Model Selection for Asymptotic Scaling Laws via Projection Residuals: Finite-Sample Guarantees and Diagnostics

We consider the recovery of a common denominator in a finite fractional-power law yi=p(xi1/n0)+εi,p(t)=∑k=0daktk, from observations with positive abscissae. For every candidate denominator n, the vectors generated by 1,x1/n,…,xd/n form a linear model space. Denominator identification is therefore a finite model-selection problem, although the spaces need not be nested and can be nearly coincident. The normalised projection residual admits a finite-sample perturbation bound. If the relative noise is at most δ<1, its change is bounded uniformly by 4δ/(1−δ). A recovery theorem then separates two roles of the ordered penalty λn/N: an upper bound prevents the penalty from overriding a genuine residual gap, whereas a lower bound is needed only when several candidates fit the noiseless sample exactly. Thus, the result is a conditional finite-sample guarantee, not an asymptotic consistency theorem. Because every model contains the constant vector, ordinary smallest principal angles are zero; the appropriate geometric diagnostics are nontrivial Friedrichs angles, condition numbers, and signal-dependent residual gaps. Reproducible experiments compare the criterion with unpenalised selection, cross-validation over the same spaces, a continuous-exponent variable-projection fit, and a fractional-power dictionary of equal dimension. Further experiments vary the assumed degree, grid size, sampling design, sample size, and conditioning. An application to planetary orbital data recovers the denominator n=2 associated with Kepler’s a3/2 law. The results also identify regimes in which no reliable denominator claim should be made.