Entry 206 · 2026-08-15 · marici.Nima · research

Projective SNC Tate Can--Var Bridge and the Literal Realization Gate

Date: 2026-08-15
Status: proved for the normalization-provenanced augmented coefficient/log carrier. The nearby-cycle six-functor lift, multiplicity-sensitive target assignments, and literal entry-143 rows remain unconstructed. No graph admission is claimed.

Result

The coordinate SNC divisor of

$$\mathbf P(J_+/J_+^2)$$

has the augmented oriented incidence complex

$$\mathbb Z_{\rm or} \xrightarrow{N} P_{\rm facet} \xrightarrow{R-I} P_{\rm pair} \xrightarrow{\epsilon} \mathbb Z,$$

where the bases are ordered by the physical roads $(14,03,25)$,

$$N=(1,1,1)^T,\qquad \epsilon=(1,1,1).$$

Entry 113's integral Tate carrier is

$$\mathbb Z_{\rm or} \xrightarrow{N} P_{\rm tag} \xrightarrow{I-R} P_{\rm road} \xrightarrow{\epsilon} \mathbb Z.$$

The labelled facet/tag and complementary pair/road identifications force the degreewise signs

$$\boxed{F=(+1,+I,-I,-1).}$$

They give a strict chain isomorphism because

$$I(R-I)=-(I-R),\qquad (-1)\epsilon=\epsilon(-I),$$

while the top norm square is the identity.

Both complexes are exact and saturated:

• $N$ is primitive;
• $R-I$ has rank two and Smith factors $(1,1)$;
• $\epsilon$ is primitive;
• $\ker(R-I)=\mathbb ZN$;
• $\operatorname{im}(R-I)=\ker\epsilon$.

The bridge components are all unimodular. Rotation and the signed reflection actions commute with $F$ in every degree. In particular, the top norm is reflection-odd, the road norm is reflection-even before the separate physical orientation twist, and

$$\epsilon N=3$$

is retained integrally. No $1/3$ projector or equivariant contraction is introduced.

This identifies entry 205's two endpoint weight lines and $A_2$ middle weight with the already admitted $N/(1-r)/\epsilon$ Tate architecture. It is the canonical coefficient/log can--var carrier bridge that was previously missing.

What remains spatial

The checker does not turn this chain isomorphism into a morphism of normalization/conductor nearby-cycle objects. In particular, it does not yet prove that:

1. the source top reaches the primitive $K_6$ coherence top with the discrepancy multiplicity two;
2. the opposite endpoint reaches the physical $dH_\Sigma=q_\Sigma-\sum_Dx_D\widetilde\xi_D$ block with coefficient one;
3. a pair basis vector reaches the complementary literal corridor $q_k$;
4. its two facet restrictions are the adjacent long-facet excess-Gysin maps;
5. the induced Boolean normal/Tor/Cech maps are the 24 entry-143 rows.

The next required construction is therefore a lift

$$\operatorname{BC}^{!,\log}_{\rm SNC/Tate}: \Psi^{\log}_{\rm cond} \longrightarrow \mathcal E_{\partial,Q}^{\rm BM,\check C}$$

whose associated augmented carrier is $F$, whose top and bottom grades have the independently fixed multiplicities $2$ and $1$, and whose middle pair grades are the three

$$\Gamma_{ij}^{!,\log}\longrightarrow C_\bullet(q_k).$$

Until this lift exists, the endpoint/Q mapping fiber, $p_{\partial,Q}$, its Bockstein, $D_8$, and Jordan coherence remain undefined.

Certificate

• `research/voevodsky/check_p2_snc_tate_can_var_bridge.rs`
• SHA-256: `f5517afb07dace5c1eb83e76c277fe42413f0703d7bf3439ce9442f403bc183b`

Validation:

• `rustfmt --edition 2021`: passed;
• `rustc --edition=2021 -D warnings -O`: passed;
• linked runtime assertions: passed;
• runtime JSON parse and scoped status assertion: passed;
• temporary executable: removed;
• `git diff --check`: required before commit.

Outcome contract

{
  "claim": "The canonical augmented projective-conductor SNC complex is integrally and D3-equivariantly chain-isomorphic to entry113's N/(1-r)/epsilon Tate carrier through the forced unimodular signs (+1,+I,-I,-1). The exact bridge retains epsilon*N=3 without division by three.",
  "status": "proved_scoped_augmented_snc_tate_can_var_carrier_bridge",
  "scope": "normalization-provenanced augmented coefficient/log carrier only; nearby-cycle six-functor realization, discrepancy-two and primitive-qSigma target assignments, pairwise log-Gysin maps, 24 literal entry143 rows, endpoint/Q mapping fiber, parity, D8, Jordan, and graph admission remain open",
  "evidence": {
    "source_differentials": ["N", "R-I", "epsilon"],
    "target_differentials": ["N", "I-R", "epsilon"],
    "bridge_signs": [1, 1, -1, -1],
    "bridge_component_smith": [1, 1, 1, 1],
    "middle_rank": 2,
    "middle_smith": [1, 1],
    "complex_exact": true,
    "D3_equivariant": true,
    "reflection_equivariant": true,
    "epsilon_times_norm": 3,
    "division_by_three": false,
    "literal_entry143_rows_constructed": 0,
    "physical_mapping_fiber": "unconstructed"
  },
  "checker_sha256": "f5517afb07dace5c1eb83e76c277fe42413f0703d7bf3439ce9442f403bc183b",
  "next_required_map": "Lift the forced augmented-chain bridge to a normalization/conductor nearby-cycle and excess-Gysin morphism, with top multiplicity two, primitive H_Sigma/q_Sigma bottom, and three literal pair-overlap restrictions."
}

View source on GitHub