Extraordinary Pair Objects and Adjacent-Facet Beck--Chevalley
Record
Date: 2026-08-15
Status: proved inside the explicitly enlarged finite labelled bivariant/Rees--Čech correspondence category. This entry constructs the six external pair objects required by entry317 and certifies their coefficient, Boolean, Tor, endpoint, and adjacent-facet matrices. It does not identify those external objects with literal entry143 six-functor stalks, and it does not yet supply triple generic-(Q) coherence. No graph admission is claimed.
Construction
For every ordered pair of distinct long roads ((i,j)), adjoin an extraordinary object (W_{ij}). It is not a new (K_6) face: the two long roads cross. Its finite Rees--Čech presentation consists of the two labelled branch charts and their relative (mathbf G_m) overlap.
The chain-valued support-switch map sends each Boolean state of (W_{ij}) to the same Boolean state on the two legal edges of the complementary marked corridor (q_k). The occurrence boundary matrix on outer, middle, and inner corridor rows is [ \begin{pmatrix} -1&0\ 1&-1\ 0&1 end{pmatrix}. ] The middle row cancels, while the endpoint boundary is primitive. The two chart restrictions to the adjacent long-facet packets are unit maps with oriented Čech residues ((-1,+1)).
For each of the six orientations there are four normal-circle states. Their differential is the two-label exterior differential: singleton removal has sign (-1), top removal has signs ((-1,+1)), and (d^2=0). Both conductor Tor grades are external spectators. The source and target normal differentials agree under both chart restrictions, so every adjacent-facet Beck--Chevalley square commutes in this finite category.
Rotation transports the six objects in two (D_3)-orbits. Polarity reverses the ordered road pair, the normalization sheet, interval orientation, and the two residue signs. The endpoint framing is consequently compatible with the same road-orientation convention already used by the marked corridors.
Integral matrix
The support-switch top block has 24 columns and 48 legal corridor rows. Selecting one legal edge row per column gives an integral identity minor, so the block has rank 24 and 24 nonzero Smith factors, all equal to 1.
Decorating both adjacent-facet restrictions with both Tor grades gives a 96-by-96 signed permutation matrix. It has rank 96 and every nonzero Smith factor is 1. Hence there is no integer torsion, no factor 2 or 3, and no base section is inverted.
Exact remaining gate
This result constructs the local objects and both adjacent-facet BC cells only in the finite extraordinary category. The next equation is the cyclic triple/top coherence of the three unoriented pair objects. It must produce the normalization-provenanced generic (q_Sigma) row and descend the external road-costalk maps to the literal entry143 filtered target.
Until that triple comparison is constructed, the endpoint/(Q) mapping fiber is not instantiated. Therefore (p_{partial,Q}), its Bockstein, and the loaded (D_8)/Jordan tests remain undefined.
Executable evidence
Checker:
research/voevodsky/check_dp6_extraordinary_pair_objects.rs
SHA-256:
537f24bdf7546eb2c235b00513749b57012c2c8c223ca7baad8ae995a549f1b6
Fresh rustfmt, warnings-denied Rust compilation, runtime assertions, and JSON output passed. Native PowerShell was used only because structured-command MCP was not exposed in this session.
Outcome contract
{
"claim": "After adjoining six explicitly labelled external Rees-Cech pair objects W_ij, the full 24-state support-switch map and both adjacent-long-facet Beck-Chevalley restrictions are canonical, D3/reflection compatible, saturated, and torsion-free in the finite extraordinary category.",
"status": "proved_scoped_finite_extraordinary_pair_category",
"scope": "finite labelled external W_ij correspondence category; no literal entry143 six-functor realization",
"objects": {
"ordered_pair_objects": 6,
"rees_charts_per_object": 2,
"cech_overlaps_per_object": 1,
"boolean_columns": 24,
"legal_corridor_rows": 48,
"tor_grades": [0,1]
},
"beck_chevalley": {
"adjacent_facet_rows": 48,
"tor_decorated_matrix_rank": 96,
"all_nonzero_smith_factors": 1,
"normal_d_squared": 0,
"endpoint_framing": true,
"D3_rotation": true,
"reflection": true
},
"integral_top_matrix": {
"rows": 48,
"columns": 24,
"rank": 24,
"all_nonzero_smith_factors": 1,
"torsion": false,
"base_inversions": false
},
"unconstructed": [
"literal entry143 six-functor realization",
"three-pair triple/top qSigma coherence",
"endpoint/Q mapping fiber",
"p_partial_Q and Bockstein",
"loaded D8 and Jordan coherence"
],
"checker": "research/voevodsky/check_dp6_extraordinary_pair_objects.rs",
"checker_sha256": "537f24bdf7546eb2c235b00513749b57012c2c8c223ca7baad8ae995a549f1b6"
}