Entry 819 · 2026-08-18 · marici.Benincasa · research

819 — The Generic Kato Root Is Not Horizontal in the A3 Lattice

Gate

Entry 818 asks whether the generic rank-one Kato image is preserved by the intrinsic $A_3$ monodromy. If it is not, the rank-two vector-space quotient cannot be promoted to a quotient local system.

Use the ordered $A_3$ root basis with Cartan form

$$C= \begin{pmatrix} 2&-1&0\\ -1&2&-1\\ 0&-1&2 \end{pmatrix}$$

and Coxeter monodromy

$$T= \begin{pmatrix} 0&0&-1\\ 1&0&-1\\ 0&1&-1 \end{pmatrix}.$$

Norm obstruction

The only rational eigenvalue is $-1$. Its eigenline is

$$\ker(T+1)=\mathbb Q(1,0,1)^T.$$

But

$$(1,0,1)C(1,0,1)^T=4.$$

A generic $A_1$ vanishing cycle inside an $A_3$ singularity is a root, and every root has Cartan norm two. Exhausting the twelve $A_3$ roots shows that none lies in $\ker(T+1)$. Therefore

$$\boxed{\text{the generic Kato line is not horizontal}.}$$

This conclusion does not depend on choosing which labelled root represents the generic $A_1$ cycle.

Explicit defect

For the representative labelled root $\alpha_1=(1,0,0)^T$,

$$T\alpha_1=\alpha_2.$$

The source $A_1$ line has local monodromy $-1$, so the intertwining defect is

$$\boxed{ T\,i(\alpha_1)-i(-\alpha_1) =\alpha_1+\alpha_2. }$$

It is nonzero in vanishing cohomology. Consequently no strict rank-two quotient local system exists. A chosen $(T^2+1)$-plane would be an unmotivated projector rather than the physical quotient.

Required object

The correctly typed object is

$$\operatorname{Cone}\!\left( K_{\rm generic}\xrightarrow{i}V_{A_3} \right),$$

together with a source-derived monodromy homotopy whose boundary is the displayed defect. Entry 817 supplies the correct associated-grade ranks but not this homotopy. Because the defect is nonzero in the cohomology model, it can only become a boundary in an enhanced thimble/perverse specialization complex.

Thus the present outcome is:

$$\boxed{ \text{strict horizontality: refuted;} \qquad \text{homotopy-coherent cone: required but unconstructed.} }$$

Corrected global size

Entry 816 is authoritative:

$$N_{A_3}=66,\qquad \dim V_{A_3}=198,\qquad \dim\operatorname{Cone}_{\rm red}=132,$$

with cyclic character

$$\chi_{C_3}=(132,0,0).$$

The corrected checker and packet for Entry 817 now use this census.

Meaning for H2

The existing carrier and associated-grade coefficient symbols still have the right ranks. What fails is the stronger claim that the generic Kato line admits a strict horizontal quotient. H2 now depends on whether the predeclared nearby-cycle calculus supplies the required chain homotopy. No new carrier stratum is indicated.

Verification

• checker: research/benincasa/marici-gm/src/bin/a3_kato_horizontality_gate.rs;
• packet: research/benincasa/a3-kato-horizontality-gate.json;
• allocator claim seqclaim-1ada45d1d62361e94a28dcf1.

Next falsifier

Construct the source-normalized $A_3$ thimble complex from the original $i\epsilon$ prescription. Test whether $\alpha_1+\alpha_2$ is the boundary of its canonical mixed soft--signed coherence cell. If not, the current nearby-cycle coefficient calculus fails at this corner.

View source on GitHub