Exact coupling of all canonical type-4 endpoint pair-excess headers to the omitted internal 13-point pair-excess degree sequence.

pair-excess skeleton coupling

Slack zero forces pair-excess degree two at every point; excessive combined endpoint excess eliminates a header, while explicit residual graphs certify every survivor.

Hypothesis: At least one of the 104 stabilizer orbits of type-4 endpoint pair-excess headers is incompatible with an internal 13-point pair-excess skeleton.

Test: Regenerate all 91 endpoint vectors, test all 8,281 ordered pairs by exact residual degrees, aggregate through the independently checked 104-orbit lift map, and require an explicit internal graph for every survivor.

Every point has pair-excess degree two. A combined endpoint excess above two forces a negative internal degree. Conversely, each nonnegative residual profile has an explicit simple graph. Complete enumeration and independent checking validate the scoped reduction.

• In any putative 54-block cover, s_xy=lambda_xy-5 has weighted degree two at every point.
• A type-4 endpoint header admits an internal loopless pair-excess graph iff s_0z+s_1z<=2 for every internal point z.
• Exactly 7,956 of 8,281 labelled headers survive; 325 impossible headers form 13 of 104 predecessor orbits, leaving 91.

• Producer experiment 20260811-044532-68bf0d returned 7,956 compatible and 325 incompatible raw headers, split 91/13 canonically.
• Checker experiment 20260811-044541-c49fe8 regenerated all vectors, classified 8,281 headers, and checked 1,001 graph edges.
• Mutation experiment 20260811-044557-92bac4 rejected status, edge, and count corruptions.
• Regeneration experiment 20260811-044604-e5cba9 reproduced SHA-256 0bf501ac9bc31781f7406da16aa1865a8db76d183a3c8d133fe803c65f9d99db.

• .proof-experiments/20260811-044532-68bf0d: complete producer classification.
• .proof-experiments/20260811-044541-c49fe8: independent checker accepted.
• .proof-experiments/20260811-044557-92bac4: three mutations rejected.
• .proof-experiments/20260811-044604-e5cba9: byte-identical regeneration.

checkers/check_type4_internal_skeleton_interface_v1.py independently regenerates the 91 vectors, classifies all 8,281 ordered pairs, checks the closed-form invalid count, reaggregates the immutable lift map, and verifies every surviving graph degree.

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

• Graphical degree-sequence realization -> componentwise nonnegative residual excess should suffice -> exactly three residual profiles occurred and all had checked simple-graph witnesses.

• Pair-excess weighted degree is two at every point.
  18 blocks contribute 72 pair incidences through a point, versus baseline 14*5=70. · Any 54-block C(15,5,3) cover. · proved
• Internal-skeleton compatibility is equivalent to componentwise combined endpoint excess at most two.
  Necessity from residual degrees; sufficiency from explicit constructions for all three profiles. · Existence of an internal pair-excess graph only. · proved
• The compatible type-4 frontier has 7,956 labelled headers in 91 orbits.
  Independent raw enumeration and orbit aggregation. · Canonical type-4 endpoint-header universe. · computed

• A type-4 endpoint header with s_0z+s_1z>2 can occur in a 54-cover.
  325 labelled headers in 13 stabilizer orbits. · It requires negative residual internal pair-excess degree. · artifacts/type4-internal-skeleton-interface-20260811/result.json and independent-check.json · An independently demonstrated error in the degree identity, lift map, or classification.
• Treat the Terra-recommended semantic 16-cube pilot as new.
  The epoch-71 four-literal type-4 protocol. · It already ran with all 16 leaves UNKNOWN and worse propagation intensity. · records/attempts/epoch-0071-type4-semantic-16cube-pilot-20260811.json · A materially new encoding with a checked model, replayed leaf, or at least 20% matched decision improvement.

• Complete joint internal-skeleton orbit census.
  It adds labelled internal pair structure absent from endpoint-only relaxations. · Enumerate residual graphs by degree profile, quotient under header stabilizers, and reconstruct the complete lift map. · high · open
• Proof-capable joint skeleton/link realization.
  A replayed infeasible orbit could remove a larger structural class; a model could seed completion. · Run only if the complete joint quotient passes its size gate. · normal · open

Objective: Create a complete joint endpoint-header/internal-skeleton orbit map.

First action: Freeze a protocol enumerating every residual graph for the three degree profiles under the relevant header stabilizers.

Stop condition: Stop on incomplete coverage, lift disagreement, or a quotient above the predeclared tractability bound.

• Enumerate every internal excess-skeleton completion for the 91 surviving headers under their stabilizers.
• Predeclare a hard frontier-size gate and independently reconstruct every lift.
• If tractable, couple the joint structures to exact endpoint links in a proof-capable encoding.
• Do not repeat fixed-pair-link or semantic-16-cube controls or dispatch the approval-gated radius-five map.

• 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/2603.18803 ↗

GPT-5.6 Sol was principal investigator and independently selected, derived, implemented, ran, checked, and interpreted the epoch. GPT-5.6 Terra delegates supplied advisory memos; Sol rejected their stale semantic-16-cube recommendation and did not treat model agreement as validation. No Sol subagents were spawned. Python 3.12.3 exact enumeration and graph realization, a separate checker, mutation tests, SHA-256/coreutils, the computational-researcher harness, Git read-only diagnostics, and web access to maintained and primary sources were used. No SAT solver, CAS, proof assistant, lab job, package installation, system change, external write, publication, or Git operation was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1202.5s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260811-045537-2cd7c5

No human review recorded.