cwCumulativeTorsion
plain-language theorem explainer
Cumulative CW-filtration torsion for fermion generations: generation g couples to all Q₃ subcells of CW dimension at most g−1. Ground generation has torsion 0; the second adds passive edge coupling; the third adds face coupling. Mass and generation-ordering arguments cite this as the geometric source of the schedule {0,11,17} at D=3. Pure case split on Generation via passiveCoupling.
Claim. For each dimension parameter $d\in\mathbb{N}$ and fermion generation $g$, the cumulative CW torsion $\tau_{\mathrm{CW}}(d,g)\in\mathbb{Z}$ is $\tau_{\mathrm{CW}}(d,1)=0$, $\tau_{\mathrm{CW}}(d,2)=c_{\mathrm{pass}}(d,\mathrm{edge})$, and $\tau_{\mathrm{CW}}(d,3)=c_{\mathrm{pass}}(d,\mathrm{edge})+c_{\mathrm{pass}}(d,\mathrm{face})$, where $c_{\mathrm{pass}}$ counts passive-coupling subcells at that CW level (vertices $0$, edges the passive field edges, faces all cube faces).
background
The module derives edge-before-face excitation ordering for fermion generation torsion from the CW structure of the $D=3$ cube $Q_3$, together with $J$-cost monotonicity on $\varphi$-power ratios. The natural CW-filtration is: $0$-skeleton (8 vertices), $1$-skeleton (12 edges, 11 passive), $2$-skeleton (6 faces).
Passive coupling counts subcells available at each CW level: vertices contribute nothing (ground couples trivially); edges contribute $\mathrm{passive_field_edges}(d)$ (one edge is the active transition); all faces participate. Generations are the three fermion modes (first/second/third).
The filtration principle says generation $g$ couples to all subcells of CW dimension $\le g-1$. That yields the cumulative schedule: Gen 1 (ground) $\tau=0$; Gen 2 adds dim-1 edge coupling; Gen 3 adds dim-2 face coupling. At $D=3$ this is the familiar ${0,11,17}$.
proof idea
Definition by cases on Generation, not a proved statement. First generation is the integer $0$. Second is the integer cast of passive edge coupling at dimension $d$. Third is the sum of passive edge and face couplings, again cast to $\mathbb{Z}$. No lemmas are applied; the body is the schedule itself.
why it matters
This is the geometric schedule that replaces ad hoc mode labels (ground/edge/face) in cube-admissible torsion with a single CW-dimensional principle. Downstream, equality to canonical generation torsion is proved by case analysis and native decision at $D=3$; cube-admissibility, incremental filtration, and the full cube-generation filtration package all reduce through that equality.
It feeds the excitation-ordering theorem: if excitations couple in CW-dimension order, then the first nontrivial excitation is edge-supported, the next is face-supported, the torsion schedule matches the canonical one, and $J(\varphi^0)=0<J(\varphi^{11})<J(\varphi^{17})$. That supplies the structural reason edges precede faces on $Q_3$ (T8 forces $D=3$). The remaining premise is still the filtration principle itself: a structural claim about coupling, not yet a derived consequence.
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