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

[jonathanlam]

PR620's archived extension-ring screen leaves the asymmetric copy cost of the 40 Eisenstein E8 lines unresolved. This research note proves cost at least 22 Eisenstein pairing-rank units (44 rational units) for any finite number of scalar binary totals. A matching dual certificate also proves that 22 is the exact limit of every nonnegative weighted-clique certificate, including nonuniform weights on smaller cliques.

The actual decoder optimum remains in [22,40]. No cost-22 decoder, paid supplier or new κ is claimed. Under the inherited Dfc accounting, positive three-stage deficit would require integer copy cost 22–26; existence and all remaining implementation costs stay open.

The change adds only research/eisenstein-clique-copy-bound/: the proof, exact certificate, standard-library replay, separate modular/Fraction cross-check, rejection tests, source pins, attribution and completed local claim. The finite domain is exhausted: 772 flats, 298,378 unordered flat pairs including empty supports, and 40,054 nonempty cliques. Closure extends the finite inequality to arbitrary source/reader subsets; rank subadditivity handles every finite number of totals. No solver-status or unproved symmetry premise is used in verification.

Reproduce from the repository root (Python 3.10+, standard library):

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

Expected:

PASS_EISENSTEIN_CLIQUE_COPY_BOUND: cost >= 22; clique-method optimum 22; 298378 flat pairs; 40054 cliques; 8 negative controls; no new kappa

Additional checks:

python3 -B research/eisenstein-clique-copy-bound/crosscheck.py
python3 -B research/eisenstein-clique-copy-bound/test_verify.py

The main receipt passes on Linux with Python 3.12.3 and optimized Python 3.11.15. The independent arithmetic implementation checks every primal inequality via modular rank lower bounds and all 241 dual costs with exact Fraction elimination. Three packaging tests pass, including relocated optimized replay, corrupted-certificate rejection and source-pin rejection with output preserved on failure. Selected-record integrity, package links, exported receipt equality and staged whitespace pass. Full repository/cold supplier suites, hosted CI and proof-assistant verification were not run or claimed; existing construction files are unchanged.

Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Structural-decoder assignment seed 48. The selected #626/#543 κ remains 51110230636731/31250000000000000. Physical interpretation inherits PR446's pairing-rank premise and the separate Dfc accounting; compiler, hosting, precision, tape and analytic obligations remain outside this finite result.

Credit Joel Pulikkan / GamingPuzzled's PR620, using its archived head 419c7374c3bf5f2eee28f08b1e50c1d57934081c, for the geometry and prior screen. Credit Douglas R. Colkitt's PR703 (358383e2f1b0f292d4d6eb897c06fd8558a4f5e6) and Sagar Thakrar's PR738 (9671c5d72f3119ba26f3774f40aa7130c67bcf77) for the weighted-clique method on Hurwitz and Gaussian geometries. Their code is not imported. DreamingOfClouds and the earlier E8/community lineage retain their credit. Exact inspected-byte hashes and review-status distinctions are in SOURCES.json.

No pinned coordination issue existed, so the claim and one result update are local. The final full metadata check covered 614 open PRs through #795; a subsequent check through #796 found only an unrelated prime-power audit using “Eisenstein.” PR767 and PR785 concern other geometries/models. Main remained at the recorded base.

Prepared by Jonathan Lam (@jonathanlam) with substantial OpenAI Codex assistance in mathematics, implementation, verification and documentation. Apache-2.0. No independent human review or full multiplication theorem proof is claimed.

[Mister-Meeseeks]

100-PR batch review at 0aea3382e49abd9a59d3cb6676cdb26fd15b04e5.

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

Reviewed scope: Eisenstein scalar-total clique floor 22 with exact dual certificate; actual optimum remains [22,40], three-stage construction not excluded.

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.