domainCost

The module treats superposition cost in Recognition Science: for a two-level state $\psi=\alpha|0\rangle+\beta|1\rangle$, the recognition cost is $J(|\alpha|/|\beta|)$. Equal superposition gives ratio $1$, hence $J(1)=0$, the global minimum; a maximally coherent balanced state therefore carries zero recognition cost.

The underlying functional is the unique J-cost forced by the Recognition Composition Law (T5): $J(x)=\frac{x+x^{-1}}{2}-1$ for $x>0$. Upstream definitions across Cost, Cosmology, Gravity, and Spiral all instantiate the same formula and record that genuine distinctions (ratio not one) have strictly positive cost and that $J$ is nonnegative on positives.

Domain cost simply specializes that functional to a ratio of two real parameters $m$ and $e$, which the module uses as stand-ins for magnitude or energy weights in a domain.

Pure definitional abbreviation: domainCost m e is defined to be Jcost (m / e). No proof obligations; the body is the single application of the imported J-cost functional to the quotient.

This is the primitive scalar of the Superposition Cost RS module (structural, zero sorry). Sibling lemmas immediately specialize it: evaluation at equal arguments, nonnegativity, and comparison against a canonical positive threshold that certifies when a superposition is costly. Those feed the module certificate SuperpositionCostCert.

In the broader forcing chain it sits under T5 J-uniqueness: every recognition imbalance is measured by the same $J$. The physical reading is that balanced coherent superpositions are free (cost zero) while any amplitude or domain skew pays a positive J-penalty, consistent with the module claim that maximally coherent states minimize recognition cost.