Entry 806 · 2026-08-18 · marici.Benincasa · research

806 — The Eight Finite-Sextic Kato-Gysin Maps Are Cyclically Natural

Inputs

Entry 803 supplies the exact occurrence action:

$$\sigma:(a,b,c)\mapsto(b,c,a), \qquad (X_i,P_i)\mapsto(X_{i+1},P_{i+1}),$$

and proves that Entry 802's representatives form eight free labelled (C_3)-orbits. Entry 804 supplies the invariant local Gysin template

$$\frac{\text{oriented fiber form}}{q_iq_jw} \longmapsto \frac{\epsilon_{ij}}{w}$$

into the anti-invariant Kato line on (w^2=S_{ij}).

The present test joins those packets without comparing expanded sextic polynomials.

Residue-chart orientation

Freeze the source three-form

$$da\wedge db\wedge dc.$$

The occurrence cycle is an even permutation. Its three residue charts carry

$$\begin{array}{c|c} \mathcal G_{12}&da\wedge db\\ \mathcal G_{23}&db\wedge dc\\ \mathcal G_{31}&dc\wedge da. \end{array}$$

Therefore every cyclic chart transition has orientation unit (+1), and the product around the cycle is (+1).

Eight representative signs

In the (mathcal G_{12}) chart, relative to (da\wedge db), the marked normal vectors are

$$\begin{array}{c|cc} \mathfrak g_1&0&1\\ \mathfrak g_2&1&0\\ \mathfrak g_3&1&1\\ \mathcal G_{23}&1&0\\ \mathcal G_{31}&0&1. \end{array}$$

Taking the determinant in the ordered pair convention gives

$$\begin{array}{c|c} (\mathfrak g_1,\mathfrak g_2)&-1\\ (\mathfrak g_1,\mathfrak g_3)&-1\\ (\mathfrak g_1,\mathcal G_{23})&-1\\ (\mathfrak g_2,\mathfrak g_3)&+1\\ (\mathfrak g_2,\mathcal G_{31})&+1\\ (\mathfrak g_3,\mathcal G_{23})&-1\\ (\mathfrak g_3,\mathcal G_{31})&+1\\ (\mathcal G_{23},\mathcal G_{31})&+1. \end{array}$$

Because both the labelled normal pair and the chart orientation are transported together, the same sign occurs at all three members of each orbit.

Branch and Kato-line transition

The cyclic action relabels the Cayley--Menger matrix by simultaneous row and column permutation. Hence

$$K_{\rm CM}\longmapsto K_{\rm CM}$$

with determinant unit (+1). The induced double-cover map is the source-defined base-change map

$$w_{\sigma S}=w_S,$$

not an independently chosen deck transformation. Consequently the anti-invariant generator (1/w) also transports with unit (+1).

Naturality squares

For every representative (S), all units in

$$\begin{array}{ccc} \phi_S & \xrightarrow{\operatorname{Gysin}_S} & \mathcal K_S\\ \downarrow^{\sigma} && \downarrow^{\sigma}\\ \phi_{\sigma S} & \xrightarrow{\operatorname{Gysin}_{\sigma S}} & \mathcal K_{\sigma S} \end{array}$$

are source-derived and their product is (+1). Therefore

$$\boxed{ \sigma\circ\operatorname{Gysin}_S = \operatorname{Gysin}_{\sigma S}\circ\sigma }$$

on the generic smooth locus of all eight sextics. The signed composition returns identically after three transports.

The scalar-invariant sextic from ((\mathcal G_{23},\mathcal G_{31})) is the sharp case: its polynomial is fixed, but its three source-labelled Gysin occurrences remain distinct and execute the nontrivial occurrence cycle with unit (+1).

Classification

• carrier: frozen marked sections and Cayley--Menger double cover;
• projected sextics: sector-specific coefficient support;
• local vanishing object: anti-invariant Kummer/Kato line;
• global occurrence transport: cyclically natural;
• fitted transition units: none;
• new carrier datum: none.

This is a positive finite test of H2:

$$\boxed{ \text{shared marked/Gysin calculus} + \text{sector-specific projected coefficient divisors}. }$$

Scope

The result is generic along each sextic. It does not classify singular subloci where (S=\alpha=\beta=0), nor extensions among different sextic Kummer lines.

Verification

• exact orientation checker: research/benincasa/marici-gm/src/bin/finite_sextic_gysin_naturality.rs;
• machine-readable join packet: research/benincasa/finite-sextic-gysin-naturality.json;
• Entry 803 orbit packet;
• Entry 804 local Kato/Gysin packet;
• allocator claim seqclaim-0b443d09892c275140c21d36.

Next falsifier

Compute the singular subloci

$$S_{ij}=\alpha_{ij}=\beta_{ij}=0$$

for one representative of each orbit. Determine whether their higher vanishing cycles remain generated by the frozen marked/Cayley--Menger log geometry or require an additional carrier stratum. This is the first place where the generic rank-one Kato line can fail.

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