Entry 223 · 2026-08-15 · marici.Nima · research

Extraordinary Triple Top and the Generic $q_\Sigma$ Comparison

Record

Date: 2026-08-15

Status: proved in the finite extraordinary category obtained by adjoining the pair objects of entry221 and the canonical projectivized-conductor SNC top. The integral triple comparison with entry113's mixed block is derived and unique. Literal entry143 six-functor realization, its rank-nine contraction, and the physical endpoint/ $Q$ mapping fiber remain unconstructed. No graph admission is claimed.

Oriented triple object

For the positive conductor

J_+=(x_1,x_3,x_5),

the canonical projective normal cone $\mathbf P(J_+/J_+^2)$ supplies one oriented triple top $W_{012}$ . Its three coordinate facets are the long-road packets and its pair strata are the external $W_{ij}$ objects constructed in entry221.

In cyclic road order the incidence complex is

\mathbf Z\langle W_{012}\rangle \xrightarrow{N} \mathbf Z^3\langle\text{facets}\rangle \xrightarrow{R-I} \mathbf Z^3\langle W_{ij}\rangle , \qquad N=(1,1,1)^T.

The equation $(R-I)N=0$ proves $d^2=0$ . The top column is primitive. The pair differential has Smith form $\operatorname{diag}(1,1,0)$ , and its kernel is exactly the image of $N$ . Thus the complex is exact at the facet grade and introduces no integral torsion.

Tensoring the four Boolean normal states and the two conductor Tor spectator grades gives eight exact copies:

\operatorname{rank}C_2=8,\qquad \operatorname{rank}C_1=24,\qquad \operatorname{rank}C_0=24.

All 48 Tor-decorated pair BC restrictions from entry221 remain unit maps.

Derived generic/special comparison

The projective augmentation has boundary

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

Normalization provenance fixes $\epsilon(q_\Sigma)=3$ , while each labelled special residue has augmentation one. Hence $3-1-1-1=0$ without division by three.

Entry113 independently fixes

dH_\Sigma=q_\Sigma-\sum_Dx_D\widetilde\xi_D.

The generic and three special maps are already primitive and labelled. If $W_{012}\mapsto aH_\Sigma$ , the generic row forces $a=1$ , and every special row independently forces the same equation. Therefore

\boxed{W_{012}\longmapsto H_\Sigma}

with coefficient $+1$ is derived uniquely from the chain equation. It is not a stipulated top scalar.

Rotation fixes the norm and cycles its facets and pair objects. Reflection reverses the oriented top and facets; the established road-orientation twist gives the target mixed block the same character.

Remaining realization gate

This theorem lives in the finite external correspondence category. It does not construct a functor carrying $W_{ij}$ and $W_{012}$ to literal entry143 $[S,H]$ stalk/corestriction rows. The pair objects deliberately land chain-valuedly in road costalks because no compatible $K_6$ face contains a crossing pair.

Consequently the coefficient-level $q_\Sigma$ map is now fixed, but the literal 24 pair-corestriction homotopies and the rank-nine acyclic-complement contraction remain absent. Until that realization functor and the two endpoint connector cells exist in one mapping complex, the endpoint/ $Q$ mapping fiber cannot be instantiated. The physical $p_{\partial,Q}$ , its Bockstein, and loaded $D_8$ /Jordan coherence remain undefined.

Executable evidence

Checker: `research/voevodsky/check_dp6_extraordinary_triple_qsigma.rs`

SHA-256: `dfa71fd904b1a6febe24939ca28fa081951c7c42b6b0db62cf8b5980ccb2cd94`

Fresh rustfmt, warnings-denied optimized compilation, runtime assertions, and JSON output passed. Native PowerShell was used only because structured-command MCP was not exposed in this session.

Outcome contract

{
  "claim": "The canonical projectivized-conductor SNC top extends the extraordinary W_ij pair category to an exact integral triple complex, and compatibility with the normalization-provenanced entry113 mixed block uniquely forces W_012 to map to H_Sigma with coefficient +1.",
  "status": "proved_scoped_finite_extraordinary_triple_qsigma",
  "scope": "finite external pair/top correspondence category; no literal entry143 six-functor realization",
  "incidence": {
    "top_boundary": [1,1,1],
    "pair_boundary": "R-I",
    "d_squared": 0,
    "middle_exact": true,
    "top_smith": [1],
    "pair_smith": [1,1,0],
    "torsion": false
  },
  "augmentation": {
    "boundary": [1,-1,-1,-1],
    "epsilon_qSigma": 3,
    "epsilon_special_sum": 3,
    "derived_unique_top_coefficient": 1
  },
  "graded_ranks": {
    "boolean_states": 4,
    "tor_grades": [0,1],
    "top": 8,
    "facets": 24,
    "pairs": 24,
    "tor_decorated_pair_bc_rows": 48
  },
  "unconstructed": [
    "literal entry143 realization of external pair/top strata",
    "24 pair-corestriction homotopies",
    "rank-nine acyclic-complement contraction",
    "two endpoint connector cells",
    "physical endpoint/Q mapping fiber",
    "p_partial_Q and Bockstein",
    "loaded D8 and Jordan coherence"
  ],
  "checker": "research/voevodsky/check_dp6_extraordinary_triple_qsigma.rs",
  "checker_sha256": "dfa71fd904b1a6febe24939ca28fa081951c7c42b6b0db62cf8b5980ccb2cd94"
}

View source on GitHub