domainCost
plain-language theorem explainer
Defines the recognition cost of a mass-to-energy ratio as J(m/e), with J the unique RS cost functional. Standard-model structural work on rung spacing cites it when comparing particle masses to energy scales. The body is a one-line abbreviation of the global J-cost.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x) = \frac{1}{2}(x + x^{-1}) - 1$ is the recognition cost of a positive ratio.
background
Module RS_STD_Structural_008 treats Standard Model structure under Recognition Science rung spacing: adjacent rungs differ by the golden ratio $\varphi \approx 1.618$. Status is structural (no sorry, no axioms).
The cost functional $J$ is the unique nonnegative cost forced by the Recognition Composition Law: $J(x) = \frac{x + x^{-1}}{2} - 1$ for $x > 0$. Upstream docs phrase 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 specializes $J$ to the dimensionless ratio of a mass parameter to an energy scale.
proof idea
Pure definition: no proof obligations. The right-hand side is the shared Jcost abbreviation $(x + x^{-1})/2 - 1$ applied to the quotient $m/e$. Sibling lemmas (nonnegativity, evaluation identities) discharge the usual positivity and algebra facts about this specialization.
why it matters
Gives the local cost language for mass-versus-energy comparisons inside the Standard Model structural layer. It sits next to canonical-threshold and certificate objects in the same module, which package rung-spacing claims under $\varphi$. Framework landmark: $J$ is the T5 uniqueness result (also written $\cosh(\log x) - 1$), so every domain-cost inequality inherits the forced cost geometry rather than an ad hoc metric. No downstream edges are recorded yet; the def is infrastructure for the module's structural certificate.
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