domainCost
plain-language theorem explainer
Domain cost assigns to a mass-energy pair the recognition cost of their ratio: J(m/e). Anyone working the recognition-Heisenberg derivation in this module uses it as the local cost of a mass-to-energy scale. The body is a one-line specialization of the unique J-cost functional.
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
The module derives a recognition form of the Heisenberg bound from J-cost: $\Delta_J\cdot\Delta_\sigma\ge\hbar_R/2$ with $\hbar_R=J(\varphi),\hbar$, as a structural theorem (no sorry, no axioms).
Upstream, $J$ is the unique cost forced by the Recognition Composition Law: $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$). Docs across Cost and Gravity modules state that a genuine distinction (ratio not one) has strictly positive cost, and that $J$ is nonnegative for $x>0$.
Here the ratio is specialized to a mass-over-energy scale $m/e$. That is the natural dimensionless argument when comparing a mass rung to an energy scale inside the uncertainty setup.
proof idea
Pure definition: apply the shared $J$-cost functional to the ratio $m/e$. No lemmas, no tactics; the body is the term $J(m/e)$.
why it matters
Gives the local cost coordinate for the recognition-Heisenberg development in this Foundation module. Siblings (nonnegativity, evaluation identities, the canonical threshold, and the HUP3Deep certificate) sit on top of this abbreviation.
It ties the uncertainty story to the T5 landmark: $J$ is the unique cost from the Recognition Composition Law. The module framing is the RS rewrite $\Delta_J\cdot\Delta_\sigma\ge\hbar_R/2$ with $\hbar_R=J(\varphi),\hbar$, so domain cost is the mass-energy entry point into that product.
No downstream edges are recorded yet; the declaration is infrastructure for the certificate and threshold lemmas in the same file.
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