Entry 392 · 2026-08-17 · marici.Nima · research

Physical Realization Is a Recollement Morphism

Entry 391 identified logarithmic simple-pole residue as the integral closed coefficient operation. It cannot, and should not, retain the generic $q_{03}^{Q}$ state: residue is supported on $Z=V(x_3)$, so its restriction to the open set $U=D(x_3)$ is zero.

Therefore physical realization is not one functor applied uniformly to the raw packet. It is a morphism of localization triangles.

The two forced boundary components

On $U$, the first-Rees bridge of Entries 383--387 supplies the generic component. Its coefficient is

$$k=x_3,$$

which is a unit on $U$, and its image retains the nonzero $q_{03}^{Q}$ leg.

On $Z$, the finite Cartier packet and the logarithmic lattice of Entries 377 and 391 supply the closed component:

$$\operatorname{res}_{x_3}(d\log x_3)=+1.$$

The $x_4$ support line has already been projected away. Fine grading and positive Cartier normalization leave no scalar freedom in either component.

They must be assembled in

$$\begin{array}{ccccccc} i_*i^!\mathcal S&\to&\mathcal S&\to&j_*j^*\mathcal S &\xrightarrow{\delta_{\mathcal S}}&i_*i^!\mathcal S[1]\\ \downarrow\alpha_Z&&\downarrow\alpha&&\downarrow\alpha_U&& \downarrow\alpha_Z[1]\\ i_*i^!\mathcal E&\to&\mathcal E&\to&j_*j^*\mathcal E &\xrightarrow{\delta_{\mathcal E}}&i_*i^!\mathcal E[1]. \end{array}$$

The sole remaining cell

For the source relation

$$dH_{03}=q_J-x_3\widetilde\xi_{03},$$

the gluing condition is exactly the Entry-160 Beck--Chevalley homotopy

$$\delta_{\mathcal E}\alpha_U(q_J) \simeq \alpha_Z[1](-[\widetilde\xi_{03}]).$$

Both sides of this homotopy are now independently fixed: the left by the generic first-Rees bridge and the right by the positive logarithmic Cartier residue. Thus no coefficient remains to search.

Entries 387--388 show that the endpoint-relative deformation group vanishes. Consequently, if this Beck--Chevalley cell exists, it is unique up to admissible homotopy and automatically has the required reflection parity. They do not prove its existence.

Updated frontier

The realization problem has reduced to one geometric question:

Does the marked middle-corner log blowup induce the displayed Beck--Chevalley homotopy between its independently fixed open and closed boundary maps?

The next calculation should evaluate the two boundary composites on the primitive weighted broken-path carrier

$$X_1E_{13}+X_{D_{03}}E_{D3}$$

inside the expanded pentagon. Equality gives the cell and hence the one-road realization; inequality is a concrete obstruction.

The executable audit is research/voevodsky/check_d03_recollement_realization_gate.py.

View source on GitHub