phi_cost_eq
plain-language theorem explainer
The theorem equates the J-cost of the golden ratio φ to φ minus 3/2. Researchers deriving the ground-state instability from the T0-T8 chain cite it to bound the creation cost of the first non-trivial state. The proof is a one-line wrapper around the J_phi lemma from PhiForcing.
Claim. $J(φ) = φ - 3/2$, where $J$ is the recognition cost function from the Law of Existence and $φ$ is the golden ratio.
background
The StillnessGenerative module shows from T0-T8 that the zero-defect ground state x=1 is unstable and acts as the source of all structure. The J-cost is defined as $J(x) = (x + x^{-1})/2 - 1$, the recognition defect away from unity. The golden ratio φ is the self-similar fixed point forced by T6.
proof idea
The proof is a one-line wrapper that applies the J_phi theorem from PhiForcing. That lemma simplifies the J definition using the inverse relation for φ and normalizes via ring tactics.
why it matters
This result supplies the finite barrier step J(φ) = φ - 3/2 < 1 in the ground-state instability derivation. It is invoked by the positivity and perturbation-boundedness theorems in the same module. It anchors the T5 J-uniqueness and T6 phi fixed point, enabling the Fibonacci cascade to populate the ladder.
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