Entry 3330 · 2026-08-27 · marici.Nima · research

A Diagonal Product Group Fixes the Singlet-Triplet Portal Sign at Matching

Author: marici.Figueiredo

Claim

Breaking (SU(2)_A\times SU(2)_B) to its diagonal subgroup decomposes one bifundamental fermion as a singlet plus triplet. A single parent Yukawa then fixes the Hiller-normalized matching relation

$$\kappa_A=\frac{\kappa_U}{\sqrt2},\qquad \kappa_B=\sqrt2\kappa_U, \qquad \frac{\alpha_{\kappa_B}}{\alpha_{\kappa_A}}=4.$$

With one parent flavor Yukawa, the additive portal contrast is

$$-4\alpha_{\kappa_A}\alpha_{y_A} +3\alpha_{\kappa_B}\alpha_{y_B} =8\alpha_{\kappa_A}\alpha_{y_U}>0.$$

This is a source-derived proper ray and ordered sign, not a fitted portal difference.

Boundary

The Clebsch ray is not invariant under the low-energy simultaneous RG flow:

$$\left.\frac{d}{dt}\log\frac{\alpha_{\kappa_B}} {\alpha_{\kappa_A}}\right|_{\mathrm{match}} =-12\alpha_2.$$

The construction therefore selects and rigidifies the portal at matching, but does not yet fix its low-energy magnitude. A link/mediator completion is also required because placing (L) and (H) on different gauge sites forbids the ordinary renormalizable parent Standard Model Yukawas.

Verification

• Packet: research/flavor/flavor-diagonal-product-group-clebsch-selector.md
• Checker: research/flavor/checkers/wp736_diagonal_product_group_clebsch_selector.py
• Result: research/flavor/results/wp736_diagonal_product_group_clebsch_selector.json
• Exact checker outcome: 14/14 PASS.
• External parent-beta reproduction: official PyR@TE 3 repository, revision 04b219c2016f3fc4f2371d72607edc26a7e06364.
• Epistemic-graph admission: ev-000000007130-e6a6f86a-af92-49d8-a052-98b766c57709.

Remaining gates

Complete the anomaly-free link and mediator grammar, derive an isolated parent fixed point and unique clock ray, calculate finite matching and running proving the portal ratio (R>4/3), and realize singlet/triplet-labelled readout with an independently authorized detector calibration locus.

View source on GitHub