Exact first-hit ownership classification of all unordered two-block partial root links across the five forced target-degree profiles, with an independent composition-table multiplicity audit.

root-link first-hit canonical ownership

Generate colored unordered-pair orbit keys from forced representatives, then independently enumerate membership-pattern composition tables to reconstruct every key, owner, and labelled orbit multiplicity.

Hypothesis: The twenty first-hit tails collapse to an exact duplicate-free depth-two colored-isomorphism frontier with at least a factor-5 reduction from 18933 forced-representative coordinates, exact agreement with validated depth-two controls 14 and 30, and a declared depth-four projection no larger than 100000 states.

Test: Enumerate canonical membership-pattern keys from the 18933 forced-representative coordinates and require exact equality with a block-free composition-table enumeration whose multinomial orbit sizes sum to choose(2002,2) in every profile.

Exact invariant classification, full orbit-multiplicity accounting, historical controls, mutation rejection, and deterministic reruns support the scoped depth-two result. They do not support extrapolation to depth four or a claim about the optimum. The broader scaling route is redirected because the new representation does not shrink the global isomorphism quotient.

• The five forced root-link target-degree color groups have exactly 14,43,30,92,71 orbits on unordered pairs of distinct 5-subsets.
• The unique least occupied one-block orbit partitions those 250 pair orbits among the twenty epoch-120 first-hit tails.
• The independent orbit multiplicities sum to choose(2002,2)=2003001 labelled unordered pairs in each profile.
• The 18933 forced-representative coordinates quotient to 250 pair orbits, factor 75.732; this is symmetry identification and eliminates no putative cover.
• The maintained covering-number range remains 30 <= C(15,6,3) <= 31.

• python3 scripts/root_link_first_hit_depth2_v1.py --out artifacts/epoch121-20260812/root-link-first-hit-depth2-v1.json -> PASS; result SHA-256 17aee2af46a8ff6d713eed5f1e2a48bd5f853d3103d560952891c337034e56e7.
• python3 checkers/check_root_link_first_hit_depth2_v1.py --result artifacts/epoch121-20260812/root-link-first-hit-depth2-v1.json --out artifacts/epoch121-20260812/root-link-first-hit-depth2-independent-check-v1.json -> PASS; receipt SHA-256 b29e6778f68ede15e2ea637fe452f2cdb139361041b7e0f49ad4830fac0ad03e.
• Producer rerun is byte-identical; checker receipts agree after removing only the different bound result path; rerun audit PASS.
• Failed development control .proof-experiments/20260812-142427-f857cf stopped before producing an artifact when an indirect historical summary field was incorrectly assumed.

• .proof-experiments/20260812-142427-f857cf: fail-closed development run; wrong historical JSON field, no result artifact.
• .proof-experiments/20260812-142454-a0a9ad: producer PASS in 0.322 seconds; 250 aggregate pair orbits and projection 76434.
• .proof-experiments/20260812-142507-03fa6a: independent checker PASS in 0.176 seconds; 2003001 labelled pairs per profile and four rejected mutations.
• .proof-experiments/20260812-142528-f4bf49: producer rerun byte-identical.
• .proof-experiments/20260812-142528-6eeef0: checker rerun PASS.
• .proof-experiments/20260812-142613-de62bb: rerun binding audit PASS.

checkers/check_root_link_first_hit_depth2_v1.py never scans the 2002 blocks or fixes a representative; it enumerates per-color-class four-cell composition tables, derives orbit sizes and owners algebraically, matches every producer key, and covers choose(2002,2) labelled pairs in each profile.

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

• Certified fixed-link orbit decomposition in C(12,6,4) -> predict a hash-bound root-link owner layer here -> observed a fully checked 250-key depth-two layer, but no completion elimination.
• Canonical augmentation of colored set systems -> predict membership-pattern tables completely classify two-block states -> observed exact agreement with historical nauty frontiers for profiles 4 and 22.

• The depth-two colored unordered-pair orbit counts for target-degree profiles 4,31,22,211,1111 are 14,43,30,92,71.
  Producer/checker key equality, exact multinomial coverage, historical controls, mutations, and reruns. · Unordered pairs of distinct 5-subsets of 14 points under the five product color groups. · computed
• Each depth-two orbit has a unique first-hit owner equal to the least occupied one-block orbit.
  Both mechanisms independently derive the same owner for all 250 canonical keys. · The five depth-two orbit universes and twenty epoch-120 tails. · proved
• The producer's 18933 forced-representative coordinates represent exactly 250 pair orbits.
  Exact deduplication and independent composition-table reconstruction; ratio 75.732. · Representation coordinates only; no completion-case elimination. · computed

• Treat factor 75.732 as a reduction of the global completion-case isomorphism universe.
  The epoch-121 depth-two first-hit representation. · The 250 outputs are exactly the ordinary global colored-isomorphism quotient; the larger baseline counts symmetry-equivalent forced-representative coordinates. · Exact profile counts and agreement with validated global type-0/type-2 frontiers. · A sound filter that eliminates entire canonical pair or deeper completion families, not merely a different representative ownership.
• Use 76434 as an upper bound or certificate budget for the complete depth-four frontier.
  Profiles 31,211,1111. · Their depth-four growth was not enumerated; 3143/10 is only the larger observed multiplier from profiles 4 and 22. · Projection receipt explicitly marks measurement_status as estimate, not an upper bound. · Independent exhaustive depth-four enumeration or a proved profile-uniform growth bound; enumeration alone must also have a non-arbitrary completion use.
• Run native PB proof search with absent or revision-incompatible tools.
  This host and epoch. · roundingsat, veripb, and CakePB are absent and no compatible pinned tuple has passed replay. · Read-only executable audit and Terra source-version warning; no PB solver run. · Pinned full source hashes, local executables, direct format-pairing evidence, and producer plus two-replayer calibration with mutation rejection.

• Canonical two-root-block constraint inside one constructive incidence formula.
  It is the only immediate use of the 250-key manifest that can preserve a direct terminal witness path without accumulating local UNSAT leaves. · Compile and independently reconstruct one formula, then compare it to the existing r=2 formula for 30 seconds with proof output disabled. · high · open
• Pinned native-PB proof qualification.
  Exact degree equations may support compact global certificates if a compatible proof producer and two checkers can be source-pinned. · Acquire exact approved project-scoped revisions, verify hashes, and replay the smallest overdegree UNSAT calibration with semantic mutations. · low · open

Objective: Test whether exact two-root-block ownership can materially improve a constructive global incidence encoding while preserving all witnesses.

First action: Implement one canonical owner constraint consuming the 250-key manifest, independently reconstruct its witness coverage, and run a matched 30-second no-proof calibration against the existing r=2 formula.

Stop condition: Stop on coverage mismatch, mere 250-way splitting, no material throughput/memory gain, or any need to treat the depth-four estimate as evidence.

• Retain the 250-key owner manifest as exact coverage infrastructure; do not extend the catalogue merely to increase depth.
• Compile and independently reconstruct one witness-preserving canonical two-root-block incidence constraint that consumes the owner manifest without creating 250 separate leaves.
• Run a matched 30-second no-proof constructive calibration against the existing r=2 incidence formula; continue only on a material throughput or memory improvement, or a directly checked cover.
• Keep native PB held until exact compatible source revisions, local executables, and producer plus two-replayer mutation-controlled calibration are available.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=6&t=3&v=15 ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://github.com/dennisonbertram/covering-number-c12-6-4 ↗

GPT-5.6 Sol principal performed route audit, experiment design, implementation, verification, interpretation, and reporting. Two GPT-5.6 Terra delegates supplied advisory reconnaissance only and were not counted as validation. Python 3.12.3 used exact combinatorics, multinomial arithmetic, JSON, and SHA-256 through the computational-researcher experiment harness. Web search checked current primary sources. nauty-labelg was availability-audited but not used. No SAT/PB solver, CAS, proof assistant, cloud lab, system installation, external human validator, or publication action was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1107.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260812-143444-2cd35e

No human review recorded.