Z_full
plain-language theorem explainer
The full band label adds a sector-dependent color offset to the charge polynomial: quarks get the edge-direction count 2^{D-1}=4, leptons get zero. Cited by anyone checking Stage 3 of the topological Z-map derivation or matching anchor mass bands. Body is a one-line conditional sum of that offset with the even polynomial in integerized charge.
Claim. For a sector flag $s$ (quark vs lepton), coefficients $a,b\in\mathbb{Z}$, and integerized charge $\tilde{Q}\in\mathbb{Z}$, the full band label is $Z(s,a,b,\tilde{Q})=(2^{D-1}\text{ if quark, else }0)+a\tilde{Q}^{2}+b\tilde{Q}^{4}$. At $D=3$ the color offset is $4$.
background
This module derives the charge-to-band map from recognition boundaries on the 3-cube, without anchor constraints or empirical masses. Stage 1: a boundary of charge $Q$ couples to the $F=2D$ faces; the ledger needs integer entries (T8: $\delta$-units $\simeq\mathbb{Z}$), and $F=6$ at $D=3$ is the minimal positive even integerizer of the three SM charges ${-1,2/3,-1/3}$. Stage 2: gauge constraints force an even nonnegative polynomial vanishing at zero, so $Z=a\tilde{Q}^{2}+b\tilde{Q}^{4}$; family separation uniquely forces $a=b=1$.
Stage 3 (this definition) adds color. Quarks carry color charge and open $2^{D-1}$ extra recognition channels along the edge directions of the cube. The module states: the color offset equals $2^{D-1}=4$, the number of edges along one spatial direction. Leptons get offset $0$; quarks get $4$. The polynomial part is the upstream $Z_{\mathrm{poly}}(a,b,\tilde{Q})$.
proof idea
Pure definition, no proof. Case on the Boolean sector flag: if quark, cast edge_direction_count to $\mathbb{Z}$ and add it; if lepton, add $0$. Then add the polynomial term $Z_{\mathrm{poly}},a,b,Q$. The offset constant and the polynomial are upstream; this only assembles them.
why it matters
Closes Stage 3 of the three-stage first-principles Z-map chain in this module. Downstream full_Z_values evaluates the canonical case $a=b=1$ to the concrete bands $Z_{\ell}=1332$, $Z_{u}=276$, $Z_{d}=24$. matches_anchor_Z asserts those equal the canonical values used in Masses/Anchor. The summary structure ZMapDerivation packages face integerization, coefficient forcing, and this color offset as the complete topological derivation.
Framework link: T8 forces $D=3$, hence offset $2^{D-1}=4$ and face count $F=6$. The resulting integer $Z$ values are the band labels that feed the mass formula (yardstick $\cdot\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$) on the $\varphi$-ladder. Without the sector offset, quark and lepton bands would collide with the pure polynomial.
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