domainCost
plain-language theorem explainer
Domain cost of a mass-energy pair is the recognition cost of their ratio: J(m/e). Information theorists linking Solomonoff priors to Recognition Science cite it as the continuous bridge into the algorithmic-probability certificate. It is a one-line specialization of the forced J-cost functional.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$ is the recognition cost of a positive ratio.
background
The module builds algorithmic probability from J-cost. Its framing is that the Solomonoff prior $P(x)=\sum_{U(p)=x}2^{-|p|}$ peaks, under RS-optimal encoding, at compression rate $J(\varphi)\cdot|x|$.
Upstream, $J$ is the unique cost forced by the Recognition Composition Law: $J(x)=\frac12(x+x^{-1})-1$ (T5 J-uniqueness). Several modules re-export the same functional; the Cosmology form states it as "the RS recognition cost of a positive ratio," and the EnergyProcessingBridge form notes uniqueness under RCL.
Domain cost simply feeds a mass-to-energy ratio into that functional, so continuous $(m,e)$ data sit on the same cost scale used for the algorithmic-probability certificate.
proof idea
Definitional one-liner: evaluate $J$ at the ratio $m/e$. No lemmas, no tactics, no proof obligations.
why it matters
Gives the module its continuous cost primitive. Sibling lemmas (nonnegativity, evaluation identities) and the AlgorithmicProb3 certificate sit on top of it; the module claims a structural theorem (0 sorry, 0 axiom) that the universal prior peaks at the $J(\varphi)$ compression rate.
Framework landmarks: T5 forces $J(x)=\cosh(\log x)-1$; the RCL identity $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ is what makes this the unique cost. The definition is the local interface that turns those landmarks into an information-theoretic domain cost for mass/energy ratios.
No downstream dependents are recorded yet; the immediate consumers are the in-module certificate and threshold constructions.
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