Z
plain-language theorem explainer
Z maps each particle sector and rational charge Q to an integer charge index for the anchor relation in mass derivations. Ablation studies and atomic radius calculations cite this map when testing consistency of the parameter-free constants. The definition first scales Q to the integer q = 6Q and then evaluates a sector-dependent quartic polynomial in q.
Claim. For sector $s$ in {Lepton, UpQuark, DownQuark, Electroweak} and rational charge $Q$, the integer $Z(s,Q)$ equals $q^2 + q^4$ for Lepton, $4 + q^2 + q^4$ for UpQuark or DownQuark, and 0 for Electroweak, where $q = 6Q$ is taken as an integer.
background
The Anchor module derives all sector constants from D=3 cube geometry: total edges equal 12, passive field edges equal 11, and wallpaper groups equal 17. Sector is the inductive type whose constructors are Lepton, UpQuark, DownQuark, and Electroweak. Upstream results fix active_edges_per_tick = 1 and cube_edges(d) = d * 2^(d-1), which together set passive_field_edges = 11 for D=3; wallpaper_groups = 17 is the Fedorov crystallographic count.
proof idea
The definition computes the integerized charge Q6 := 6 * Q.num / Q.den and then matches on the sector constructor to select the corresponding polynomial expression in Q6.
why it matters
Z supplies the integer map required by AnchorEq in the Ablation module, which verifies that transformed maps preserve the gap function Fgap. It realizes the charge-based integer map from Paper 1 and connects directly to the cube-derived constants (E_total = 12, passive_field_edges = 11) together with forcing-chain steps T8 (D=3) and T7 (eight-tick octave). Downstream applications include atomic-radii inequalities for lithium and sodium.
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