phi_cost_pos
plain-language theorem explainer
The theorem establishes that the J-cost of the golden ratio φ is strictly positive. Researchers tracing the Recognition Science T0-T8 chain would cite it to confirm the creation barrier is positive yet finite. The proof is a one-line wrapper that rewrites the cost expression via an explicit identity and applies a lower bound on φ.
Claim. $0 < J(φ)$ where $J(x) = (x + x^{-1})/2 - 1$ is the cost function and $φ > 1.6$ is the golden ratio satisfying $φ^2 = φ + 1$.
background
The StillnessGenerative module derives that the unique zero-defect ground state x=1 is unstable and must generate non-trivial structure. The J-cost, introduced in LawOfExistence, quantifies deviation from this state via the Recognition Composition Law. PhiForcing supplies the golden ratio φ as the self-similar fixed point forced by T6 closure on geometric sequences, along with the identity J(φ) = φ - 3/2.
proof idea
One-line wrapper that applies the equality phi_cost_eq to replace J(φ) with φ - 3/2, then invokes linear arithmetic on the upstream bound φ > 1.6 from phi_gt_onePointSix.
why it matters
This result feeds directly into origin_question_resolved and stillness_is_creative. It supplies the missing positivity half of the finite-barrier claim in the module derivation chain: J(φ) < 1 is crossable precisely because J(φ) > 0. It closes the T6 step on φ while linking to T4 recognition forcing and T7 eight-tick non-degeneracy.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.