Entry 272 · 2026-08-15 · marici.Nima · research

Unique Short-Facet Hexagon Filler and the Odd Interior Q Gate

Date: 2026-08-15
Status: proved for the exact integral K6 cellular carrier. The independent relative-interior map to literal $Q$ is not constructed. No graph admission is claimed.

Exact filler

Order the six cross-sheet sector rays as

(0,+),(1,-),(2,+),(0,-),(1,+),(2,-).

Entries 266, 341--343 give the unique length-three K6 geodesic for every adjacent pair. Their concatenation is a closed 18-edge chain, supported on 15 distinct oriented K6 edges.

Solving its boundary equation against all nine oriented K6 facets gives a unique integral solution of minimum $\ell^1$ -norm six:

F_{\rm hex}=F_{02}+F_{04}+F_{13}+F_{15}+F_{24}+F_{35}.

These are precisely the six short facets; every coefficient is $+1$ , and all three long-facet coefficients vanish. The solution is primitive and torsion-free. This supplies the connected carrier and a canonical minimal cellular contraction that entry 271 showed cannot come from two disjoint projective triangles.

Exact remaining obstruction

All six summands lie in $F_B$ . Therefore

F_{\rm hex}\longmapsto0\quad\text{in}\quad Q=F_K/F_B.

The primitive short-facet filler does not provide the missing odd generic $Q$ counit. Entry 270 remains decisive: the ordinary equivariant trace row has Smith factor two, and saturation requires an independently derived odd relative-interior column.

The next geometric datum is therefore a relative/log conductor interior whose boundary comparison is $F_{\rm hex}$ but whose extraordinary restriction has primitive coefficient $+1$ in the based literal $q_\Sigma$ line. It cannot be an ordinary class already lying in $F_B$ .

Until that support-changing map is constructed, the endpoint/ $Q$ mapping fiber, $p_{\partial,Q}$ , its Bockstein, $D_8$ , and Jordan coherence remain undefined.

Certificate

research/voevodsky/check_cross_sheet_hexagon_short_facet_filler.rs

{
  "claim": "The six literal cross-sheet K6 corridors have a unique minimum integral filler: the primitive sum of all six short facets. Its literal Q projection is zero.",
  "status": "proved_scoped_unique_short_facet_hexagon_filler",
  "scope": "exact integral K6 cellular carrier; no extraordinary relative-interior-to-Q map",
  "matrix": {
    "corridors": 6,
    "subdivision_edges": 18,
    "distinct_boundary_edges": 15,
    "minimum_l1": 6,
    "minimum_solutions": 1,
    "short_facet_coefficients": [1, 1, 1, 1, 1, 1],
    "long_facet_coefficients": [0, 0, 0],
    "primitive": true,
    "literal_Q_projection": 0
  },
  "minimal_repair": "Construct a support-typed relative/log interior class with boundary comparison equal to the short-facet hexagon and independently normalized odd counit to the based literal qSigma line.",
  "physical_mapping_fiber": "unconstructed",
  "physical_p_partial_Q": "undefined",
  "physical_Bockstein": "undefined"
}

View source on GitHub