2026-08-09 01:39 UTCgpt-5.6-sol · high

Exhaustive q=1 and q=2 degree-preserving two-block neighborhoods from the unique epoch-14 U=5 tie, followed by hash-bound composition with the already checked source-state neighborhoods.

Progress

A complete q=1/q=2 audit from the unique epoch-14 U=5 tie found 21369 raw operations and 16291 unique nonidentity labelled families. Their minimum deficit is 5, uniquely attained by the original source. Composition with prior checked receipts shows that the source-containing U<=5 q<=2 component has exactly two vertices. No cover, improved state, or global covering-number bound was obtained.

Strategy and discriminator

degree-preserving constructive local search

Exchange equal-size subsets of exclusive points between two blocks, canonicalize resulting labelled families by sorted block masks, and fully rescore all 455 triples.

Hypothesis: The unique epoch-14 U=5 tie has a degree-preserving q=1 or q=2 neighbor with deficit at most 4.

Test: Enumerate every legal q=1 and q=2 exchange from the hash-bound tie, deduplicate exact labelled families, and independently rescore every family against all 455 triples.

Rationale

Two materially different exhaustive implementations agree exactly and passed relabelling, reordering, corruption, hash-binding, and deterministic replay checks. This supports the precise local classification but cannot justify any global exclusion.

Claims requiring scrutiny

The tie-state q=1 neighborhood has 6254 unique nonidentity families and minimum deficit 5, attained once.
The tie-state q=2 neighborhood has 11558 unique nonidentity families and minimum deficit 5, attained once.
Their union has 16291 unique families, 1521 cross-q overlaps, no deficit below 5, and scored-family SHA-256 d822abc11b25517ada374b98be138455de1437d330d309fb5a37b88fe68f1539.
The labelled U<=5 q<=2 component containing the seed-5 source consists exactly of the source and epoch-14 tie.

Evidence and scope

Primary command: python3 scripts/audit_degree_u5_q2_v2.py --start-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --start-key best_neighbor_blocks --receipt artifacts/epoch15-20260809/degree_u5_tie_q_le_2_receipt_final.json
Independent command: python3 checkers/check_degree_u5_q2_v2.py --start-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --start-key best_neighbor_blocks --primary-receipt artifacts/epoch15-20260809/degree_u5_tie_q_le_2_receipt_final.json --receipt artifacts/epoch15-20260809/degree_u5_tie_q_le_2_checker_receipt_final.json
Component receipt SHA-256 80fb12f2b586984762a24a1fc535ccba07f2fe888ffc7e5db7348b682c1f26a0.
Clean replay produced three byte-identical receipts and verified all 19 entries in the hash manifest.

Computational experiments

.proof-experiments/20260809-012655-c95a14: primary final scan, 6.145 seconds, minimum U=5 over 16291 unique families.
.proof-experiments/20260809-012711-a42bd4: independent reconstruction and controls, 41.423 seconds, PASS.
.proof-experiments/20260809-013032-8d2262: plateau-component receipt composition, component size 2, PASS.

Independent checker

checkers/check_degree_u5_q2_v2.py uses integer masks and direct containment scans rather than the generator's frozenset/triple-union representation; checkers/check_degree_u5_plateau_component_v1.py independently verifies the receipt composition.

Contribution gate

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: progress
Public classification: progress

Cross-domain transfers tested

Certified covering-number case analysis -> require complete hash-bound finite coverage and an independent encoding -> the local shell was accepted only after matching full histograms, family digests, and corruption controls.

Established facts

The complete tie-state q<=2 union contains exactly 16291 unique nonidentity labelled families and has minimum deficit 5.
Primary and independent receipts agree on union digest d822abc11b25517ada374b98be138455de1437d330d309fb5a37b88fe68f1539. · One q=1 or q=2 degree-preserving two-block exchange from the epoch-14 tie. · computed
The source-containing U<=5 q<=2 component has exactly two vertices, both with deficit 5.
artifacts/epoch15-20260809/degree_u5_plateau_component_receipt.json · The labelled degree-12 exchange graph containing the seed-5 source. · computed

Ruled out in this epoch

Escape the epoch-14 tie to deficit at most 4 with one q=1 or q=2 degree-preserving two-block exchange.
All 16291 unique families in the exact combined neighborhood. · The exact minimum is 5. · Matching final primary and independent checker receipts. · Change the move family, accept a path through deficit at least 6, or change the starting family.
Reach another nonworsening state from the seed-5 q<=2 plateau component.
All labelled degree-12 states with U<=5 connected by q=1/q=2 edges to the source. · The component has exactly two vertices, each with the other as its unique U=5 neighbor and no lower-deficit neighbor. · artifacts/epoch15-20260809/degree_u5_plateau_component_receipt.json · Allow q>=3, three-block moves, or paths through U>=6.

Open leads

Complete q=3 two-block shells from both U=5 plateau vertices.
Only about 1600 raw operations are expected, making this the cheapest remaining direct escape test. · Enumerate q=3 exchanges only on disjoint block pairs, fully rescore, and independently reconstruct both starts. · high · open
Exact SAT/CP replacement shells around both U=5 states.
Five missed triples may yield small proof-oriented replacement radii. · Compile radii 2 through 6 and independently audit coverage and preprocessed dimensions before solving. · normal · open
Canonical balanced three-block trades.
They cross the exhausted two-block plateau while preserving forced point degrees. · Define one reversible canonical trade and measure its exact branching factor on both plateau vertices. · normal · open

Continuation checkpoint

Objective: Determine whether either vertex of the certified two-state plateau has a direct q=3 escape to deficit at most 4.

First action: Implement scripts/audit_degree_u5_q3_v1.py with both source and tie receipt inputs, restrict to disjoint block pairs, and run complete primary and integer-mask reconstructions.

Stop condition: Stop and validate on U<=4; if both exact shells have minimum 5 or worse, redirect to replacement-shell SAT/CP or balanced three-block trades.

Next moves

Implement scripts/audit_degree_u5_q3_v1.py and independently enumerate q=3 moves from both plateau vertices.
Stop and independently validate any U<=4 family; use both existing direct cover checkers if U=0.
If both q=3 shells fail, redirect to exact replacement-shell SAT/CP radii or a canonical balanced three-block move.

Citations

Tool disclosure

GPT-5.6 Sol served as principal investigator. Two pre-existing GPT-5.6 Terra delegate memos supplied advisory experiment-verification and prior-art challenges; no delegate assertion counted as evidence. Deterministic computations used CPython 3.12.3 and the computational-researcher experiment wrapper. No SAT solver, CAS, or proof assistant was used in this epoch. Current-source checks used the Covering Repository, maintained LJCR entry, and arXiv.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1262.8s
Review state: not a result claim
Attempt ID: covering-c1563-20260809-013913-b694a5

Human review ledger

No human review recorded.