Residual-normalizer-reduced, node-capped C5-orbit bitset DFS with independent reconstruction and fail-closed mutation controls.

C5-invariant orbit-level constructive search

Fix a least-supported C5 triple orbit, quotient its 167 covering block-orbit choices into eight stabilizer cases, then search with 91-bit coverage masks, exact residual-degree profile DP, rarest-uncovered-triple branching, and memoization.

Hypothesis: A deterministic residual-normalizer-reduced C5-orbit DFS finds a directly checkable 30-block cover within 1000000 nodes or 60 seconds.

Test: Search exactly eight normalized root cases with equal 125000-node caps; accept only six C5 block orbits developing to 30 distinct blocks covering all 455 labelled triples with every point degree 12.

The group and orbit counts are independently reproducible and yield a legitimate scoped reduction. The search miss cannot support an exclusion because no case was exhaustively certified, but its telemetry decisively identifies late tail rejection as the next algorithmic target.

• For the specified fixed-point-free C5 action, the normalizer has order 3000 and the stabilizer of the triple orbit represented by {0,1,2} has order 500.
• The 167 C5 block orbits covering that triple orbit partition into eight stabilizer cases of sizes 2,10,10,10,10,25,50,50.
• The bounded DFS expanded exactly 1000000 nodes across all eight cases and found no witness; this proves no C5 or global exclusion.

• python3 scripts/c5_orbit_dfs_v1.py --output artifacts/epoch31-20260809/c5_orbit_dfs_receipt.json --node-limit 1000000 --time-limit 60
• python3 checkers/check_c5_orbit_dfs_v1.py --input artifacts/epoch31-20260809/c5_orbit_dfs_receipt.json --output artifacts/epoch31-20260809/c5_orbit_dfs_checker_receipt.json
• python3 checkers/test_c5_orbit_dfs_fail_closed_v1.py --input artifacts/epoch31-20260809/c5_orbit_dfs_receipt.json --checker checkers/check_c5_orbit_dfs_v1.py --output artifacts/epoch31-20260809/c5_orbit_dfs_fail_closed_receipt.json
• Hash-manifest audit checked 17 files with zero mismatches.

• .proof-experiments/20260809-203039-cd3c46: primary DFS expanded 1000000 nodes in 2.7739718169905245 seconds and returned LIMITED_NO_WITNESS.
• .proof-experiments/20260809-203132-0f88e9: independent reconstruction returned PASS with witness=false and exclusion=false.
• .proof-experiments/20260809-203141-bd3cee: four fail-closed mutations were all rejected.

checkers/check_c5_orbit_dfs_v1.py uses frozenset orbit partitions and explicit enumeration of all 3000 normalizer maps, materially differing from the primary tuple/bitmask search; it validates no exclusion for a bounded miss.

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

• Finite permutation-group normalizers -> predict a smaller exhaustive first branch after fixing a required triple orbit -> observed 167 choices partitioning into eight cases.
• Knapsack-style residual DP -> predict sound bulk pruning of impossible degree completions -> implemented exact three-coordinate profile feasibility, though late coverage failure remained dominant.

• The normalizer of the fixed C5 action has order 3000 and effective order 600 modulo C5.
  artifacts/epoch31-20260809/normalizer_root_reduction_lemma.md and independent complete enumeration · The specified action on three labelled five-point cycles · proved
• The fixed least-supported triple orbit has 167 covering block orbits in eight stabilizer orbits with size histogram {2:1,10:4,25:1,50:2}.
  Primary receipt SHA-256 4d05ae87ae79eb3364d762ca8a471d9296bec04f58fdadbef1759bf21e7073c7 and checker receipt · C5 block-orbit model · computed
• No witness occurred among the exact 1000000 DFS nodes traversed under the recorded deterministic order.
  artifacts/epoch31-20260809/c5_orbit_dfs_receipt.json · Only the recorded capped traversal prefix; no family exclusion · computed

• Scale the unchanged current C5 DFS merely by increasing its node cap.
  The present rarest-static-triple DFS with only profile DP and the coarse 20-times-slots coverage bound · 888391 of 1000000 expanded nodes failed only after selecting all six orbits, demonstrating dominant late rejection. · artifacts/epoch31-20260809/c5_orbit_dfs_receipt.json · A sound one- or two-orbit completion index, stronger admissible coverage bound, or matched pilot showing materially earlier rejection

• Exact one- and two-orbit C5 tail-completion lookup
  Directly attacks the measured 88.8391% late-terminal failure mode while preserving deterministic witness extraction. · Implement exact one-orbit residual-profile/mask lookup and repeat the fixed one-million-node protocol. · high · open
• Proof-producing C5 profile SAT
  The eight root cases and 6036 feasible profile multisets offer a bounded scoped certificate frontier if independent replay becomes available. · Generate and replay one difficult profile-conditioned leaf under a 60-second proof cap. · normal · open
• Global incidence/PB cube decomposition
  This is the only active negative route capable of settling the unrestricted problem. · Qualify hash-pinned CakeLPR on two retained intact proofs and two final-line-deletion mutants. · normal · open

Objective: Determine whether exact tail completion can turn the eight normalized C5 cases into an efficient witness search or complete scoped classification.

First action: In scripts/c5_orbit_dfs_v1.py, replace ordinary branching when slots==1 by an exhaustive lookup over unused candidates having exactly the residual profile and a coverage mask containing all uncovered bits; then rerun the saved run_experiment command with the same limits.

Stop condition: Stop immediately on an independently checked witness; redirect if late depth-six failure remains dominant or if a claimed complete C5 traversal cannot be independently replayed.

• At DFS states with one remaining orbit, directly index candidates by the exact residual profile and require their mask to contain all uncovered triples.
• Rerun the same eight-case, one-million-node protocol and compare terminal-failure depth and cases completed.
• If one-orbit lookup helps materially, precompute profile-keyed two-orbit coverage unions; otherwise hold C5 DFS and test one proof-producing C5 SAT leaf.
• Do not reopen global LRAT scale-up until hash-pinned CakeLPR accepts retained intact proofs and rejects their mutations.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?v=15&k=6&t=3 ↗
• https://arxiv.org/abs/math/9502238 ↗
• https://arxiv.org/html/2607.23766v1 ↗
• https://github.com/tanyongkiam/cake_lpr ↗

GPT-5.6 Sol was the principal investigator and independently designed, implemented, ran, and audited the epoch. GPT-5.6 Terra delegates supplied advisory prior-art and verification memos; their agreement was not validation, and used suggestions were promoted with provenance and independently reconstructed. Python 3.12.3 performed exact enumeration, integer-bitset DFS, SHA-256 hashing, and mutation controls. No SAT solver, CAS, proof assistant, or external compute lab was used. Web search checked the maintained source and nearby literature.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1221.8s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260809-203817-7f885a

No human review recorded.