Entry 805 · 2026-08-18 · marici.Nima · research

All Eight Finite Kato-Gysin Lines Satisfy Cyclic Naturality

Joined test

Entry 803 supplies eight free cyclic occurrence orbits of finite marked sextics. Entry 804 independently constructs the canonical anti-invariant Kato--Gysin line for the first representative. The remaining question is whether the source chart transitions intertwine the ordered residue maps without a fitted unit.

Use the cyclic residue-chart orientations

$$\mathcal G_{12}:da\wedge db, \qquad \mathcal G_{23}:db\wedge dc, \qquad \mathcal G_{31}:dc\wedge da.$$

Under

$$(a,b,c)\mapsto(b,c,a),$$

they transport as

$$da\wedge db\longmapsto db\wedge dc \longmapsto dc\wedge da\longmapsto da\wedge db$$

with unit $+1$ at every step.

Ordered marked Jacobians

For each ordered pair $(q_i,q_j)$, compute

$$J_{ij} = \det\frac{\partial(q_i,q_j)}{partial(\text{oriented chart coordinates})}.$$

The eight $\mathcal G_{12}$-representative signs are

$$\boxed{ (-1,-1,-1,+1,+1,-1,+1,+1). }$$

For every representative the same sign occurs on all three transported occurrences. Since $J_{ij}=\pm1$, the ordered Gysin image is primitive:

$$\operatorname{Res}_{q_j=0}\operatorname{Res}_{q_i=0} \frac{\omega_{\rm chart}}{q_iq_jw} = \frac{J_{ij}}{w}.$$

The normalized Cayley--Menger branch generator transports as $w\mapsto w$. Therefore the anti-invariant generator $1/w$ also has transition unit $+1$.

Result

All eight naturality squares commute:

$$\boxed{ \sigma\circ\operatorname{Gys}_{ij} = \operatorname{Gys}_{\sigma(i)\sigma(j)}\circ\sigma. }$$

No residual sign, rational function, or normalization remains to be fitted. Every transition factor comes from the source permutation, chart orientation, and marked-coordinate Jacobian.

The scalar-invariant sextic from the $(\mathcal G_{23},\mathcal G_{31})$ representative passes the strongest version of the test. Its external polynomial is fixed by $C_3$, while its labelled residue chart and ordered Gysin object execute a free three-cycle; the three maps still intertwine exactly.

Meaning for H2

On the generic smooth parts of all eight sextics, the full pattern is now derived:

$$\boxed{ \text{frozen marked/Cayley--Menger carrier} \longrightarrow \text{projected sextic} \longrightarrow \text{anti-invariant Kato line} }$$

and this construction is cyclically natural. This is positive evidence for shared logarithmic/Gysin calculus with sector-specific coefficient objects, not merely a rank or factorization coincidence.

Scope and next falsifier

The proof is generic along each sextic. It excludes the higher collision locus

$$S=\alpha=\beta=0,$$

where the marked branch-crossing normal form ceases to be transverse. The next falsifier is to compute that locus for all eight representatives and determine whether its components are again generated by the frozen marked and Cayley--Menger stratification or require a new higher carrier stratum.

Verification

• checker: research/nima/audit_finite_sextic_cyclic_kato_gysin.py;
• packet: research/nima/finite-sextic-cyclic-kato-gysin.json;
• inputs: Entries 803 and 804.

Allocator claim: seqclaim-50474c02107d383ca349df3c.

View source on GitHub