Double-count pair excess over all 30 blocks, choose a block of maximum internal excess as the normalized anchor, and independently reconstruct the resulting fixed-first arithmetic profile frontier.

pair-excess averaging anchor normalization

The identity sum_F q(F)=120+sum e_xy^2 forces some block to have q(F)>=5; block and point relabelling then removes all q=0..4 profiles from an existence-complete fixed-first search.

Hypothesis: Every putative 30-block C(15,6,3) cover has a block with internal pair excess at least five, allowing existence-complete normalization to profiles with q>=5.

Test: Derive the excess-moment identity, independently enumerate all 278256 weak profiles, and require exactly 249 profiles before and 228 after the q>=5 normalization, with identity and mutation controls.

Exact degree 12 and pair coverage make the pair excesses nonnegative integers of total mass 30. Their second moment raises the average internal block excess to at least five. Independent reconstruction exactly reproduced the predicted profile reduction and rejected all declared controls.

• Every hypothetical simple 30-block C(15,6,3) cover has a block F with q(F)>=5.
• Choosing such a block as normalized first block leaves exactly 228 necessary epoch-44 profiles in 19 cells.
• If no block has q>5, the pair-excess multigraph is simple 4-regular and every selected block induces exactly five excess edges.

• python3 scripts/anchor_excess_average_v1.py --manifest artifacts/epoch44-20260810/fixed-first-profile-manifest-v3/profile-manifest.json --output artifacts/epoch54-20260810/anchor-excess-average-receipt.json
• python3 checkers/check_anchor_excess_average_v1.py --manifest artifacts/epoch44-20260810/fixed-first-profile-manifest-v3/profile-manifest.json --result artifacts/epoch54-20260810/anchor-excess-average-receipt.json --output artifacts/epoch54-20260810/anchor-excess-average-check.json
• Primary receipt SHA-256 a98bfc4c89fcd596b1f11bbeb2fc62c885c7ae500c7949e82eec05d0d252efcb; checker receipt SHA-256 63277d99d161474a2cfa326c2ae5cec47c562136a41959eee035683dced3bd48.

• .proof-experiments/20260810-133352-fa1280: primary lemma/profile reduction returned PASS with 249 profiles before and 228 after.
• .proof-experiments/20260810-133400-7f1cc9: independent checker returned PASS after 278256 compositions, 5604 partitions, 276 incidence controls, and four mutations.

checkers/check_anchor_excess_average_v1.py uses a separate bar-position composition enumeration, direct cyclic incidence systems, integer partitions, and four fail-closed controls.

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

• Standard excess-graph machinery from pair coverings -> predict a block-average identity for the pair excess of this triple-cover problem -> identity yields the verified q>=5 anchor.

• Every hypothetical 30-cover has an anchor block with q>=5.
  Algebraic identity in artifacts/epoch54-20260810/anchor-excess-average-receipt.json and independent PASS receipt. · All hypothetical simple 30-block C(15,6,3) covers. · proved
• Exactly 228 of the 249 necessary arithmetic profiles satisfy q>=5.
  Independent bar-position enumeration in artifacts/epoch54-20260810/anchor-excess-average-check.json. · The hash-bound epoch-44 fixed-first profile universe. · computed
• The equality case forces a simple 4-regular pair-excess graph and q=5 at every block.
  Equality characterization for the unique square-sum minimizer (1^30). · Hypothetical covers with maximum block excess at most five. · proved

• Repeat the rank-1 unique-owner LRAT calibration as this epoch's discriminator.
  The exact epoch-42 profile (0,1,25,0,0,3). · Epoch 42 already reconstructed the CNF, dual-replayed the complete proof, and rejected truncation plus four encoding mutations. · artifacts/epoch42-20260810/epoch-receipt.json · A materially different proof-prefix, solver, or encoding question rather than another replay.
• Repeat the unchanged 10000-node canonical root-link catalogue.
  Unfiltered partial links under the epoch-19 canonical augmenter. · Epoch 19 independently verified 10001 distinct nodes and breached the gate; epoch 20's naive exact-completion filter returned UNKNOWN. · artifacts/epoch19-20260809/root_link_canonical_checker_receipt.json and artifacts/epoch20-20260809/root_link_completion_checker_receipt.json · A sound bulk filter, distinct decomposition, or two-augmenter ownership proof with measured frontier compression.
• Strengthen max q>=5 using only the first two excess moments.
  Abstract nonnegative integer pair excesses of total mass 30. · The lower bound is sharp when the 30 positive excesses all equal one. · All 5604 integer partitions checked; unique square-sum minimizer is (1^30). · Use additional block-incidence constraints to exclude or restrict the equality case.

• Exceptional equality-branch incidence encoding
  It isolates the only obstruction to proving an anchor with q>=6 and adds global exact-pair structure across every block. · Compile one independently reconstructed q=5 equality CNF and run a five-second matched proof-producing calibration. · high · open
• Constructive q>=5 anchor search
  The new normalization safely removes eleven arithmetic cells before witness search without requiring a complete excess-graph catalogue. · Add the q>=5 assumption to the strongest retained constructive incidence model and compare fixed-node coverage. · normal · open
• Two-augmenter complete root-link catalogue
  A genuinely independent augmentation order could eventually certify ownership, but the unfiltered frontier remains too large. · Only resume after deriving a sound completion filter with at least an order-of-magnitude measured dead-node reduction. · low · open

Objective: Determine whether the equality branch is empty or substantially more tractable than unrestricted q=5.

First action: Extend the epoch-44 shared incidence encoding with exact pair multiplicities 4/5 and an exact-five induced-excess condition for every selected column, then independently reconstruct the delta.

Stop condition: Redirect on reconstruction mismatch, proof replay failure, formula growth without propagation gain, or less than a 2x matched conflict-rate improvement.

• Encode the equality branch lambda_xy in {4,5} with exactly five excess edges induced by every selected block.
• Independently reconstruct all equality-branch clauses and test semantic boundary mutations.
• Run a matched five-second CaDiCaL/LRAT calibration against the unrestricted q=5 union.
• Retain q>=6 constructive search as the complementary branch and directly validate any witness against all 455 triples.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://www.dmgordon.org/papers/cover.pdf ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://ajc.maths.uq.edu.au/pdf/15/ocr-ajc-v15-p91.pdf ↗

GPT-5.6 Sol was principal investigator and designed, implemented, executed, audited, and interpreted the discriminator. GPT-5.6 Terra delegates supplied advisory prior-art and verification memos; their agreement was not treated as validation, and stale repeated-control suggestions were rejected. CPython 3.12.3, exact integer arithmetic, SHA-256, and the Proof Factory experiment harness generated the deterministic evidence. Web search checked primary status, construction, certificate-method, and excess-graph sources. No SAT solver, CAS, proof assistant, cloud lab, external proof service, or human validator was used in this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

879.5s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260810-134026-1270e6

No human review recorded.