step_up_gen2
This definition assigns the integer 11 to the E_pass generation step in the sector-dependent torsion chain for up-type quarks. Mass spectrum constructors on the recognition manifold would cite it when placing the second rung for up quarks. The assignment follows directly from the difference between the D-cube edge count and the active edge count per tick.
claimThe second-generation up-quark step constant equals the integer 11, obtained as the difference between the edge count of the three-dimensional cube and the active edge count per fundamental tick.
background
The module introduces sector-dependent generation torsion (SDGT) for fermion sectors. The four step values form a cyclic chain V+F-C=13 to E_pass=11 to F=6 to V=8, with each sector taking two consecutive values that partition N_3 = 2D^D +1 =55. Upstream, the edge count is given by E(D) = D * 2^(D-1) and the active edge count per tick is A=1.
proof idea
The definition is a direct constant assignment. The inline comment derives the value as the difference E minus A using the upstream edge count and anchor definitions.
why it matters in Recognition Science
It supplies the E_pass slot to tau_sdgt, the canonical cumulative generation torsion for all fermion sectors. This supports rung construction for masses on the phi-ladder and aligns with the eight-tick octave in the unified forcing chain.
scope and limits
- Does not compute cumulative torsion for any sector.
- Does not reference the J-cost function or recognition composition law.
- Does not depend on the spatial dimension D.
- Does not incorporate the mass formula or phi-ladder rungs.
formal statement (Lean)
61def step_up_gen2 : ℤ := 11 -- E_pass = E - A = 12 - 1
proof body
Definition body.
62
63/-- Down-quark generation steps: {F=6, V=8}. HYPOTHESIS. -/