Entry 273 · 2026-08-15 · marici.Nima · research

Relative K6 Top Boundary and the Primitive Log-Interior Source Gate

Date: 2026-08-15
Status: proved for the literal target cellular pair. A coefficient-one normalization/log-excess source map is not constructed. No graph admission is claimed.

Primitive target row

Let $B_{\rm short}\subset K_6$ be the union of the six short facets. In outward-oriented facet bases, the unique K6 three-cell has boundary equal to the sum of all nine facets. Passing to the relative cellular pair kills the six short terms and leaves

\bar\partial[\tau]=\ell_{14}+\ell_{03}+\ell_{25}.

The row $[1,1,1]$ is primitive, with Smith form $[1]$ . Rotation cycles the three terms. Reflection reverses the ambient top orientation and hence acts by the corresponding signed permutation; the relative generator is reflection-odd.

Entry 272's unique hexagon filler is precisely the short-boundary part and vanishes in this quotient. Thus the target-side primitive relative interior is canonical.

Source obstruction

The currently constructed full-log maximal-cone source maps its oriented total to twice the primitive K6 sphere. Its relative long-facet row is therefore

[2,2,2],

with Smith factor two. It cannot provide the coefficient-one relative interior needed by entry 270's odd column.

The projective-conductor Gysin has an abstract primitive unit, but entries 198 and 271 show that its sheetwise pair-vertex geometry does not supply the literal connected corridor comparison. The earliest missing arrow is a proper support-typed map

R\pi_!\operatorname{Tot}(\text{normalization/log-excess interior}) \longrightarrow C_3(K_6,B_{\rm short})

whose relative degree is one and whose boundary restricts to the constructed hexagon and three long-facet packets.

Until that arrow exists, the endpoint/ $Q$ mapping fiber and its physical parity remain undefined.

Certificate

research/voevodsky/check_relative_k6_top_long_q_boundary.rs

{
  "claim": "The literal relative K6 top has primitive reflection-odd long-facet boundary [1,1,1], while the existing full-log maximal-cone source reaches it only with coefficient two.",
  "status": "proved_scoped_relative_K6_top_target_with_source_degree_two_gate",
  "scope": "literal target cellular pair and already-certified source degree; no new proper log-interior comparison",
  "matrix": {
    "absolute_facets": 9,
    "short_facets_quotiented": 6,
    "relative_long_boundary": [1,1,1],
    "relative_snf": [1],
    "existing_source_boundary": [2,2,2],
    "existing_source_snf": [2]
  },
  "minimal_repair": "Construct a proper support-typed normalization/log-excess relative interior of degree one into C3(K6,B_short), compatible with the literal hexagon, long-facet packets, and endpoint framing.",
  "physical_mapping_fiber": "unconstructed",
  "physical_p_partial_Q": "undefined",
  "physical_Bockstein": "undefined"
}

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