perturbation_cost_positive
plain-language theorem explainer
The declaration shows that any nonzero real perturbation ε of the ground state, satisfying 1 + ε > 0, produces strictly positive J-cost. Researchers modeling the generative instability of the zero-defect state in Recognition Science would reference this when deriving structure from departures from unity. The proof consists of a single exact application of the J-cost positivity lemma after verifying the input conditions.
Claim. Let ε ∈ ℝ satisfy ε ≠ 0 and 1 + ε > 0. Then the J-cost of 1 + ε is strictly positive: 0 < J_cost(1 + ε).
background
The StillnessGenerative module establishes that x = 1 is the unique zero-defect ground state via the Law of Existence (T5) and shows it functions as an unstable generative source rather than passive equilibrium. The J-cost function, imported from the Cost module, quantifies recognition cost and obeys J_cost(x) > 0 for all x > 0 with x ≠ 1, as stated in the upstream lemma Jcost_pos_of_ne_one. The local setting derives from T0–T8 that a uniform ledger at x = 1 carries zero information and violates the eight-tick cycle, forcing non-trivial content on the phi-ladder.
proof idea
The proof is a one-line wrapper that applies the lemma Jcost_pos_of_ne_one to the argument 1 + ε. It passes the positivity hypothesis 0 < 1 + ε directly and obtains 1 + ε ≠ 1 from ε ≠ 0 via a short linarith tactic inside an intro.
why it matters
This result supports the module's claim that x = 1 is the maximally creative source of structure by showing any departure incurs positive J-cost. It instantiates the core positivity property required for the finite barrier J(φ) < 1 and the subsequent Fibonacci cascade on the phi-ladder. The theorem fills a direct step in the T5-to-T8 derivation chain without introducing external assumptions.
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