step_up_gen1
plain-language theorem explainer
The definition assigns the integer 13 to the first-generation step for up quarks in the sector-dependent torsion scheme. Mass-ladder constructions in Recognition Science cite it to set the initial rung increment for up-type quarks. It is introduced by direct numerical assignment from the arithmetic V + F - C = 8 + 6 - 1 using cube geometry counts.
Claim. The up-quark first-generation step is the integer $13$, obtained from the combination of vertex count $V=8$, face count $F=6$, and correction $C=1$ in the three-dimensional cube.
background
The module defines sector-dependent generation torsion (SDGT) via a cyclic chain of four constants: V+F-C=13, E_pass=11, F=6, V=8. Up quarks employ the pair {13, 11} whose sum is 24; leptons use {11, 6} summing to 17; down quarks use {6, 8} summing to 14. These three spans partition the total N_3 = 2D^D + 1 = 55 for D=3. The legacy universal torsion {0, 11, 17} is retained only for backward compatibility with leptons.
proof idea
This is a direct definition that hard-codes the integer 13. The accompanying comment records the arithmetic origin from the cube counts V=8, F=6, C=1. No lemmas or tactics are invoked.
why it matters
The constant is consumed by the sector-dependent cumulative generation torsion function to initialize the up-quark rung increment. It completes the cyclic chain that partitions N_3=55 and supplies the D=3 geometry required by the forcing chain T8. The downstream tau_sdgt declaration applies it to produce generation-dependent torsion values for the phi-ladder mass formula.
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