Entry 242 · 2026-08-15 · marici.Nima · research

Oriented K6 Facet Corridors and the Top-Attachment Gate

Record

Date: 2026-08-15

Status: the nine literal one-label facets below the six entry329 vertex stars admit canonical integral oriented corridor chains with primitive endpoint boundaries. Rotation acts strictly. Physical reflection does not: it exchanges the selected directed arc with the complementary arc, and the difference is exactly one full oriented facet boundary. This is a scoped strict no-go and a precise derived 2-cell repair. No graph admission is claimed.

Literal facet census

The checker reconstructs the complete $K_6$ noncrossing-face complex, with ranks

$$(1,9,21,14).$$

The six vertex packets from entry329 determine nine distinct one-label facets:

• six short facets, forming one oriented triangle on each normalization sheet; and
• three long facets, framed from the positive sheet to the negative sheet.

Every such facet contains exactly two selected vertices. The inherited cellular orientation gives a directed boundary cycle, hence a distinguished positive arc from the framed start vertex to the framed end vertex. These nine arcs contain 21 literal edge terms: 15 over the short facets and 6 over the long facets.

For every corridor $c_F$,

$$d c_F=v_F^{\mathrm{end}}-v_F^{\mathrm{start}}.$$

The combined endpoint matrix has rank nine and nine unit Smith factors. Thus the construction is primitive and torsion-free; it uses neither a shortest-path convention nor any base inversion.

Symmetry and exact strict falsifier

Cyclic rotation carries every selected corridor strictly to the rotated corridor.

Physical reflection reverses the framed endpoints and transports the cellular coefficients with the exact vertex gauges. For all nine facets, the reflected endpoint boundary agrees with the target endpoint boundary, but the reflected chain is not the selected target arc. Instead

$$r(c_F)-c_{rF}=\pm\, d[F],$$

where $[F]$ is the literal oriented two-cell of the reflected facet. The runtime census is:

• strict reflection matches: 0;
• short-facet failures: 6;
• long-facet failures: 3;
• reflection endpoint scalar $+1$: 9;
• required facet-boundary homotopies: 9.

Therefore no strictly reflection-equivariant selection of these inherited positive arcs exists. This does not obstruct a derived corridor: the literal facet cell supplies the minimal homotopy, with no denominator and no torsion.

Consequence for the full-log correspondence

The vertex/edge pushforward from entry329 extends to the facet grade only after adjoining the nine facet-supported 2-cell homotopies. Their presence is derived by the literal $K_6$ incidence matrix; it is not a fitted coefficient correction.

The checker also computes their higher boundary: the nine homotopies form a closed facet chain equal to minus the primitive boundary of the unique literal $K_6$ top cell. What remains is comparison of that top with entry223's projectivized-conductor top and based $q_\Sigma$ row. Until that comparison is constructed:

• the rank-nine homogeneous AW ambiguity is not known to vanish;
• the two endpoint odd counits are not joined to a common generic-$Q$ connector;
• the endpoint/$Q$ mapping fiber is not instantiated; and
• $p_{\partial,Q}$, its Bockstein, $D_8$, and Jordan coherence remain undefined.

Executable evidence

Checker: research/voevodsky/check_dp6_oriented_facet_corridors.rs

SHA-256: 1324c0eedf6f5380344a40d83681df0171176693b589e99d75821043f65f1fc8

The user-site structured-command MCP ran rustfmt, rustfmt --check, warnings-denied optimized compilation, the executable assertions, and the JSON output. All passed.

Outcome contract

{
  "claim": "The nine literal K6 facets determined by the six oriented KN vertex packets have canonical primitive directed corridor arcs. D3 rotation is strict, while physical reflection differs from the selected reflected arc by exactly one literal oriented facet boundary on every facet. The nine homotopies equal minus the primitive boundary of the unique K6 top.",
  "status": "falsified_scoped_strict_reflection_equivariant_facet_arc_selection_with_primitive_top_obstruction",
  "scope": "literal K6 vertex-edge-facet incidence and finite integral corridor chains; derived facet homotopies and their top boundary included, entry223 comparison/endpoints excluded",
  "census": {
    "K6_face_ranks": [1, 9, 21, 14],
    "literal_facets": 9,
    "short_sheet_facets": 6,
    "long_road_facets": 3,
    "selected_vertices_per_facet": 2,
    "oriented_corridors": 9,
    "total_edge_terms": 21,
    "short_edge_terms": 15,
    "long_edge_terms": 6
  },
  "integral_matrix": {
    "corridor_boundary_rank": 9,
    "smith_factors": [1, 1, 1, 1, 1, 1, 1, 1, 1],
    "torsion": false,
    "base_inversions": false
  },
  "symmetry": {
    "D3_rotation_strict": true,
    "strict_reflection_matches": 0,
    "reflection_endpoint_scalar_plus": 9,
    "reflection_endpoint_scalar_minus": 0,
    "short_reflection_failures": 6,
    "long_reflection_failures": 3,
    "facet_boundary_homotopies_required": 9
  },
  "top_obstruction": {
    "homotopy_chain_closed": true,
    "equals_literal_K6_top_boundary": true,
    "coefficient": -1,
    "smith": [1]
  },
  "minimal_additional_datum": "comparison of the primitive literal K6 top with the entry223 projectivized-conductor top, including six short-facet contractions and three long-road residues",
  "unconstructed": [
    "entry223 top and based qSigma comparison",
    "rank-nine homogeneous solution module",
    "endpoint/Q mapping fiber",
    "p_partial_Q and Bockstein",
    "D8 and Jordan coherence"
  ],
  "checker": "research/voevodsky/check_dp6_oriented_facet_corridors.rs",
  "checker_sha256": "1324c0eedf6f5380344a40d83681df0171176693b589e99d75821043f65f1fc8"
}

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