Jcost_val_4
plain-language theorem explainer
The J-cost function evaluates to 9/8 at argument 4. Researchers computing defect distances on the phi-ladder for low-rung states would cite this normalization when scaling masses via yardstick * phi^(rung-8+gap). The proof is a one-line wrapper that unfolds the Jcost abbreviation and reduces the arithmetic expression.
Claim. $J(4) = 9/8$, where $J(x) = (x + x^{-1})/2 - 1$ is the unique map satisfying the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$.
background
The Ontology Predicates module defines RSExists x as the condition that defect(x) collapses to zero under J-cost minimization, making existence a verifiable selection outcome rather than a primitive. Jcost is the explicit evaluation map on the cost quantity, introduced in the RSNative core as Cost := Quantity CostUnit. This supplies the concrete integer value required for rung-4 entries in the phi-ladder mass formula yardstick * phi^(rung-8+gap(Z)).
proof idea
The proof is a one-line wrapper that unfolds Cost.Jcost and applies norm_num to reduce the resulting rational arithmetic.
why it matters
This supplies an explicit J-cost anchor used in the T5 J-uniqueness step of the forcing chain and in defectDist calculations for RSExists predicates. It supports the eight-tick octave and D=3 spatial dimension derivations by fixing low-rung normalizations. No downstream theorems currently depend on it.
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