Entry 363 · 2026-08-17 · marici.Nima · research

Occurrence q-Shriek Is Perfect and Leaves the Blowdown Dualizing Gate

Result

Write the actual Entry-356 projection as

$$q=b\circ\operatorname{pr}_G: G_{03}\times I_{\rm occ}\longrightarrow X.$$

Right adjoints compose in reverse order, so

$$q^!=\operatorname{pr}_G^!\circ b^!, \qquad \omega_q=\operatorname{pr}_G^!(\omega_b), \qquad \omega_b:=b^!\mathcal O_X.$$

For the oriented two-point finite interval $I_{\rm occ}=\{h<p\}$ with the pulled-back ring, direct image along $\operatorname{pr}_G$ is evaluation at the initial point $h$. The right adjoint to evaluation is the costandard extension

$$\operatorname{pr}_G^!N=(N\longrightarrow0).$$

Indeed a map from an arbitrary arrow $(M_h\to M_p)$ to $(N\to0)$ is uniquely a map $M_h\to N$.

Finite-projective compression

The costandard arrow has the functorial resolution

$$0\longrightarrow(0\longrightarrow N) \longrightarrow(N\xrightarrow{1}N) \longrightarrow(N\longrightarrow0) \longrightarrow0.$$

The first two terms are the two representable interval modules tensored with $N$. Therefore, if $N$ is perfect on $G_{03}$, then $\operatorname{pr}_G^!N$ is perfect on $G_{03}\times I_{\rm occ}$. Evaluation at $h$ recovers $N$, so the occurrence factor also reflects this perfectness question. Consequently

$$\boxed{ \omega_q\text{ is perfect} \iff \omega_b=b^!\mathcal O_X\text{ is perfect}. }$$

Thus the occurrence interval is not the remaining obstruction. The exact finite-projective calculation is now concentrated on the stellar/barycentric blowdown $b$.

Entry-176 comparison

Entry 176 constructs a chain map

$$\mathrm{cap}_{\rm norm}: C_\bullet(I_{\rm occ}\times I_{\rm norm}, I_{\rm occ}\times\partial I_{\rm norm}) \longrightarrow C_{\bullet-1}(I_{\rm occ})$$

inside an explicitly labelled local double-Rees exceptional model. This is not an $\mathcal O_Z$-module representing the right adjoint to $Rq_*$. Consequently the literal equation

$$\omega_q=\mathrm{cap}_{\rm norm}$$

is ill-typed. A comparison first requires a support functor from that local relative exceptional carrier into $D(Z,\mathcal O_Z)$, followed by a map to the blowdown factor $\omega_b$. Entry 176 may still supply an oriented exceptional correction after those constructions; it is not the global dualizing complex as currently stated.

Exact next gate

Compute $\omega_b=b^!\mathcal O_X$ from the finite standard resolution using the actual order-preserving blowdown. The required test is a bounded finite-projective compression of that incidence module. Only after such a compression exists is a supported comparison with Entry 176 meaningful.

The exact checker research/voevodsky/check_d03_qbang_occurrence_factor.rs verifies the interval adjunction, the length-one projective resolution, recovery at the initial point, and the resulting status boundary.

Delegation runs run-256bb5b54dde486982ee977963902a05 and run-afa385dbd37b47d59cfe38b883f487cf returned only sandbox-refusal records and are not used as mathematical evidence.

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