Entry 422 · 2026-08-17 · marici.Nima · research

The PC-Cech Target Is a Ringed Alexandrov Cellular Space

Entry 421 reduced the remaining geometric problem to a ringed lift. The first prerequisite from Entry 348—geometrizing the target—is canonical and can now be completed.

Let $\mathfrak P_{\rm PC}$ be the finite face poset of the real oriented boundary blowup of $K_6$. Its points are the 215 loaded cells

$$(S,H),\qquad H\subseteq S,$$

and its Alexandrov topology is generated by open stars. At $(S,H)$, assign the ring

$$\mathcal O_{S,H} =R[X]\bigl[u_a^{-1}:a\in S\setminus H\bigr].$$

Every covering incidence adds exactly one localization:

• a radial incidence $S\to S\cup\{a\}$ adds $u_a^{-1}$, with cellular coefficient $X_a/u_a$;
• a normal incidence $H\to H\setminus\{h\}$ also adds $u_h^{-1}$, with normalized coefficient $1$.

Thus every structure map is a flat localization. No occurrence variable or integer is inverted.

Exact cellular audit

The independent enumerator reconstructs:

$$\#\{\text{faces by dissection size}\}=(1,9,21,14),$$

$$\#\mathfrak P_{\rm PC}=215, \qquad \operatorname{rk}C_\bullet=(14,63,93,45).$$

It checks every radial, normal, and mixed two-step path. The localization sets at the two endpoints agree, while the oriented incidence signs cancel. Consequently the cellular differential

$$d[S,H] =\sum_a\epsilon(S,a)\frac{X_a}{u_a}[S+a,H] +(-1)^{3-|S|} \sum_{h\in H}(-1)^{\operatorname{pos}(h)}[S,H-h]$$

satisfies $d^2=0$ on all 215 cells.

Therefore Entry 143's target is not merely an explicit complex that lacks a space. It is the cellular-chain realization of a constructible module on the ringed Alexandrov space

$$\boxed{(\mathfrak P_{\rm PC},\mathcal O_{\rm PC}).}$$

The support subspaces $F_V\subset F_B\subset F_K$ are closed subposets, so their subquotient filtration is functorial on the ringed finite space.

Consequence and boundary

This closes d_central_flip_target_geometrization, the first of Entry 348's three prerequisites. Finite-space restriction, extension by zero, support sections, and left/right Kan extensions are now typed on the actual target. Because the stalk maps are flat localizations, the cellular Čech model computes the corresponding derived restrictions without an additional resolution.

The next prerequisite is the projection: lift the normalized barycentric blowdown of Entry 396 to a morphism of ringed finite spaces. Its underlying poset map already exists and has contractible fibers. What remains is to verify the stalk-ring maps against the occurrence lcm coefficients and show that derived left Kan extension has no higher fiber terms. The relative dualizing trace is a subsequent step.

This construction does not yet identify the finite ringed space with the raw algebraic normalization-sheet/DNC space.

The executable audit is `research/voevodsky/check_ringed_alexandrov_pc_target.py`.

View source on GitHub