J_phi
J_phi defines the J-cost evaluated at the golden ratio φ as a real constant. Researchers deriving mass hierarchies along the phi-ladder or establishing coherence costs in Recognition Science would reference this value. The definition is a direct one-line application of the Jcost function to phi.
claimDefine the constant $J_φ$ to be the J-cost of the golden ratio $φ$, that is $J(φ) = ½(φ + φ^{-1}) - 1$.
background
The Spectral Emergence module starts from D = 3 spatial dimensions to obtain the binary cube Q₃ with eight vertices and derives the gauge group SU(3) × SU(2) × U(1), three generations, and the φ-ladder mass hierarchy. The J-cost function, supplied by the Cost module, quantifies departure from unity for any ratio x via the formula J(x) = (x + x^{-1})/2 - 1. This definition specializes the cost to φ, the self-similar fixed point forced earlier in the PhiForcing module.
proof idea
This declaration is a one-line definition that applies the Jcost function directly to the constant phi.
why it matters in Recognition Science
J_phi supplies the base coherence cost at φ that is unfolded to prove positivity in J_phi_pos and to obtain the algebraic identity J(φ) = φ - 3/2 in phi_cost_eq. It anchors the mass formula yardstick · φ^(rung - 8 + gap(Z)) and links the forced φ (T6) to the spectral emergence of gauge content and generations from Q₃ (T8).
scope and limits
- Does not prove that J(φ) is positive.
- Does not derive the value of φ from the forcing axioms.
- Does not map J_phi onto specific particle masses.
- Does not address alternative fixed points or dimensions.
Lean usage
unfold J_phiformal statement (Lean)
263def J_phi : ℝ := Jcost phi
proof body
Definition body.
264
265/-- **THEOREM**: J(φ) > 0. Departure from unity always costs. -/