down_rung_gen2
plain-language theorem explainer
This definition sets the second down-quark generation rung to 10. Mass ladder modelers in Recognition Science cite it when sequencing down-quark rungs for the phi-ladder mass formula. It is a direct arithmetic sum of the D=3 baseline rung and the 3-cube face count.
Claim. The second down-quark generation rung is the natural number $r_{d,2} := r_{d,1} + s_{d,12}$, where $r_{d,1} = 2^{3-1} = 4$ is the baseline rung and $s_{d,12} = 6$ is the step equal to the number of faces of the three-dimensional cube.
background
The Sector-Dependent Generation Torsion module verifies algebraic constraints on the integers {13, 11, 6, 8} as Q3 cell counts, with down-quark assignments {6, 8} treated as data-supported hypotheses rather than first-principles derivations. The upstream definition states: Down-quark generation rungs (with baseline r_q = 4). The companion upstream definition states: Down-quark generation steps from cube geometry, fixing the step at the face count of the 3-cube.
proof idea
This is a one-line definition that adds the first down-quark generation rung to the down-quark generation step from cube geometry.
why it matters
This definition supplies the middle term for the down-generation ordering theorem establishing 4 < 10 < 18 and for the equality theorem proving the value 10. It contributes rung positions to the mass formula yardstick * phi^(rung - 8 + gap(Z)) on the phi-ladder. The module distinguishes fully derived lepton torsion from hypothesized quark assignments and notes the D=3 coincidence W = 2V + 1 = N0 as a genuine derivation.
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