non_existence_has_positive_cost
plain-language theorem explainer
Non-existence of a recognition pattern at scale x carries strictly positive J-cost for every x > 0 with x ≠ 1. Workers on the pre-Big-Bang cost-minimisation principle cite this result to show that only the fixed point x = 1 is stable under recognition evolution. The proof is a direct one-line application of the core positivity lemma for the J-cost function.
Claim. For every real number $x > 0$ with $x ≠ 1$, the J-cost satisfies $J(x) > 0$.
background
The Cost-First Existence module formalises the selection principle that stable patterns are those minimising recognition cost. A pattern x > 0 exists in the recognition sense precisely when its J-cost vanishes, which occurs if and only if x = 1. The J-cost function is the derived cost of a multiplicative recognizer, shown positive away from unity by the upstream lemma Jcost_pos_of_ne_one: for x > 0 and x ≠ 1, 0 < Jcost x holds by rewriting Jcost as a square over a positive denominator and invoking positivity of squares.
proof idea
The proof is a one-line wrapper that applies the lemma Jcost_pos_of_ne_one from the Cost module (also present in JcostCore), passing the hypotheses 0 < x and x ≠ 1 directly to obtain the strict inequality.
why it matters
This theorem supplies the non_existence_costly field of the CostFirstExistenceCert, which bundles the core certificates for the cost-first selection principle. It directly supports the claim that all non-unit positive scales are transiently unstable, aligning with the T5 J-uniqueness and the pre-geometric cost landscape in the Recognition Science framework. The parent certificate appears in the same module and closes the structural argument for existence emerging from cost minimisation.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.