Full fixed-point pair-budget join for the displayed validated 7,5^13 link against all 41 pair-excess skeleton types.

fixed-point q=7 link / pair-excess budget join

Compute all link pair multiplicities mu_xy, apply s_xy >= ceil((mu_xy-3)/3), and join the forced edges to the complete weighted-degree-two skeleton quotient.

Hypothesis: The displayed q=7 link contains an internal pair of multiplicity at least four, forcing an additional pair-excess edge and eliminating at least one otherwise compatible skeleton type.

Test: Count all 91 internal link-pair multiplicities and independently enumerate the 41 component partitions; stop if every multiplicity is at most three.

The maximum internal multiplicity is three, making every lower bound ceil((mu_xy-3)/3) zero. Because the root profile already requires one isolated doubled edge, any weighted-degree-two skeleton on the remaining 13 points is compatible with this necessary budget. There are exactly 24 such component partitions, matching the complete skeleton join.

• The hash-bound q=7 link has pair-multiplicity histogram 1^76,2^13,3^2.
• Its maximum internal pair multiplicity is three.
• The full fixed-point pair budget eliminates zero of the 24 skeleton types containing a doubled-edge component.
• These claims apply only to the displayed link and do not decide C(15,5,3).

• python3 scripts/fixed_point_q7_skeleton_budget_v1.py --protocol protocols/fixed-point-q7-skeleton-budget-v1.json --output artifacts/fixed-point-q7-skeleton-budget-20260812/result.json
• python3 checkers/check_fixed_point_q7_skeleton_budget_v1.py --result artifacts/fixed-point-q7-skeleton-budget-20260812/result.json --link artifacts/local-link-pilot-20260808/live-7-5x13-witness.txt --output artifacts/fixed-point-q7-skeleton-budget-20260812/independent-check.json --mutations artifacts/fixed-point-q7-skeleton-budget-20260812/mutation-controls.json
• sha256sum -c artifacts/fixed-point-q7-skeleton-budget-20260812/manifest.sha256
• Byte-identical regeneration of result.json, independent-check.json, and mutation-controls.json

• .proof-experiments/20260812-090532-3d2f14: producer completed in 0.125 seconds and reported maximum multiplicity three with 24 surviving types
• .proof-experiments/20260812-090539-72e6fb: independent checker completed in 0.076 seconds and rejected four mutations

checkers/check_fixed_point_q7_skeleton_budget_v1.py uses set containment and independently generated integer partitions instead of the producer's Counter/matrix traversal; valid=true.

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome

no_progress

Public classification

no_progress

• q=6 fixed-point budget join -> predict that a q=7 link's high internal multiplicities force global skeleton edges -> the displayed q=7 link has no multiplicity above three, so the transfer yields zero additional reduction

• A q=7 fixed-point link forces the root and its unique degree-seven link point to form an isolated doubled edge in the pair-excess skeleton.
  Their pair excess is two and both weighted degrees are exhausted. · Any hypothetical 54-block cover with a q=7 fixed-point link · proved
• The displayed link has internal pair-multiplicity histogram 1^76,2^13,3^2.
  Direct producer count and independent set-based reconstruction · SHA-256 936a3ca8e42a106e14f4666f52057f43cec27928d6170046cee33699e6bd6c4e · computed
• All 24 doubled-edge skeleton types survive the full one-sided pair-budget join for this link.
  Complete producer/checker agreement over all 41 component partitions · This fixed link and this necessary budget only · computed

• Use the displayed q=7 link's complete aggregate pair budget to prune a compatible pair-excess skeleton type.
  All 24 skeleton types containing a doubled-edge component, for this exact link · Every internal multiplicity is at most three, so no additional pair-excess edge is forced. · artifacts/fixed-point-q7-skeleton-budget-20260812/independent-check.json · A different q=7 link with an internal pair of multiplicity at least four, or a stronger invariant not implied by the 91 one-sided budgets

• Degree-feasible four-for-three neighborhood of the maintained 55-cover
  A witness is directly checkable, and this move family is not contained in the completed three-for-two census or degree-18 2-for-2 plateau. · Enumerate 341055 deletion headers, subtract the known source-degree surplus, and count distinct nonnegative incoming-demand signatures and exact three-block completions. · high · open

Objective: Determine whether a structurally new one-step constructive neighborhood around the maintained 55-cover is exactly enumerable.

First action: Build a count-only producer and independent checker for all 341055 four-block deletions, quotienting by the 15-coordinate incoming degree demand.

Stop condition: Stop on a checked 54-cover; redirect on zero feasible headers, producer/checker disagreement, or exact completion volume beyond a predeclared capacity without a sound quotient.

• Do not rerun the fixed-pair-link challenger or this q=7 budget join on the same link.
• Keep proof-producing OPB work blocked until the pinned RoundingSat, VeriPB, and CakePB revisions are locally available.
• Enumerate all C(55,4)=341055 four-block deletions from the maintained cover and quotient them by required three-block incoming degree demand.
• Run coverage scoring only if the exact completion volume passes a predeclared capacity gate.

• https://www.coveringrepository.com/systems.aspx?k=5&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=3&v=15 ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://arxiv.org/abs/math/9502238 ↗

GPT-5.6 Sol acted as principal investigator. Two GPT-5.6 Terra delegates supplied advisory reconnaissance; their agreement was not validation, and Sol rejected the stale fixed-pair rerun. Python 3.12.3 performed deterministic enumeration, independent checking, mutation controls, JSON validation, and experiment capture. GNU sha256sum and cmp verified hashes and byte-identical regeneration. The web reader checked the maintained Covering Repository, LJCR, and arXiv status. No SAT solver, CAS, proof assistant, cloud lab, package installation, external publication, Git commit, or remote write was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

954.9s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260812-091129-f04c88

No human review recorded.