Entry 324 · 2026-08-15 · marici.Nima · research

Rees--Čech Tor Vertex Cone and the Spatial Push--Pull Gate

Record

Date: 2026-08-15

Status: proved in the finite labelled product-Rees/Čech/Tor totalization and literal entry143 incidence/Boolean complex. The vertex-supported cone repairs entry323's direct third-edge chain failure. Proper log-BM six-functor provenance and the literal occurrence-line radial comparison remain unconstructed. No graph admission is claimed.

Correct totalization

The Rees overlap cannot map directly to the third edge because

$$d_{\check C}U_\times=-U_0+U_1.$$

Instead, map the overlap/Tor total generator to the common legal triangulation vertex.

After the primitive log-excess cap, the three axes are

$$(\tau,n_0,n_1).$$

Here $\tau$ is the conductor Tor direction and $n_0,n_1$ are the two chart-normal directions. With the fixed ordered-pair orientation, the total boundary is

$$D=\iota_\tau+\iota_{n_0}-\iota_{n_1}.$$

The last two terms are the two Čech chart restrictions $(-U_0,+U_1)$; the first is the wall/third-edge restriction. Since exterior contractions anticommute,

$$D^2=0.$$

The unique axis dictionary of entry322 sends $\tau$ to the persistent corridor label and $n_0,n_1$ to the two moving labels. Therefore the cone has the literal entry143 three-label Boolean packet, while its three boundary terms land on exactly the three codimension-one faces of that vertex.

Integral certification

Across six ordered pairs:

• the cone-to-vertex map is a $48\times48$ labelled identity;
• it has rank 48 and 48 unit Smith factors;
• the total boundary has 72 rows, split into 48 chart rows and 24 wall rows;
• its rank is 42, with 42 unit Smith factors;
• all 72 two-step boundary equations cancel integrally;
• no base section or integer is inverted.

The empty state is the sole zero column in each $12\times8$ boundary block. Each of the seven nonempty states has a distinct unit pivot obtained by contracting its least present axis.

Degree and symmetry

The wall term combines overlap Čech degree $+1$ with Tor contraction degree $-1$, hence has total degree zero. Physical reflection reverses both orientation lines, so the loaded wall sign is $(-1)(-1)=+1$.

Rotation and physical reflection preserve all six completed vertex stars. They may exchange chart and wall roles, but the combined totalization retains degree zero and the fixed loaded sign. This is the finite derived mechanism absent from the two-edge truncation.

Remaining geometric gate

The finite cone is not yet a spatial six-functor theorem. A genuine correspondence must construct a proper/log-BM kernel whose exceptional vertex stratum realizes this cone and whose three boundary maps are the literal entry143 extraordinary corestrictions.

The decisive unverified rows are the occurrence directions. Entry143 radial maps carry principal occurrence sections, whereas the Čech boundary has unit coefficients. The spatial kernel must provide the corresponding principal-line duals and Gysin evaluations and prove proper Beck--Chevalley naturality; these evaluations may not be replaced by scalar identities in advance.

Once that push--pull is constructed, its endpoint restrictions can be compared with the established odd endpoint counits, and its three-road top can be attached to the based $q_\Sigma$ row. Until then the pointed endpoint/$Q$ mapping fiber and all downstream parity and coherence classes remain undefined.

Executable evidence

Checker: research/voevodsky/check_dp6_rees_cech_tor_vertex_cone.rs

SHA-256: 60ed787ae2baf4e73aeeb983d35f39cb22737b631ec2bb13537b6fa3a13d71d1

Fresh rustfmt --check, warnings-denied optimized compilation, runtime assertions, and JSON field checks passed. Native PowerShell was used for Rust verification because no repository-scoped structured-command MCP capable of invoking rustc is exposed.

Outcome contract

{
  "claim": "The correct finite derived repair maps the Rees-Cech/Tor overlap totalization to the common literal vertex cone, with total boundary i_tau+i_n0-i_n1. This gives a primitive 48-row vertex realization and a saturated 72-row full-star boundary with d squared zero, D3 covariance, and physical-reflection closure.",
  "status": "proved_scoped_finite_rees_cech_tor_vertex_cone",
  "scope": "finite labelled Rees-Cech/Tor totalization and literal entry143 incidence/Boolean complex; proper log-BM six-functor realization excluded",
  "matrix": {
    "ordered_pairs": 6,
    "vertex_rows": 48,
    "vertex_rank": 48,
    "vertex_smith_factors": 48,
    "boundary_rows": 72,
    "chart_rows": 48,
    "wall_rows": 24,
    "boundary_rank": 42,
    "boundary_smith_factors": 42,
    "d_squared": 0,
    "base_inversions": false
  },
  "degree_and_symmetry": {
    "cech_degree": 1,
    "tor_contraction_degree": -1,
    "wall_total_degree": 0,
    "reflection_cech_sign": -1,
    "reflection_tor_sign": -1,
    "reflection_loaded_sign": 1,
    "D3_full_star": true,
    "physical_reflection_full_star": true
  },
  "unconstructed": [
    "proper log-BM six-functor realization",
    "literal occurrence-line radial Gysin maps",
    "endpoint extensions",
    "based qSigma connector",
    "endpoint/Q mapping fiber",
    "p_partial_Q and Bockstein",
    "D8 and Jordan coherence"
  ],
  "checker": "research/voevodsky/check_dp6_rees_cech_tor_vertex_cone.rs",
  "checker_sha256": "60ed787ae2baf4e73aeeb983d35f39cb22737b631ec2bb13537b6fa3a13d71d1"
}

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