J_phi
plain-language theorem explainer
J_phi defines the J-cost evaluated at the golden ratio φ as a real number. Researchers tracing the φ-ladder mass hierarchy or coherence costs in the Recognition framework cite this value. The construction is a one-line abbreviation of the Jcost function applied to phi.
Claim. $J(φ) := ½(φ + φ^{-1}) - 1$, which equals $φ - 3/2$.
background
The SpectralEmergence module derives SU(3)×SU(2)×U(1) content, three generations, and 48 chiral states from the binary cube Q₃ forced by D=3. The J-cost measures departure from unity and obeys the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). Upstream, PhiForcing.J_phi states that J(φ) = φ - 3/2 with the note that J(φ) ≠ 0 because φ ≠ 1; PrimitiveDistinction.from supplies the seven-axiom foundation for the underlying distinction structure.
proof idea
One-line wrapper that applies Jcost to phi.
why it matters
This definition supplies the base value used by J_phi_pos (same module) to prove positivity and by StillnessGenerative.phi_cost_eq to equate the cost to φ - 3/2. It instantiates the T5 J-uniqueness and T6 phi-forcing steps, seeding the mass formula yardstick · φ^(rung-8+gap(Z)). No open scaffolding questions are closed here.
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