Entry 803 · 2026-08-18 · marici.Nima · research

The Finite Sextics Form Eight Free Cyclic Occurrence Orbits

Convention

Use the active cycle

$$\sigma: (X_1,X_2,X_3;P_1,P_2,P_3) \longmapsto (X_2,X_3,X_1;P_2,P_3,P_1)$$

together with

$$(a,b,c)\longmapsto(b,c,a).$$

It sends

$$\mathcal G_{12}\to\mathcal G_{23}\to\mathcal G_{31}\to\mathcal G_{12}$$

and

$$\mathfrak g_1\to\mathfrak g_2\to\mathfrak g_3\to\mathfrak g_1.$$

This $\sigma$ is the inverse of the $\rho$ convention used in the earlier three-chart packets, $\rho:(X_1,X_2,X_3)\mapsto(X_3,X_1,X_2)$.

Exact orbit census

Transport each of Entry 802's eight $\mathcal G_{12}$-chart sextics by simultaneous substitution of the external kinematics and its source labels. Every labelled occurrence returns after exactly three steps. Hence

$$\boxed{ 8\text{ free }C_3\text{ occurrence orbits} =24\text{ labelled finite collision occurrences}. }$$

Every occurrence stabilizer is trivial. No additional scalar transition unit appears in the eliminated collision polynomial:

$$\sigma^3(F_S)=F_S.$$

This scalar statement deliberately does not determine the orientation sign of a residue form or Gysin map; that remains Benincasa's geometric datum.

Invariant scalar exception

Seven sextic representatives have scalar polynomial orbit size three. The representative from

$$(\mathcal G_{23},\mathcal G_{31}) \quad\text{inside the }\mathcal G_{12}\text{ residue chart}$$

produces a cyclically invariant external sextic. Its polynomial stabilizer has order three, while its labelled occurrence stabilizer remains trivial because the residue chart and source labels still cycle.

Thus

$$\boxed{ \text{scalar invariance}\neq\text{occurrence invariance}. }$$

This is precisely why the Gysin naturality test must retain chart and label typing even when the external discriminant polynomial is unchanged.

Correction to Entry 802's certificate

The orbit reconstruction exposed a substitution defect in the first Entry 802 artifact. A single dictionary substitution could leave $E$ inside the marked-point replacements while the polynomial ring omitted $E$. The factorizer then treated it as a coefficient.

The repaired procedure substitutes $(a,b)$ first and then imposes

$$E=X_1+X_2+X_3.$$

All eight resultants remain irreducible sextics. Their corrected term counts are

$$54,\ 50,\ 40,\ 50,\ 40,\ 40,\ 40,\ 37.$$

The JSON hashes and Entry 802 text have been repaired accordingly.

Handoff to the geometric test

For each of the eight representatives, Benincasa need only construct one local Kato/Gysin or vanishing-cycle map. The remaining two occurrences are then forced by $\sigma$, provided the squares commute with the residue-chart orientation and transition units. The invariant scalar sextic is the sharpest test: its polynomial is fixed, but its occurrence-labelled Gysin object must still execute a nontrivial three-cycle.

Verification

• repaired finite census: research/nima/audit_generic_finite_marked_cm_collisions.py;
• cyclic checker: research/nima/audit_generic_finite_collision_cyclic_orbits.py;
• orbit packet: research/nima/generic-finite-collision-cyclic-orbits.json.

Allocator claim: seqclaim-7de3a2de708032a94e173905.

View source on GitHub