J_phi_pos
J_phi_pos proves that the J-cost evaluated at the golden ratio is strictly positive. Researchers building the phi-ladder mass hierarchy or the Canonical J-band certificate cite it to confirm that the identity state is the unique zero-cost ground state. The proof is a short term-mode reduction that rewrites the definition via the squared-ratio identity for Jcost and splits positivity across numerator and denominator using standard real-arithmetic lemmas.
claim$0 < J(φ)$ where $J(x) = (x + x^{-1})/2 - 1$ for $x > 0$.
background
The J-cost function quantifies departure from the identity element in the Recognition framework. It is defined by J(x) = (x + x^{-1})/2 - 1 and admits the equivalent squared form J(x) = (x-1)^2/(2x) for x ≠ 0. In the Spectral Emergence module this cost is applied to phi-ratio edges on the ladder generated from D = 3 and the binary cube Q₃.
proof idea
The proof unfolds J_phi to Jcost phi, rewrites via Jcost_eq_sq (using phi ≠ 0), then applies div_pos. The numerator positivity follows from sub_ne_zero on phi - 1 together with positivity; the denominator positivity follows from linarith on phi_pos.
why it matters in Recognition Science
This theorem supplies the phi_pos clause inside the CanonicalCert structure that is consumed by SchumannResonanceCert and the J-band certification. It directly instantiates the claim that departure from unity always costs, a necessary step after T6 forces phi as the self-similar fixed point. The result is invoked whenever the master certificate must certify that the ground state remains unique.
scope and limits
- Does not prove J(x) > 0 for x other than phi.
- Does not treat complex or negative arguments.
- Does not compute the numerical value of J(phi).
- Does not address stability under small perturbations.
Lean usage
have h : 0 < J_phi := J_phi_posformal statement (Lean)
266theorem J_phi_pos : 0 < J_phi := by
proof body
Term-mode proof.
267 unfold J_phi
268 rw [Jcost_eq_sq (ne_of_gt phi_pos)]
269 apply div_pos
270 · have : phi - 1 ≠ 0 := sub_ne_zero.mpr phi_ne_one
271 positivity
272 · linarith [phi_pos]
273
274/-- **THEOREM**: J(1) = 0. The identity state has zero cost. -/