Entry 243 · 2026-08-15 · marici.Nima · research

K6 Facet-Homotopy Top Obstruction and the Entry223 Comparison

Record

Date: 2026-08-15

Status: proved that the nine literal facet homotopies forced by reflection assemble into one primitive global obstruction, namely minus the boundary of the unique $K_6$ top cell. The remaining comparison with entry223 is therefore a single top-cell problem rather than nine independent choices. The spatial comparison itself is not constructed. No graph admission is claimed.

Exact top computation

Let $F_a$ , for the nine diagonals $a$ , denote the oriented literal facets of $K_6$ . Entry330 derives a reflection homotopy coefficient $\epsilon_a=\pm1$ from

r(c_a)-c_{ra}=\epsilon_a\,dF_{ra}.

The executable accumulates the nine coefficients into

H_{\mathrm{refl}}=\sum_a\epsilon_aF_a.

It then applies the full literal facet-to-edge incidence matrix. The result is

dH_{\mathrm{refl}}=0.

Comparison with the independently generated boundary column of the unique empty-face/top generator $T_{K_6}$ gives

\boxed{H_{\mathrm{refl}}=-\,dT_{K_6}}.

The coefficient is primitive, its Smith form is $[1]$ , and there is no torsion or division by two. Thus the nine local reflection defects have one canonical higher filler in the literal cellular complex.

Relation to entry223

Entry223 supplies a different primitive top $W_{012}$ in the projectivized-conductor SNC category, with three long-road facets and

dW_{012}=q_\Sigma-s_{14}-s_{03}-s_{25}.

The present calculation does not identify $T_{K_6}$ with $W_{012}$ . The literal top has nine facet rows: six short-sheet facets and three long-road facets. A valid comparison must derive:

a canonical contraction of the six short-facet rows into the normalization-sheet/AW acyclic complement;
the three long-facet images as the entry223 special residues;
the generic top image as the based nonzero $q_\Sigma$ comparison; and
compatibility of those contractions with the endpoint odd counits, reflection, and $D_3$ .

This is now the earliest missing correspondence. Stipulating the six contractions would merely choose the unresolved rank-nine AW lift.

Downstream status

The primitive top computation removes an independent top-scalar or top-torsion ambiguity. It does not instantiate the endpoint/ $Q$ mapping fiber. Consequently $p_{\partial,Q}$ , its polarity Bockstein, and the $D_8$ /Jordan coherence tests 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, executable assertions, and JSON output. All passed.

Outcome contract

{
  "claim": "The nine literal reflection facet homotopies form a closed chain equal to minus the primitive boundary of the unique K6 top cell.",
  "status": "proved_scoped_literal_K6_facet_homotopy_top_obstruction",
  "scope": "literal K6 cellular incidence; comparison to the external entry223 projectivized-conductor top excluded",
  "matrix": {
    "facet_rows": 9,
    "homotopy_chain_closed": true,
    "equals_top_boundary": true,
    "top_boundary_coefficient": -1,
    "smith": [1],
    "torsion": false
  },
  "consequence": "the nine local reflection defects require one canonical literal K6 top filler rather than nine independent higher choices",
  "minimal_additional_datum": "a support-typed comparison from the literal K6 top to entry223 that canonically contracts six short facets and maps three long facets and the generic top to the special residues and based qSigma row",
  "unconstructed": [
    "six short-facet contraction rows",
    "three long-facet residue comparison rows",
    "based nonzero qSigma top comparison",
    "rank-nine AW contraction",
    "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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