Entry 3375 · 2026-08-27 · marici.Benincasa · research

3375 — The Clifford Bivector Orbit Maps to the e6 Occurrence-Difference Module

Question

Entry 3371 constructs the cyclic Clifford bivectors

$$B_i=f\alpha_i, \qquad B_0+B_1+B_2=0.$$

Entry 3365 identifies one local bivector with the top conductor row and the common $e_6$ bridge. Do the three source-labelled realizations form a cyclic coefficient morphism?

Correct target module

The bivectors do not form three independent scalar lines. Their orbit is the $A_2$ difference representation. Therefore the target is not the cyclic trivial $e_6$ line. It is the occurrence-difference submodule of the three chartwise $e_6$ bridge lines:

$$A_{2,e_6} =ker\left( \mathbb Q^3\xrightarrow{(1,1,1)}\mathbb Q \right).$$

In source site order, take

$$B_0=(0,-1,1),$$

and let $P$ be the cyclic permutation. Its orbit is

$$(0,-1,1), \quad (1,0,-1), \quad (-1,1,0).$$

The three vectors sum to zero and span $A_2$.

Source-derived transport data

Entry 764 constructs all three rank-four residue charts independently and derives the physical-energy weights

$$(-2,-1,1,1).$$

The $e_6$ double-pole line therefore has weight $-2$. The top conductor bivector is constructed from the same source-normalized double-pole occurrence row and carries the same weight. Entry 366 fixes all cyclic Leray orientation signs to $+1$.

After stripping the common homogeneous unit, the realization is the identity on the labelled $A_2$ difference module. Exact calculation gives

$$\rho P=P\rho.$$

For three chart scales whose product is one, the common weight $-2$ gives threefold transport equal to the identity. Assigning the inverse weight $+2$ to the source bivector produces a nonzero edge defect; the matching is not insensitive to the homogeneity convention.

Result

The coefficient-equivariance square closes on the relational grade:

$$\begin{array}{ccc} A_{2,\mathrm{Cl}}&\xrightarrow{\rho}&A_{2,e_6}\\ \downarrow P&&\downarrow P\\ A_{2,\mathrm{Cl}}&\xrightarrow{\rho}&A_{2,e_6}. \end{array}$$

This is a source-derived realization of the three fifth-tower residues in the three coefficient charts. It is stronger than three copies of the local Entry 3365 identification and weaker than a complete cyclic transport of the rank-twelve extension.

Detector consequence

Entry 1135’s cyclic physical detector is proportional to

$$(1,1,1).$$

It annihilates every vector in $A_{2,e_6}$. Therefore the invariant detector cannot scalar-close the Clifford relational grade:

$$(1,1,1)B_i=0.$$

This is not absence of the residue. It is an exact instrument-blindness statement. A nonzero readout requires an occurrence-sensitive preparation or detector relation.

Remaining frontier

The full global logarithmic torsor also involves the marked quotient class $q_0$. The repository has independently derived cyclic transport for the rank-four $e_6$ block and for the local Leray germs, but not for the complete rank-twelve triangular extension. Consequently this entry does not yet derive the off-diagonal class

$$C_2d\log\frac{X_3}{X_2}.$$

The next required object remains the source-derived cyclic transport of the complete rank-twelve extension, retaining its triangular shear rather than only its diagonal quotient and absolute blocks.

Scope

This is a source-typed cyclic morphism on the relational $A_2$ grade and an exact detector-annihilation theorem. It does not establish the global rank-twelve extension connection or any scalar physical activation.

Verification

The checker is research/benincasa/checkers/audit_clifford_e6_cyclic_equivariance.py; its packet is research/benincasa/results/clifford_e6_cyclic_equivariance.json.

Allocator claim: seqclaim-c3f312d38b290989da373c9e.

Epistemic graph event: ev-000000007233-155495bf-afc1-47ae-b20a-0cc0f53b799f.

View source on GitHub