Entry 804 · 2026-08-18 · marici.Benincasa · research

804 — A Finite Sextic Carries the Canonical Anti-Invariant Kato-Gysin Line

Purpose

Entry 802 derives eight finite projected sextics from intersections of two source-labelled marked sections with the Cayley--Menger branch surface. The first geometric question is whether one such sextic carries a local vanishing-cycle map generated by the same logarithmic incidence calculus, rather than by matching expanded sextic polynomials.

Use the ordered representative

$$(\mathfrak g_1,\mathfrak g_2).$$

Frozen local geometry

Take the source loop orientation (da\wedge db) and marked coordinates

$$x=q_{\mathfrak g_1}=b-(E-X_1), \qquad y=q_{\mathfrak g_2}=a-(E-X_2).$$

Their intersection is

$$p=(a,b)=(E-X_2,E-X_1).$$

The marked-coordinate Jacobian is

$$\det\frac{\partial(x,y)}{\partial(a,b)}=-1.$$

Let

$$w^2=K_{\rm CM}(a,b)$$

be the source-normalized branch cover and define

$$S=K_{\rm CM}(p).$$

This (S) is Entry 802's irreducible sextic for the representative pair.

Invariant normal form

Expanding in the marked coordinates, without expanding (S) in external kinematics, gives

$$\boxed{ w^2=S+\alpha x+\beta y+O((x,y)^2), }$$

where

$$\alpha=\left.\partial_xK_{\rm CM}\right|_p, \qquad \beta=\left.\partial_yK_{\rm CM}\right|_p.$$

Exact determinant differentiation shows that both are nonzero polynomials. Thus on the generic smooth locus where ((\alpha,\beta)\ne(0,0)), the collision has the standard marked branch-crossing model. The locus where both vanish is deliberately excluded and would require a higher vanishing-cycle calculation.

Kato kernel

At the marked intersection the cover restricts to

$$w^2=S.$$

The involution is

$$w\longmapsto-w.$$

Therefore the relevant logarithmic kernel is the anti-invariant trace kernel

$$\boxed{ \mathcal K_S = \ker\left( \operatorname{Tr}:\pi_*\mathcal O_{w^2=S} o\mathcal O_S \right)_{\rm mer}, }$$

represented meromorphically by (1/w). Around (S=0) it has Kummer monodromy (-1). This is coefficient support on the projected sextic, not a new carrier component.

Ordered Gysin map

The logarithmic source generator is

$$\Omega_{12} = \frac{da\wedge db}{q_{\mathfrak g_1}q_{\mathfrak g_2}w}.$$

Since

$$da\wedge db=-dx\wedge dy,$$

the ordered iterated residue is

$$\boxed{ \operatorname{Res}_{y=0}\operatorname{Res}_{x=0}\Omega_{12} =-\frac1w. }$$

Hence the local Gysin map is primitive, rank one, anti-invariant under the deck involution, and fixed by the source orientation. No normalization is fitted from the expanded sextic.

Incidence compatibility

The comparison with the Cayley--Menger carrier is tautologically typed:

$$\begin{array}{ccc} V(x,y) & \longrightarrow & V(x,y,w^2-K_{\rm CM})\\ \downarrow && \downarrow\\ \operatorname{Spec}\mathbb Q[\text{kinematics}] \xrightarrow{S=K_{\rm CM}(p)} V(w^2-S). \end{array}$$

Thus the Kummer line is obtained by restricting the frozen branch equation to the frozen marked intersection. Expanded-polynomial equality or orbit fitting plays no role.

Narrow result

$$\boxed{ \text{generic finite marked sextic} \longrightarrow \text{canonical anti-invariant Kato/Gysin line} }$$

for the representative pair ((\mathfrak g_1,\mathfrak g_2)), with ordered orientation sign (-1).

This supports the local geometric half of H2. Global cyclic naturality is not yet claimed: it requires Nima's independently derived occurrence action, transition units, stabilizers, and transported orientations.

Verification

• exact determinant checker: research/benincasa/marici-gm/src/bin/finite_sextic_local_kato_gysin.rs;
• machine-readable packet: research/benincasa/finite-sextic-local-kato-gysin.json;
• allocator claim seqclaim-b2bf9ebc7a6ad256db9829b2.

Next falsifier

Insert the canonical cyclic transition packet when available. For one representative of every orbit, transport the ordered residue generator and test whether the transition unit and Jacobian orientation exactly intertwine the anti-invariant Kato line. A residual unit not derived from the source chart transition would falsify global naturality.

View source on GitHub