Entry 256 · 2026-08-15 · marici.Nima · research

Octahedral Equivariant Top-Trace No-Go and the Odd Interior Excess Gate

Scoped obstruction

Entry336 supplies the canonical octahedral carrier. Its eight faces are indexed by sign triples

$$(s_{14},s_{03},s_{25})\in\{\pm1\}^3.$$

Rotation cyclically permutes the coordinates. Physical reflection sends

$$(s_0,s_1,s_2)\longmapsto(-s_0,-s_2,-s_1).$$

The faces therefore split into two $D_3$-orbits: the all-equal pair and the remaining six faces.

Geometric reflection reverses each oriented triangular face. The single already-retained source orientation twist contributes a second minus, so the loaded face basis is reflection even, whereas the loaded entry143 generic target line is reflection odd. Consequently an ordinary equivariant trace has rotation-invariant, reflection-odd coefficients determined by two integers:

• $a$ on the all-positive face, with value $-a$ on the all-negative face;
• $b$ on every one-negative face, with value $-b$ on the reflected two-negative faces.

Pairing this trace with the primitive octahedral fundamental cycle gives

$$2a-6b.$$

Thus every ordinary equivariant face trace has even image. The equation $2a-6b=1$, required to map the primitive carrier top to the normalized generic $q_\Sigma$ top, has no integral solution. The presentation row $[2,-6]$ has Smith factor $[2]$, so the obstruction is exactly $\mathbb Z/2$.

This is an intrinsic no-go for an ordinary degree-zero trace on the octahedral faces. It does not rule out the requested derived log-excess correspondence.

Minimal derived repair

One independently derived reflection-odd interior excess/Tor generator with primitive counit changes the presentation to

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

whose Smith factor is $[1]$. The normalized equation then has the integral solution given by the unit interior column.

This column cannot be stipulated from the carrier. It must arise geometrically as a shifted interior log-excess class whose boundary is the eight loaded Beck--Chevalley face transformations and whose restriction to the literal entry143 generic top is $+1$. The central exceptional $[2,1]$ correction of entry176 has the correct local arithmetic, but its support is not presently connected to this interior carrier. Using it requires a new support-typed central-to-interior BC map; otherwise the use would be circular.

Consequently the earliest remaining datum is now sharper than “eight face maps”: the face maps and one primitive odd interior excess cell must be constructed together in a single normalization-provenanced six-functor kernel.

Evidence

• research/voevodsky/check_cross_polytope_equivariant_top_trace_no_go.rs
• entries 143, 176, 249, 251, 253, 333, and 255.

{
  "status": "falsified_scoped_ordinary_equivariant_octahedral_top_trace",
  "geometric_face_reflection_sign": -1,
  "source_orientation_twist_sign": -1,
  "loaded_face_reflection_sign": 1,
  "loaded_target_reflection_sign": -1,
  "face_orbits": [2, 6],
  "trace_parameters": ["a", "b"],
  "fundamental_pairing_row": [2, -6],
  "smith": [2],
  "obstruction_group": "Z/2",
  "primitive_top_value_one_solution": "EMPTY",
  "minimal_repair_row": [2, -6, 1],
  "repaired_smith": [1],
  "odd_interior_excess_column_required": true,
  "odd_interior_column_geometrically_constructed": false,
  "global_correspondence_no_go": 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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