Independent verification: P112 root-star cost 64 via maximal-clique census

.github/workflows/root-star-integer-cover.yml

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+name: P112 root-star integer cover
+on:
+  pull_request:
+    paths:
+      - 'research/root-star-integer-cover-seed46/**'
+      - '.github/workflows/root-star-integer-cover.yml'
+  push:
+    branches: [main]
+    paths:
+      - 'research/root-star-integer-cover-seed46/**'
+      - '.github/workflows/root-star-integer-cover.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  verify:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.12'
+      - name: Both exact implementations and failure controls
+        run: python -I -O -B research/root-star-integer-cover-seed46/test_package.py
+      - name: Committed artifacts remain unchanged
+        run: git diff --exit-code

research/root-star-integer-cover-seed46/CLAIM.json

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+{
+  "assignment_seed": 46,
+  "randomness": "Python secrets.randbelow(100), OS-backed; generated 2026-10-13T20:41:26.209187+00:00",
+  "track": "Structural decoder research (40-49)",
+  "base_commit": "49b3157d1d89adcef16df8be98445451672be5ba",
+  "candidate_tasks": [
+    "Classify the ten-term both-outside-code residual of PR1224; deferred because coefficient and cost geometry is too broad for one session.",
+    "Close PR1226 cost-63 root-star assembly window using integer frame coverage and the nonorthogonality clique number; selected, seed mod 3 = 1.",
+    "Remove PR1222 quadratic restriction from common three-dimensional five-cube codes; deferred because it needs a different symbolic rank argument."
+  ],
+  "object": "PR1226 P112 masked E-graph decoder, support pairs C(alpha) or N(alpha), 120 root normals; unrestricted coefficients and finite term count, plus free odd singleton corrections.",
+  "changed_mechanism": "Replace averaged fractional frame coverage by pairwise compatibility of missed-frame sets. The support alphabet is unchanged; no repeated optimizer search.",
+  "hypothesis": "Every decoder in this root-star support family has cost at least 64; eight coordinate-star terms attain 64.",
+  "method": "Independent integer E8 generation, exact F2 pairing ranks for 57600 support pairs, complete 1260-frame enumeration, exact 112-vertex nonorthogonality clique census and a separate coloring-bound oracle.",
+  "protected_invariant": "A lower bound must cover every allowed support pair and every finite term count, with exact arithmetic; it must not be promoted to arbitrary supports or a physical supplier.",
+  "ways_evidence_can_fail": [
+    "Wrong punctured frame or root-line conventions",
+    "Confusing support containment with exact support",
+    "Omitting rank-7 atoms or frames",
+    "Assuming averaged coverage implies simultaneous compatibility",
+    "Mistaking bounded search for complete enumeration",
+    "Relying on assertions disabled by optimized Python"
+  ],
+  "optimistic_budget": "If cost>=64 in the unchanged copied-information model, this family cannot reduce the existing coordinate decoder charge; source access, helpers, frame changes, restoration, precision and recursive realization remain unprovided. No kappa claim.",
+  "expected_deliverable": "Scoped proof, deterministic certificate, independent check and meaningful rejection controls in one new research directory.",
+  "expected_finish_utc": "2026-10-13T21:41:26Z",
+  "budget": "60 minutes per AGENTS.md; orientation at most 12 minutes, final 15 reserved for verification/handoff; earlier completion allowed.",
+  "coordination": "No pinned issues at initial live inspection; local claim per donor instructions. Discussion 1210 is PR1226 census, a predecessor; live search found no integer missed-frame packing claim.",
+  "predecessor_prs": {
+    "1226": "8b10989e4e45f0f9ce41004d7898a21a32f87e11",
+    "1212": "072ae2f43d5c249a31e5d6ed266bfef3924417f3",
+    "1213": "8867cf20fbe0d3960f68bcffca118a5ac0624f7f",
+    "1236": "ee8437bd9879272094e20c024cd34a53c4e84cd8"
+  },
+  "predecessor_credit": [
+    "Navinor, PR1226: full trimmed-star census, covering lemma and residual assembly question",
+    "DreamingOfClouds, PR717/595: masked E-graph model and binary label pairing",
+    "Joel Pulikkan / GamingPuzzled, PR620: coordinate decoder and geometric lineage",
+    "David Arias / DavidAriasC, PR1236: concurrent theorem and same endpoint argument; this package independently corroborates it."
+  ],
+  "status": "COMPLETE",
+  "scope_status": "refined",
+  "scope_reason": "Exact integer compatibility closes the assembly gap explicitly left by the full fixed-family census; this is not a search inside an already excluded class.",
+  "final_contribution_type": "Independent verification of concurrent PR1236; no theorem priority claim.",
+  "overlap_refinement": {
+    "discovered_utc": "2026-10-13T20:57:44Z",
+    "pr": 1236,
+    "head": "ee8437bd9879272094e20c024cd34a53c4e84cd8",
+    "reason": "The same structural endpoint theorem arrived during the investigation. Preserve distinct independent maximal-clique verification and exact five-section source-certificate binding; do not submit as a new theorem."
+  },
+  "result": "Independently corroborated PR1236 exact copy floor 64, with complete maximal-clique/subset and five-section certificate binding.",
+  "result_artifact": "README.md",
+  "local_final_handoff": "The submitted PR URL is recorded in the external sprint completion receipt."
+}

research/root-star-integer-cover-seed46/NOTICE.md

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+# Attribution and provenance
+
+Independent verification contribution: Hanson Sun (@bbMisk), with substantial OpenAI Codex
+assistance in research, proof, implementation, verification and prose.
+Both checkers are AI-assisted. Separate algorithms are not independent human
+review. New work is supplied under the repository's Apache-2.0 license.
+
+Navinor's PR1226 at `8b10989e4e45f0f9ce41004d7898a21a32f87e11` supplies
+the precise root-star alphabet, the complete mixed-cost census, the averaged
+frame-cover lemma and the residual cost-63 assembly question. Its README,
+proof, notice, results and verifier are retained verbatim under `sources/`,
+with Git blob and SHA256 bindings in SOURCES.json. Those files preserve their
+own attribution to opencode and predecessor authors. The new executables do
+not import or execute that verifier.
+
+The model and free odd-entry corrections belong to DreamingOfClouds'
+PR717/595 and retained E8/common-label lineage. The coordinate-star cost-64
+decoder is predecessor work, including Joel Pulikkan / GamingPuzzled's PR620.
+This package does not claim a new attaining decoder, frame geometry, or rank
+histogram. Its contribution is independent evidence for the integer
+compatibility obstruction that closes PR1226's fixed-family assembly window.
+
+David Arias' concurrent PR1236 (@DavidAriasC), prepared with substantial
+OpenAI Codex assistance, had already submitted the same theorem and endpoint
+argument when the final overlap refresh found it. The exact audited head is
+`ee8437bd9879272094e20c024cd34a53c4e84cd8`. Its certificate is retained
+verbatim and source-bound under `sources/pr1236-CERTIFICATE.json`. The separate
+checker reconstructs and matches all five geometric sections, including the
+full atom-stream hash. It uses a pivoted maximal-clique/subset proof instead
+of PR1236's lattice/quadratic-form method. PR1236's control-execution claims
+and its independent lattice receipt are not being replayed or certified.
+There is no theorem-priority claim over PR1236.
+
+Concurrent PR1212 (@burntout76) and PR1213 (@DavidAriasC) address broader
+eight-term masked decoder rigidity. They are credited as related work, not
+imported mathematical dependencies. PR1191 (@mightyOil) treats a root-linear
+factor at arbitrary term count. None of these hypotheses is assumed by the
+nine-cost-seven endpoint argument.
+
+Selected PR1094 (Rohan Arun), Jeremy Kothe's packing/construction lineage,
+Hao Yan's controller and the unchanged PR589/PR555 bit lineage are the
+reviewed background at the base. This package makes no change to that result.

research/root-star-integer-cover-seed46/OVERLAP.json

@@ -0,0 +1,100 @@
+{
+  "status": "POSSIBLE_OVERLAP_NOT_A_NOVELTY_CERTIFICATE",
+  "searched_at": "2026-10-13T20:43:49.152543+00:00",
+  "terms": [
+    "five-cube",
+    "nine-term",
+    "mixed-beta"
+  ],
+  "coverage": {
+    "open_prs": {
+      "fetched_at": "2026-10-13T20:43:47.202163+00:00",
+      "complete_pagination": true,
+      "records": 839
+    },
+    "discussions": {
+      "fetched_at": "2026-10-13T20:43:49.110585+00:00",
+      "complete_pagination": true,
+      "records": 11
+    }
+  },
+  "warnings": [],
+  "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission.",
+  "matches_total": 19,
+  "nearest_work": [
+    {
+      "number": 1236,
+      "title": "Research: close the masked root-star assembly gap with sharp cost 64 (no new kappa)",
+      "head": "ee8437bd9879272094e20c024cd34a53c4e84cd8",
+      "updated_at": "2026-10-13T20:46:58Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1236",
+      "distinction": "Same theorem and endpoint argument arrived concurrently. Final submission is independent verification: direct root generation, low-column elimination, pivoted maximal-clique/subset coverage and byte-bound reproduction of all five geometric sections of this exact PR certificate. No priority claim."
+    },
+    {
+      "number": 1226,
+      "title": "Research: masked root-star trimmed-star census on P112; exact covering bound 784/13, 64 falsified for beta-7 atoms (no new kappa)",
+      "head": "8b10989e4e45f0f9ce41004d7898a21a32f87e11",
+      "updated_at": "2026-10-13T20:29:08Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1226",
+      "distinction": "Direct predecessor. Adds simultaneous missed-frame compatibility and excludes every cost-63 assembly; source census is independently reconstructed."
+    },
+    {
+      "number": 1224,
+      "title": "Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa)",
+      "head": "00e6ed0810b8ebc7f6133e2c036d0f1c9a3a4162",
+      "updated_at": "2026-10-13T20:26:31Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1224",
+      "distinction": "Nine-term all-pairs normal form; not used. Here exactness is only on the E-graph."
+    },
+    {
+      "number": 1213,
+      "title": "Research: classify the P112 masked decoder coordinate component (no new kappa)",
+      "head": "8867cf20fbe0d3960f68bcffca118a5ac0624f7f",
+      "updated_at": "2026-10-13T20:22:41Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1213",
+      "distinction": "Eight-term coordinate-component classification; not used."
+    },
+    {
+      "number": 1212,
+      "title": "Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa)",
+      "head": "072ae2f43d5c249a31e5d6ed266bfef3924417f3",
+      "updated_at": "2026-10-13T20:14:55Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1212",
+      "distinction": "Broader eight-term masked rigidity; not used. New step concerns nine cost-seven terms in a fixed alphabet."
+    },
+    {
+      "number": 1191,
+      "title": "Research: one-sided E8 mask rigidity on 1,373 punctured units (no new kappa)",
+      "head": "640b43bb13d3d476a2d83b74ef0d33f14d1b5711",
+      "updated_at": "2026-10-13T19:54:49Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1191",
+      "distinction": "One root-linear factor at arbitrary term count; not used. Root-star coefficients here are unrestricted on their exact supports."
+    }
+  ],
+  "coordination": "Initial GraphQL pinnedIssues inventory empty. All active discussions inspected; local claim follows user instructions when no pinned issue exists.",
+  "final_refresh": {
+    "searched_at": "2026-10-13T20:57:44.369567+00:00",
+    "terms": [
+      "root-star",
+      "trimmed-star",
+      "cost-63",
+      "missed-frame",
+      "integer cover"
+    ],
+    "coverage": {
+      "open_prs": {
+        "fetched_at": "2026-10-13T20:57:40.322905+00:00",
+        "complete_pagination": true,
+        "records": 843
+      },
+      "discussions": {
+        "fetched_at": "2026-10-13T20:57:44.311345+00:00",
+        "complete_pagination": true,
+        "records": 16
+      }
+    },
+    "warnings": [],
+    "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission.",
+    "matches_total": 10
+  }
+}

research/root-star-integer-cover-seed46/PROOF.md

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+# Independent finite proof of the P112 root-star cost-64 floor
+
+This is a computer-assisted theorem about one finite support alphabet. It is
+not a physical supplier, a formal proof-assistant result, or a multiplication
+theorem. The finite premises below are reconstructed by two separate programs.
+The result and endpoint argument independently corroborate David Arias'
+concurrent PR1236 at `ee8437bd9879272094e20c024cd34a53c4e84cd8`, found during
+the submission overlap refresh. The distinct second implementation uses
+pivoted maximal-clique enumeration, not PR1236's lattice/quadratic-form
+obstruction. No new theorem priority is claimed.
+
+## Model and statement
+
+Use the E8 roots of squared norm two and identify antipodal roots. Remove the
+eight lines of the coordinate frame
+`f = e_(2a) +/- e_(2a+1)`, `a=0,...,3`, leaving P112. Write
+`G_ij = <r_i,r_j>`. The binary label pairing of PR717/PR1226 is
+
+```text
+S_ij = (G_ij + 1) mod 2.
+```
+
+The E-graph includes the diagonal and the orthogonal pairs. A masked decoder
+has finitely many real or complex rank-one terms `d_k t_k^T` whose sum equals
+`2G` on that graph. Transpose is ordinary transpose, not conjugate transpose.
+In particular, on any eight-line orthogonal frame F the sum is `4 I_8`.
+
+For each of the 120 root-line normals alpha, permit the supports
+
+```text
+C(alpha) = {i in P112 : <r_i, alpha> != 0},
+N(alpha) = C(alpha) minus alpha's own line, if that line is live.
+```
+
+Each positive-cost term must have **exactly** one of these supports on each
+side. Its cost is `beta_k = rank_F2 S[supp(d_k), supp(t_k)]`. Arbitrary nonzero
+coefficients on those supports are allowed; arbitrary proper sub-supports are
+not included. Additional terms of pairing rank zero may have arbitrary
+supports. In particular, the free odd-entry singleton corrections in PR717
+are allowed. Zero terms can be discarded.
+
+**Theorem.** Every such decoder has `sum beta_k >= 64`. Eight coordinate-star
+terms attain 64. This closes the fixed-family assembly window left by
+Navinor's PR1226; it does not strengthen the fractional inequality itself.
+
+## Exact finite premises
+
+Both implementations independently generate the 120 lines and the same
+punctured frame. Their internal root order differs from PR1226's order; the
+certificate specifies this package's order by a hash and explicit witness roots.
+
+1. P112 has exactly **1,260** orthogonal eight-line frames. Every live line is
+   in **90** frames. Every orthogonal pair of distinct live lines extends to a
+   live frame: 2,856 such pairs belong to ten frames each, and 448 to fifteen.
+2. The 240 indexed supports and all **57,600 ordered support pairs** have
+   cost histogram `7: 7616`, `8: 37440`, `9: 12544`. Indexing retains the eight
+   duplicates `C(f)=N(f)` at punctured normals, as in PR1226.
+3. A support misses a live frame precisely when it is `N(alpha)` with alpha
+   live and the frame contains alpha. Every cost-seven atom has at least one
+   such live trimmed endpoint. There are 1,792 atoms with one and 5,824 with
+   two; these meet 1,170 and 1,080 frames respectively.
+4. The graph on P112 joining distinct **nonorthogonal** lines has clique
+   number **eight**. The complete clique counts, including the empty clique,
+   are:
+
+   | Size | Count |
+   |---:|---:|
+   | 0 | 1 |
+   | 1 | 112 |
+   | 2 | 2,912 |
+   | 3 | 25,088 |
+   | 4 | 88,704 |
+   | 5 | 161,280 |
+   | 6 | 145,152 |
+   | 7 | 55,296 |
+   | 8 | 4,032 |
+
+   The primary checker visits all **482,577** cliques by increasing-vertex
+   recursion, with no size cutoff. The second checker finds all **27,968**
+   maximal cliques by pivoted Bron-Kerbosch recursion, then takes the union of
+   all their subsets. Every clique of a finite graph extends to a maximal
+   clique, so this is a separate complete coverage argument. Both recover the
+   entire displayed census and an explicit size-eight witness.
+
+The support/frame assertion in item 3 also has a direct explanation. A frame
+is a real basis, so a nonzero root normal cannot be orthogonal to every frame
+line. Removing the normal's own line can remove the final intersection only
+when that line belongs to the frame. If it does belong, it is orthogonal to
+the other seven lines. The programs check this identity on every indexed
+support and every frame; no generic-position assumption is used.
+
+## From the premises to the floor
+
+Call a term a *hit* on F when both supports meet F. Its restricted outer
+product has rank at most one and vanishes unless it hits F. Since the target
+on F is `4 I_8`, at least eight terms must hit every frame. A cost-zero term
+cannot hit a frame: every binary pairing between two members of F is one,
+whereas a matrix of binary rank zero has all entries zero.
+
+Let n count positive-cost terms. Each costs at least seven, by item 2.
+Suppose the total cost were below 64. Then `8 <= n <= 9`.
+
+* If `n=8`, every term must hit every frame. But cost below 64 forces at
+  least one cost-seven term. Its live trimmed endpoint belongs to 90 missed
+  frames, contradicting the eight-hit requirement.
+* If `n=9`, every term must cost seven: the integer total is exactly 63.
+  Select one live trimmed endpoint alpha_k from each term, which item 3
+  permits. A frame containing alpha_k is missed by that term. No frame can
+  contain two selected endpoints, since it would receive at most seven
+  hits. Repeated endpoints are therefore impossible. Two orthogonal
+  endpoints are also impossible, by item 1. The nine selected endpoints
+  would consequently form a nonorthogonality nine-clique, contradicting
+  item 4. This holds for any choice when a term has two eligible endpoints.
+
+All finite term counts have been covered. This is an integer compatibility
+argument: the averaged coverage bound `784/13` from PR1226 stays valid, but
+does not account for two terms missing the same frame.
+
+## Attainment and accounting limits
+
+For the eight removed frame roots f_k take
+`d_k(i)=t_k(i)=<r_i,f_k>`. Orthogonality and norm two give
+
+```text
+sum_k <r_i,f_k><r_j,f_k> = 2 <r_i,r_j>.
+```
+
+All 12,544 entries are checked exactly, and each term costs eight. This is
+the existing coordinate decoder, credited to the predecessor lineage; the
+result being corroborated is its optimality in the specified mixed-cost root-star family.
+
+The result prevents a reduction of the copied-information constant below 64
+within this alphabet, even before charging source access, gather arithmetic,
+helper frames, dirty restoration, recursive realization, precision, or
+hosting. It does not assert that all other costs are fixed, nor rule out
+improvements to them. No complete new invoice, native word, improved moment,
+or kappa is claimed. Selected PR1094 and its compiler, all-size, mutable-source,
+whole-call hosting, precision/tape, analytic and bit-interface assumptions
+are unchanged.
+
+Further trimming, unions of stars, other punctures, different label pairings,
+and positive-cost terms outside this alphabet remain outside the theorem.
+The assertion is for the displayed coordinate-frame puncture; no unverified
+frame-transitivity argument is needed. It covers complex coefficients because
+the only coefficient-field fact used is the rank of `4 I_8`.