Research: Gaussian E8 weighted maximal-clique ceiling 40 and independent replay

research/gaussian-e8-copy-bound/CLAIM.md

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+# Sprint claim
+
+- Track: bit supplier; OS-random assignment seed **6** (one-byte rejection sampling).
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Started: 2026-10-13 04:46:52 UTC; planned handoff by 06:16 UTC.
+- Object: #620's 60 Gaussian E8 lines, scalar binary fitting decoders with
+  arbitrary reader/source supports and any number of totals.
+- Hypothesis: averaging maximum nonorthogonality-clique rank inequalities
+  yields a stronger exact pairing-rank COPY lower bound. Close supports to
+  their rational Gaussian spans and exhaust all flat pairs; check independently.
+- Deliverable: scoped proof or counterexample, offline exact verifier,
+  deterministic certificate, meaningful negative controls and source pins.
+  No new kappa or physical supplier is promised.
+
+Three candidates considered: (1) the fixed P111 nine-total decoder basis,
+(2) Gaussian/Eisenstein extension-unit pairing-rank bounds, and (3) the
+fixed-plan P111 mini-node helper floor. #620 already computes candidate (1).
+Candidate (2) directly addresses an open report question, using #703's rank
+averaging approach on a different geometry. Seed parity chooses Gaussian
+(6 is even); its maximum nonorthogonal clique has 12 ports, compared with
+7 for Eisenstein, making its optimistic bound more promising.
+
+Before computation, two orthogonal-root hyperplanes show that averaging
+maximum 12-cliques can prove at most 40 Gaussian COPY units. Thus even a
+successful certificate will not prove the reported approximately 42.6-unit
+threshold needed to exclude this unit as a possible improvement. A stronger
+partial lower bound and a precise limitation of the method remain useful.
+
+No pinned sprint coordination issue was available: the open issue search
+returned only #650 (requesting Discussions). This claim is therefore local.
+Initial overlap review includes current selected #626/#543, #620, #631, #703, #715,
+and recent bit circuit/frame work through #720. #703 treats ten quaternionic
+lines, while #631/#715 treat real E8 star decoders/punctures. No matching
+Gaussian asymmetric COPY-cost contribution was located at that initial check. Claims in unreviewed
+PRs are treated as source context, not accepted mathematical conclusions.
+
+Prepared for @chinmayarvind23 with substantial OpenAI Codex assistance.
+
+## Completed result and final overlap update
+
+The completed [proof and review package](README.md) establishes a ceiling of
+40 for all nonnegative weighted inclusion-maximal-clique certificates on
+the 60 Gaussian E8 lines, attained by the uniform maximum-clique certificate.
+Two separate exact implementations reproduce the corresponding copy floor
+on all 2,582,128 unordered flat pairs. The actual decoder optimum remains
+in [40,60], with no new supplier or kappa claim.
+
+The final overlap refresh found Sagar Thakrar / @thakrarsagar's
+[#738](https://github.com/CrocSwap/integer-mult-bounds/pull/738), inspected at
+`9671c5d72f3119ba26f3774f40aa7130c67bcf77`. It already reports the same floor,
+closure reduction and uniform-weight sharpness. The scope was therefore
+reframed as independent verification plus the stronger ceiling for arbitrary
+nonnegative weights on inclusion-maximal cliques. Its inspected proof gives
+48 for general clique weights; our theorem gives 40 in this specified class.
+Non-maximal cliques are outside the extension. Including empty supports
+explains the pair-count difference from #738 and is not a new mathematical
+domain. No #738 code was imported.
+
+The refresh inspected Gaussian search results and new PR metadata through
+#763, including the distinct P111 factorization work in #757/#761. A last
+Gaussian search before submission also inspected #767: its planar and
+incidence decoder subclass obstructions differ from this maximal-clique
+certificate ceiling, and no code or theorem from it is used here. Main and
+the inspected #703/#738 heads remained unchanged. The only open issue was
+still #650; no separate reservation issue or claim comment was posted.
+
+The 21-case reproducibility and rejection run completed on 2026-10-13 at
+05:14:34 UTC; [VALIDATION.md](VALIDATION.md) also records the subsequent
+replay of exact staged bytes after manifest line-ending normalization. The experiment
+stopped after exhaustive flat-pair and maximal-clique coverage. The remaining
+sprint time was used for verification, overlap review and handoff, rather
+than further speculative search. The original seed and selected object
+were retained.

research/gaussian-e8-copy-bound/INDEPENDENT.json

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+{
+  "arithmetic": "Gaussian integers; fraction-free rank and exact minors",
+  "copy_lower_bound": 40,
+  "flat_count": 2272,
+  "flat_counts": {
+    "0:0": 1,
+    "1:1": 60,
+    "2:2": 360,
+    "2:3": 320,
+    "2:6": 30,
+    "3:15": 60,
+    "3:4": 960,
+    "3:6": 480,
+    "4:60": 1
+  },
+  "geometry_self_checks": "mandatory; distinct lines and principal two-line Gram ranks",
+  "inclusion_maximal_clique_count": 8704,
+  "inclusion_maximal_clique_weighting_ceiling": 40,
+  "line_count": 60,
+  "maximal_clique_counts": {
+    "10": 2304,
+    "11": 1920,
+    "12": 640,
+    "7": 1920,
+    "8": 1920
+  },
+  "maximum_clique_count": 640,
+  "maximum_clique_size": 12,
+  "maximum_clique_weight": "1/192",
+  "maximum_clique_weighting_ceiling": 40,
+  "negative_control_count": 10,
+  "negative_controls": [
+    "overstrong_denominator_191",
+    "false_rank_one_for_sharp_gram_block",
+    "orthogonal_pair_called_clique",
+    "changed_gram_normalization",
+    "nonhermitian_gram",
+    "dependent_flat_basis",
+    "omitted_flat_member",
+    "corrupt_bound_receipt",
+    "corrupt_clique_count_receipt",
+    "modified_pinned_source_bytes"
+  ],
+  "per_clique_sharp_pair_incidence": 270,
+  "root_count": 240,
+  "sharp_hyperplane_pairs": 450,
+  "sharp_pair_clique_incidence": 384,
+  "source_manifest_sha256": "cc694727a7f6b8705549918f2bb991be5968f37cbf73ae15df0642e30a7b7191",
+  "source_pin_count": 7,
+  "status": "PASS_INDEPENDENT_GAUSSIAN_COPY_BOUND",
+  "unordered_flat_pairs_with_repetition": 2582128
+}

research/gaussian-e8-copy-bound/NOTICE.md

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+# Attribution, source status and license
+
+Prepared for **@chinmayarvind23** with substantial **OpenAI Codex** assistance
+in research selection, proof development, implementation, automated checking,
+code review and documentation. Separate AI agents developed the primary and
+independent replays. This does not constitute independent human review, and
+no human-only authorship or human review is claimed.
+
+**Concurrent result.** During the final overlap refresh, after this package's
+proof and code had been developed, we located **Sagar Thakrar / @thakrarsagar's
+[PR #738](https://github.com/CrocSwap/integer-mult-bounds/pull/738)** at
+`9671c5d72f3119ba26f3774f40aa7130c67bcf77`. That contribution already states
+the same Gaussian copy floor of 40, the closure reduction, the uniform
+640-clique certificate, its uniform-weight sharpness and the interval
+[40,60]. It was prepared with substantial OpenAI Codex assistance. Those
+overlapping conclusions are credited here; this package does not claim a
+new discovery of the floor or treat its inclusion of empty-support pairs
+as a mathematical advance. No #738 executable code was imported or used
+to produce our replays.
+
+The resulting scope is **independent verification plus an extension**.
+The inspected #738 proof gives a ceiling of 48 for general nonnegative
+clique weights. Section 5 of our [proof](PROOF.md) strengthens that ceiling
+to 40 for all nonnegative weighted inclusion-maximal-clique certificates,
+including mixtures of different maximal-clique sizes. It does not address
+certificates using non-maximal cliques. The pinned
+[#738 proof](https://github.com/CrocSwap/integer-mult-bounds/blob/9671c5d72f3119ba26f3774f40aa7130c67bcf77/research/gaussian-clique-copy-bound/PROOF.md)
+documents the comparison rather than relying on its PR summary alone.
+
+The Gaussian E8 line configuration, extension-ring reduction, decoder
+question, symmetric cost-60 result and open copy-floor problem come from
+**Joel Pulikkan / GamingPuzzled's
+[PR #620](https://github.com/CrocSwap/integer-mult-bounds/pull/620)**, prepared
+with Claude Opus 5.5 assistance. Its retained extension-ring files are pinned
+in [SOURCES.json](SOURCES.json). This contribution does not claim that
+geometry, problem formulation or upper bound as new.
+
+The weighted-clique rank-averaging approach follows **Douglas R. Colkitt /
+Mister-Meeseeks' [PR #703](https://github.com/CrocSwap/integer-mult-bounds/pull/703)**
+for the ten Hurwitz E8 lines. The inspected head is
+`358383e2f1b0f292d4d6eb897c06fd8558a4f5e6`. Its
+[original proof](https://github.com/CrocSwap/integer-mult-bounds/blob/358383e2f1b0f292d4d6eb897c06fd8558a4f5e6/research/hurwitz-asymmetric-copy-floor/PROOF.md)
+and [original notice](https://github.com/CrocSwap/integer-mult-bounds/blob/358383e2f1b0f292d4d6eb897c06fd8558a4f5e6/research/hurwitz-asymmetric-copy-floor/NOTICE.md)
+are retained byte-for-byte under [vendor/pr703](vendor/pr703), including
+their attribution and license statements. Relative links within those
+unchanged copies refer to the original package context.
+
+The source/reader pairing-rank model and associated accounting derive from
+**DreamingOfClouds' [#446](https://github.com/CrocSwap/integer-mult-bounds/pull/446)
+and [#595](https://github.com/CrocSwap/integer-mult-bounds/pull/595)** work as
+used by #620. Their use here is explicit and conditional. Standard E8 root
+coordinates, Gram-rank identities and clique rank subadditivity are not
+claimed as new.
+
+The scoped contribution is the independent exact replays and the proof that
+nonnegative weighting of inclusion-maximal-clique rank inequalities cannot
+exceed 40 in this geometry. The lower bound and closure reduction overlap
+with #738. We neither assert global novelty nor claim to settle the remaining
+charge interval [40,60]. No predecessor implementation is imported by either
+verifier. Pinned source files are read as data for provenance checks.
+
+The repository base is
+`66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+[CLAIM.md](CLAIM.md) records the bounded overlap review; unreviewed PR claims
+are source context, not adopted mathematical conclusions. The existing
+selected construction and its predecessor lineage retain their credit and
+assumptions. This package adds no supplier and changes no selected kappa.
+
+New files are provided under the repository's
+[Apache License 2.0](../../LICENSE). The original authorship and license
+notices of the retained #703 documents remain in their unchanged copies;
+this notice supplements them.