Prove sharp cost 40 for Eisenstein binary decoders

research/eisenstein-reduction-copy-floor/README.md

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+# Sharp pairing cost for binary decoders on the 40 Eisenstein lines
+
+We prove **C >= 40** for the scalar rectangle model in Jonathan Lam's
+[PR #798](https://github.com/CrocSwap/integer-mult-bounds/pull/798), and use
+his row-star construction to attain equality. We allow arbitrary supports,
+asymmetric readers and sources, any finite number of terms, and XOR
+cancellations. We do not require clean rectangles or a minimum-rank decoder.
+
+We obtain the lower bound by reducing a normalized Hermitian Gram matrix
+modulo 2 and applying rank subadditivity to a Hadamard product. DreamingOfClouds
+used the same-field rank argument in Lemma B of
+[PR #1590](https://github.com/CrocSwap/integer-mult-bounds/pull/1590).
+We add the good-reduction certificate and pairing-rank comparison for this
+Eisenstein geometry. With these steps, we close
+the scalar cost interval left by the floor 22 in #798, the floor 23 in
+[PR #1977](https://github.com/CrocSwap/integer-mult-bounds/pull/1977), and
+the non-clean cost screen in
+[PR #1794](https://github.com/CrocSwap/integer-mult-bounds/pull/1794).
+We retain the selected multiplication exponent.
+
+## Exact model
+
+Use the 240 doubled E8 roots and the order-three orthogonal action J from
+Joel Pulikkan / GamingPuzzled's PR #620, as reconstructed by Jonathan Lam
+in `research/eisenstein-clique-copy-bound/geometry.py`. We pin that file in
+`SOURCES.json`. Its six-root orbits under +/-J give 40 lines over
+K = Q(omega), where omega^2 + omega + 1 = 0 and multiplication by omega
+acts as J. Write r_i for a representative of line i.
+
+For a set S of lines, write X_S for the rational span of r_i and Jr_i.
+For any rational row bases V_R and V_A of X_R and X_A, define
+
+    c(R,A) = rank_Q(V_R V_A^T)/2.
+
+Choose any binary matrix B satisfying
+
+    B_ii = 1;
+    B_ij = 0 when i != j and lines i,j are nonorthogonal.
+
+The entries at orthogonal pairs may differ in the two directions. Choose
+any finite decomposition over F2,
+
+    B = sum_j 1_Rj 1_Aj^T,
+    C = sum_j c(R_j,A_j).
+
+**Theorem.** For every such B and decomposition, C >= 40. For each fitting
+B, a row-star decomposition has C = 40. Thus each fitting B has minimum
+abstract cost 40, regardless of its binary rank.
+
+We prove a finite algebraic statement. A researcher who interprets C as a
+physical COPY charge must establish the corresponding implementation
+premise. We do not establish the unreviewed physical premise that earlier
+authors traced to PR #446.
+
+## Proof
+
+### 1. An integral Hermitian Gram matrix at the prime 2
+
+For x,y in Q^8, put d0 = x dot y and d1 = x dot Jy. Define
+
+    h(x,y) = (d0 - d1 - omega*(d0 + 2*d1))/12.
+
+Using J^2+J+I = 0 and orthogonality of J, we obtain
+
+    h(x,Jy) = omega*h(x,y),
+    h(Jx,y) = conjugate(omega)*h(x,y),
+    h(y,x) = conjugate(h(x,y)),
+    x dot y = 4*Tr_(K/Q)(h(x,y)).
+
+Thus we may regard h as a Hermitian form on the four-dimensional K-space.
+For a root r, we have r dot r = 8 and r dot Jr = -4, hence h(r,r) = 1.
+Two lines are orthogonal exactly when both d0 and d1 vanish, which is
+equivalent to h(r_i,r_j) = 0.
+
+Write H_ij = h(r_i,r_j). The doubled-root inner products d0 and d1 are
+multiples of 4, including after J permutes the roots. Therefore
+
+    H_ij in S = Z[omega,1/3].
+
+We can reduce S modulo 2. Since t^2+t+1 is irreducible over F2, we obtain
+S/(2) = F4. Write Hbar for the entrywise reduction. We retain the diagonal
+ones and the zeros at orthogonal pairs. We do not need to assume that a
+nonzero off-diagonal entry stays nonzero, although the finite replay also
+checks that property for this geometry.
+
+We reduce this fixed Gram matrix. We never reduce an arbitrary rational
+row basis, whose denominators might include 2.
+
+### 2. Pairing cost bounds the reduced submatrix rank
+
+We first identify
+
+    rank_K H[R,A] = c(R,A).
+
+The representatives r_i span X_R and X_A over K. A vector x in X_R
+annihilates the rational pairing with X_A exactly when h(x,y) = 0 for
+all y in X_A. Indeed, rational annihilation gives both
+Tr(h(x,y)) = 0 and Tr(omega*h(x,y)) = 0, because Jy also lies in X_A.
+For z=a+b*omega these two traces are 2a-b and -a-b; over Q their joint
+vanishing forces a=b=0. The converse follows from the trace formula.
+We have therefore identified the two radicals. Their K-dimensions are
+half their rational dimensions, proving the rank equality.
+
+Next, if rank_K H[R,A] = c, every (c+1)-minor vanishes in K. Each minor
+lies in S, a subring of K, so it vanishes in S and after reduction to F4.
+Consequently
+
+    rank_F4 Hbar[R,A] <= rank_K H[R,A] = c(R,A).       (1)
+
+This proof covers arbitrary line subsets, including empty sets. We do not
+need to enumerate flats or assume nonsingular restricted spans.
+
+### 3. The fitting identity forces total rank 40
+
+Embed B and its binary decomposition in F4. Denote the entrywise product
+by "o". The fitting conditions and the Gram zeros give
+
+    Hbar o B = I_40.
+
+Distributing over the rectangle sum gives
+
+    I_40 = sum_j diag(1_Rj) Hbar diag(1_Aj).
+
+The j-th summand has rank rank_F4 Hbar[R_j,A_j]. Rank subadditivity and
+(1) imply
+
+    40 <= sum_j rank_F4 Hbar[R_j,A_j] <= C.
+
+The equality before the rank estimate holds in characteristic two, so we
+have included overlapping supports and XOR cancellations.
+
+For the matching upper bound, decompose B by rows. In row i choose
+R_i={i} and A_i={j:B_ij=1}. The set A_i contains i and otherwise contains
+only lines orthogonal to i. Hence H[{i},A_i] has one nonzero entry and
+rank one. The 40 rows give C=40. This construction and its cost already
+appear in #798; our contribution is the matching lower bound.
+
+### 4. Binary minrank and minimum-rank support rigidity
+
+We also recover the known binary minrank 10 without a SAT premise. If B
+has binary rank h, factor it into h binary rank-one matrices. Each support
+pair costs at most four, since K has ambient dimension four. The theorem
+gives 40 <= 4h, hence h >= 10. The known partition into ten orthogonal
+four-frames attains h=10: put an all-ones block on each frame. We replay a
+literal partition and verify its fitting conditions and rank.
+
+If B has rank 10, every nonzero vector in its row space and every nonzero
+vector in its column space has support spanning the full four-dimensional
+K-space. For example, suppose a row-space vector had support of K-rank
+at most three. Extend it to a binary row-space basis. The resulting
+ten-term factorization would cost at most 3+9*4=39, a contradiction.
+Use a column-space basis for the other assertion. Thus a researcher
+cannot build a rank-10 decoder from even one hyperplane-supported basis
+vector, regardless of the other nine basis vectors.
+
+## Fixed-support block-linear corollary
+
+Suppose port t holds an independent b_t-bit vector. Message j consists of
+q_j fixed binary linear forms in the inputs at A_j, and each receiver in
+R_j receives that message. Receiver t also knows the full inputs at ports
+orthogonal to t. Assume exact recovery on every input and the additive
+pricing premise
+
+    C = sum_j q_j*c(R_j,A_j).
+
+Then **C >= sum_t b_t**, with equality from direct messages. In particular,
+block-linear coding at equal demand b costs at least 40b.
+
+To prove this, choose a linear recovery map for each demanded bit. Such a
+map exists even if the declared decoder was nonlinear: exact recovery of
+a linear coordinate from linear observations means that the observation
+kernel lies in the coordinate kernel. The coordinate therefore belongs
+to the observation row space.
+
+Stacking these maps gives a block matrix B=sum_j D_j E_j over F2, with
+diagonal blocks I_(b_t) and zero blocks at distinct nonorthogonal ports.
+Here D_j has q_j columns, E_j has q_j rows, and their nonzero port blocks
+lie in R_j and A_j. Lift Hbar by replacing Hbar_st with the constant
+b_s by b_t block of that value. Call the resulting matrix G. Then
+
+    G o B = I_(sum_t b_t).
+
+Expand D_j E_j into its q_j rank-one terms. Each term after entrywise
+multiplication has rank at most rank_F4 Hbar[R_j,A_j]: repeating rows and
+columns to form G does not increase rank, and diagonal weighting cannot
+increase it. Sum these bounds and apply (1).
+
+We have not extended this argument to nonlinear encoders. The earlier
+Michael Fischler's `eisenstein-message-copy-bound` package treats a broader nonlinear message
+model and proves a different lower bound. We leave its nonlinear claims
+and their attribution unchanged. Adaptive supports and pricing shared
+physical work below the declared additive charge also remain outside our
+theorem.
+
+## Consequences and limits
+
+Under the specific accounting in Maximooch's #1794 screen, a supplier
+that beats the selected construction would need C <= 31, or C <= 29 at
+window h=10. Our lower bound excludes those costs for the stated scalar
+model. We do not rerun that screen here; we pin its incorporated receipt
+as provenance and state this consequence conditional on its accounting.
+The proof C=40 does not depend on the screen or the selected exponent.
+
+The ten-source proposal that motivated this check also fails. If each
+source lies in one orthogonal 12-line hyperplane, each term costs at most
+three. Ten such terms would cost at most 30, contradicting the theorem,
+even with unrestricted asymmetric reader sets.
+
+Researchers can still change the geometry, scalar field, support model,
+message nonlinearity, or physical pricing premise. We have not proved a
+barrier for arbitrary integer multiplication algorithms or for the native
+paired-log constructions that motivated our separate investigation.
+
+## Reproduction and evidence
+
+Run from the repository root with Python 3.12 and its standard library:
+
+```sh
+python3.12 scripts/research_package.py --package research/eisenstein-reduction-copy-floor
+```
+
+For a direct replay under an optimized interpreter:
+
+```sh
+python3 -I -O -B research/eisenstein-reduction-copy-floor/verify.py
+```
+
+We reconstruct the pinned geometry, check all 1,600 Hermitian entries,
+6,400 rational trace identities, the quotient-field arithmetic, and all
+Gram-value reduction identities. We also compare rational, K and F4
+ranks on 209 boundary or deterministic sample blocks, check 40 hyperplane
+column ranks, and replay two cost-40 row-star witnesses plus a non-clean
+cost-48 cancellation witness. We verify the rank-10 frame partition, a
+two-bit sharp frame code and 16 mixed-bit block estimates. Seven corruption
+controls exercise diagonal, orthogonality, fitting, demand, denominator
+and per-bit pricing assumptions. Acceptance
+checks use explicit exceptions and remain active under `-O`.
+
+The sample rank comparisons corroborate the proof in Section 2. We use
+the radical and minor arguments there to cover arbitrary supports; we do
+not infer that universal statement from samples. We claim a written proof
+with exact finite checks, without proof-assistant formalization.
+
+## Open next questions
+
+- Can nonlinear encoding lower cost below 40b under the fixed-support,
+  per-bit pairing charge? A researcher can start with one orthogonal
+  frame and one non-clean message, then test exact decoding and compare
+  with the existing nonlinear floor. The Schur-rank proof requires linear
+  encoders, so it leaves this question open.
+- Can a native supplier share physical work across messages and escape
+  the additive pairing charge? A researcher must specify a complete
+  source/reader/frame contract and price one small instance. We do not
+  have such a contract, and a cheaper abstract rectangle search within
+  the present model cannot supply one.
+
+## Attribution
+
+Jose Manuel (@jmbarrancoml), with substantial OpenAI Codex assistance in
+the proof, implementation and prose. A read-only mathematical agent
+identified the reduction argument during a ten-source construction
+search; the primary agent derived the trace/rank proof and implemented
+the replay. We have not obtained independent human review.
+
+Joel Pulikkan / GamingPuzzled (#620) supplied the Eisenstein geometry.
+Jonathan Lam (#798) supplied the exact pairing-cost model, geometry code,
+floor 22 and row-star upper bound. Josie Elliston (#1977) obtained floor
+23 by principal-submatrix minrank, and Maximooch (#1794) derived the
+cost screen and the clean-class floor. DreamingOfClouds (#1590, Lemma B)
+gave the same-field Hadamard rank argument for another decoder model.
+Michael Fischler (@shiftybitshiftr) treated fixed-support nonlinear messages
+in `eisenstein-message-copy-bound`. We use standard specialization and
+trace-pairing arguments with the Hadamard bound to obtain the sharp cost.
+We preserve their packages and licenses. See `SOURCES.json` for exact pins.

research/eisenstein-reduction-copy-floor/SOURCES.json

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+  ],
+  "predecessors": [
+    {"author": "Joel Pulikkan / GamingPuzzled", "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/620", "source_head": "419c7374c3bf5f2eee28f08b1e50c1d57934081c", "scope": "Eisenstein E8 geometry and the prior ten-frame construction; this archived source head is identified in PR798's SOURCES.json."},
+    {"author": "Jonathan Lam (@jonathanlam)", "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/798", "head": "0aea3382e49abd9a59d3cb6676cdb26fd15b04e5", "scope": "Exact pairing-cost model, imported geometry code, clique floor 22 and matching row-star upper bound 40."},
+    {"author": "DreamingOfClouds", "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1590", "head": "08e134b4a0c7f04869dae703f2a2e688e53c44f0", "scope": "Lemma B: same-field Hadamard-product rank argument for another decoder model. We add the Eisenstein good-reduction and pairing-rank bridge."},
+    {"author": "Maximooch", "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1794", "head": "148762722bb39c7ce2fd3fd0d42ef25d945aae4d", "scope": "Certified necessary cost screen and clean-class cost 40; contextual consequence only, not executed or required for the sharp theorem."},
+    {"author": "Josie Elliston (@JosieElliston)", "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1977", "head": "ea021c002e61fea7f2f9cd1b613f65b88f7eb7e9", "readme_git_blob": "1c4f3f947f3bf4047215ca1b14a3fc84f3e72d49", "readme_sha256": "63324a1bda6445fad12e5d68a0a34a88729980cca3cd8f87f44bfb19d7a6f19f", "scope": "Principal-submatrix floor 23 and the remaining non-clean interval; no code imported."},
+    {"author": "Michael Fischler (@shiftybitshiftr)", "path": "research/eisenstein-message-copy-bound/PROOF.md", "scope": "Fixed-support nonlinear message model and per-bit pairing pricing. Our block-linear corollary leaves nonlinear encoders open."}
+  ],
+  "attribution": "The new result is the sharp cost bound from the normalized Eisenstein Gram matrix, its good reduction at 2 and the pairing-rank conversion. Rank subadditivity, minor specialization and trace-pairing arguments are standard. We retain predecessor files and notices unchanged."
+}