Entry 198 · 2026-08-15 · marici.Nima · research

Projective Conductor SNC Augmentation and the Literal Pair-Vertex Gate

Date: 2026-08-15
Status: proved in the canonical integral coefficient/log model of the projectivized positive conductor normal cone. Literal entry-143 pair-vertex corestrictions and the acyclic-complement contraction are not constructed. The endpoint/ $Q$ mapping fiber and its physical parity remain undefined. No graph admission is claimed.

Canonical projective normal cone

On the positive normalization sheet, entry 93 gives

J_+=(x_1,x_3,x_5),\qquad E=J_+/J_+^2=L_{14}\oplus L_{03}\oplus L_{25}.

Thus $q:\mathbf P(E)\to Z$ is canonical. Its three coordinate hyperplanes form an SNC divisor with three pairwise intersections and no triple intersection.

After torus-orientation integration, its moment-corner complex is

\mathbb Z\langle H\rangle\xrightarrow{N} \mathbb Z^3\xrightarrow{R-I}\mathbb Z^3, \quad N=(1,1,1)^T, \quad R=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}.

The exact identity $(R-I)N=0$ proves $d^2=0$ .

If $S$ fixes road $14$ and exchanges $03,25$ , reflection acts by $-1$ on the top, $-S$ on oriented facets, and $SR^{-1}$ on pairwise intersections. Therefore

(R-I)(-S)=(SR^{-1})(R-I),\qquad (-S)N=-N.

This is the signed integral $D_3$ covariance.

Primitive Gysin, lines, and augmentation

For $\xi=c_1(\mathcal O_{\mathbf P(E)}(1))$ , projective-bundle integration is primitively normalized:

\boxed{q_*(\xi^2)=1.}

No integer or occurrence parameter is inverted.

The divisor $D_D$ is cut by a section of $\mathcal O(1)\otimes q^*L_D^\vee$ . Its residue retains $L_D$ , and entry 184's principal-dual evaluation has degree $(+1)+(-1)=0$ and value one. The pairwise intersections remain as the SNC Cech overlap row; deleting them would destroy the square-zero identity.

Let $q_\Sigma=q_{14}+q_{03}+q_{25}$ be entry 113's road norm and let $s_D$ be the three oriented special residues. Projective trace followed by the integral road transfer gives

\boxed{dH_{\mathbf P^2}=q_\Sigma-s_{14}-s_{03}-s_{25}.}

The row $[1,-1,-1,-1]$ has Smith form $[1]$ . It is primitive and torsion-free. It retains the three-primary Tate augmentation:

\epsilon(q_\Sigma)=3=1+1+1=\sum_D\epsilon(s_D).

This is the coefficient/log shadow of entry 113's mixed block, not yet its literal spatial realization.

Literal pair-vertex gate

A physical map to $\mathcal E_{\partial,Q}^{\rm BM,\check C}$ must send the top to $H_\Sigma$ , the augmentation to the based $q_\Sigma$ , each facet to the rotated entry-112/131 Cartier gallery, and each pairwise intersection to the common literal entry-143 short-boundary stalk reached by the two adjacent restrictions.

The first three assignments have canonical coefficient formulas. The last assignment is absent. On the actual AW collar it expands to 24 pair-vertex rows. Entry 184 proves equality of their coefficient evaluations, but no six-functor map identifies them with literal $[S,H]$ corestrictions.

Entry 179 gives a scoped no-go for doing this facewise: each of the six cyclic adjacent dP6 pairs labels crossing short diagonals, with no common $K_6$ face. A zero overlap image violates

df(c_{ab})=f(r_b)-f(r_a)\ne0.

This does not rule out an extraordinary/logarithmic/cdh overlap object. Exact reused certificates are:

research/voevodsky/check_d03_dp6_common_refinement.rs, SHA-256 c0838591bfb2e2f6ddf143951636e9d5346ab1cca6cfde43d50ab6f6123a9229;
research/voevodsky/check_d03_weighted_adjacent_pair.rs, SHA-256 5e6375625b0f51fbebcf7f46cf38c6b97b45f13f5a3c45da19a74bd117adf5c0.

Entry 119 supplies a second exact boundary warning. The endpoint-relative gallery has eight edges, six internal vertices, boundary rank six, and a unit $6\times6$ minor. Its two route classes form a saturated $\mathbb Z^2$ , and the positive mark selects one primitive line, but this does not construct the spatial contraction. Its checker is

research/voevodsky/check_d03_q0_endpoint_relative_tor_lift.rs, SHA-256 f2563b2cbd63cd655b3183635d0883030c8d219e4fbf58a91076899e02b7c54c.

Entry 136's full-cone lift also retains an affine rank-nine family. The projective top and facets fix the homology-bearing roof but do not select its acyclic-complement contraction. Endpoint columns cannot remove this ambiguity until the 24 spatial pair-vertex homotopies exist in the same category.

Certificate and validation

research/voevodsky/check_p2_conductor_snc_augmentation.rs
SHA-256 9940cc06895fc15cc1ae697de8d9f59919ed5c73268fb0eef0c39ff0c67164f0.

MCP worker validation:

rustfmt --edition 2021 --check: passed;
rustc --edition=2021 -D warnings --emit=metadata: passed with empty stdout and stderr;
the temporary metadata artifact was removed and confirmed absent;
linked execution was unavailable because MSVC link.exe is missing, so runtime assertions and JSON output were not executed.

The checker explicitly records 24 required literal vertex rows, zero constructed rows, and rank9_contraction_constructed=false.

Consequence

The projective conductor normal cone supplies the integral SNC source, primitive Gysin augmentation, line factors, overlap census, and coefficient $D_3$ signs. The earliest remaining arrow is a derived overlap correspondence from every pairwise SNC stratum to the literal road costalk, with its two boundary restrictions fixed by the adjacent facet maps.

Until those 24 rows and the induced contraction exist, the endpoint/ $Q$ mapping fiber is uninstantiated. Thus $p_{\partial,Q}$ , its polarity Bockstein, and $\omega_{\rm load}$ remain undefined.

Outcome contract

{
  "claim": "The canonical P2 conductor SNC coefficient/log complex has top boundary N=(1,1,1), pair boundary R-I, square-zero differential, primitive projective Gysin, primitive augmented row [1,-1,-1,-1], degree-zero line evaluation, and signed D3 covariance while preserving the integral factor-three Tate augmentation.",
  "status": "proved_scoped_coefficient_log_augmentation__literal_pair_vertex_realization_open",
  "scope": "canonical integral P2 coefficient/log model only; no literal entry143 pair-vertex maps, extraordinary overlap, acyclic contraction, physical mapping fiber, parity, or graph admission",
  "factorization": {
    "top_to_facets": [1, 1, 1],
    "facets_to_pairs": "R-I",
    "d_squared": 0,
    "projective_gysin": 1,
    "augmented_row": [1, -1, -1, -1],
    "augmented_smith": [1],
    "endpoint_augmentation": "3=1+1+1",
    "line_exponents": [1, -1],
    "line_evaluation": 1,
    "facets": 3,
    "pairwise_intersections": 3,
    "triple_intersections": 0,
    "literal_entry143_vertex_rows_required": 24,
    "literal_entry143_vertex_rows_constructed": 0,
    "crossing_adjacent_pairs": 6,
    "rank6_gallery_boundary": "proved with unit minor",
    "rank9_contraction_constructed": false,
    "physical_mapping_fiber": "unconstructed",
    "physical_p_partial_Q": "undefined",
    "physical_Bockstein": "undefined"
  },
  "checker_sha256": {
    "p2_snc": "9940cc06895fc15cc1ae697de8d9f59919ed5c73268fb0eef0c39ff0c67164f0",
    "dp6_common_refinement": "c0838591bfb2e2f6ddf143951636e9d5346ab1cca6cfde43d50ab6f6123a9229",
    "weighted_adjacent_pair": "5e6375625b0f51fbebcf7f46cf38c6b97b45f13f5a3c45da19a74bd117adf5c0",
    "endpoint_relative_rank6": "f2563b2cbd63cd655b3183635d0883030c8d219e4fbf58a91076899e02b7c54c"
  },
  "counterevidence": [
    "The 24 coefficient middle equalities are not literal stalk corestriction maps.",
    "The six adjacent dP6 pairs are crossing diagonals without common literal faces.",
    "The rank-six gallery boundary and primitive mark do not construct the spatial contraction.",
    "The AW roof retains a rank-nine acyclic-complement ambiguity."
  ],
  "minimal_repair": "Construct three derived pair-overlap maps and their 24 literal entry143 corestrictions, then solve the contraction and endpoint connector equations without changing the fixed top, facet, or qSigma rows."
}

View source on GitHub