Pair-Overlap Carrier Contractibility and the Mixed-Variance Realization Gate
Date: 2026-08-15
Status: proved for the integral unloaded carrier and external spectator tensors.
The support-changing Tor/Čech realization, endpoint/Q mapping fiber, and physical
parity remain unconstructed. No graph admission is claimed.
Correction to the rank-six interpretation
Entry 198 correctly records that the six short-facet boundaries form a saturated rank-six sublattice of the twenty-one edge chains. It treated the corresponding choice of corridor representative as a remaining strict ambiguity. In the homotopy-coherent carrier category this ambiguity is contractible.
Let [ d_2:mathbb Z^6longrightarrowmathbb Z^{21} ] be the actual oriented short-facet boundary matrix. The established unit maximal minor implies that (d_2) is a split integral monomorphism. Therefore, if two endpoint-pointed corridor representatives determine the same fixed class in (H_1(B_{ m short},V)), their difference is (d_2z) for a unique integral (zinmathbb Z^6). There is no comparison automorphism because (ker d_2=0).
For three cyclic comparisons, an alternating sum whose edge boundary vanishes lies in (ker d_2), hence is zero. Thus carrier triple coherence is automatic. Rotation and reflection transport the unique comparison, so no additional carrier choice can break (D_3) covariance.
Because the monomorphism is integrally split, tensoring with an external spectator complex preserves this conclusion degreewise. This statement does not cover a support-changing Tor or Čech map: such a map is not an external tensor factor.
What remains missing
The three pairwise long-road intersections are still not objects of the literal entry-143 face category. Distinct long diagonals cross, so no compatible (K_6) face contains a pair. The canonical projective-conductor SNC points therefore require an extraordinary comparison, not an ordinary face corestriction.
The exact first missing arrow is a mixed-variance realization [ Gamma_{ij}^{!,log}longrightarrow C_\bullet(q_k)subset F_B/F_V, \qquad {i,j,k}={14,03,25}, ] whose two restrictions agree with the adjacent long-facet packets and whose image is the fixed complementary marked corridor. It must identify the occurrence duals, both normal-circle removals, Tor grades, and Čech signs on the 24 pair-vertex incidences. Carrier uniqueness will then make its comparison and cyclic coherence canonical; it does not construct those vertical maps.
Consequently this result removes only the rank-six carrier-choice objection. It does not remove entry 135's larger rank-nine butterfly ambiguity, produce the based (q_Sigma) comparison, instantiate the endpoint/Q mapping fiber, or define (p_{partial,Q}).
Certificate
A focused executable source is present at
research/voevodsky/check_p2_pair_overlap_homotopy_contractibility.rs.
Current source SHA-256:
617be8dc031a0dcb499bc7e8a6f1db324165de96abfffb616e0091263595280e.
It reconstructs the actual (K_6) face census ((1,9,21,14)), the oriented
short-boundary matrix, rank six, a unit maximal minor, (d_1d_2=0), zero
kernel, and the three crossing long-road pairs. It also maps all six dP6
cones to primitive differences of legitimate short-facet chains and verifies
that their cyclic residue sum telescopes to zero. Its mathematical input is
also independently covered by entry 119's validated rank-six/unit-minor
certificate. The new source has been completely reread through filesystem
MCP and passes rustfmt --edition 2021 --check. A bounded validation did produce the checker metadata artifact before its
worker report timed out, proving rustc --edition=2021 -D warnings --emit=metadata succeeded. Linking and runtime execution remain unavailable
because the inherited Windows worker environment has no MSVC linker.
Consequence
The minimal additional geometry is not a choice of a carrier 2-chain. It is a support-typed log/nearby-cycle correspondence realizing the external SNC pair points inside the literal entry-143 mixed-variance target. Once one such realization exists with the prescribed boundary class, the carrier comparison, reflection transport, and cyclic 2-coherence are forced integrally.
Outcome contract
{
"claim": "For each fixed endpoint-pointed road class, the actual K6 short-boundary rank-six ambiguity is contractible in the integral homotopy-coherent carrier category: comparison 2-chains exist integrally, are unique, have no automorphisms, and satisfy cyclic coherence.",
"status": "proved_scoped_carrier__mixed_variance_realization_open",
"scope": "unloaded integral carrier and external spectator tensor factors only; no support-changing Tor/Cech realization, literal entry143 pair map, endpoint/Q mapping fiber, parity, or graph admission",
"evidence": {
"k6_face_census": [1, 9, 21, 14],
"short_boundary_rank": 6,
"short_boundary_kernel_rank": 0,
"unit_maximal_minor": true,
"cokernel_torsion": false,
"crossing_long_pairs": 3,
"dp6_cone_residue_rows": 6,
"dp6_cyclic_residue_sum": 0,
"carrier_comparison_groupoid": "contractible for a fixed homology class",
"external_spectator_tensor": "preserves the split injection",
"support_changing_tor_cech": "unconstructed",
"literal_pair_vertex_rows": "unconstructed",
"rank9_butterfly_contraction": "unconstructed",
"physical_mapping_fiber": "unconstructed",
"physical_p_partial_Q": "undefined"
},
"checker_sha256": "617be8dc031a0dcb499bc7e8a6f1db324165de96abfffb616e0091263595280e",
"validation": {
"filesystem_mcp_full_reread": "PASS",
"rustfmt": "PASS via worker MCP run-3e7ac8218dec4f1cbcbee0c69b3ec942",
"rustc_metadata": "PASS: warnings-denied metadata artifact produced by bounded MCP validation",
"runtime": "NOT EXECUTED: inherited MCP worker environment has no MSVC linker"
},
"minimal_geometry": "Construct a support-typed log/nearby-cycle pair correspondence Gamma_ij whose two facet restrictions and complementary marked-corridor image satisfy all occurrence, normal, Tor, Cech, endpoint, reflection, and D3 Beck-Chevalley squares."
}