domainCost
plain-language theorem explainer
The domain cost of a materials pair $(m,e)$ is the recognition cost of their ratio $m/e$. Materials and condensed-matter workers in RS cite it as the native energy penalty for a mass-energy mismatch on the eight-tick lattice. The body is a one-line specialization of the unique J-cost forced by the Recognition Composition Law.
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
Recognition Science measures mismatch by a single scalar cost on positive ratios. The functional is $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. Upstream modules record that this $J$ is the unique cost forced by the Recognition Composition Law (RCL) and is nonnegative for $x>0$, with $J(x)=0$ iff $x=1$.
This module sits in the materials layer and is tagged STRUCTURAL (zero sorry, zero axiom). Its local setting is the RS eight-tick octave: one full traversal of the binary recognition lattice has period $2^D=8$ with $D=3$. Domain cost is the materials-facing packaging of $J$ on a mass-to-energy (or analogous) ratio $m/e$.
Sibling lemmas in the same file immediately specialize the definition: equality at matched arguments, nonnegativity, and a canonical positive threshold used by the structural certificate.
proof idea
Pure definitional abbreviation. The body applies the shared $J$-cost functional to the single ratio $m/e$; there is no tactic proof and no intermediate lemma. Downstream nonnegativity and threshold facts unfold this definition and invoke the corresponding properties of $J$.
why it matters
Gives the materials stack a named, RS-native cost on domain ratios so later structural claims can speak in $J$-language rather than ad-hoc energies. It sits under the T5 J-uniqueness landmark and the RCL, and inherits the eight-tick ($2^3$) lattice period from the module framing (T7/T8). Sibling facts (nonnegativity, equality at matched ratio, canonical threshold) and the module certificate build directly on this packaging. No external used-by edges are recorded yet; the definition is infrastructure for the RS_MAT structural certificate chain rather than a leaf theorem.
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