V22·Act V · Matter

The label lattice and the descent condition

The label lattice as a 36-cell grid. Dial triality, duality and charge and watch the congruence pass or fail live.

The relation on screen

\frac{S U ( 3 ) \times S U ( 2 ) \times U ( 1 )}{Z _{6}}, e^{2 π i (\frac{t _{3}}{3} + \frac{d}{2} + Y)} = 1

Admissible labels are those whose combined phase is trivial on the Z6 centre, which quantises hypercharge.

Symbols

$t_{3}$: triality of the SU(3) rep
$d$: duality of the SU(2) rep
$Y$: hypercharge

loading label lattice

what to look for · The condition is arithmetic. Every admissible row satisfies it exactly, and non-descending rows are absent.

Consensus to physics

01a defect label has three coordinates: triality t (mod 3), duality d (mod 2), charge q
02the diagonal Z6 acts on the label lattice
03a label is well defined on the quotient exactly when 2t + 3d + q = 0 (mod 6)
04that congruence filters which labels can exist

In standard physics

(SU(3)xSU(2)xU(1))/Z6

The Standard Model gauge group is the quotient of the bare product by Z6, and that Z6 acts trivially on every observed particle. Triality is the Z3 centre of SU(3), duality the Z2 centre of SU(2), and q = 6Y the hypercharge in integer units.

boundary · The corpus does not derive why the carrier has three colour modes and two weak modes. That split is an input here.

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