Research note: Eisenstein E8 copy cost ≥22 and sharp clique-method limit

research/eisenstein-clique-copy-bound/CLAIM.md

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+# Sprint claim
+
+- Track: structural decoder research; OS assignment seed **48**, drawn with
+  `secrets.randbelow(100)` at 2026-10-13 05:22:15 UTC.
+- Exact base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Object: scalar binary asymmetric decoders on the **40 Eisenstein E8 lines**
+  of PR620, charged by rational pairing rank divided by two.
+- Method: reconstruct the rational order-three geometry, enumerate its
+  maximum nonorthogonality cliques and all line-matroid flats, then certify
+  the strongest uniform clique weight on every flat pair. Use the weighted
+  clique argument credited to PR703 and PR738, in this different geometry.
+- Deliverable: a scoped copy-charge bound or exact limitation of this
+  method, a complete finite certificate, negative controls and a paid-format
+  optimistic screen. No new kappa is promised.
+- Three candidates considered: (0) Eisenstein asymmetric copy cost;
+  (1) the remaining Gaussian decoder gap; (2) P112 decoders beyond frame
+  partitions. Recent Gaussian/P111 work makes (0) less crowded. Seed 48
+  fixes the independent rank-control samples and distinguishes line 8
+  (48 modulo 40) for the saturated-singleton dual rejection obligations.
+- Prior work: PR620's extension-ring solver reports leave the asymmetric
+  Eisenstein cost unresolved. This changes the proof method and covers any
+  finite number of totals, rather than extending a timed-out solver run.
+  PR703 handles ten Hurwitz lines; PR738 handles 60 Gaussian lines; PR767
+  handles Gaussian subclasses. None supplies the Eisenstein certificate.
+- Coordination: no pinned issue existed at the fresh pre-claim check;
+  no separate reservation issue was opened. Open PR metadata through #769
+  and the review ledger were inspected, including PR620's partial status
+  and PR446's unreviewed status. No matching Eisenstein/Witting claim was
+  found in the open PR search.
+- Budget: 45 minutes from 05:22:15 UTC; stop investigation by 05:56 UTC
+  and reserve the final quarter for verification and handoff. Expected
+  finish: **2026-10-13 06:07 UTC**. Stop after the complete finite clique/flat
+  domain; do not broaden to other geometries or claim decoder optimality.
+- Contributor: Jonathan Lam (@jonathanlam), with substantial OpenAI Codex
+  assistance in research, implementation and verification; Apache-2.0.
+
+## One result update — 2026-10-13 05:48 UTC
+
+Completed: **C ≥ 22 Eisenstein rank units**, and an exact matching dual proves
+that *all* nonnegative clique-weight certificates have optimum 22. The actual
+decoder optimum remains in [22,40]; no new κ is claimed. The same finite
+domain gave one refinement: the maximum-clique-only bound 280/13 saturated,
+so singleton inequalities were added, and the resulting all-clique dual
+closed this proof method at 22. No other geometry or supplier was searched.
+
+Coverage: 772 flats, 298,378 unordered flat pairs, 40,054 cliques, eight
+mathematical controls, and an independent modular/Fraction arithmetic check.
+Python 3.11/3.12 replay, three packaging tests and selected-record integrity
+passed. Final PR metadata covered 614 open PRs through #795 with no matching
+Eisenstein contribution. Origin/main remains at the claimed base.
+
+Deliverable: [proof and reproduction](README.md), [validation](VALIDATION.md),
+and the [focused contribution branch](https://github.com/jonathanlam/integer-mult-bounds/tree/research/eisenstein-clique-bound-seed48/research/eisenstein-clique-copy-bound).
+One research-note PR is the intended handoff; no frontier edit or merge.

research/eisenstein-clique-copy-bound/PROOF.md

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+# Eisenstein E8: copy charge at least 22, and a sharp limit of clique certificates
+
+This is a theorem about a specified finite decoder model. It is not a new
+supplier, a proof of decoder optimality, or a new multiplication exponent.
+The weighted-clique argument follows the distinct Hurwitz and Gaussian
+geometries of [PR703](https://github.com/CrocSwap/integer-mult-bounds/pull/703)
+and [PR738](https://github.com/CrocSwap/integer-mult-bounds/pull/738).
+The new result is the exact primal/dual certificate for the **40-line
+Eisenstein geometry**, including a bound on *all* clique-weight certificates.
+
+## Model and geometry
+
+Start with the 240 E8 roots in doubled coordinates: vectors with two entries
+equal to ±2 and all other entries zero, and the vectors in {±1}^8 with an even
+number of minus signs. Each root has squared norm eight.
+
+Following Joel Pulikkan / GamingPuzzled's archived PR620, choose four mutually
+orthogonal A2 planes, with roots (a,b) satisfying a·b = -4, and let J be the
+product of their rotations s_a s_b. The literal pairs are in the certificate.
+The independent implementation uses the same root order and greedy choices
+as the pinned predecessor, so it defines the same rational action, not merely
+a configuration with similar counts. The reflection formula is
+
+    s_a(x) = x - (x·a)a/4.
+
+The exact checks establish J^3 = I, J^2 + J + I = 0, and preservation of the
+inner product and root set. Thus Q^8 is a four-dimensional vector space over
+K = Q(ω), with multiplication by ω realized by J. The root orbits under ±J
+partition the roots into 40 K-lines. For a line set S, let X_S be the rational
+span of r_i,Jr_i for i in S. Every X_S is J-invariant.
+
+Two distinct lines are nonorthogonal if r_i·r_j or r_i·Jr_j is nonzero. Each
+line has 27 such neighbors. Let B be an arbitrary 40×40 matrix over F2 with
+
+    B_tt = 1;
+    B_ts = 0 if distinct lines t,s are nonorthogonal.
+
+The other directed entries are unconstrained. Consider any finite decomposition
+
+    B = Σ_j 1_(R_j) 1_(A_j)^T       over F2.
+
+There is no symmetry, minimal-rank, or number-of-totals restriction. For any
+row bases V_R,V_A of the rational spans, define the abstract copy charge
+
+    c(R,A) = rank_Q(V_R V_A^T)/2,       C = Σ_j c(R_j,A_j).
+
+This is a *pairing* rank; it need not equal either support's span dimension.
+The rank is even: the kernel X_R ∩ X_A^⊥ is J-invariant, because J is an
+isometry and X_A is J-invariant. Both domain and kernel are K-spaces.
+Consequently each c is an integer. Translating this abstract charge into a
+physical copy lower bound inherits PR446's premise, which the base review
+ledger marks unreviewed. The finite algebraic result needs no such premise.
+
+## General clique bound and the lower certificate
+
+A clique Q is a set of pairwise nonorthogonal lines. The fitting rules give
+B[Q,Q] = I. A binary rectangle restricted to Q×Q has rank one exactly when
+both supports meet Q. Rank subadditivity gives
+
+    |Q| ≤ Σ_j [Q∩R_j ≠ ∅ and Q∩A_j ≠ ∅].                 (1)
+
+For each nonempty clique assign a nonnegative rational weight λ_Q. Suppose
+these weights satisfy, for all supports R,A,
+
+    Σ_Q λ_Q [Q meets R and A] ≤ c(R,A).                  (2)
+
+Multiplying (1) by λ_Q and summing proves C ≥ Σ_Q λ_Q |Q|.
+
+The certificate uses weight **1/40 on each singleton** and **1/960 on each
+of the 2,880 seven-line cliques**, with weight zero elsewhere. The resulting
+objective is exactly
+
+    40/40 + 2,880·7/960 = 22.                            (3)
+
+To certify (2), close R and A in the line matroid: cl(S) contains every
+configuration line lying in X_S. Closure leaves X_S, hence the pairing rank,
+unchanged. It can only increase the clique-hit indicators. Therefore it is
+sufficient to check every pair of flats F,G. Write H_7(F,G) for the number
+of seven-line cliques meeting both flats. The required integer inequality is
+
+    24 |F∩G| + H_7(F,G) ≤ 960 c(F,G).                    (4)
+
+The replay verifies (4) by exact rational rank on all **298,378 unordered
+pairs of the 772 flats**, including the empty flat. Symmetry of the pairing
+rank and hit count covers both orders. This is a complete reduction from
+arbitrary supports; the theorem does not assume supports were originally
+flats. Equations (1)–(4) prove **C ≥ 22** for every finite decomposition.
+
+## Why no weighting of cliques can prove more
+
+The following upper bound concerns the certificate method, not decoder cost.
+Let P consist of the **240 unordered pairs** F,G of twelve-line, rank-three
+flats with c(F,G)=2. Each such pair has |F∩G|=2 and H_7(F,G)=1,872, so (4)
+is tight. Give each of these pairs dual weight 1/24; give the full/full
+pair dual weight 1/2. Its cost is
+
+    (240·2)/24 + 4/2 = 22.                              (5)
+
+For *every literal nonempty clique* Q, the certificate verifies
+
+    1/2 + #{(F,G) in P : Q meets F and G}/24 ≥ |Q|.      (6)
+
+The complete incidence census is below. In particular, no unproved
+transitivity assumption or orbit-averaging argument is needed.
+
+| Clique size | Dual pairs hit | Number of cliques |
+|---:|---:|---:|
+| 1 | 12 | 40 |
+| 2 | 48 | 540 |
+| 3 | 81 | 2,880 |
+| 3 | 108 | 360 |
+| 4 | 106 | 6,480 |
+| 4 | 123 | 2,880 |
+| 4 | 192 | 90 |
+| 5 | 125 | 5,184 |
+| 5 | 134 | 8,640 |
+| 6 | 145 | 10,080 |
+| 7 | 156 | 2,880 |
+
+Now take *any* nonnegative clique weights satisfying (2), including arbitrary
+nonuniform weights on smaller cliques. Multiply (6) by λ_Q, sum over Q,
+and apply (2) to the full/full pair and each pair in P. Equation (5) gives
+
+    Σ_Q λ_Q |Q| ≤ 22.
+
+Together with (3), this proves that the optimum of the entire weighted-clique
+certificate method is exactly **22**. A stronger copy lower bound will need
+an ingredient beyond inequalities (1)–(2). In particular, using more clique
+weights or a different clique-weight optimizer alone cannot improve it.
+
+## Decoder bounds and the optimistic accounting screen
+
+Every fitting B has an abstract cost-40 decomposition: for each row t,
+take R={t} and A equal to that row's nonzero positions. A contains t; every
+other member of A is orthogonal to t. The rational Gram block therefore has
+rank two, so this row costs one. These 40 rectangles reproduce B. Hence the
+minimum charge lies in **[22,40]**. The lower endpoint is attained by a
+*certificate*, and no cost-22 decoder is constructed.
+
+For the separate Dfc stage accounting in the pinned format notes, v=40,
+each line has rational rank k=2, and cst=2C. Thus, conditionally on that
+accounting and the physical interpretation of the charge,
+
+    D/T = (s-1)vk - s·cst ≤ 80(s-1) - 44s = 36s-80.
+
+At s=3 the upper deficit is 28, which is positive. This lower bound **does
+not exclude three stages**. A positive three-stage deficit would require
+C < 80/3; integrality narrows the unresolved possibilities to C=22,...,26.
+All helpers, source access, frame changes, copies beyond this lower bound,
+restoration and recursive children would still have to be paid. None is
+implemented or assumed free in an achieved construction. At the cost-40
+row decomposition the deficit is -80 for every stage count, so that abstract
+upper-bound witness itself cannot supply a saving under Dfc.
+
+The selected #626/#543 composition remains complex-bound. This research
+does not provide a replacement supplier or change any assembly constraint.
+
+## Completeness and proof boundary
+
+Flats are generated from the empty span by adding one root line and taking
+exact rational span/closure. Any flat has a line basis, so induction on basis
+size proves this generates every flat in four K-dimensions. The verifier
+also checks all 30,880 one-line extensions remain in the enumerated set.
+The flat counts by (K-rank, number of lines) are
+
+    (0,0):1; (1,1):40; (2,2):240; (2,4):90;
+    (3,5):360; (3,12):40; (4,40):1.
+
+All nonempty cliques are generated once by successively adjoining larger
+vertices adjacent to every previous vertex. This recursion exhausts every
+ordered increasing clique. A separate Bron–Kerbosch maximal-clique census
+corroborates the 2,880 maximum cliques and 90 smaller maximal cliques.
+
+The committed certificate contains the literal flat generators, maximum
+cliques, all 240 dual pairs, complete pair/incidence censuses, and hashes of
+the regenerated root orbits and all 40,054 cliques. The replay uses no solver.
+An ordinary Fraction elimination on original root generators checks 258
+seeded/boundary Gram blocks independently of the integer RREF implementation.
+Eight mathematical negative controls cover fitting identities, clique
+admission, rank accounting, and both primal and dual weights/coverage.
+
+The separate `crosscheck.py` imports neither implementation. It checks all
+298,378 primal inequalities on original integer generators using rank modulo
+101. Such a rank is a lower bound on rational rank, which is the sufficient
+direction for (4). It checks all 240 dual-pair costs and the full-pair cost
+by exact Fraction elimination. Geometry and enumeration completeness remain
+the responsibilities of the main replay and the preceding argument.
+
+This is a written proof with exhaustive finite premises, not a proof-assistant
+formalization or a finite replay claimed to prove the full multiplication
+theorem. Whole K-lines, scalar binary totals and the stated fitting pattern
+are essential to the scope. Sub-line/vector-valued operations, different
+decoder patterns or geometries, physical realization, all-size compilation,
+hosting, precision, tape and analytic transfer remain outside it.
+
+Prepared by Jonathan Lam (@jonathanlam) with substantial OpenAI Codex
+assistance. See [SOURCES.json](SOURCES.json) for exact pins and predecessor
+credit. New material is Apache-2.0 under the repository license.

research/eisenstein-clique-copy-bound/README.md

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+# Research note: Eisenstein E8 copy floor and clique-method limit
+
+**Every scalar binary decoder on the 40 Eisenstein E8 lines costs at least
+22 Eisenstein pairing-rank units (44 rational units).** The best certificate
+using arbitrary nonnegative weights on nonorthogonality cliques proves
+exactly 22. This identifies a precise limit of that proof method.
+
+The abstract decoder optimum remains in **[22,40]**. No cost-22 decoder,
+paid supplier or new κ is claimed. Under the inherited Dfc accounting, a
+three-stage construction with positive deficit would need integer copy
+cost 22–26. This result does not decide whether such a decoder exists.
+
+From the repository root, using Python 3.10+ and only the standard library:
+
+```sh
+python3 -B research/eisenstein-clique-copy-bound/verify.py
+```
+
+Expected output:
+
+```text
+PASS_EISENSTEIN_CLIQUE_COPY_BOUND: cost >= 22; clique-method optimum 22; 298378 flat pairs; 40054 cliques; 8 negative controls; no new kappa
+```
+
+The replay takes about 16 seconds on the development Linux host, verifies
+the committed certificate, and leaves the checkout unchanged. Optional
+`--write /tmp/eisenstein-replay.json` exports the receipt only after checking
+it. Optimized Python is supported; checks do not depend on `assert`.
+
+Read [PROOF.md](PROOF.md) for the model, completeness argument and exact
+primal/dual proof. [certificate.json](certificate.json) gives the literal
+finite evidence. [SOURCES.json](SOURCES.json) records source hashes and
+attribution; [CLAIM.md](CLAIM.md) records seed, scope and sprint handoff.
+
+Additional verification commands:
+
+```sh
+python3 -B research/eisenstein-clique-copy-bound/crosscheck.py
+python3 -B research/eisenstein-clique-copy-bound/test_verify.py
+```
+
+The first independently checks every primal inequality using ranks modulo
+101 and all 241 dual costs using Fraction arithmetic; it imports no package
+implementation. The second runs three tests for relocated optimized replay,
+corrupt-certificate rejection and source-pin rejection. See
+[VALIDATION.md](VALIDATION.md) for the checks actually run and their scope.
+
+The experiment exhausted all 772 line-spanned flats, all 298,378 unordered
+flat pairs including empty supports, and all 40,054 nonempty cliques. It
+stopped after this finite domain and its exact certificate. Seed **48**
+controls the additional Fraction cross-checks and the distinguished line
+for dual rejection controls. A discovery LP nominated the rational coefficients; the
+reproduction command has no optimizer or numerical-infeasibility dependency.
+
+PR620's archived extension-ring report supplied the geometry and reported
+the earlier lower bound of 9. Its solver/barrier claims were only partially
+reviewed; they are not premises of this proof. Douglas R. Colkitt's PR703
+and Sagar Thakrar's PR738 supplied the weighted-clique method in the distinct
+Hurwitz and Gaussian geometries. This package adds the Eisenstein certificate
+and its matching upper certificate over *all* clique weights. None of their
+implementations is imported or executed. DreamingOfClouds' PR446 pairing-rank
+premise and PR595 format accounting are inherited only for the conditional
+physical interpretation and stage screen. The E8 and wider community lineage
+retains its predecessor credit.
+
+Exact base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. The selected conditional
+κ there is `51110230636731/31250000000000000`, from #626 with #543. This
+package changes no existing supplier, generator, selected pointer or
+`upstream/` file. The finite algebraic result does not discharge the selected
+construction's compiler, realization, hosting, precision, tape or analytic
+assumptions. No full multiplication proof or independent human review is
+claimed.
+
+Prepared by Jonathan Lam (@jonathanlam), with substantial OpenAI Codex
+assistance in mathematics, implementation, verification and documentation.
+New material is licensed under the repository's Apache-2.0 license.