Mixed-Cone Toroidal Map No-Go and the Minimal Log Expansion
Record
Date: 2026-08-16
Status: definitive scoped no-go for a direct label-preserving toroidal map from the six mixed cones of the full-log conductor source into the literal entry143 (K_6) cone complex. The minimal equivariant boundary expansion is proved. Maximal-cone Beck--Chevalley cells, the global six-functor kernel, and the based (q_Sigma) comparison remain unconstructed. No graph admission is claimed.
First obstruction in the log category
The full-log toric source has six mixed cross-sheet two-cones. The two labelled endpoint chambers of each mixed cone are literal (K_6) triangulations, but their common compatible face is empty.
A toroidal/log morphism induces a map of cone complexes and must carry each source cone into one target cone. A direct label-preserving map would therefore require a common target cone for the two endpoint labels of every mixed source cone. The executable census proves that no such cone exists for any of the six mixed cones.
This obstruction is stronger in scope than the ordinary face-poset observation: it rules out the unexpanded fs-log/toroidal source map itself. It does not rule out a logarithmic expansion, derived correspondence, or proper extraordinary push--pull.
Minimal boundary expansion
The (K_6) flip graph has fourteen triangulation chambers. For each mixed source cone, the two endpoint chambers have distance three and admit exactly one shortest gallery. Thus the minimal cone-complex repair subdivides the source-link interval into three chambers by inserting two intermediate strata.
Across the six mixed cones this gives:
- eighteen expanded chambers;
- twelve inserted vertices; and
- six uniquely determined galleries.
Rotation and physical reflection permute the complete six-gallery system. No road multiplicity is divided and no occurrence or normal line is identified with another.
Remaining construction
The expansion fixes the first missing support map but does not yet construct the extraordinary correspondence. The next object must extend these six expanded boundary galleries over the eight maximal cones of (prod_Dmathbf P(mathcal Ooplus L_D)), with:
- the independently defined occurrence, normal-circle, Tor, and Čech differentials;
- the twenty-four maximal-cone Beck--Chevalley homotopies;
- proper pushforward to the literal entry143 facet and endpoint stars; and
- the hemisphere/top comparison to the based (q_Sigma) generator.
Only after those vertical maps are constructed can the integral global matrix, pointed endpoint/(Q) mapping fiber, physical (p_{partial,Q}), its Bockstein, (D_8), and Jordan coherence be computed.
Executable evidence
Checker:
research/voevodsky/check_full_log_mixed_cone_toroidal_expansion_gate.rs
SHA-256:
0ab2fbeb2287acbe078c3ea828a1f6643004bf741c97873c6c3ba012824e6103
Fresh rustfmt --edition 2021 --check, warnings-denied optimized
compilation, runtime assertions, and JSON emission passed.
Outcome contract
{
"status": "falsified_scoped_direct_full_log_toroidal_map",
"scope": "direct label-preserving cone-complex/toroidal map from the six unexpanded mixed source cones; logarithmic expansions and extraordinary correspondences remain open",
"mixed_edges": 6,
"endpoint_common_target_cones": 0,
"k6_triangulations": 14,
"unique_shortest_gallery_length": 3,
"unique_galleries": 6,
"minimal_log_expansion": {
"chambers": 18,
"inserted_vertices": 12
},
"D3_rotation": true,
"physical_reflection": true,
"integer_inverted": false,
"unconstructed": [
"occurrence/normal/Tor/Cech vertical transformations on the expansion",
"maximal-cone Beck-Chevalley realization",
"global six-functor kernel",
"based qSigma attachment",
"pointed endpoint/Q mapping fiber",
"physical p_partial_Q and Bockstein",
"D8 and Jordan coherence"
],
"checker": "research/voevodsky/check_full_log_mixed_cone_toroidal_expansion_gate.rs",
"checker_sha256": "0ab2fbeb2287acbe078c3ea828a1f6643004bf741c97873c6c3ba012824e6103",
"graph_admission_claimed": false
}