su3SquaredU1Anomaly6
plain-language theorem explainer
Integer coefficient of the SU(3)^2 U(1) mixed gauge anomaly for one left-handed SM generation, written in sixth-units of hypercharge as 2 Y_Q + Y_{u^c} + Y_{d^c}. Cited by anyone checking anomaly freedom of the cube-completion hypercharge layer. Body is the direct linear combination of the three quark multiplet sixth-charges.
Claim. Define the integer $\mathrm{SU}(3)^2\mathrm{U}(1)$ anomaly coefficient in sixth-units by $2 Y_{Q,6} + Y_{u^c,6} + Y_{d^c,6}$, where $Y_{\bullet,6}=6Y$ and the three values are the canonical SM assignments $Y_{Q,6}=1$, $Y_{u^c,6}=-4$, $Y_{d^c,6}=2$.
background
The module continues the cube-completion gauge skeleton (compact factors SU(3)×SU(2)×U(1) with recognition-axis counts (3,2,1)) by asking whether SM fermion multiplets and hypercharges fit the same units. Every hypercharge is stored as the integer $Y_6=6Y$.
The six Weyl multiplets of one left-handed generation (including a sterile neutrino) carry sixth-charges $1,-4,2,-3,6,0$. The quark sector alone enters the mixed color-hypercharge anomaly: the left-handed quark doublet contributes with multiplicity weight 2 (color×isospin bookkeeping in the anomaly trace), while the two right-handed conjugates each contribute once.
Upstream, hypercharge6 is the piecewise map sending each multiplet to its $Y_6$; the quark-doublet label is the ordinary $(u,d)$ weak doublet of multiplicity 2.
proof idea
Pure definition, not a proved equality. The body is the three-term integer linear form $2\cdot Y_6(Q)+Y_6(u^c)+Y_6(d^c)$ evaluated on the canonical sixth-charge table. No tactics, no lemmas: just the anomaly combinatorics written in cube units.
why it matters
Supplies the raw integer that the certificate structure SMHyperchargeCert requires to be zero (su3_anomaly_zero), and that the companion theorem su3SquaredU1Anomaly6_eq_zero discharges by native_decide. Downstream it is wired into the forcing-chain bridge T8_To_GaugeStandardModel_Bridge, which routes the T8 dimension result ($D=3$) into the gauge/SM layer.
In the Recognition framework this is the exact anomaly-free SM hypercharge layer expressed in the cube completion's $1/6$ unit (punchlist item P0-S2-01). It does not claim the charges are uniquely forced; it only exhibits the integer cancellation that any later uniqueness argument must recover. Landmark context: T8 forces three spatial dimensions and thereby the cube whose gauge skeleton this hypercharge layer sits on.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.