full_Z_values
plain-language theorem explainer
With canonical coefficients a=b=1, the full Z-map (color offset included) evaluates to 1332 for the lepton, 276 for the up quark, and 24 for the down quark. Mass-ladder and anchor audits cite these three integers as the topological band labels. The proof unfolds the polynomial and offset definitions, then closes by integer arithmetic.
Claim. With coefficients $a=1$, $b=1$, the full band map $Z_{\mathrm{full}}$ (color offset $2^{D-1}=4$ on quarks, zero on leptons) satisfies $Z_{\mathrm{full}}(\ell)=1332$, $Z_{\mathrm{full}}(u)=276$, and $Z_{\mathrm{full}}(d)=24$, where the integerized charges are $\tilde Q_\ell=-6$, $\tilde Q_u=4$, $\tilde Q_d=-2$.
background
This module derives the charge-to-band polynomial from recognition topology on the 3-cube, without anchor or empirical mass input. Stage 1 integerizes SM charges by the face count $F=2D=6$ (forced at $D=3$), giving $\tilde Q=6Q\in{-6,4,-2}$. Stage 2 forces the minimal even non-negative polynomial $Z=a\tilde Q^2+b\tilde Q^4$ with family separation to $a=b=1$. Stage 3 adds a quark-only color offset equal to the edge-direction count $2^{D-1}=4$.
The full map is therefore $Z_{\mathrm{full}}=\mathrm{offset}+\tilde Q^2+\tilde Q^4$, with offset $0$ for leptons and $4$ for quarks. Explicitly: lepton $0+36+1296=1332$; up $4+16+256=276$; down $4+4+16=24$. Spatial dimension $D=3$ is the T8 landmark used for both the face count and the color offset.
proof idea
Term-mode reduction by simplification, then omega. Unfold the full map into offset plus the degree-4 even polynomial, substitute the three integerized charges and the edge-direction count (via $D=3$), discharge the Boolean sector branch, and finish with pure integer arithmetic on the three conjunctions. No external lemmas beyond the local definitions.
why it matters
Closes Stage 3 of the topological Z-map derivation by exhibiting the three concrete band integers that the mass ladder consumes. Downstream, matches_anchor_Z is a one-line alias of this statement, tying the first-principles values to the canonical anchors in Masses/Anchor. The same triple feeds derivation_complete, the packaged certificate that face integerization, gauge invariance, neutral vanishing, and family separation all hold with these coefficients.
In the broader framework this is the bridge from T8 ($D=3$, eight-tick cube geometry) to the phi-ladder mass formula: yardstick times $\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$ uses exactly these $Z$ labels. Without the match, the topological derivation would not certify the anchor band numbers.
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