[Matarener]

Requested by Mark

Scoped negative result; no new κ.

Before: a7-a1-decoder-floor (#1224 frame) proves the separate-term COPY floor 72 for every complex shift, but only for linear messages. Its scope list puts arbitrary nonlinear decoder architectures outside the theorem.

After: on the same fixed nine-term supports, every deterministic zero-error nonlinear message (AND gates allowed) priced by its width still costs at least 72. That holds for every shift without reader side information. On both pinned sharp witnesses it also holds when readers know every orthogonal input.

How: Lemma 1 uses a fooling block, an invertible B[S,R] of size rho_k per term, so any message needs 2^rho_k values. Lemma 2 turns side information into index coding on the induced orthogonality graph of each term's demand. The checker finds an induced Petersen + (rho_k − 5) isolated-line witness (rate rho_k; the rho=9 case is the 14-vertex witness of p112-characteristic-two-minrank) or a rho_k-line non-orthogonal set in every term of the (0,9^8) and (1,8,9^7) witnesses. Both totals are 72.

New package: research/a7-a1-nonlinear-copy-floor (base ff3cc58e3, card complex-rank-shape-decoder, scope changed). It uses the standard library only and runs in about 2 s.

PASS_A7_A1_NONLINEAR_COPY_FLOOR roots=120 witnesses=2 no_side_info=72 orthogonal_side_info=72 petersen_terms=16 certificate=09a3e17032f727cb no_new_kappa
PASS_A7_A1_NONLINEAR_COPY_FLOOR_CONTROLS rejected=9
PASS_REGISTERED_RESEARCH_PACKAGE research/a7-a1-nonlinear-copy-floor outcome=negative-result

Limits: other architectures, shared COPY billing and receivers that combine terms nonlinearly are not covered. The side-information floor for every shift is left open in the package's open questions. With side information even linear codes can drop below a term's rank on small demand graphs, so that question is a contract change and not a nonlinear effect. The frozen predecessor is unchanged.

Prepared with substantial Claude Code assistance.

🤖 Generated with Claude Code

[jwr404]

Independent review (James Reed, with Claude Code). Head 73a53f08b44d1fe7b3f3fe91064ad4c31f4d2f95.

Verdict: both lemmas are correct, and the stored certificates check out under independent code. It is correctly scoped, with no κ.

Proof check

• Lemma 1. If B[S,R] is invertible, the 2^ρ assignments on S (inputs elsewhere zero) give distinct reader-value vectors B[S,R]ᵀ x_S. With no side information each is a function of m_k, so |range m_k| ≥ 2^ρ. Summing over terms and applying the predecessor's Σ ρ_k ≥ 72 gives the every-shift claim. ✓
• Lemma 2. Reader j's value is [j ∈ X_k] x_j + Σ_{i ∈ X_k, i ⊥ j} x_i. With orthogonal side information, only x_j for j ∈ X_k ∩ Y_k is unknown. Restricting to inputs supported on W = X_k ∩ Y_k gives index coding on G[W], and restriction can only lower the rate. ✓
• The rate bounds are entropic, as they must be for nonlinear codes. I checked that p112-characteristic-two-minrank certifies an exact Shannon-inequality LP dual for Petersen (rate 5), with entropy additivity for the four isolated receivers, so 9d is a nonlinear bound, not a minrank bound. Your isolated-vertex step H(M) ≥ d + H(M | x_v) is the same argument.

Independent certificate check

I wrote separate code from the model definitions and read only the stored RESULT.json certificates. It does the following:

• rebuilds the 120 lines, B, the orthogonality relation, the A7+A1 readers and both pinned shift decoders, with the all-pairs identity D·Tᵀ = 32·G;
• recomputes every term rank;
• checks every fooling block for invertibility and support, and every Petersen + isolated witness for exact Kneser adjacency, isolation and membership in X_k ∩ Y_k;
• checks the independent set in the rank-1 term.

witness	costs	Σ	fooling	side info	predecessor
eight-active	0, 9×8	72	72	72	equal
nine-active	1, 8, 9×7	72	72	72	equal

scripts/research_package.py also passes locally (outcome=negative-result).

Small suggestions

1. Wording in Lemma 2. "Fix the inputs outside X_k to zero" should also fix the non-demanded sources X_k \ Y_k. Those inputs are nobody's demand but can be side information. The reduction to G[X_k ∩ Y_k] needs them fixed, and this is again a restriction that only lowers the rate.
2. The Petersen + 3 term (ρ = 8, nine-active witness) does not need the BKL citation. It follows from the certified Petersen + 4 ≥ 9, since adding one isolated receiver raises the rate by at most d (send it raw). That would make the side-information row depend only on the in-repo LP certificate.
3. Scope note. The side-information model gives reader j the orthogonal inputs but not its own x_j. If readers also held their own line's input, every demand would be trivially met, so "strongest natural side information" is strongest short of own-input knowledge. A clause saying so would pre-empt the question.

This review does not cover the every-shift side-information question, which the PR leaves open.