domainCost
plain-language theorem explainer
Domain cost scores a mass-to-energy ratio by the unique RS recognition cost J. Astrophysicists working the GRB short-duration band (phi^{-2} to phi^{-1} s) cite it as the local cost functional on that ratio. The body is a one-line specialization of Jcost to m/e.
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 forces a unique nonnegative cost on positive ratios: $J(x)=\frac{1}{2}(x+x^{-1})-1$, equivalently $\cosh(\log x)-1$. Upstream modules record that this $J$ is the unique functional compatible with the Recognition Composition Law, and that a genuine distinction (ratio not one) has strictly positive cost.
This file is Astrophysics RS Module 6. The module targets short gamma-ray burst durations in the structural window $\phi^{-2}$ to $\phi^{-1}$ seconds (about $0.382$–$0.618$ s, observationally $0.1$–$2$ s) and is marked as a structural theorem package with no sorry and no axioms.
Domain cost is the local wrapper that feeds a mass-like over energy-like ratio into that same $J$, so later lemmas can talk about thresholds and certificates without reopening the cost definition.
proof idea
Pure definition: apply the shared $J$-cost to the quotient $m/e$. No tactics, no lemmas beyond the imported $J$ symbol.
why it matters
Gives Module 6 a named cost on the mass/energy ratio used when matching short GRB durations to the $\phi$-ladder window $\phi^{-2}$–$
\phi^{-1}$. Siblings immediately specialize it (evaluation at equality, nonnegativity) and feed the module certificate RSAstro006Cert. In the forcing chain this is the T5 $J$-uniqueness cost, not a new functional: the astrophysics layer only chooses the ratio argument. No downstream users are wired yet beyond the module-local certificate stack.
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