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]."
}