no_scale_change_is_free
plain-language theorem explainer
The J-cost function satisfies Jcost(1 · x) = Jcost(x) for every positive real x, supplying the zero-cost reference point at unit scale inside the scale-invariance selection certificate. This fact is cited by scaleInvarianceCert to anchor the cost-of-scale-change principle. The proof is a one-line simplification that reduces the equality directly from the definition of multiplication by 1.
Claim. For every real number $x > 0$, $J(x) = J(x)$, where $J$ is the J-cost function obeying the Recognition Composition Law.
background
The Scale-Invariance Selection module shows that J(cx) equals J(x) only in special cases; in general the ratio J(cx)/J(x) is bounded above by J(c). The Recognition Composition Law supplies the identity J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The present theorem records the trivial equality that holds when the scale factor c is exactly 1, establishing the base case of the cost-of-scale-change principle from the pre-Big-Bang paper §5.
proof idea
The proof is a one-line wrapper that applies the simp tactic to unfold Jcost and reduce 1 * x to x.
why it matters
It supplies the free_at_unit field of the ScaleInvarianceCert definition, which assembles the full certificate for scale-invariance selection. The result closes the unit-scale reference point required by the cost-minimisation argument in paper §5 and aligns with T5 J-uniqueness in the forcing chain. It is invoked directly by scaleInvarianceCert alongside rcl_equality and scale_change_cost.
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