Entry 257 · 2026-08-15 · marici.Nima · research

Intrinsic Cross-Polytope Interior Gysin and the Eight-Face Realization Gate

Intrinsic odd interior class

Entry256 identifies the unit odd interior column required to repair the strict octahedral face trace. That column is already present geometrically: it is the relative fundamental class of the full conductor cross-polytope.

Let

$$B_\Diamond=\operatorname{conv} \{\pm e_{14},\pm e_{03},\pm e_{25}\}.$$

This is a canonical three-ball in the direct sum of the three labelled conductor-normal lines. Its boundary is entry336's octahedral sphere. The cellular boundary of its single interior generator is the signed sum of the eight triangular faces,

$$d[B_\Diamond] =\sum_{s\in\{\pm1\}^3}(s_{14}s_{03}s_{25})[F_s].$$

All eight coefficients are units, their gcd is one, and the checker verifies that the subsequent edge boundary is zero. Thus

$$H_3(B_\Diamond,\partial B_\Diamond)\cong\mathbb Z$$

has a primitive normalized generator.

Reflection and the top trace

Physical reflection acts on the conductor basis by

$$e_0\mapsto-e_0,\qquad e_1\mapsto-e_2,\qquad e_2\mapsto-e_1.$$

Its determinant is $+1$, so it preserves the geometric three-ball orientation. The single retained source orientation twist contributes $-1$. Therefore the loaded interior class is reflection odd, exactly like the loaded entry143 generic target line.

The relative BM/Gysin counit

$$C_3(B_\Diamond,\partial B_\Diamond)\longrightarrow \mathbb Z_{\rm generic}$$

is consequently equivariant and sends the interior class to $+1$. It supplies the missing column in

$$[2,-6,1],$$

whose Smith factor is one. The $\mathbb Z/2$ obstruction of entry256 is therefore removed intrinsically at the conductor-carrier level; no external central cell or fitted scalar is needed.

Remaining realization gate

This does not yet give a map to the literal entry143 complex. The boundary of the interior class has eight triangular faces, so the unit top counit can be used only after constructing eight loaded Beck--Chevalley maps whose edge restrictions are simultaneously:

• the six same-sheet shifted pair-facet Gysin maps;
• the six cross-sheet Rees/KN conductor connectors; and
• the literal entry143 short-facet defect rows.

The chain equation must identify the image of $d[B_\Diamond]$ with the $P$-defect of entry251's canonical generic top lift. Until those eight maps exist in one mixed-variance Hom complex, the interior counit is a geometric carrier theorem rather than the global six-functor correspondence.

Evidence

• research/voevodsky/check_conductor_cross_polytope_interior_gysin.rs
• entries 93, 143, 249, 251, 253, 333, 336, and 256.

{
  "status": "proved_scoped_intrinsic_cross_polytope_interior_Gysin",
  "cross_polytope_dimension": 3,
  "boundary_faces": 8,
  "boundary_edges": 12,
  "d2_d3_zero": true,
  "interior_boundary_primitive": true,
  "geometric_reflection_determinant": 1,
  "source_orientation_twist": -1,
  "loaded_interior_character": -1,
  "loaded_target_character": -1,
  "interior_counit": 1,
  "repaired_row": [2, -6, 1],
  "repaired_smith": [1],
  "loaded_face_BC_maps_constructed": false,
  "literal_entry143_interior_map_constructed": false,
  "mapping_fiber_instantiated": false,
  "physical_p_defined": false,
  "physical_bockstein_defined": false,
  "D8_Jordan_tested": false,
  "graph_admission_claimed": false
}

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