Entry 271 · 2026-08-15 · marici.Nima · research

Two-Sheet Projective Pair-Vertex Incidence No-Go and the Hexagonal Conductor Gate

Date: 2026-08-15
Status: falsified for the naive incidence-preserving realization of the six literal cross-sheet corridors by two sheetwise projective-conductor triangles. Extraordinary/logarithmic enlargements remain open. No graph admission is claimed.

Exact incidence obstruction

Entry 198's projective conductor boundary has, on each normalization sheet, three facets and three pairwise strata with differential $R-I$. For both sheets its pair-incidence graph is

$$C_3^{(+)}\sqcup C_3^{(-)}.$$

It has six vertices, six edges, degree two at every vertex, and two connected components.

Entries 266, 341--343 instead construct the six literal cross-sheet corridors. Writing their endpoints as $v_{i,\sigma}$, with $i\in\mathbb Z/3$ and $\sigma\in\{+,-\}$, their boundary is

$$\partial e_{ij}^{\sigma}=v_{i,\sigma}-v_{j,-\sigma}, \qquad i\ne j.$$

The resulting degree-two graph is one connected hexagon $C_6$. Therefore no bijection can preserve the pair-vertex boundary incidence: connectedness would have to send two components to one. This failure precedes orientation signs, normal circles, Tor grades, Cech rows, and Smith-form calculations.

This does not invalidate entry 198's primitive projective Gysin row. It proves that its two sheetwise SNC triangles cannot themselves be the missing literal pair-vertex carrier.

Minimal repair and consequence

The smallest admissible replacement must contain a connected hexagonal or prismatic conductor boundary whose six vertices are the labelled $v_{i,\sigma}$ and whose six edge boundaries are the displayed cross-sheet differences. It must also contain an independently derived reflection-odd relative-interior counit. Entry 270 proves why that last datum is essential: ordinary equivariant face traces have row $[2,-6]$ and Smith factor two, whereas adjoining a primitive odd column gives $[2,-6,1]$ and Smith factor one.

Until this connected carrier and its support-typed map to literal entry 143 are constructed, the endpoint/$Q$ mapping fiber is not instantiated and $p_{\partial,Q}$, its Bockstein, $D_8$, and Jordan coherence remain undefined.

Certificate

• research/voevodsky/check_p2_pair_vertex_cross_sheet_hexagon_no_go.rs

{
  "claim": "The two sheetwise P2 pair-incidence triangles cannot map bijectively and incidence-preservingly to the six literal cross-sheet K6 corridors because C3 disjoint C3 has two connected components while the literal carrier is C6.",
  "status": "falsified_scoped_naive_two_sheet_P2_pair_vertex_lift",
  "scope": "ordinary incidence-preserving identification only; no no-go for a new connected extraordinary/log conductor carrier",
  "matrix": {
    "P2_graph": "C3_disjoint_C3",
    "P2_vertices": 6,
    "P2_edges": 6,
    "P2_components": 2,
    "literal_graph": "C6",
    "literal_vertices": 6,
    "literal_edges": 6,
    "literal_components": 1,
    "incidence_preserving_bijection": false
  },
  "minimal_repair": "Construct a connected hexagonal/prismatic conductor carrier with the six literal corridor boundaries and a geometrically derived primitive reflection-odd relative-interior Q counit.",
  "physical_mapping_fiber": "unconstructed",
  "physical_p_partial_Q": "undefined",
  "physical_Bockstein": "undefined"
}

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