cubeGeometricTorsion
plain-language theorem explainer
Charged-generation torsion is assigned from Q₃ cube combinatorics alone: ground generation gets 0, the second gets the passive-edge count at D=3, and the third gets the endogenous face-plus-edge count. Mass and generation-bridge authors cite it as the numeral-free source of the schedule {0,11,17}. The body is a three-branch pattern match on generation, each arm a named cube function of D.
Claim. Define charged-generation torsion $\tau_{\mathrm{cube}}:\{\mathrm{gen}_1,\mathrm{gen}_2,\mathrm{gen}_3\}\to\mathbb{Z}$ by $\tau_{\mathrm{cube}}(\mathrm{gen}_1)=0$, $\tau_{\mathrm{cube}}(\mathrm{gen}_2)=E_{\mathrm{passive}}(D)$, and $\tau_{\mathrm{cube}}(\mathrm{gen}_3)=W_{\mathrm{endo}}(D)$, where $D=3$ is the forced spatial dimension, $E_{\mathrm{passive}}(D)$ is the number of passive field edges of the $D$-cube, and $W_{\mathrm{endo}}(D)=E_{\mathrm{passive}}(D)+F(D)$ with $F(D)$ the face count.
background
The Generation Torsion Bridge module is the authoritative geometric source for charged-generation torsion ${0,11,17}$. It replaces raw numerals by $Q_3$ cube arithmetic at the forced spatial dimension $D=3$ (T8/T9). The three generation modes come from parity patterns on the cube: ground, edge-dressed, and face-plus-edge-dressed.
Passive field edges $E_{\mathrm{passive}}(d)$ are total cube edges minus the single active edge per tick; at $D=3$ this is 11. The endogenous count $W_{\mathrm{endo}}(d)=E_{\mathrm{passive}}(d)+F(d)$ adds the face count; at $D=3$ one gets $11+6=17$. That integer coincides with the crystallographic wallpaper-group count, but the primary route here is $E_{\mathrm{passive}}+F$, forced by cube arithmetic alone.
The remaining structural premise is how fermion generations couple to ground / passive-edge / face+edge modes. Until that coupling is derived from the Recognition Composition Law, the module upgrades hardcoded numerals to an explicit geometric schedule with uniqueness under a cube-admissibility predicate.
proof idea
Definitional pattern match, not a proof. On the first generation the value is the integer 0. On the second it is the coercion of passive_field_edges D to $\mathbb{Z}$. On the third it is the coercion of W_endo D to $\mathbb{Z}$. No tactics or lemmas fire at the definition site; downstream simp lemmas unfold each branch by rfl.
why it matters
This is the single geometric source of truth for charged-generation torsion in the masses sector. Downstream, it feeds pointwise equality with RecogSpec generationTorsion, matching against Anchor Integers.tau at indices 0,1,2, the cube-admissibility certificate, the uniqueness theorem that any cube-admissible schedule equals the canonical one, and the incremental filtration view.
Framework landmarks: $D=3$ from the forcing chain (T8), and the eight-tick / cube combinatorics that produce $E_{\mathrm{passive}}=11$ and $W_{\mathrm{endo}}=17$. The module certifies no raw numerals in the schedule and agreement across every other representation in the codebase.
Open gap: CubeAdmissibleTorsion still encodes a structural assignment rule (ground / edge / face+edge coupling) not yet derived from the RCL alone. A stronger route via CW-filtration of $Q_3$ appears in ExcitationOrdering.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.