Character Moduli, Regular Extension, and Rational Rank for Reciprocal Cost

We study non-negative solutions to the reciprocal cost law on positive rationals and determine which of them admit regular extensions to positive reals. Using the substitution H=1+F, we reduce the reciprocal composition law to d’Alembert’s functional equation. We prove that every non-negative solution on Q>0 is determined by one real weight αp for each prime p, with only a global sign identification. Thus, the rational solution space is infinite-dimensional. We prove that every non-negative rational solution has an algebraic extension to R>0, but a regular extension exists exactly when the prime weights satisfy αp=λlogp for some λ∈R. In this case, the extension is unique and belongs to the one-parameter family Fλ(x)=cosh(λlogx)−1. Otherwise, the rational solution is unbounded on every nonempty open subset of Q>0, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of R>0, where the alternative is governed by rational rank. Finally, we show that the unit logarithmic curvature selects the canonical reciprocal cost J(()x)=(x+x−1)/2−1, whereas, for a carrier G whose logarithm is dense in R, the single asymptotic condition F(es)∼s2/2 implies both regularity and calibration.