J
plain-language theorem explainer
Recognition Science defines the J-cost explicitly as half of x plus its reciprocal minus one. Researchers modeling multiplicative recognition and observer forcing cite this definition to compute event costs under the composition law. The declaration is a one-line alias to the Jcost primitive from the Cost module.
Claim. $J(x) = ½(x + x^{-1}) - 1$ for real $x$, the unique cost functional satisfying the Recognition Composition Law.
background
The J-cost is the cost function induced by a multiplicative recognizer, given by derivedCost on its comparator. It satisfies the Normalization axiom: the cost at unity is zero, which encodes that perfect balance (ratio = 1) has no cost. Any cost functional measuring deviation must vanish at the reference point.
proof idea
The definition is a one-line wrapper that aliases Jcost x from the imported Cost module. No tactics or reductions are applied beyond the direct assignment.
why it matters
This supplies the explicit form for J-uniqueness in the forcing chain at T5, where J(x) equals cosh(log x) minus one. It anchors cost calculations that feed the eight-tick octave and the emergence of three spatial dimensions. Sibling results such as SatisfiesRCL and costCompose depend on this concrete expression to verify the Recognition Composition Law.
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