Entry 205 · 2026-08-15 · marici.Nima · research

Projective Conductor Log-Torus Two-Top Source and the Spatial Can--Var Gate

Date: 2026-08-15
Status: proved for the normalization-provenanced integral coefficient/log two-top source. The spatial nearby-cycle can--var realization and literal entry-143 rows remain unconstructed. No graph admission is claimed.

Result

Let

$$E=J_+/J_+^2=L_{14}\oplus L_{03}\oplus L_{25}, \qquad \overline U=\mathbf P(E),$$

and let $D_{\rm SNC}$ be its three coordinate divisors. The canonical open log stratum is

$$U=\overline U\setminus D_{\rm SNC}\simeq(\mathbf G_m)^2.$$

Its compactly supported integral cohomology has ranks

$$\operatorname{rk}(H_c^2(U),H_c^3(U),H_c^4(U))=(1,2,1).$$

The middle lattice is the $A_2$ road lattice. In a simple-root basis, rotation and reflection act by

$$R=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}, \qquad S=\begin{pmatrix}0&1\\1&0\end{pmatrix},$$

with

$$R^3=S^2=1,\qquad SRS=R^{-1}.$$

There are no nonzero integral $R$-fixed vectors in $H_c^3$, while

$$\operatorname{SNF}(R-I)=(1,3).$$

Thus the middle weight retains, rather than rationally splits, the integral three-primary Tate datum.

Both endpoint weights are primitive lines. The $H_c^4$ line is $D_3$-invariant. Rotation fixes $H_c^2$, while reflection reverses its log orientation. Tensoring with the independently fixed road-orientation line contributes the second minus sign, so

$$H_c^2(U)\otimes\operatorname{or}_{\rm road}$$

is also $D_3$-invariant. Hence the canonical normalization-provenanced log source contains two distinct invariant weight lines:

$$\boxed{ H_c^4(U)\cong\mathbb Z,\qquad H_c^2(U)\otimes\operatorname{or}_{\rm road}\cong\mathbb Z. }$$

After fixing the admitted positive log orientation, the primitive logarithmic volume and Poincare duality give a unit comparison between the appropriately shifted lines. Its matrix is $[1]$, with Smith form $[1]$. The direct two-line lattice has Smith form $(1,1)$.

This identifies an intrinsic source for entry 204's required separation:

• the coherence weight can carry $C_\Sigma$ and the discrepancy-two boundary multiplicity;
• the other weight can carry the primitively normalized physical $H_\Sigma/q_\Sigma$ boundary;
• the $A_2$ middle weight retains the order-three extension information.

The multiplicity-two and primitive-one target assignments are not asserted here. They must be derived from a spatial can--var realization.

Boundary

This theorem is not yet the requested full-log excess-Gysin correspondence. The primitive log-volume comparison is a coefficient/log Gysin shape. It has not been realized as a morphism of normalization/conductor nearby-cycle objects, nor identified with the literal entry-143 support filtration.

The next required map is

$$\operatorname{CanVar}^{!,\log}_\Sigma: \left(H_c^2(U)\otimes\operatorname{or}_{\rm road}\right)[2] \longrightarrow H_c^4(U) \longrightarrow \mathcal E_{\partial,Q}^{\rm BM,\check C},$$

with associated grades satisfying all of the following:

1. the coherence grade maps with discrepancy multiplicity two to the primitive $K_6$ top boundary;
2. the physical grade maps primitively to $dH_\Sigma=q_\Sigma-\sum_Dx_D\widetilde\xi_D$;
3. the $A_2$ middle lattice restricts to the three pair-overlap objects;
4. those restrictions derive the 24 literal $[S,H]$ rows;
5. endpoint framing and reflection commute with the comparison.

Until this map exists, the pairwise $\Gamma_{ij}^{!,\log}$, endpoint/Q mapping fiber, $p_{\partial,Q}$, Bockstein, $D_8$, and Jordan tests remain open.

Certificate

• `research/voevodsky/check_p2_log_torus_two_top_can_var.rs`
• SHA-256: `985ae176f3ef88c4216891a01777c47902fe957572786a72ee31c27261b33c6e`

Validation:

• `rustfmt --edition 2021 --check`: passed after formatting;
• `rustc --edition=2021 -D warnings -O`: passed;
• linked executable and all runtime assertions: passed;
• runtime JSON parsed and scoped status asserted: passed;
• temporary executable: removed.

The structured-command user-site binding currently admits only `node`, `pnpm`, and `npm`; Rust validation therefore used the disclosed native-shell exception after the sandbox helper failed to launch.

Outcome contract

{
  "claim": "The canonical open log stratum U=(G_m)^2 of the projective conductor supplies two distinct primitive D3-invariant top-weight lines after the established road-orientation twist. Its A2 middle lattice has coinvariant Smith form [1,3], and the positively oriented log-volume/Poincare bridge between the shifted endpoint lines is primitive of Smith form [1].",
  "status": "proved_scoped_normalization_provenanced_two_top_coefficient_log_source",
  "scope": "normalization-provenanced coefficient/log source only; spatial nearby-cycle can-var, discrepancy-two and qSigma target assignments, pair-overlap maps, literal entry143 rows, endpoint/Q mapping fiber, parity, D8, Jordan, and graph admission remain open",
  "evidence": {
    "open_log_stratum": "(G_m)^2",
    "compact_support_ranks_hc2_hc3_hc4": [1, 2, 1],
    "middle_rotation_fixed_rank": 0,
    "middle_coinvariant_smith": [1, 3],
    "loaded_d3_invariant_top_weight_lines": 2,
    "log_volume_bridge_smith": [1],
    "two_top_basis_smith": [1, 1],
    "spatial_can_var_comparison": "unconstructed",
    "literal_entry143_rows_constructed": 0,
    "physical_mapping_fiber": "unconstructed"
  },
  "checker_sha256": "985ae176f3ef88c4216891a01777c47902fe957572786a72ee31c27261b33c6e",
  "next_required_map": "Realize the primitive two-weight log bridge as a normalization/conductor nearby-cycle can-var map whose associated grades carry discrepancy-two coherence and primitive H_Sigma/q_Sigma, then derive the three pair-overlap restrictions and 24 literal rows."
}

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