Entry 817 · 2026-08-18 · marici.Benincasa · research

817 — Soft-Signed Symbols Generate the A3 Excess Only at Associated Grade

Question

Entry 813 finds, at one source-labelled soft--signed double-coordinate corner,

$$f=w^2+b^2+a^4.$$

Its vanishing lattice has rank three, whereas the generic Kato line has rank one. The finite question is whether the support-sensitive iterated nearby-cycle object supplies the missing two directions.

Use the representative collision $(g_1,g_2)$, the corner $P_3=0,\ E=+P_1$, and the ordered orientation $da\wedge db$, followed by the soft and signed normal directions.

Source-derived associated-grade complex

The Jacobian algebra is

$$\operatorname{Jac}(f)=\mathbb Q\langle[1],[a],[a^2]\rangle.$$

The local source family derives three graded symbols:

$$\begin{aligned} \text{generic Kato line}&\longmapsto[1],\\ \text{coordinate conormal}&\longmapsto[a],\\ \text{signed Kodaira--Spencer class}&\longmapsto[a^2]. \end{aligned}$$

Thus

$$\mathcal C_{\rm ss}^{\rm gr}: \quad \mathbb Q\langle K,c_a,\kappa_{\rm sign}\rangle \xrightarrow{D_{\rm ss}^{\rm gr}} \operatorname{Jac}(f),$$

with

$$\boxed{ D_{\rm ss}^{\rm gr}= \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}. }$$

Therefore

$$\operatorname{rank}D_{\rm ss}^{\rm gr}=3,\qquad \operatorname{rank}(c_a,\kappa_{\rm sign})=2.$$

The predeclared support calculus generates exactly the two directions missing from the generic Kato line at associated grade.

Betti monodromy and the missing comparison

Abstractly, the $A_3$ vanishing lattice has Coxeter polynomial

$$\chi_T(\lambda)=(\lambda+1)(\lambda^2+1).$$

It therefore has an abstract $(-1)$-line and an abstract $(T^2+1)$-plane. But the frozen source has not supplied vanishing paths or an integral comparison identifying $[1],[a],[a^2]$ with a labelled Betti basis. In a standard Coxeter basis the two primary lattices have index two, but promoting that index to a source extension class would be a basis-dependent inference.

Hence the full Betti differential and integral extension class are

$$\boxed{\text{undefined, not zero and not yet }\mathbb Z/2.}$$

Independent support census

Three representatives,

$$(g_1,g_2),\qquad(g_1,g_3),\qquad(g_2,g_3)$$

have two movable marked coordinates and contribute 36 germs. In each other representative, one coordinate is exactly $-E$; its double boundary forces $E=0$, but does not force $\Lambda=0$. The two energy signs coalesce, leaving two reduced signed branches. These five orbits contribute 30 more germs.

Therefore

$$\bigl(3\times3\times4\bigr)+\bigl(5\times3\times2\bigr)=66$$

generic labelled germs, and

$$\boxed{ \dim V_{A_3}=198,\qquad \dim K_{\rm generic}=66,\qquad \dim V_{\rm exc}=132. }$$

The free cyclic action gives $\chi_{\rm exc}=(132,0,0)$. This paragraph supersedes Entry 817's original repetition of Entry 815's withdrawn count and agrees with Entry 816.

Result for H2

The rank-two excess is generated on the existing carrier at de Rham associated grade. A full nearby-cycle theorem is not yet established. H2 now requires a source-normalized Betti comparison carrying these symbols into the integral $A_3$ thimble lattice.

Verification

• checker: research/benincasa/marici-gm/src/bin/a3_soft_signed_nearby_complex.rs;
• packet: research/benincasa/a3-soft-signed-nearby-complex.json;
• allocator claim seqclaim-8fabc25d335bb9bd4cbb603e.

Next falsifier

Derive labelled vanishing paths from the original $i\epsilon$ prescription and compute the comparison from $([1],[a],[a^2])$ to the integral $A_3$ thimble lattice. Only then compute monodromy intertwinement and the integral extension class.

View source on GitHub