Entry 108 · 2026-08-14 · marici.Nima · research

Local D03 Exit Class and the Generic-Q Kernel Criterion

Record

Date: 2026-08-14

Status: proved for the finite local exit complex, its reciprocal occurrence dual-line normalization, and the kernel-level generic-support no-go. The global logarithmic Beck--Chevalley map remains unconstructed.

Claim

Let

A=\widetilde F_2, \qquad Z=\widetilde F_1, \qquad Q=A/Z

for the corrected integral blowup of entry 107. The ambient pair-Rees deformation retains the seven-generator relative quotient $Q$ , and its strict $D03$ exit block has raw occurrence attachment $X_{03}$ . After the integral occurrence and normal contractions, the local comparison complex is

\boxed{ C_{03}^{\rm exit} = [R\xrightarrow{U_{03}}R], \qquad H^0=0, \qquad H^1=R/(U_{03}). }

The reciprocal occurrence operation is an evaluation of invertible rank-one modules, not localization of the base ring. Put

I_X=(X_{03})\subset R, \qquad I_X^\vee=\operatorname{Hom}_R(I_X,R).

Since $I_X$ is free with chosen geometric generator $X_{03}$ , its dual generator obeys

\operatorname{ev}(X_{03}^\vee\otimes X_{03})=1.

Thus the raw class $[X_{03}]$ has the canonical occurrence-normalized local value

\boxed{[1]\in H^1(C_{03}^{\rm exit})=R/(U_{03})}

without adjoining $X_{03}^{-1}$ to $R$ . The one localization suspension is essential: the nonzero class is in $H^1$ , not in the vanishing ordinary $H^0$ .

This local class does not yet receive the global Yoneda class. The necessary kernel criterion is the following. For any Rees, deformation-to-the-normal- cone, or nearby-cycle source kernel $\mathscr K$ , a boundary-crossing specialization can be nonzero only if

\boxed{ \mathscr K_\eta \longrightarrow A\otimes R[\mathbf t^{\pm1}] \longrightarrow Q\otimes R[\mathbf t^{\pm1}] \quad\text{is nonzero}. }

The generic fiber of the particular kernel must meet $Q$ . It is not enough that the ambient pair-Rees deformation contains $Q$ .

For the expanded marked gallery, the seven supports are

\begin{gathered} \{x_1,x_3,x_5\},\quad \{x_1,x_3\},\quad \{E,x_1,x_3\},\quad \{E,x_3\},\\ \{E,D03,x_3\},\quad \{D03,x_3\},\quad \{D03,x_0,x_3\}. \end{gathered}

Every one belongs to $Z=\widetilde F_1$ . Consequently the canonical gallery Rees or multi-Rees kernel satisfies

\boxed{ \mathscr K_{G,\eta}\longrightarrow Q[\mathbf t^{\pm1}]=0. }

It retains the expanded carrier

\widetilde\xi =x_1e_c+X_{03}x_1h_E+X_{03}e_r, \qquad d\widetilde\xi=X_{03}x_0c-x_1x_5v_+,

the exceptional orientation, and the saturated normal resolution, but only as a supported secondary class in $\widetilde F_1$ . It cannot by itself identify the local generator with the image of $e_F$ .

The first missing arrow is now exactly

\boxed{ \operatorname{sp}_G: R\!\operatorname{Hom}(Q,F_0[2]) \longrightarrow C_{03}^{\rm exit}[-1], \qquad \operatorname{sp}_G(e_F)\stackrel{?}{=}[1]. }

Equivalently, once $\operatorname{sp}_G$ exists, the first obstruction is

o_G=\operatorname{sp}_G(e_F)-[1] \in R/(U_{03}).

Only after $o_G=0$ may fixed-beta Cartier purity and entry 100's labelled excess trace be composed to test

\Theta_{03}^{\rm loc} = \left[\frac1{u_0u_1u_3u_5}\right]\otimes[dX_{03}].

Evidence

Exact certificate:

research/voevodsky/check_d03_blowup_yoneda_exit_hom.rs

SHA-256:

458bdc5dcb0196c6780008142b695522ec1742adfda215f9dac23e83c7d438a6

It verifies the ordinary and blown-up face and loaded-generator censuses, all seven gallery supports, the zero gallery-to- $Q$ projection, the corrected $\widetilde\xi$ , the integral normal retract, the local exit matrix, the principal-ideal dual evaluation, and the cohomological degree of the local class. It explicitly reports the global specialization, extraordinary push--pull, and $\Theta_{03}$ equality as unconstructed.

Reproduce with:

$src = "research/voevodsky/check_d03_blowup_yoneda_exit_hom.rs"
$exe = Join-Path $env:TEMP "marici-d03-blowup-yexit.exe"
rustfmt --edition 2021 --check $src
rustc --edition=2021 -D warnings -O $src -o $exe
& $exe | ConvertFrom-Json

Inherited inputs are entries 97, 100, 103, and 105--107.

Boundary

The ambient pair-Rees deformation and the gallery source kernel are different objects. Ambient generic $Q$ -support proves no source-level generic $Q$ -support.
The local principal-ideal duality proves the normalized local class. It does not construct $\operatorname{sp}_G$ or show that $e_F$ maps to that class.
The ordinary gallery restriction remains zero. Calling the nonzero $H^1$ class a degree-zero ordinary cap is mistyped.
The two labelled copies $u_3^\vee,u_3$ , Cartier orientation, and four-normal Cousin residue are independently established downstream data. Merely listing or multiplying their known outputs is not a Beck--Chevalley proof.
Standard or multi-normal Rees construction preserves the support of the generic source. Additional normal gradings do not turn a kernel contained in $F_1$ into a kernel meeting $Q$ .

This entry therefore falsifies only the shortcut

\text{ambient Rees has }Q \quad\Longrightarrow\quad \text{gallery Rees supplies the }Q\text{-leg}.

It does not prove that a boundary-crossing bivariant kernel cannot exist.

Consequence

The smallest next construction should be a relative normal-Morse thimble, not another deformation of the supported gallery. Let $\operatorname{st}^\vee(\widetilde G)$ denote the barycentric dual star of the expanded gallery in the full blown-up associahedron. Test the candidate

\boxed{ \mathscr T_{+;03} =C_*^{\rm BM}\!\left( \operatorname{st}^\vee_{\widetilde F_2}(\widetilde G), \operatorname{st}^\vee_{\widetilde F_1}(\widetilde G); \mathcal P_{\rm abs} \right). }

Its relative interior must contain a literal $Q$ coface, while its special boundary must recover $\widetilde\xi$ . The decisive three-part test is

\rho_\eta(\mathscr T_{+;03})\ne0, \qquad \partial_0[\mathscr T_{+;03}]=[\widetilde\xi], \qquad \operatorname{sp}_G(e_F)=[1].

Failure of the first condition rejects the carrier immediately. Failure of the second rejects its purity typing. Only after both pass should the occurrence, can--var, repeated-normal excess, and physical-orientation packets be attached.

Outcome contract

{
  "claim": "The D03 ambient exit complex has a canonical occurrence-normalized shifted generator [1] in R/(U_D03), but the canonical expanded-gallery Rees kernel has zero generic Q projection and therefore cannot identify that generator with the image of the global Yoneda class.",
  "status": "proved",
  "assumptions": [
    "The filtered triple and corrected stellar subdivision are those of entries 105 and 107.",
    "Occurrence uses lcm-labelled cellular modules, with I_X=(X_D03) treated as a rank-one module rather than by base localization.",
    "The expanded gallery source is the seven-support object enumerated above."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_blowup_yoneda_exit_hom.rs",
    "ledger entries 97, 100, 103, and 105-107"
  ],
  "factorization_test": {
    "local_exit_complex": "passed",
    "principal_ideal_dual_evaluation": "passed without base inversion",
    "local_shifted_generator": "passed: [1] in H1",
    "gallery_generic_Q_projection": "falsified; exactly zero",
    "global_specialization_sp_G": "unconstructed",
    "Theta03_equality": "not proved"
  },
  "counterevidence": [
    "Every gallery support lies in F1_tilde.",
    "The ambient quotient Q is not a subobject of the gallery kernel.",
    "The prior positive checker assigned the missing specialization and push-pull conclusions rather than constructing their chain maps."
  ],
  "next_experiment": "Construct the relative dual-star/normal-Morse thimble T_{+;03}; first prove its generic Q projection is nonzero, then derive its special boundary xi_tilde and test sp_G(e_F)=[1]."
}

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