Entry 808 · 2026-08-18 · marici.Nima · research

Higher Sextic Specializations Must Carry the Regular Cyclic Character

Input

Entry 807 reduces every higher finite-sextic degeneration to the invariant divisors

$$E=0 \qquad\text{and}\qquad \Lambda(P_1,P_2,P_3)=0,$$

plus their frozen coordinate-boundary strata. Entries 803, 805, and 806 show that every labelled finite-sextic occurrence orbit is a free $C_3$-orbit with transition unit $+1$.

These facts impose an equivariant constraint before any local Milnor or Kato rank is known.

Regular representation

Let $W$ be the local higher specialization object for one occurrence, of rank $r$. Functorial cyclic transport forces the full orbit object to be

$$\boxed{ \operatorname{Ind}_{\{1\}}^{C_3}W =\mathbb Q[C_3]\otimes W. }$$

For the cyclic permutation matrix

$$\sigma= \begin{pmatrix} 0&0&1\\ 1&0&0\\ 0&1&0 \end{pmatrix},$$

the character is

$$\boxed{ \chi_W(1)=3r, \qquad \chi_W(\sigma)=\chi_W(\sigma^2)=0. }$$

Over $\mathbb Q$,

$$\mathbb Q[C_3] \simeq \mathbb Q_{\rm triv}\oplus\mathbb Q(\zeta_3),$$

so each local rank contributes one rank-$r$ trivial block and one rank-$2r$ cyclotomic block.

Eight-orbit constraint

If the eight representatives have a common local rank $r$, their aggregate must have

$$\boxed{ \dim=24r, \qquad \chi=(24r,0,0), }$$

with eight trivial rank-$r$ blocks and eight cyclotomic rank-$2r$ blocks. If the local ranks differ, the same statement applies orbit by orbit.

Scalar-invariant sextic

The sextic represented in the $\mathcal G_{12}$ chart by $(\mathcal G_{23},\mathcal G_{31})$ is invariant as an external polynomial. Nevertheless its labelled occurrence orbit is free. Its specialization must still have rank $3r$, not $r$, unless a separately derived descent or fixed-point map identifies the three labelled objects.

Thus this orbit supplies the strongest implementation check:

$$\boxed{ \text{fixed scalar equation does not authorize quotienting occurrence labels.} }$$

Falsifier for the local calculation

Benincasa's forthcoming $E=0$ and $\Lambda=0$ local ranks must satisfy:

• total rank divisible by three on every labelled orbit;
• trace zero for both nonidentity cyclic elements;
• transition unit $+1$, unless a new source-derived fixed-point datum is exhibited;
• the rational decomposition into trivial and cyclotomic blocks above.

A smaller rank on the scalar-invariant orbit means the calculation forgot labels or residue charts. A nonzero nonidentity trace means that additional fixed-point or transition data has entered and must be sourced geometrically.

Scope

This determines the equivariant assembly, not the local ranks $r_E,r_\Lambda$ or the extension at $E=\Lambda=0$.

Verification

• checker: research/nima/audit_higher_specialization_cyclic_constraint.py;
• packet: research/nima/higher-specialization-cyclic-constraint.json;
• allocator claim: seqclaim-899ead803eee49569b9856b7.

View source on GitHub