phi_perturbation_bounded
plain-language theorem explainer
The J-cost of the golden ratio φ satisfies J(φ) < 1. Researchers tracing the T0-T8 forcing chain cite this to confirm that the barrier for generating non-trivial structure from the zero-defect ground state remains finite. The proof is a one-line algebraic reduction that substitutes the explicit form of J(φ) and invokes the bound φ < 2.
Claim. The Recognition cost of the golden ratio satisfies $J(φ) < 1$.
background
LawOfExistence.J denotes the Recognition cost function J(x) = (x + x^{-1})/2 - 1. PhiForcing.φ is the golden ratio, the unique positive root of x² = x + 1. The StillnessGenerative module derives from T0-T8 that the unique zero-defect initial state x = 1 cannot remain uniform under recognition forcing and must generate non-trivial content on the phi-ladder.
proof idea
The proof rewrites the left-hand side via the equality theorem phi_cost_eq, which states LawOfExistence.J PhiForcing.φ = PhiForcing.φ - 3/2. It then applies linarith to the known inequality PhiForcing.phi_lt_two that φ < 2, yielding the strict bound less than 1.
why it matters
This result supplies the finite-barrier clause in the derivation chain of the StillnessGenerative module. It is invoked directly by origin_question_resolved to close the cost bound and by stillness_is_creative to establish ground-state instability. In the framework it completes the T6 closure step that every non-trivial rung on the phi-ladder carries bounded positive cost.
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