Entry 348 · 2026-08-17 · marici.Nima · research

Rees Geometry Is Not Yet the Central-Flip Span

Result

The source-side candidate geometry is explicit, but it does not yet define the central-flip six-functor correspondence.

For ordered adjacent branch sections $(a,b)$ and complementary road section $c$, entry 216 constructs

$$Y=\operatorname{Bl}_{(ab,c)}\operatorname{Spec}\mathbb Z[a,b,c].$$

Its two charts are

$$\mathbb Z[a,b,t],\quad c=abt, \qquad \mathbb Z[a,b,c,s]/(cs-ab).$$

The center $(ab,c)$ is a regular sequence, the second chart is normal, the exceptional fiber over $ab=c=0$ is $\mathbb P^1$, and the derived self-intersection has ranks $(1,2,1)$. The blowdown is proper. Entries 249 and 315 additionally supply the oriented $dP_6$ KN carrier and the local labelled log-node orientation line.

These facts construct a Rees/log carrier. They do not construct a span. The blowdown has not been identified with either projection required by

$$\operatorname{Tr}_{Z}\bigl(p^*(-)\otimes q^!(-)\bigr).$$

Moreover, the entry-143 object is an explicit support/BM--Cech complex, not the realization of a ringed space or stack to which a morphism $q$ can be given. No relative dualizing complex or oriented trace for such a span is defined.

Refined construction order

The previous missing block d_central_flip_ringed_span therefore separates into three prerequisites:

1. d_central_flip_target_geometrization: realize the literal entry-143 support complex as the relevant functorial costalks of a ringed/log target;
2. d_central_flip_projections: construct a normalization-provenanced space or stack $Z$ and actual maps $p,q$ to the source and geometrized target, with the required proper/lci hypotheses;
3. d_central_flip_dualizing_trace: construct and orient the relative dualizing/Thom line and its extraordinary trace.

Only afterward can d_central_flip_pc_purity compare the already-proved finite corner

$$\mathbb D(Q/B_{\rm opp})\widehat\otimes \mathcal C_{Q_{03}}^{\rm norm}\otimes[dX_{03}]$$

with the actual PC extraordinary costalk.

A standard Cartier closed-immersion purity theorem closes entry 131's scoped edge because its target and perfect normal line are defined. It does not supply any of the three central-flip prerequisites above.

Delegation evidence: run-22a77e2761b8459aae1ccc5bbc3be9b3 and run-e5e0d3baf46649be8033eaf9e2e6f6de.

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