dP6 Boundary Adjacency No-Go and the Octahedral Log-Correspondence Gate
Exact adjacency comparison
The entry329 labelled vertex dictionary determines, without an extra assignment, which two short facets meet each of the six ordered long-road sectors. It gives twelve sector--short incidences. Every short facet occurs twice, so these incidences determine six edges between sector vertices.
Those six literal facet-sharing edges are
They are the two parity-sheet triangles. By contrast, entry249's oriented dP6 boundary edges are
and every one joins opposite parity sheets. The two edge sets are disjoint. Therefore no dP6 boundary node connector is itself the pair-facet corestriction required by the literal entry143 short-facet defect.
This falsifies the proposed identification of the oriented dP6 boundary cycle with the missing facet comparison. It does not falsify a larger log-excess correspondence.
Minimal combinatorial completion
The union of the six dP6 boundary edges and the six literal sheetwise edges is
the octahedral graph. It has twelve edges and a canonical completion by eight triangular faces; every edge occurs in two faces and
Thus the smallest symmetry-compatible carrier containing both already certified incidence systems is the octahedral two-sphere. Geometrically, the next datum must be a normalization-provenanced log/excess correspondence whose one-skeleton contains:
- the cross-sheet dP6 conductor hexagon;
- the positive and negative sheetwise facet triangles; and
- eight Beck--Chevalley two-cells comparing their alternating composites.
Its sheetwise edges must carry the shifted pair-facet Gysin maps of entry333, while its cross-sheet edges carry the Rees/KN endpoint and center connectors of entries249, 334. Only such a two-dimensional comparison can map the interior top to entry251's canonical lift while deriving all six short defect rows.
The octahedral completion is presently a forced finite carrier, not yet an admitted log space or six-functor kernel. Constructing that spatial realization and its eight loaded two-cells is now the earliest remaining geometric gate.
Evidence
- research/voevodsky/check_dp6_literal_facet_adjacency_obstruction.rs
- entries 143, 249--251, 329, 333--334, and 252.
{
"status": "falsified_scoped_dp6_boundary_as_literal_facet_clutching",
"ordered_sectors": 6,
"sector_short_incidences": 12,
"literal_short_facet_edges": 6,
"dp6_boundary_edges": 6,
"common_edges": 0,
"literal_edges_same_sheet": true,
"dp6_edges_cross_sheet": true,
"minimal_union_graph": "octahedral_K6_minus_matching",
"union_edges": 12,
"octahedral_faces": 8,
"euler_characteristic": 2,
"octahedral_log_correspondence_constructed": false,
"global_correspondence_no_go": false,
"mapping_fiber_instantiated": false,
"physical_p_defined": false,
"physical_bockstein_defined": false,
"D8_Jordan_tested": false,
"graph_admission_claimed": false
}