domainCost
plain-language theorem explainer
Domain cost scores a mass–energy pair by the recognition cost of their ratio: J(m/e). Astrophysicists working the RS pulsar-period module use it as the local cost functional on (m,e). The body is a one-line abbreviation of the forced J-cost.
Claim. For real numbers $m$ 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
Recognition Science measures mismatch of positive ratios by the unique cost $J(x) = \frac{x + x^{-1}}{2} - 1$, equivalently $\cosh(\log x) - 1$. Upstream modules (Cost, CoherenceCollapse, EnergyProcessingBridge, RefineTrigger, SpiralField) all expose this same $J$ as the functional forced by the Recognition Composition Law (T5 in the forcing chain).
This file is Astrophysics RS Module 10. The module doc fixes the structural setting: pulsar periods on the phi ladder, with typical scale $\phi^{-1},\mathrm{s} \approx 0.618,\mathrm{s}$ and millisecond scale $\phi^{-10},\mathrm{s} \approx 8.1,\mathrm{ms}$. Status is structural (zero sorry, zero axiom).
Domain cost specializes $J$ to a mass–energy ratio $m/e$, giving a scalar cost for astrophysical (m,e) pairs inside that module.
proof idea
Pure definitional abbreviation: domainCost m e is definitionally equal to Jcost (m / e). No tactics, no lemmas, no proof obligations.
why it matters
Gives the module a named cost on mass–energy pairs so later certificates (domainCost_nonneg, canonicalThreshold, RSAstro010Cert and siblings) can reason about nonnegativity and thresholds without reopening the J formula. It sits on the T5 J-uniqueness landmark: the same forced cost used across cosmology, gravity, and spiral modules. No downstream used_by edges are recorded yet; the immediate consumers are the sibling lemmas in this file that close the Module 10 structural certificate on pulsar period scales.
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