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IndisputableMonolith.Foundation.SMHyperchargeFromCube

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Packages one left-handed Standard Model generation as Weyl multiplets with hypercharges in units of 1/6, built from the 3-cube gauge completion. Proves sixteen Weyl states per generation, forty-eight for three generations, and vanishing of the mixed SU(3)^2 U(1)_Y anomaly. Higgs hypercharge is fixed in the same normalization. Cited by the unified forcing chain; proofs are enumeration plus direct arithmetic on tabulated charges.

claimOne left-handed SM generation is a finite list of Weyl multiplets with hypercharges $Y$ stored as integer multiples of $1/6$. The generation has $16$ Weyl states; three generations have $48$. The mixed anomaly coefficient $\mathrm{Tr}_{\mathrm{color}}(T^a T^b Y)$ vanishes. The Higgs hypercharge is the standard value in the same $1/6$ units.

background

Recognition Science forces $D=3$ and the eight-tick period from the cost foundation (T7–T8). Upstream, gauge Lie completion from the 3-cube records the forced $B_3$ layer counts: three axis permutations and an even sign-flip completion of size two. Those combinatorial factors are the seed for the gauge data specialized here to hypercharge and chiral fermion content.

A Weyl multiplet is a left-handed chiral multiplet of the SM gauge algebra, carrying an $\mathrm{SU}(3)$ representation, weak isospin, and hypercharge. Hypercharges are integers in units of $1/6$ (so ordinary fractions $Y=n/6$). The module also fixes the Higgs hypercharge in those units and evaluates the cubic mixed color-hypercharge anomaly against zero.

Local setting: foundation layer that turns cube combinatorics into SM-facing quantum numbers before the unified forcing chain assembles T0–T8.

proof idea

Definition-led module with short equality lemmas rather than a long derivation. Multiplet lists, multiplicity, and hypercharge tables (including Higgs) are introduced as data. Generation state counts are sums of multiplicities over that list, reducing to the classical $16$ and $48$ by arithmetic. The mixed $\mathrm{SU}(3)^2\mathrm{U}(1)_Y$ anomaly is an explicit weighted sum of hypercharges times color indices; it evaluates to zero by direct computation on the table. Higgs hypercharge equality is a one-line normalization check against the same $1/6$ convention.

why it matters in Recognition Science

Bridges cube gauge completion to the SM fermion and hypercharge sector that the unified forcing chain imports. Downstream UnifiedForcingChain claims all of T0–T8 are forced inevitabilities from the cost foundation (Recognition Composition Law); this module supplies the chiral multiplet count, $Y$ assignment, and anomaly cancellation that make the $B_3$ story match one left-handed generation. Without vanishing $\mathrm{SU}(3)^2\mathrm{U}(1)_Y$ and a consistent Higgs $Y$, the cube-derived gauge layer would not close on Standard Model quantum numbers. Anchors the foundation side of generation counting ($16$ and $48$) before higher chain steps cite the gauge package.

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