domainCost
plain-language theorem explainer
Domain cost assigns the Recognition Science cost of a mass-to-energy ratio: J(m/e) with the unique J-cost forced by the composition law. Standard-Model structural arguments in the D=3 channel-count module use it as the local cost of a mass scale against a reference energy. The declaration is a one-line definition wrapping J on the 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
Recognition Science measures mismatch of positive ratios by the J-cost $J(x)=\frac{1}{2}(x+x^{-1})-1$. Upstream modules record that this is the unique cost forced by the Recognition Composition Law, and that a genuine distinction (ratio not one) has strictly positive cost.
This module is Standard Model structural module 3. Its setting is the RS count law $2^D-1=7$ independent channels, exact once spatial dimension is fixed at $D=3$. Domain cost is the local cost functional used when comparing a mass parameter $m$ to an energy reference $e$ inside that structural story.
Notation is RS-native: arguments are real; positivity and non-negativity of $J$ on positive ratios are handled by sibling lemmas, not by this definition.
proof idea
Pure definition: evaluate the shared J-cost functional on the ratio $m/e$. No tactics, no lemmas beyond the imported definition of $J$.
why it matters
Gives the Standard Model structural layer a named cost of mass-versus-energy so later claims (non-negativity, equality at matched scales, canonical thresholds, and the module certificate) can speak in RS units rather than ad hoc norms.
It sits under the T5 J-uniqueness landmark and the T8 $D=3$ forcing that produces the seven-channel count in this module. Downstream use edges are empty in the graph snapshot; sibling results in the same file (non-negativity, threshold positivity, and the structural certificate) are the intended consumers.
No open scaffold: the module claims structural status with zero sorry and zero axioms. This def is the cost primitive those certificates quantify over.
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