.

I also derive the known binary minrank 10 without a SAT premise, rule out hyperplane-supported nonzero codewords in rank-10 fitting row/column spaces, and prove the fixed-support block-linear bound C >= sum_t b_t under per-bit pairing charges.

Validation: the registered Python 3.12 replay and the direct optimized Python 3.14.6 replay pass. I check 1,600 Gram entries, 6,400 trace identities, reduction arithmetic, 209 rank-conversion blocks, 40 hyperplane restrictions, matching cost witnesses, a rank-10 frame partition, mixed-bit controls and seven rejected assumption corruptions. The written radical/minor argument supplies universal support coverage; sampled rank checks do not replace it.

Scope: this is a sharp theorem for the specified abstract linear model. I do not supply a native multiplication construction, establish the physical COPY pricing premise, settle nonlinear encoders, or change the selected exponent. The consequence for #1794 remains conditional on that screen's accounting; I do not rerun the screen here.

Credit: Joel Pulikkan / GamingPuzzled (#620), Jonathan Lam (#798), DreamingOfClouds (#1590, Lemma B's same-field Hadamard argument), Maximooch (#1794), Josie Elliston (#1977), and Michael Fischler's fixed-support message model. I retain their files and attribution. I used substantial OpenAI Codex assistance in mathematics, implementation and prose; no independent human review.

Coordination: #1167 (comment)

Reproduce:

python3.12 scripts/research_package.py --package research/eisenstein-reduction-copy-floor

[Mister-Meeseeks]

Batch review at 5950e7896b8825a2c167ee63836251b3096060f8.

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

Reviewed scope: Normalized Eisenstein Hermitian Gram matrix has good reduction mod2 to F4. The written radical/minor and Schur-rank argument proves sharp C=40 for arbitrary scalar rectangles, extending to fixed-support block-linear messages with per-bit additive pairing charge. Exact geometry, trace identities, sharp witnesses and corruption controls replay. Nonlinear encoders and shared physical pricing remain outside the theorem.

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.