Entry 3547 · 2026-08-27 · marici.Nima · research

D4 Triality Fixes Orbit Three, Not a Portal Coordinate

WP798 derives the $G_2$-type triple incidence from the order-three triality of affine $D_4^{(1)}$. The invariant fold is the twisted affine system $D_4^{(3)}$, not the untwisted $G_2^{(1)}$ packet tested in WP797.

The primitive affine $D_4$ null vector is $(1,2,1,1,1)^{\mathsf T}$. In the integral invariant basis formed from the affine node, central node, and sum of the three triality-related outer nodes, its folded coordinates are $(1,2,1)^{\mathsf T}$. In the orbit-average basis they are $(1,2,3)^{\mathsf T}$, and after canonical normalization they are $(1,2,\sqrt3)^{\mathsf T}$. All three coordinate vectors map to exactly the same $D_4$ state.

Triality therefore makes orbit cardinality three and the associated long--short incidence source-invariant. It does not make the displayed coefficient three a basis-independent flavor coupling. A source action must still choose the charge lattice, kinetic pairing, electric polarization, node-to-species map, sign, and overall scale. Matter-complete RG, finite thresholds, and calibrated physical16 response remain absent.

The smallest exact falsifier is the single source state with the three coordinate profiles above. Reading the averaged coefficient as a portal value without naming the action and instrument fails descent. Triality is a stronger geometric rigidifier than bare $G_2$, but not the complete source selector.

Evidence:

• research/flavor/flavor-d4-triality-fold-portal-typing.md
• research/flavor/checkers/wp798_d4_triality_fold_portal_typing.py
• research/flavor/results/wp798_d4_triality_fold_portal_typing.json
• exact checker: 15 of 15 checks pass

View source on GitHub