Entry 321 · 2026-08-15 · marici.Nima · research

Literal Corridor-Vertex Wall Realization and the Endpoint Extension Gate

Record

Date: 2026-08-15

Status: proved in the literal entry143 face/Boolean incidence complex. The six ordered pair packets admit a primitive 48-row realization on six legal common triangulation vertices, with both adjacent-edge relative residues and the physical endpoint reflection. The three-pair unoriented quotient has the expected 24 rows. Identification with the normalization/log source, endpoint extension, and the generic-Q connector remain unconstructed. No graph admission is claimed.

The missing literal wall was already a vertex

For $W_{03,25}$ , the two positive $q_{14}$ corridor edges are

S_0=\{x_5,x_1\},\qquad S_1=\{D_{14},x_1\}.

Their union

V_{14}=\{x_1,D_{14},x_5\}

is a legal noncrossing triangulation vertex of $K_6$ . Rotation gives legal vertices $V_{03}$ and $V_{25}$ . Thus the shifted wall need not be an external face: it can land on three positive literal entry143 vertices. Physical reflection $v\mapsto3-v$ produces three disjoint negative vertices, matching entry221's six ordered pair objects.

For $|V_D|=3$ , the Boolean packet has profile

1+3t+3t^2+t^3.

After the uniform codimension-one Gysin shift it becomes

t+3t^2+3t^3+t^4=P(t)(1+t),

exactly the two-normal packet with Tor grades zero and one. The labelled map is the identity on the resulting three Boolean axes. Across all six ordered pairs it is a $48\times48$ identity matrix, so its rank is 48 and every Smith factor is one. The unoriented three-pair quotient retains 24 rows.

Adjacent-edge Beck--Chevalley maps

Each corridor edge is obtained from $V_D$ by deleting one label. The relative quotient of the vertex Boolean cube by states not containing that label is canonically isomorphic to the edge Boolean cube by oriented residue. The checker derives all 48 residue rows and all 48 nonzero normal-removal squares. The entry143 sign difference between $|S|=3$ and $|S|=2$ is exactly canceled by the contraction sign. Each residue block has a unit minor; no occurrence or normal section is inverted.

The earlier two-edge rank obstruction remains correct for its stated target $P\oplus P$ . It is bypassed—not contradicted—by using the literal common vertex and one uniform Gysin shift. Likewise the odd two-boundary suspension cocycle of entry228 is the truncation shadow of changing which vertex axis is treated as the Tor direction; the full three-axis vertex packet carries the ordinary entry143 reflection action.

Physical symmetry

Rotation $v\mapsto v+2$ preserves and cycles each three-vertex sheet. Physical reflection $v\mapsto3-v$ exchanges the positive and negative endpoint sheets, their vertices, and their exact selected edge pairs. The induced exterior permutation signs commute with every normal differential. Thus the six oriented vertex packets and all 48 adjacent-edge residues are strictly reflection closed. The target-only reflection $v\mapsto2-v$ used in the target Cech checker is a different convention and must not be substituted for the physical endpoint reflection.

Remaining gate

This closes the literal oriented pair-wall incidence problem. It does not yet prove that the normalization/log-excess object $W_{ij}$ pushes forward to this vertex relative quotient. The required next comparison must:

identify each of the six ordered source packets and its Tor axis with the corresponding oriented vertex packet and omitted edge label;
restrict to the already normalized endpoint odd counits;
extend the three vertex packets through the entry223 top to the based $q_\Sigma$ row in literal entry143.

Until those source and endpoint/generic comparisons are constructed, the pointed endpoint/ $Q$ mapping fiber and physical parity remain undefined.

Executable evidence

Checker: research/voevodsky/check_dp6_literal_vertex_wall_realization.rs

SHA-256: 2785b5cee56bb15fd94fa7c4d8fd5db799ed7e0bb1a2f9bcdec714b391b86301

Fresh rustfmt --check, warnings-denied optimized compilation, runtime assertions, and JSON output passed. Native PowerShell was used only because the user-site structured-command surface cannot access this repository or invoke rustc.

Outcome contract

{
  "claim": "The six oriented common legal triangulation vertices of the complementary two-edge corridors supply literal entry143 three-label Boolean wall packets. After one uniform Gysin shift they realize the full two-normal times Tor packets with a primitive 48-by-48 matrix; the three-pair unoriented quotient has 24 rows, and the two relative residues derive all 48 adjacent-edge BC rows.",
  "status": "proved_scoped_literal_oriented_vertex_wall_realization",
  "scope": "literal entry143 face incidence and Boolean normal complex; source normalization/log pushforward and endpoint/Q extension excluded",
  "matrix": {
    "source_states": 48,
    "literal_vertex_rows": 48,
    "unoriented_quotient_rows": 24,
    "rank": 48,
    "all_smith_factors": 1,
    "adjacent_edge_residue_rows": 48,
    "relative_normal_bc_squares": 48
  },
  "uniform_gysin_shift": 1,
  "D3_rotation": true,
  "physical_reflection": "v -> 3-v exchanges endpoint sheets",
  "oriented_edge_blocks_reflection_closed": true,
  "base_inversions": false,
  "unconstructed": [
    "normalization/log-excess source identification",
    "literal endpoint extensions",
    "based qSigma connector",
    "endpoint/Q mapping fiber",
    "p_partial_Q and Bockstein",
    "D8 and Jordan coherence"
  ],
  "checker": "research/voevodsky/check_dp6_literal_vertex_wall_realization.rs",
  "checker_sha256": "2785b5cee56bb15fd94fa7c4d8fd5db799ed7e0bb1a2f9bcdec714b391b86301"
}

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