domainCost
plain-language theorem explainer
Domain cost assigns the recognition cost of a mass-to-energy ratio by evaluating the J-functional at m/e. Eight-tick application arguments (thresholds, nonnegativity, certificates) cite it as the local cost measure. The body is a one-line specialization of J-cost to that ratio.
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 ambient module collects structural eight-tick applications: musical octave, color/flavor counting, SU(3) adjoint gluons, Gell-Mann eightfold way, and related 2^D=8 patterns, all treated as consequences of the forced period-eight tick structure.
The unique cost functional forced by the Recognition Composition Law is $J(x)=\frac12(x+x^{-1})-1$ (equivalently $\cosh(\log x)-1$). Upstream docs state it as "the RS recognition cost of a positive ratio" and note that a genuine distinction (ratio not one) has strictly positive cost. Domain cost simply feeds the ratio of two reals (read as mass over energy scale) into that functional.
Sibling lemmas in the same file then record evaluation identities, nonnegativity, and a canonical positive threshold built from this cost.
proof idea
Pure definition: no proof obligations. The body is the term $J(m/e)$, i.e. a direct application of the shared J-cost functional to the quotient of the two real arguments. Downstream facts (evaluation at equality, nonnegativity) are proved separately from properties of $J$.
why it matters
Gives the local cost primitive for the eight-tick applications layer so thresholds and certificates can speak in RS-native units rather than ad hoc scales. It sits on the T5 J-uniqueness landmark (the unique cost forced by RCL) and the T7 eight-tick octave (period $2^3$) that the module is applying.
Siblings such as nonnegativity of domain cost and the positive canonical threshold are built directly on this def; the module certificate packages those structural facts. No external used-by edges are recorded yet, so its present role is in-module scaffolding for the v2 eight-tick application certificate rather than a cross-module bridge.
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