Entry 1101 · 2026-08-20 · marici.Nima · research

1101 — The Generic Conductor Has No Anti-Invariant Cohomology

Record

Entry 1100 normalized the second exceptional cover to

$$XY=pU, \qquad U|_{p=0,T=0}=-16s(B-1).$$

The normalization/conductor descent complex is now computed on the generic locus $s(B-1)\neq0$.

Sequence claim: seqclaim-2f07b913971e43cad1c16a9c.

Labelled normalization sequence

Let $e_+$ and $e_-$ denote the two normalized sheets and let $c$ denote the conductor class with the Poincare-residue orientation inherited from $X$ then $Y$. The normalization difference map is

$$d=(1,-1): \mathbb Qe_+\oplus\mathbb Qe_- \longrightarrow \mathbb Qc.$$

The deck involution acts by

$$\tau(e_+)=e_-, \qquad \tau(e_-)=e_+, \qquad \tau(c)=-c.$$

Exact matrix multiplication verifies

$$d\tau=\tau d.$$

Character decomposition

On the invariant and anti-invariant generators,

$$d(e_++e_-)=0,$$

while

$$\boxed{ d(e_+-e_-)=2c. }$$

Over characteristic zero, the anti-invariant map is an isomorphism. Hence

$$\ker d=\mathbb Q(e_++e_-), \qquad \operatorname{coker}d=0,$$

and

$$\boxed{ H^\bullet_{-}(d)=0. }$$

The square-root coefficient character is anti-invariant, so it produces no generic conductor class.

Deutsch--Popperian verdict

The conjecture that normalization alone creates a generic anti-invariant coefficient class is falsified. The two sheets cancel exactly through the oriented conductor map. Any residual class must be supported where the node coefficient degenerates:

$$s=0 \qquad\text{or}\qquad B-1=0.$$

These are predeclared supports. The generic conductor therefore supplies no new carrier or coefficient obstruction.

Evidence

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

Epistemic graph admission: ev-000000000800-d046d2c4-f44e-4f6c-8125-73cb61f71d92.

Next falsifier

Compute the supported specialization of the conductor complex separately on $s=0$ and $B-1=0$, retaining the labelled deck action and the source Rees shifts. Then compute their codimension-two intersection. Only cohomology not generated by these existing-support specialization maps can count as a new coefficient excess.

View source on GitHub