Entry 597 · 2026-08-17 · marici.Nima · research

Entry-143 Cellular Cosheaf Geometrization

Result

The literal entry-143 BM--Cech target has an exact geometric realization in the finite cellular category. It is not yet a ringed/log six-functor object.

Let $P_{K_6}^{\rm or}$ be the finite state poset whose elements are pairs $(S,H)$, where $S$ is a noncrossing dissection of the labelled hexagon and $H\subseteq S$ is the retained normal-circle set. Give this poset its Alexandrov topology and dimension

$$\deg(S,H)=3-|S|+|H|.$$

Define the coefficient cosheaf $\mathscr E^{\check C}$ by the costalks

$$\mathscr E^{\check C}_{S,H} =R[X]\bigl[u_a^{-1}:a\in S\setminus H\bigr].$$

There are two kinds of covering relation. For radial addition $(S,H)\prec(S\cup\{a\},H)$, corestriction is multiplication by $X_a/u_a$. For normal deletion $(S,H)\prec(S,H\setminus\{h\})$, it is the canonical localization map, whose coefficient in the chosen generator bases is one.

These maps are functorial. Two radial additions commute because the coefficient ring is commutative; two normal deletions commute as iterated localizations; and radial addition commutes with normal deletion. The signs in entry 143 are therefore the cellular incidence and exterior-fiber signs, not failures of cosheaf functoriality.

Consequently the compact cellular chain complex of this coefficient cosheaf is exactly

$$\mathcal E_{\partial,Q}^{\rm BM,\check C} =\bigoplus_{(S,H)\notin F_V} \mathscr E^{\check C}_{S,H}[S,H]$$

with entry 143's radial $X_a/u_a$ and signed normal unit differential. The existing global promotion checker verifies all 215 states, every square, $d^2=0$, denominator support, and $D_3$-covariance.

The support subsets $F_V\subset F_B\subset F_K$ are closed under the cellular boundary and hence define coefficient subcosheaves. Their quotient realizes the strict filtration

$$0\to F_B/F_V\to F_K/F_V\to Q\to0$$

with ranks $(12,57,84,39)$, $(12,57,87,43)$, and $(0,0,3,4)$.

Remaining boundary

This proves cellular cosheaf geometrization, not a constructible-sheaf identification on an algebraic or log ringed space. The semilinear finite dual $\mathbb D_\iota$ is likewise not yet identified with $R\mathcal Hom(-,\omega_X)$.

The next missing datum is therefore narrower than target geometrization: construct a ringed/log enhancement of $(P_{K_6}^{\rm or},\mathscr E^{\check C})$ whose exceptional inverse image and proper corestrictions reproduce these costalks and maps. Only such an enhancement can type $q^!$, a relative dualizing trace, and the PC-purity comparison.

Delegation evidence: run-9681aca3750a48678fa421a6ef08c27e and run-a24fa38ef2644ae280c3692b2363ae97.

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