Entry 233 · 2026-08-15 · marici.Nima · research

Rees--Čech Tor Third-Edge Packet and the Vertex-Cone Gate

Record

Date: 2026-08-15

Status: the third-edge packet and its integral residue matrix are proved, but the direct overlap-to-third-edge total chain map is falsified. The Čech overlap has a nonzero chart-difference boundary that cannot land in the literal direct sum of edge-support grades. A vertex-supported cone is the minimal additional datum. No graph admission is claimed.

The candidate third-edge packet

For each ordered product-Rees pair, the two standard charts determine the two previously selected corridor edges. Their common legal triangulation vertex has one remaining codimension-one face.

The Rees cover has a canonical relative-$\mathbf G_m$ overlap in Čech degree $+1$. After the primitive full-log cap, write the three source axes as

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

where $\tau$ is the conductor Tor axis and $n_0,n_1$ are the two moving normal axes. The chart restrictions are contraction by $n_1$ and $n_0$. Contracting the overlap packet by $\tau$ produces the remaining edge:

$$\iota_\tau: \Lambda^\bullet\langle\tau,n_0,n_1\rangle \longrightarrow \Lambda^{\bullet-1}\langle n_0,n_1\rangle.$$

The overlap Čech shift and Tor contraction cancel:

$$(+1)+(-1)=0.$$

Thus the overlap packet has the same total degree and rank as the third edge packet.

This assignment is forced by incidence. The first chart edge omits $n_1$, the second omits $n_0$, and the only remaining literal edge omits $\tau$. No additional face, midpoint, base localization, or fractional coefficient is introduced.

Integral matrix

For each ordered pair, the three residue blocks form a $12\times8$ matrix. The empty source state is its sole zero column. For each of the seven nonempty states, contraction along its least present axis selects a distinct unit pivot. Hence the block has rank seven and seven Smith factors equal to one.

Across six ordered pairs:

• the combined residue matrix has 72 rows and rank 42;
• all 42 nonzero Smith factors are one;
• the new overlap-to-third-edge block has 24 rows, rank 24, and 24 unit Smith factors;
• all 72 normal-removal squares commute;
• no base section is inverted.

Incidence and symmetry

Rotation transports all six full vertex stars. Physical reflection $v\mapsto3-v$ permutes every star as a set, including the previously missing third edges. It can exchange a chart role with the overlap role, which is why the two-edge truncation failed.

Reflection reverses both the Čech overlap orientation and the conductor Tor orientation. Their signs multiply to $+1$, while their degree shifts cancel. Consequently the packet and incidence-star data are reflection-compatible after the already fixed road-orientation convention.

The total chain equation fails

The Rees overlap is not a closed third boundary component. Its Čech differential is the primitive chart difference

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

Project literal entry143 to the direct sum of the three edge-support grades of the common vertex. There is no differential from one distinct edge summand to another in this quotient: radial maps go from an edge to the common vertex. Therefore the target differential of the proposed third-edge image is zero, while the image of the source differential is $(-1,+1,0)$.

Hence

$$d_{143}\,\Gamma(U_\times)=0 \ne (-e_0+e_1)=\Gamma(d_{\check C}U_\times).$$

The degree, sign, rank, and Smith checks are necessary but insufficient: the direct overlap-to-third-edge assignment is not a chain map.

Minimal repair

The smallest repair is a vertex-supported cone or Beck--Chevalley homotopy whose boundary realizes the primitive chart difference. Its radial maps must carry the actual occurrence sections, so principal-line Gysin evaluations—not scalar identities—must compare them with the unit Čech coefficients.

Only after adjoining and geometrically deriving that cone can one define the spatial transformation

$$R\pi_!\operatorname{Tot}(U_0\leftarrow U_\times\to U_1) \longrightarrow E_{\partial,Q}^{\mathrm{BM,\check C}}$$

whose overlap/Tor packet participates in a literal vertex-star chain map. Proper base change, local-cohomology variance, endpoint restriction, and the based $q_\Sigma$ comparison remain unconstructed.

Executable evidence

Checker: research/voevodsky/check_dp6_rees_cech_tor_vertex_star.rs

SHA-256: 87c808735bc183fce671a442142f083920038356c0a8dbe7c351b57923c73f71

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 relative-Gm overlap and Tor contraction produce the unique integral third-edge packet, but mapping that overlap directly to the third literal edge is not a total chain map: its Cech boundary is the nonzero chart difference while the target edge-support quotient has zero edge-to-edge differential.",
  "status": "falsified_scoped_overlap_to_third_edge_chain_map",
  "scope": "finite labelled product-Rees/Cech/Tor total complex and literal entry143 edge-support quotient; vertex-supported cone enlargements are not excluded",
  "matrix": {
    "ordered_pairs": 6,
    "edges_per_vertex": 3,
    "combined_rows": 72,
    "combined_rank": 42,
    "all_nonzero_smith_factors": 1,
    "third_edge_packet_rows": 24,
    "third_edge_packet_rank": 24,
    "third_edge_packet_smith_factors": 24,
    "normal_chain_squares": 72,
    "base_inversions": false
  },
  "degree_and_orientation": {
    "overlap_cech_degree": 1,
    "tor_contraction_degree": -1,
    "total_degree": 0,
    "reflection_cech_sign": -1,
    "reflection_tor_sign": -1,
    "loaded_reflection_sign": 1
  },
  "chain_falsifier": {
    "source_cech_boundary": [-1, 1, 0],
    "target_edge_quotient_differential_rank": 0,
    "direct_overlap_to_third_edge_chain_equation": false
  },
  "symmetry": {
    "D3_rotation_incidence_star": true,
    "physical_reflection_incidence_star": true,
    "reflection_mixes_chart_and_overlap_roles": true
  },
  "unconstructed": [
    "vertex-supported cone/Beck-Chevalley homotopy",
    "proper log-BM six-functor realization",
    "literal entry143 costalk/base-change comparison",
    "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_star.rs",
  "minimal_additional_geometry": "A vertex-supported cone/Beck-Chevalley homotopy whose boundary is the primitive chart difference and whose principal-line Gysin maps match the literal radial occurrence coefficients.",
  "checker_sha256": "87c808735bc183fce671a442142f083920038356c0a8dbe7c351b57923c73f71"
}

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