Unique Cross-Sheet K6 Geodesics and the Three-Segment Excess Lift Gate
Canonical carrier repair
Entry340 rules out gluing a cross-sheet edge by two existing half-gallery arms. The full literal K6 flip graph nevertheless supplies a canonical replacement.
For every admissible cross-sheet pair with different road labels and opposite signs, the two ordered-sector middle triangulations have distance three in the K6 flip graph and possess exactly one shortest path. Thus each of the six cross-sheet toric edges has a uniquely determined three-segment literal K6 subdivision. The six paths form one D3-covariant family, and reflection maps each oriented path to the corresponding reflected path.
This removes the carrier-chain choice without selecting a road, diagonal, or boundary primitive. The resulting 18 literal flip edges are all genuine K6 incidences.
Remaining excess lift
The unique carrier paths do not by themselves construct the required full-log correspondence. Along each three-segment path the exchanged normal label changes at every flip. One must lift the subdivision through the full Boolean replacement of entry339, retain both Tor grades, and prove that the two internal-vertex restrictions agree as Beck--Chevalley maps.
Equivalently, the minimal new geometric datum can now be reduced from six arbitrary bridges to one D3-seed three-segment log/DNC bridge whose literal carrier is the certified geodesic. Its stabilizer and reflection squares must be proved before transporting it to the other five pairs.
Until that lift is constructed, the eight maximal-cone equations and the endpoint/Q mapping fiber remain uninstantiated.
Evidence
- research/voevodsky/check_cross_sheet_unique_k6_geodesics.rs
- entries 143, 259, 339, and 263.
{
"status": "proved_scoped_unique_literal_K6_cross_sheet_geodesics",
"K6_vertices": 14,
"cross_sheet_pairs": 6,
"shortest_length": 3,
"shortest_paths_per_pair": 1,
"subdivision_edges": 18,
"D3_orbit": true,
"reflection_covariant": true,
"carrier_choice": false,
"full_Boolean_excess_lift_constructed": false,
"global_maximal_cone_gluing_constructed": false,
"mapping_fiber_instantiated": false,
"physical_p_defined": false,
"physical_bockstein_defined": false,
"D8_Jordan_tested": false,
"graph_admission_claimed": false
}