Full-Log Maximal-Cone K6 Fillers and the Nine Facet BC Gate
Canonical two-dimensional carrier
Entries341--342 fix the literal image of every edge in the full-log octahedral fan. The boundary of each of its eight maximal cones can therefore be evaluated in the complete 14-vertex, 21-edge K6 flip graph.
The two all-positive and all-negative cones collapse through their common sheet endpoint, so their carrier boundary is zero. Each of the six mixed cones gives an eight-edge loop. Exhaustive integral solution against the nine associahedral facet boundaries shows that every mixed loop has a unique minimum-support filler consisting of three oriented facets.
Across all six mixed cones there are 18 facet occurrences. Every one of the nine K6 facets occurs exactly twice. Identifying the two occurrences gives nine disjoint primitive BC rows of type [1,-1], hence rank nine, nine unit Smith factors, and no integer torsion.
The second carrier layer is therefore canonical after the unique-geodesic edge subdivision: two zero fillers and six unique minimal three-facet patches.
With the coherent source orientation given by the determinant of the three signed rays, the sum of the eight fillers is twice the primitive oriented K6 fundamental sphere. Thus the cellular map has absolute degree two. The checker derives the primitive rank-one facet kernel and verifies the factor two on all nine oriented facet coefficients; it is not inferred from unsigned occurrence counts.
Remaining literal and geometric lift
The checker certifies the complete cellular carrier matrix, but it does not yet distribute each oriented facet through all literal entry143 Boolean, Tor, and Cech states. That full packet must restrict to the edge matrices of entry342 and prove the nine paired-facet BC equations state by state.
After that finite lift, the still-essential geometric arrow remains the proper/log-excess comparison from the product-branch Rees Cech carrier into the literal entry143 BM-Cech diagram. Without that spatial map the carrier fillers cannot be promoted to the requested normalization-provenanced global chain map.
Evidence
- research/voevodsky/check_full_log_maximal_cone_k6_fillers.rs
- entries 143, 259--260, 338--342.
{
"status": "proved_scoped_full_log_maximal_cone_K6_fillers",
"K6_vertices": 14,
"K6_edges": 21,
"K6_facets": 9,
"maximal_cones": 8,
"same_sheet_zero_fillers": 2,
"mixed_cones": 6,
"mixed_loop_edge_support": 8,
"minimal_facets_per_mixed_filler": 3,
"minimal_fillers_per_mixed_cone": 1,
"facet_occurrences": 18,
"distinct_facets": 9,
"occurrences_per_facet": 2,
"facet_BC_equalities": 9,
"facet_BC_rank": 9,
"facet_BC_smith_unit_factors": 9,
"oriented_cellular_degree_abs": 2,
"primitive_K6_sphere_kernel_rank": 1,
"integer_torsion": false,
"literal_full_Boolean_facet_lift_constructed": false,
"global_endpoint_Q_map_constructed": false,
"mapping_fiber_instantiated": false,
"physical_p_defined": false,
"physical_bockstein_defined": false,
"D8_Jordan_tested": false,
"graph_admission_claimed": false
}