phi_perturbation_bounded
plain-language theorem explainer
The Recognition cost J of the golden ratio φ satisfies J(φ) < 1. Researchers tracing the T0-T8 forcing chain cite this to bound the creation cost of non-trivial structure from the x=1 ground state. The result supplies the finite barrier clause in the module derivation. The proof is a one-line term wrapper that rewrites via the explicit cost formula and applies linear arithmetic with φ < 2.
Claim. $J(φ) < 1$, where $J$ is the Recognition cost function from the Law of Existence and $φ$ is the golden ratio.
background
The module derives from T0-T8 that x=1 is the unique zero-defect state yet must generate non-trivial content. The J-cost function measures defect or recognition cost; φ is the self-similar fixed point forced by T6. Upstream phi_cost_eq states LawOfExistence.J PhiForcing.φ = PhiForcing.φ - 3/2. Multiple phi_lt_two lemmas establish φ < 2. The local setting is the eight-step chain ending in the finite barrier J(φ) < 1 and the Fibonacci cascade.
proof idea
Term-mode proof. It rewrites the goal using phi_cost_eq to obtain φ - 3/2 < 1, then applies linarith with the phi_lt_two lemma.
why it matters
Supplies the finite barrier step quoted in the module doc-comment. Feeds origin_question_resolved (which lists 0 < J(φ) ∧ J(φ) < 1) and stillness_is_creative. Closes the T5-T6 argument that departure from x=1 has bounded cost while satisfying T7 non-degeneracy. No scaffolding remains.
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