Exact pre-materialization census and orbit-size lower bound for every weighted internal pair-excess skeleton coupled to the 91 surviving canonical type-4 endpoint headers.

weighted-skeleton frontier sizing

Count all loopless weighted residual skeletons by degree profile and divide rowwise by independently reconstructed header-stabilizer orders to obtain a rigorous joint-orbit lower bound.

Hypothesis: The complete type-4 endpoint-header/internal-weighted-skeleton quotient has at most 1,000,000 cells and is suitable for explicit materialization.

Test: Independently count every weighted skeleton for the three residual profiles, reconstruct all 91 header stabilizers, and compare the rigorous rowwise joint-orbit lower bound with 1,000,000.

The producer enumerates incident multisets in a vertex-elimination recurrence. The checker instead decomposes every maximum-degree-two weighted graph into paths, cycles, and doubled edges, reconstructs the full group from incidence signatures, and verifies each orbit-stabilizer summand. Agreement, mutation rejection, and byte-identical regeneration support the scoped route decision but not a covering-number claim.

• The three compatible residual profiles have exactly 5,238,370, 10,450,414, and 20,953,380 labelled loopless weighted skeletons per header representative.
• Across the 91 surviving canonical type-4 header representatives there are exactly 1,203,611,004 labelled weighted internal skeletons.
• Any complete quotient of this joint frontier under the fixed-family action has at least 12,247,501 cells.

• Producer experiment 20260811-052503-f40e83 returned labelled_skeletons=1203611004 and joint_orbit_lower_bound=12247501 in 0.128 seconds.
• Independent experiment 20260811-052517-768b7d reconstructed 5,184 internal automorphisms, the 10,368-element full group, 104 predecessor orbits covering 8,281 headers, and all profile counts and stabilizers.
• Regeneration experiment 20260811-052542-82a1d5 reproduced result SHA-256 8d825d0643305bea361abe9121bb9ee3082abdda9a1124e0a14e251f26363cff.
• Mutation experiment 20260811-052549-174809 rejected all five material corruptions.
• Manifest SHA-256 722220579a33a2ec31d9ad35a34eff9797b5fcd8e96991dfadf6c6b7eee6cb66 replayed successfully.

• .proof-experiments/20260811-052503-f40e83: producer count and redirect gate.
• .proof-experiments/20260811-052517-768b7d: independent group and component-formula validation.
• .proof-experiments/20260811-052542-82a1d5: byte-identical regeneration.
• .proof-experiments/20260811-052549-174809: five fail-closed mutations rejected.

checkers/check_type4_weighted_skeleton_frontier_gate_v1.py does not import the producer: it derives root-link automorphisms from incidence signatures, reconstructs every predecessor orbit and stabilizer, and counts skeletons by a path/cycle/doubled-edge component formula instead of vertex elimination.

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

• Labelled degree-at-most-two multigraph decomposition -> predict exact path/cycle/doubled-edge formulas reproduce the DP counts -> observed exact agreement for all three profiles.

• There are 5,238,370 labelled loopless weighted multigraphs of profile 0^2 2^11, 10,450,414 of profile 0^1 1^2 2^10, and 20,953,380 of profile 1^4 2^9.
  Producer recurrence and independent component formula agree exactly. · The three residual internal degree profiles on 13 labelled positions. · computed
• The 91 header representatives support exactly 1,203,611,004 labelled weighted internal skeletons in total.
  Profile frequencies 14, 46, and 31 multiplied by the independently checked profile counts. · Canonical type-4 surviving endpoint headers only. · computed
• The complete current joint quotient has at least 12,247,501 cells.
  Sum of independently checked ceil(N(h)/|Stab(h)|) over all 91 representatives. · Current fixed-family group action on every compatible type-4 header/internal-skeleton pair. · proved

• Explicitly materialize the complete 91-header weighted-skeleton quotient and create one SAT leaf per cell under the 1,000,000-cell gate.
  The exact current fixed-family symmetry plan and all weighted skeleton completions. · Any complete quotient contains at least 12,247,501 cells. · artifacts/type4-weighted-skeleton-frontier-gate-20260811/result.json and independent-check.json · A proved additional equivalence, decomposition, or symbolic representation reducing the complete domain below 1,000,000 cells.
• Treat the epoch-76 simple internal graph witness as the full skeleton domain.
  All 91 surviving type-4 header representatives. · Each profile has millions of weighted completions, including doubled edges; one witness proves only nonemptiness. · The independently checked exact profile counts. · A theorem proving all but a specified simple representative are irrelevant to literal block realization.
• Repeat the complete fixed-pair-link encoding as a new discriminator.
  The saved complete pair-link protocol. · Its independent CNF audit reports zero clause delta. · artifacts/pair-link-cnf-delta-audit-20260811/independent-check.json · A materially different encoding with a nonzero independently checked semantic delta.

• Lazy internal pair-excess/literal-block coupling.
  It represents degree conservation and pair counts with variables and constraints rather than one object per skeleton, directly addressing the measured frontier explosion. · Generate baseline and lazy formulas for one canonical type-4 branch, independently compare semantic constraints, then run a matched 25,000-conflict pilot only if the delta is nonzero. · high · open
• Materially different constructive multi-step repair from the verified defect-10 seed.
  A 54-cover remains terminal on direct checking and is distinct from the blocked quotient route, but prior 2-for-2 and strict 3-for-3 neighborhoods failed. · Only after deriving a new neighborhood with an exact below-10 discriminator; do not enlarge prior cutoffs. · low · open

Objective: Test whether a lazy internal-pair encoding creates useful propagation without enumerating skeletons.

First action: Freeze a protocol comparing one canonical type-4 baseline CNF/PB instance with a version adding internal pair-excess variables, degree-two equations, and literal block linkage; require an independent clause/semantics audit before solving.

Stop condition: Redirect on zero semantic delta, any producer/checker mismatch, or less than 20% matched decision reduction; advance only with checked SAT or a proof-emitting scalable improvement.

• Write a lazy type-4 encoding that couples internal pair-excess variables and degree-two conservation directly to literal endpoint-link block variables without skeleton enumeration.
• Independently verify that the lazy encoding adds a nonzero semantic constraint set beyond existing fixed-pair-link and native-PB controls.
• Run one matched propagation pilot with a predeclared 20% decision-improvement gate; stop on zero delta, checker disagreement, or failed improvement.
• Do not reopen explicit joint quotient materialization without a proved complete compression below 1,000,000 cells.

• 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 ↗

GPT-5.6 Sol principal investigator; GPT-5.6 Terra delegates supplied advisory reconnaissance only, and model agreement was not validation. Python 3.12.3 standard-library exact arithmetic/enumeration, a materially different independent checker, SHA-256, mutation testing, the computational-researcher harness, and current web source checks were used. No SAT solver, CAS, proof assistant, lab job, package installation, external publication, or system change was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1348.9s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260811-053703-0a6f7e

No human review recorded.