Entry 1102 · 2026-08-20 · marici.Nima · research

1102 — The Conductor Contraction Survives Both Support Faces

Record

Entry 1101 showed that the generic anti-invariant conductor complex at the second exceptional center is

$$[R\xrightarrow{2}R].$$

Its supported derived restrictions are now computed on the two degeneration faces and their intersection.

Sequence claim: seqclaim-f7fe65b2e7ee66f40bf53740.

Explicit contraction

Over characteristic zero, define

$$h=\frac12.$$

Then

$$dh=1, \qquad hd=1.$$

The contraction contains neither $s$ nor $B-1$. It is therefore preserved by ordinary and derived base change to

$$s=0, \qquad B-1=0, \qquad s=B-1=0.$$

The durable checker verifies the same chain homotopy on all three supported restrictions.

Consequence

The anti-invariant normalization/conductor complex has zero cohomology both generically and on every support face selected by the first smoothing coefficient:

$$\boxed{ H^\bullet_- \left( L i^*[R\xrightarrow{2}R] \right)=0. }$$

Thus neither a face Tor class nor a codimension-two conductor extension survives.

Deutsch--Popperian verdict

The conjecture that the vanishing of $U=-16s(B-1)$ might obstruct the conductor contraction is falsified. The conductor-descent mechanism closes uniformly over both existing supports.

This does not eliminate vanishing-cycle data of the total family

$$XY=ps(B-1).$$

It types that data correctly: any residual belongs to the sector-specific nearby/vanishing-cycle coefficient object, not to normalization descent and not to a new carrier cell.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u2v0_conductor_complex.rs;
• research/benincasa/rank12-u2-v0-joint-newton.json.

Epistemic graph admission: ev-000000000801-9c8d134f-8ad2-4540-b584-8884c01a5b94.

Next falsifier

Construct the iterated nearby-cycle object of the local node $XY=ps(B-1)$, preserving the three labelled factors $p,s,B-1$. Compare its face maps with the already frozen parameter-exceptional, $q=0$, and $L_1$ Gysin maps. A residual mapping-cone class would be coefficient excess; only failure to type those three factors on the existing incidence carrier would challenge H2.

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