Exhaustively apply residual subset and cut-capacity inequalities to the hash-bound 2258-node type-0 depth-4 canonical root-link frontier, then independently reconstruct every result using a different uncovered-edge encoding.

residual subset/cut capacity root-link pruning

For each partial root link, maximize the internal and crossing pairs coverable by the eight remaining 5-blocks under exact residual incidence totals, rejecting a node when an uncovered-pair subset or cut exceeds that capacity.

Hypothesis: Residual subset/cut capacity rejects at least 226 of the 2258 old-filter-admissible type-0 depth-4 root-link nodes.

Test: Scan all 16382 proper internal subsets and 8191 complement-unique cuts of every recorded node; continue only if an independent checker reproduces at least 226 rejected nodes.

Each rejected node has an explicit violating cut whose uncovered crossing edges exceed the exact maximum coverable by the eight remaining 5-blocks under its residual degree sum. The complete rejected-key set was reproduced by a materially different encoding, while the limited source scope and failed scale threshold prevent any broader claim.

• If remaining 5-block intersection sizes with S sum to D_S, their total internal-pair capacity is at most 10*floor(D_S/5)+choose(D_S mod 5,2), and their total cut capacity is at most (R-r)*a*(5-a)+r*(a+1)*(4-a), where a=floor(D_S/R) and r=D_S mod R.
• Exactly 164 of the 2258 recorded type-0 depth-4 canonical root-link nodes cannot extend to a 12-block link of degree type (8,4^13).
• All 164 exclusions are cut-only; the recorded frontier contains 347 violating complement-unique cuts and zero internal-subset violations.
• The 226-node continuation gate failed, and no global covering bound changed.

• Experiment 20260810-185619-c8dcd3 ran the frozen 55485834-test primary scan and produced 164 rejected nodes.
• Experiment 20260810-185702-50a776 independently reconstructed 2258 nodes and reproduced rejected-key SHA-256 0bfc4d5470102e4536da116815ea60289efcd20af979cfe4f82efca5a0c6bb32.
• Experiment 20260810-185918-41872a used exact dynamic programming to verify both capacity closed forms for R=1,...,12 and every D=0,...,5R.
• sha256sum -c artifacts/epoch62-20260810/SHA256SUMS returned OK for every packet entry.

• .proof-experiments/20260810-185619-c8dcd3: primary exhaustive scan completed in 24.828 seconds and found 164 cut-only rejections.
• .proof-experiments/20260810-185702-50a776: independent scan completed in 26.231 seconds and reproduced all counts, keys, and mutation failures.
• .proof-experiments/20260810-185918-41872a: corrected exact formula DP passed in 0.071 seconds.
• .proof-experiments/20260810-185856-b45b47: non-evidentiary failed launch reached receipt serialization but used lowercase Python true; corrected by the fresh successful run.

checkers/check_root_link_subset_capacity_v1.py uses a 91-bit uncovered-edge representation intersected with independently precomputed internal and cut masks, imports no primary scanner code, reconstructs every node, and rejects pair-count, degree-sum, and subset-certificate mutations. checkers/check_subset_capacity_formula_v1.py separately computes exact maxima by dynamic programming.

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

• Network cut capacity -> predict residual point cuts can reject nodes surviving global and singleton capacities -> exactly 164 nodes were rejected, but the 7.26 percent rate missed the 10 percent scale gate.

• The stated internal and cut closed forms equal the exact extrema over s_i in {0,...,5} with fixed sum D.
  Elementary convexity/concavity derivation and artifacts/epoch62-20260810/subset_capacity_formula_checker_receipt.json. · R=1 through 12 was computed; the mathematical derivation applies to every positive R. · proved
• Exactly 164 recorded type-0 depth-4 nodes violate residual cut capacity.
  Primary receipt SHA-256 06c26600d8a11c021d27695952074dbd71f1061767860049d855418d6741b02a and checker receipt SHA-256 a45d6b631e9abaf350d728716ef4a042f8174f829c9138ad115e7a2ff7a5cfe8. · The 2258-node frontier in source receipt SHA-256 0ad095a6c238147d6f8f3236570ef5ad8caa99d8dce068f72efeec26fb5161eb. · computed
• The local represented survivor set contains 2094 nodes after applying the filter.
  2258 reconstructed source keys minus the independently matched 164-key rejection set. · Recorded type-0 depth-4 nodes only. · computed

• Scale the unchanged residual subset/cut filter as the sole canonical root-link reopening mechanism.
  The exact two inequalities on the 2258-node recorded type-0 depth-4 frontier. · Only 164 nodes were rejected, below the frozen 226-node threshold. · Matching primary and independent checker receipts. · A materially stronger joint relaxation must reject at least 210 of the 2094 survivors or return terminal evidence.
• Use the internal-subset inequality alone to prune this frontier.
  All 36990556 proper internal-subset tests on the 2258 recorded nodes. · Zero violations occurred. · Both implementations independently report internal=0. · A strictly stronger cardinality-aware or joint-incidence inequality with a proved safety argument.
• Apply symmetric pair-covering determinant nonexistence results.
  A hypothetical 30-block C(15,6,3) cover viewed as a pair covering. · It has 15 points and 30 blocks, violating the equal point/block-count hypothesis of symmetric-design results. · Direct parameter mismatch. · A theorem explicitly covering nonsymmetric 15-point, 30-block pair coverings with the forced multiplicity structure.

• Joint residual block-intersection b-matching or flow relaxation
  It can couple multiple cut demands that the separately maximized inequalities ignore while retaining short infeasibility certificates. · Define an exact relaxation, prove completion implies feasibility, and scan the 2094 survivor keys with a 210-rejection continuation threshold. · high · open
• Constructive no-proof r=2 incidence frontier control
  A 30-block SAT witness remains the smallest terminal certificate, and removing LRAT overhead directly tests the measured proof-output bottleneck. · Run the frozen 16 one-second leaves and one 16-second unsplit control, validating any SAT model with independent Python and C cover checkers. · normal · open
• Proof-producing global incidence encoding with material proof compression
  A complete replayable UNSAT partition would settle the value at 31, but current proof prefixes exceed their output gate. · Qualify a pinned compression or preprocessing proof map on one legitimate leaf below the existing 1 MiB cap. · low · open

Objective: Find a stronger certificate-friendly joint residual-feasibility condition for the 2094 surviving recorded root-link nodes.

First action: Design and prove a b-matching, flow, or LP relaxation over residual block intersection profiles, then run it against the survivor complement encoded in the epoch-62 primary receipt.

Stop condition: Redirect on source/rejection-set mismatch, an unsound implication, fewer than 210 incremental rejections, or any arbitrary cutoff without a structural result.

• Construct a proved b-matching, flow, or LP relaxation coupling residual block intersections across one cut or a laminar cut family.
• Apply it only to the exact 2094 surviving keys and require at least 210 independently reproduced incremental rejections before catalogue scale-up.
• If that gate fails, run the already specified constructive no-proof r=2 16-cube versus unsplit SAT timing control; accept mathematical progress only from a directly checked 30-block cover.
• Do not scale the unchanged subset/cut filter or invoke symmetric-design determinant results.

• 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/2607.23766 ↗

GPT-5.6 Sol served as principal investigator, independently deriving, implementing, checking, and interpreting the result. Pre-supplied GPT-5.6 Terra delegates provided advisory prior-art and experiment-design memos; the adopted cut-capacity suggestion was promoted with provenance and was not counted as validation. Python 3.12.3 performed deterministic exact integer scans and dynamic programming; SHA-256 bound the packet. No SAT solver, CAS, proof assistant, cloud lab, external proof service, or human validator produced evidence this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1076.7s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260810-190553-c816ac

No human review recorded.