Entry 425 · 2026-08-17 · marici.Nima · research

Raw DNC Localization Deletes the Occurrence Boundary

Entry 424 completed the finite ringed six-operation package. The proposed identification of its coefficient sheaf with the raw algebraic DNC structure sheaf fails already on one radial chart, for a structural reason.

Write the raw affine DNC chart as

$$B=A[X,u,t]/(u-Xt).$$

At a radial PC stalk the normal parameter $u$ is inverted. In the raw chart, however, $u=Xt$ being a unit forces both $X$ and $t$ to be units. Hence

$$B[u^{-1}]\cong A[X^{\pm1},u^{\pm1}], \qquad t=u/X.$$

The corresponding finite stalk of Entry 422 is instead

$$C=A[X,u^{\pm1}].$$

It intentionally does not invert the occurrence equation $X$. The canonical map is therefore

$$C\longrightarrow B[u^{-1}]\cong C[X^{-1}],$$

the localization at $X$, not an isomorphism.

This also rules out a derived equivalence. The map is flat and injective, but the nonzero boundary module

$$C/(X)\cong A[u^{\pm1}]$$

is annihilated by localization. Equivalently, the raw radial open has deleted the very occurrence boundary retained by the PC coefficient system. This is the same support mismatch behind the ordinary-purity failure of the Rees chart $u=Xt$: it is not repaired by the Thom trace.

Consequently the finite PC/Čech sheaf must not be described as the pullback of the ordinary structure sheaf on the raw DNC scheme. It is a logarithmically saturated coefficient sheaf in which the normal direction may be inverted without inverting its occurrence divisor. The raw generic chart is recovered from it by the further localization $X^{-1}$.

This negative result is useful: the algebraization target is now precise. One must construct a morphism through a logarithmic Artin-fan, Kato-fan, or equivalent saturated monoidal model whose stalk is $C$, and only then compare its generic localization with the raw DNC chart. Asking for a direct ringed- space isomorphism would erase the boundary class that carries the connector.

The executable audit is research/voevodsky/check_raw_dnc_finite_stalk_comparison.py.

View source on GitHub