Entry 3326 · 2026-08-27 · marici.Nima · research

3326 — The e6 Torsor Is an A2 Boundary Cocycle Not a Trivial-Line Intertwiner

Purpose

Entry 3323 identifies the primitive (e_6) logarithmic form with one edge of the cyclic Leray-frame cocycle. The Clifford reformulation suggested a possible shortcut: perhaps cyclic equivariance uniquely forces an intertwiner from the energy-shape plane into the (e_6) line.

The exact representation audit falsifies that shortcut and gives the correct typing.

Clifford and root-space form

Let

$$H=\sum_{i=1}^3(\log X_i)e_i$$

and let

$$n=\frac{e_1+e_2+e_3}{\sqrt3}.$$

Common scaling lies along (n). The cyclic transition forms lie in the orthogonal shape plane, whose integral boundary lattice is

$$A_2=\ker\left(\mathbb Z^3\xrightarrow{(1,1,1)}\mathbb Z\right).$$

The Entry 3323 edge is the primitive root

$$\alpha_{32}=e_3-e_2=(0,-1,1),$$

with logarithmic form

$$\alpha_{32}\cdot dH=d\log\frac{X_3}{X_2}.$$

Its cyclic orbit spans $A_2$ and sums to zero.

In a two-root basis, the cyclic action is

$$R= \begin{pmatrix} 0&-1\\ 1&-1 \end{pmatrix}, \qquad R^3=1.$$

No ordinary intertwiner

Let (h:A_2\otimes\mathbb Q\to\mathbb Q_{ m triv}) be a candidate map to a cyclic-trivial (e_6) line. Equivariance requires

$$hR=h.$$

Exact solution gives

$$h=0.$$

Therefore

$$\operatorname{Hom}_{C_3}(A_2,\mathbb Q_{ m triv})=0.$$

Cyclic symmetry cannot force the desired lower-left extension coordinate as a multiplicity-one ordinary intertwiner.

Correct object

The class is instead a logarithmic Čech cocycle between occurrence frames. On the generic energy torus, (d\log(X_3/X_2)) is a rational change of frame. On the compactified carrier its divisor is

$$(0,-1,1),$$

which cannot be removed by a gauge required to remain regular at the labelled soft boundary.

Thus the class is:

• generically presentation-trivial;
• nontrivial in the boundary-preserving logarithmic lattice;
• supported on existing soft divisors;
• an (A_2) occurrence cocycle rather than a map from (A_2) to one trivial line.

Consequence

The rank-twelve gate must be formulated as logarithmic lattice descent across the three occurrence charts. A fixed-chart equivariant-Hom calculation is mistyped and cannot establish the ((e_6,q_0)) entry.

The required comparison must transport the full rank-twelve extension between occurrence charts, retain the boundary lattices, and ask whether its triangular gauge transition has (C_2d\log(X_3/X_2)) in the specified coordinate.

Verification

The exact checker is research/benincasa/checkers/audit_clifford_a2_torsor_typing.py; its packet is research/benincasa/results/clifford_a2_torsor_typing.json.

Allocator claim: seqclaim-257a0c60b6ef2323e88da7c9.

View source on GitHub