generationWeylStateCount
plain-language theorem explainer
One left-handed Standard Model generation carries sixteen Weyl fermion states once color and weak components are counted. The total is the sum of six multiplet multiplicities: quark doublet 6, up and down conjugates 3 each, lepton doublet 2, electron and sterile-neutrino conjugates 1 each. Anyone matching the cube-completion hypercharge layer to SM fermion content, or checking the three-generation count against |B3|=48, cites this. It is a plain arithmetic definition over the multiplet table.
Claim. The Weyl-state count of one left-handed Standard Model generation is $N_{\mathrm{gen}} = 6 + 3 + 3 + 2 + 1 + 1 = 16$, namely the sum of the multiplicities of the quark doublet $Q_L$, the up-type conjugate $u^c_L$, the down-type conjugate $d^c_L$, the lepton doublet $L_L$, the electron conjugate $e^c_L$, and the sterile neutrino conjugate $\nu^c_L$.
background
This module continues the cube-completion program after the compact gauge skeleton $SU(3)\times SU(2)\times U(1)$ with recognition-axis counts $(3,2,1)$ and carrier counts $(8,3,1)$. The question is whether SM fermion multiplets and hypercharges fit the same units. Hypercharges are written in the canonical sixths $Y_6=6Y$.
For one left-handed generation (including the sterile right-handed neutrino as the $Y=0$ completion) the multiplet table is: $Q_L$ multiplicity 6 with $Y_6=1$; $u^c_L$ mult. 3, $Y_6=-4$; $d^c_L$ mult. 3, $Y_6=2$; $L_L$ mult. 2, $Y_6=-3$; $e^c_L$ mult. 1, $Y_6=6$; $\nu^c_L$ mult. 1, $Y_6=0$.
The upstream multiplicity map assigns those six sizes (including color and weak components). Weak-sector doublets supply the base factors 2 for quarks and leptons; color triples the quark entries to 6 and 3.
proof idea
Pure definition: sum the six values of the multiplet multiplicity map on the quark doublet, up conjugate, down conjugate, lepton doublet, electron conjugate, and neutrino conjugate. No tactics, no lemmas beyond evaluating that map (6+3+3+2+1+1).
why it matters
This is the integer backbone of the SM hypercharge layer in cube units. Downstream, a native-decide theorem pins the sum at 16; three generations are defined as three times this count and matched to $|B_3|=48$ (the signed-permutation order on the 3-cube). The certificate structure packages the one-generation 16-count together with six multiplets, the three-generation $|B_3|$ match, and vanishing of the $SU(3)^2U(1)$ anomaly in sixths.
The module is explicit that this is not a uniqueness proof for the hypercharges: it is the exact anomaly-free SM layer written in the cube's $1/6$ unit, with integer cancellation of $SU(3)^2U(1)$, $SU(2)^2U(1)$, gravitational-$U(1)$, and $U(1)^3$ sums. It sits in the foundation chain after gauge-Lie completion from the cube (P0-S2-01).
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