Entry-143 Ringed Incidence Enhancement
Result
The cellular cosheaf of entry 349 has a canonical enhancement to a module on a ringed finite space. This supplies an honest target derived category in which an extraordinary inverse image can be typed. It does not construct the central-flip span or compute that extraordinary inverse image.
Let (P=P_{K_6}^{ m or}), and use the opposite Alexandrov space (X=P^{ m op}). For (x=(S,H)), set
[ A_x=R[X]\bigl[u_a^{-1}:ain Ssetminus H\bigr]. ]
For every entry-143 boundary relation (xprec y), the set of inverted normal parameters only grows. Hence the canonical localization ( ho_{xy}:A_x o A_y) is a ring map. Iterated localizations commute, so the (A_x) and ( ho_{xy}) define a sheaf of rings (mathcal O_X) on the opposite Alexandrov space.
Define a rank-one diagram of (mathcal O_X)-modules (mathcal L^{check C}) by (mathcal L_x=A_x e_x). Its transition on a normal deletion is
[ e_xlongmapsto e_y, ]
and its transition on radial addition by (a) is
[ e_xlongmapsto (X_a/u_a)e_y. ]
These maps are semilinear over ( ho_{xy}). Every two-step square commutes: localization commutes with localization, the radial factors commute with each other, and localization commutes with multiplication by a radial factor. Thus (mathcal L^{check C}) is a well-defined module on the ringed incidence space, not merely a list of coefficient groups.
After the standard cellular incidence signs and the degree (deg(S,H)=3-|S|+|H|) are inserted, the compact cellular chains of (mathcal L^{check C}) are exactly (mathcal E_{partial,Q}^{ m BM,check C}). The support diagrams (F_Vsubset F_Bsubset F_K) are submodules, so the filtered quotients and their ranks are unchanged.
What this types
For a morphism (q:Z o X) of finite ringed spaces, the derived direct-image functor on the bounded constructible/perfect subcategory is a functor between small derived module categories. Whenever the chosen subcategory is preserved and the right adjoint exists, denote that right adjoint by
[ q^!:D(X,mathcal O_X)longrightarrow D(Z,mathcal O_Z). ]
The target of (q^!) is now defined and the expression (q^!mathcal L^{check C}) is type-correct once (q) is supplied. This is a typing theorem, not an existence theorem for a particular central-flip (q), and not a purity or trace calculation.
Exact remaining boundary
The enhancement is a ringed finite incidence space. It is not claimed to be the exit-path diagram of an algebraic or fs-log scheme, and no comparison with étale, analytic, or Kato--Nakayama six operations is asserted. In particular:
- no normalization-provenanced correspondence space (Z) has been constructed;
- no projections (p) and (q) from such a (Z) have been defined;
- preservation of the required perfect/constructible subcategory for those projections is unproved;
- no relative dualizing object, orientation, proper trace, or PC-purity comparison has been computed.
Consequently the first remaining central-flip obligation is now
d_central_flip_projections, followed by the dualizing trace and PC
purity. The product-branch Rees blowup of entry 216 remains useful input,
but entry 348's warning still applies: it is not itself the required span.
Delegation evidence: run-62988bfa58c046d19b3ec80df2416628. The two parallel constructive audits, run-2f4fe14ee2364adea9055de7be8cbb6d and run-786a0fc498bd41bb8fd49af8eb1ebf1e, exited without results and are not used as evidence.