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

Abstract

We consider the recovery of a common denominator in a finite fractional-power law $y_i=p\left(x_i^{1/n_0}\right)+{\epsilon }_i,p\left(t\right)={\displaystyle \sum _{k=0}^d}{a}_k{t}^k,$ from observations with positive abscissae. For every candidate denominator n, the vectors generated by $1,x^{1/n},\dots ,x^{d/n}$ 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 $\delta <1$, its change is bounded uniformly by $4\delta /(1-\delta )$. A recovery theorem then separates two roles of the ordered penalty $\lambda n/N$: 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 $a^{3/2}$ law. The results also identify regimes in which no reliable denominator claim should be made.

1. Introduction

Fractional powers occur naturally in local algebraic expansions, similarity laws, singular perturbation theory, and empirical scaling relations. The focus is a deliberately restricted question. Given observations ${\left\{(x_i,y_i)\right\}}_{i=1}^m$ with $x_i>0$, can one distinguish among models

$$y\approx p\left(x^{1/n}\right),p\left(t\right)=a_0+a_{1}t+\cdots +a_d{t}^d,$$

(1)

when the degree d and a finite set of candidate denominators have been specified? The target is the common denominator n, not an arbitrary power or an unrestricted symbolic expression.

For a candidate n, evaluating the functions $1,x^{1/n},\dots ,x^{d/n}$ at the sampled abscissae gives a matrix $X_n$. The least-squares fit is the orthogonal projection of the response vector onto ${\mathcal{V}}_n=\mathrm{col}\left(X_n\right)$. The nonlinear-looking comparison of fractional scales is consequently reduced to a finite collection of linear projections. What remains difficult is identification: different ${\mathcal{V}}_n$ may approximate one another extremely well on a bounded range, and some may fit the same finite sample exactly.

Normalising by the observed response introduces a first-order perturbation, and a noisy residual can reverse the order induced by the penalty unless $\lambda$ also satisfies a lower bound. The uniform perturbation bound is therefore derived from the normalised response vector, and both sides of the admissible interval for $\lambda$ are established.

All design spaces share the intercept vector $\mathbf{1}$, so their smallest ordinary principal angle is identically zero. Consequently, this angle cannot be assumed to be positive. The analysis uses instead the signal-dependent noiseless residual margin. Friedrichs angles after removing common directions remain useful as an observable warning about near-coincidence, but they are not substituted for the required signal-specific gap.

1.1. Relation to Existing Approaches

Classical fractional polynomial regression selects one or two powers from a prescribed set and was developed primarily to model a flexible covariate effect [1,2,3]. Its powers need not share a denominator and it does not seek the ramification index of an expansion. In contrast, model (1) selects a structured block $\{0,1/n,\dots ,d/n\}$ as one object. The two approaches overlap, but their inferential targets differ.

Dictionary methods and symbolic regression search much larger expression classes. SINDy uses sparsity over a user-specified library [4,5]; AI Feynman and modern symbolic regression systems explore compositional structure and accuracy–complexity trade-offs [6,7]. Their flexibility is valuable when the form of the law is unknown. It also changes the question: an unrestricted search may find an accurate expression without identifying a common denominator. Recent benchmarking work stresses that comparisons are meaningful only when representation capacity and tuning budgets are made explicit [7,8].

Allowing n to vary continuously produces a separable nonlinear least-squares problem: the coefficients are linear once n is fixed. This is a standard setting for variable projection [9,10,11]. Our discrete estimator can therefore be viewed as a constrained, interpretability-oriented counterpart of continuous variable projection. Cross-validation provides a natural data-driven comparison and tuning device [12]; information criteria are less informative when all candidate spaces have the same dimension.

Table 1 summarises the distinction.

Table 1. Scope of the compared methods.

Method	Candidate Representation	Primary Target	Selection Device
Common-denominator projection	$\{x^{k/n}:0\le k\le d\}$ as a block	Integer denominator n	Residual plus ordered penalty
Fractional polynomial	A small number of powers from a prescribed set	Predictive functional form	Deviance tests or information criteria
Fractional dictionary	Arbitrary d powers from the union dictionary	Sparse fractional representation	Cross-validation
Continuous variable projection	$\{x^{k/\nu }:0\le k\le d\}$, $\nu \in [1,N]$	Real exponent parameter $\nu$	Nonlinear least squares
Symbolic regression	Broad expression grammar	Predictive symbolic formula	Accuracy–complexity search

1.2. Contributions and Organisation

The contributions are the following.

• A uniform, explicit perturbation bound is proved for the normalised projection residual, including the required condition $\delta <1$.
• The recovery theorem states separate lower and upper requirements on $\lambda$ and distinguishes the unique-fit and aliased-fit cases.
• The theoretical result is described as a finite-sample stability guarantee. A Gaussian high-probability corollary is given without claiming standard $m\to \infty$ consistency.
• A complete algorithm, data-dependent diagnostics, and operational rules for d, N, and $\lambda$ are supplied.
• Reproducible comparisons use baselines with comparable fractional-power capacity, and the sensitivity analysis includes a real-data example.

Section 2 defines the model and finite-grid identifiability. Section 3 contains the perturbation and recovery results. Section 4 gives the implementation and tuning rules. Section 5 reports the experiments. Limitations and conclusions appear in Section 6 and Section 7.

2. Model, Identifiability, and Geometric Diagnostics

Let $\mathcal{G}\subset \mathbb{N}$ be a nonempty finite candidate grid and put $N=\max \mathcal{G}$. For distinct $x_i>0$, define

$$X_n=\left[\begin{array}{cccc}1& x_1^{1/n}& \dots & x_1^{d/n}\\ ⋮& ⋮& & ⋮\\ 1& x_m^{1/n}& \dots & x_m^{d/n}\end{array}\right],{\mathcal{V}}_n=\mathrm{col}\left(X_n\right).$$

(2)

We assume $m>d+1$ and $\mathrm{rank}\left(X_n\right)=d+1$ for every $n\in \mathcal{G}$. If $Q_n$ is a thin orthonormal factor obtained from a QR factorisation or an SVD of a column-scaled $X_n$, then $P_n=Q_n{Q}_n^T$ is the orthogonal projector onto ${\mathcal{V}}_n$. Forming $X_n{\left(X_n^T{X}_n\right)}^{-1}{X}_n^T$ is unnecessary and is avoided numerically.

For a nonzero vector $z\in {\mathbb{R}}^m$, write

$$R_n\left(z\right)={\displaystyle \frac{\parallel (I-P_n){z\parallel }_2^2}{{\parallel z\parallel }_2^2}}\in [0,1].$$

(3)

Let $y^0$ denote the noiseless response, $y=y^0+e$ the observation, and

$$\mathcal{M}=\{n\in \mathcal{G}:R_n\left(y^0\right)=0\}$$

(4)

the set of exact fits on the sampled abscissae.

Assumption 1 (Finite-grid identifiability). 

There is a target $n_0\in \mathcal{G}$ such that $n_0=\min \mathcal{M}$. Thus $y^0\in {\mathcal{V}}_{n_0}$ and no smaller candidate fits the noiseless sample exactly.

In an algebraic application, Assumption 1 is motivated by a minimal ramification index. It is nevertheless a statement about the chosen function class and finite sampling design; it cannot be inferred from a small noisy sample without qualification.

Proposition 1 (Generic exclusion of accidental exact fits). 

Let $f_0\left(x\right)=p\left(x^{1/n_0}\right)$, and suppose that, as functions on an open interval $I\subset (0,\infty )$, $f_0$ does not belong to $\mathrm{span}\{1,x^{1/n},\dots ,x^{d/n}\}$ for a candidate n. If $m\ge d+2$ and the $x_i$ are sampled independently from a distribution with a density on I, then

$$\mathrm{Pr}\{{(f_0\left(x_1\right),\dots ,f_0\left(x_m\right))}^T\in {\mathcal{V}}_n\}=0.$$

The statement holds simultaneously for every such n on a finite grid.

Proof. 

The $d+2$ functions $1,x^{1/n},\dots ,x^{d/n},f_0\left(x\right)$ are linearly independent. Hence, their evaluation determinant at $d+2$ points is a nonzero real-analytic function of those points. Its zero set has Lebesgue measure zero. Membership of the full response vector in ${\mathcal{V}}_n$ would force this determinant to vanish for the first $d+2$ observations. A finite union over n still has probability zero.    □

The relevant separation quantity is the signal-dependent margin

$$g=\underset{n\in \mathcal{G}\setminus \mathcal{M}}{\min }{R}_n\left(y^0\right),$$

(5)

with the convention $g=+\infty$ if every candidate is in $\mathcal{M}$. Since the grid is finite, $g>0$ whenever $\mathcal{G}\setminus \mathcal{M}$ is nonempty. The magnitude of g, rather than finiteness alone, determines whether the problem is stable under noise.

Why the Ordinary Smallest Angle Is Not Usable

Every ${\mathcal{V}}_n$ contains $\mathbf{1}$. Consequently, the smallest principal angle between any two model spaces is zero. For an x-only diagnostic one may remove their common intersection ${\mathcal{C}}_{nr}={\mathcal{V}}_n\cap {\mathcal{V}}_r$ and compute the Friedrichs angle

$$\cos {\theta }_F(n,r)=\underset{\begin{array}{c}u\in {\mathcal{V}}_n\cap {\mathcal{C}}_{nr}^{\perp },\parallel u\parallel =1\\ v\in {\mathcal{V}}_r\cap {\mathcal{C}}_{nr}^{\perp },\parallel v\parallel =1\end{array}}{\max }\left|u^{T}v\right|.$$

(6)

A small nontrivial angle warns that some directions in two spaces are hard to distinguish. It does not by itself lower-bound (5), because the actual signal may point in a better or worse separated direction. We therefore use g in the theorem and reserve ${\theta }_F$, scaled condition numbers, and observed residual gaps for diagnostics.

3. Perturbation and Finite-Sample Recovery

The noise level is measured relative to the noiseless response:

$${\parallel e\parallel }_2\le \delta {\parallel y^0\parallel }_2,0\le \delta <1.$$

(7)

No probabilistic assumption is needed in this section.

Lemma 1 (Uniform perturbation of the normalised residual). 

Under (7), $y\ne 0$ and, for every orthogonal projector $P_n$,

$$|R_n(y)-R_n(y^0)|\le \Delta (\delta ):={\displaystyle \frac{4\delta }{1-\delta }}.$$

(8)

The bound is uniform in n, m, and the rank of $P_n$.

Proof. 

Put $u=y/{\parallel y\parallel }_2$ and $u_0=y^0/{\parallel y^0\parallel }_2$. The reverse triangle inequality and (7) give ${\parallel y\parallel }_2\ge (1-\delta ){\parallel y^0\parallel }_2>0$. Moreover,

$$\parallel u-u_0{\parallel }_2\le {\displaystyle \frac{{\parallel e\parallel }_2}{{\parallel y\parallel }_2}}+{\displaystyle \frac{{|\parallel y\parallel }_2-\parallel y^0{\parallel }_2|}{{\parallel y\parallel }_2}}\le {\displaystyle \frac{2\delta }{1-\delta }}.$$

For $Q_n=I-P_n$, which is also an orthogonal projector,

$$|\parallel Q_n{u\parallel }_2^2-\parallel Q_n{u}_0{\parallel }_2^2|=|⟨Q_n(u+u_0),Q_n(u-u_0)⟩|\le 2\parallel u-u_0{\parallel }_2.$$

Combining the two inequalities proves (8).    □

The estimator minimises

$$J_{\lambda }(n;y)=R_n\left(y\right)+\lambda {\displaystyle \frac{n}{N}},n\in \mathcal{G},$$

(9)

where $\lambda \ge 0$. The term $\lambda n/N$ is an ordered preference for a smaller denominator. It is not a dimension penalty: all ${\mathcal{V}}_n$ have dimension $d+1$ under the rank assumption.

To state the exact conditions, define

$$\begin{array}{cccc}\hfill h& =\underset{r\in \mathcal{M}\setminus \left\{n_0\right\}}{\min }{\displaystyle \frac{r-n_0}{N}},\hfill & \hfill & h=+\infty \mathrm{if}\mathcal{M}=\left\{n_0\right\},\hfill \end{array}$$

(10)

$$\begin{array}{cccc}\hfill {\rho }_{-}& =\underset{n\in \mathcal{G},n<n_0}{\max }{\displaystyle \frac{n_0-n}{N}},\hfill & \hfill & {\rho }_{-}=0\mathrm{if}\mathrm{no}\mathrm{candidate}\mathrm{is}\mathrm{below}{n}_0.\hfill \end{array}$$

(11)

Theorem 1 (Finite-sample recovery on a fixed grid). 

Let Assumption 1 and (7) hold, and write $\Delta =\Delta \left(\delta \right)$. Suppose

$$g>2\Delta ,\lambda {\rho }_{-}<g-2\Delta ,$$

(12)

and, if $\mathcal{M}\ne \left\{n_0\right\}$, also

$$\lambda h>\Delta .$$

(13)

Then, $n_0$ is the unique minimiser of $J_{\lambda }(n;y)$ on $\mathcal{G}$. If $\mathcal{M}=\left\{n_0\right\}$, condition (13) is void; in particular, $\lambda =0$ is permitted whenever $g>2\Delta$.

Proof. 

For $n\notin \mathcal{M}$, Lemma 1 gives $R_n\left(y\right)\ge g-\Delta$, whereas $R_{n_0}\left(y\right)\le \Delta$. If $n>n_0$, the penalty difference is positive, so $J_{\lambda }(n;y)-J_{\lambda }(n_0;y)>0$. If $n<n_0$, then

$$J_{\lambda }(n;y)-J_{\lambda }(n_0;y)\ge g-2\Delta -\lambda {\rho }_{-}>0$$

by (12).

Now let $r\in \mathcal{M}\setminus \left\{n_0\right\}$. Assumption A1 implies $r>n_0$. Both exact-fit residuals observed under noise lie in $[0,\Delta ]$, and hence

$$J_{\lambda }(r;y)-J_{\lambda }(n_0;y)\ge -\Delta +\lambda {\displaystyle \frac{r-n_0}{N}}\ge -\Delta +\lambda h>0.$$

All competing candidates have therefore been excluded.    □

When aliases are possible, the simple but conservative interval

$$N\Delta <\lambda <g-2\Delta$$

(14)

is sufficient because $h\ge 1/N$ and ${\rho }_{-}\le 1$. It is nonempty if $g>(N+2)\Delta$. The two-sided restriction makes the trade-off explicit: too little regularisation cannot resolve noisy aliases, while too much can force a smaller but incorrect denominator.

Remark 1 (No standard consistency claim). 

Theorem 1 concerns a fixed finite grid, fixed sampled abscissae, and a bounded relative perturbation. It is not a proof that $\mathrm{Pr}(\widehat{n}=n_0)\to 1$ as $m\to \infty$. Such a result would require a sampling scheme, a margin bounded uniformly away from zero, and control of the relative noise as m changes.

Proposition 2 (Gaussian high-probability guarantee). 

If $e\sim N(0,{\sigma }^2{I}_m)$, then with probability at least $1-\alpha$,

$${\displaystyle \frac{{\parallel e\parallel }_2}{\parallel y^0{\parallel }_2}}\le {\delta }_{\alpha }:={\displaystyle \frac{\sigma \{\sqrt{m}+\sqrt{2\log (1/\alpha )}\}}{\parallel y^0{\parallel }_2}}.$$

(15)

If ${\delta }_{\alpha }<1$ and the conditions of Theorem 1 hold with $\Delta \left({\delta }_{\alpha }\right)$, recovery has probability at least $1-\alpha$.

Proof. 

The displayed norm bound is the standard Gaussian concentration inequality for the Euclidean norm. On that event, Theorem 1 applies.    □

4. Algorithm, Tuning, and Diagnostics

The arithmetic cost for a grid $\{1,\dots ,N\}$ is $O(Nm{d}^2+N{d}^3)$ using thin QR decompositions. Column scaling does not alter ${\mathcal{V}}_n$ but avoids magnitude-driven condition numbers. It cannot remove near-collinearity caused by clustered transformed abscissae (Algorithm 1).

Algorithm 1 Complete computational workflow

Finite-grid projection-residual selection Input: positive abscissae x, nonzero response y, degree d, candidate grid $\mathcal{G}$, and a grid $\mathrm{\Lambda }$ of penalty values. Check $m>d+1$, distinct positive $x_i$, and nonzero y. For each $n\in \mathcal{G}$, form $X_n$ using $x_i^{k/n}=\exp \left\{(k/n)\log x_i\right\}$; scale every column to unit norm. Compute a thin QR factorisation (SVD if rank is doubtful), retain $Q_n$, and record the scaled condition number. Evaluate $R_n\left(y\right)=\parallel y-Q_n{Q}_n^T{y\parallel }_2^2/{\parallel y\parallel }_2^2$. Select $\lambda$ either from a valid theoretical interval when an external noise bound is available, or by the K-fold procedure below. Return $\widehat{n}=\arg {\min }_{n\in \mathcal{G}}\{R_n\left(y\right)+\lambda n/N\}$ and the least-squares coefficients. Report the full residual curve, objective gap, scaled condition numbers, nontrivial Friedrichs angles, fold/bootstrap selection frequencies, and whether the selected n changes when d or the upper grid limit changes. Warning rule: do not report a unique denominator without qualification when the objective gap is small relative to the estimated perturbation, the selected design is nearly rank deficient, or resampling is unstable.

4.1. A Reproducible Rule for Choosing $\lambda$

When a credible relative noise bound and estimates of the relevant margins are available, inequalities (12) and (13) give a rule rather than a heuristic: choose $\lambda$ inside the nonempty admissible interval. These bounds are conservative, and g, h, and ${\rho }_{-}$ are not all known from a single noisy sample.

Our default practical rule is therefore K-fold cross-validation on a predeclared grid $\mathrm{\Lambda }$ that includes zero. For each fold and each $\lambda$, the exponent is selected on the training part by (9); its coefficients are fitted there and the squared prediction error is evaluated on the validation part. The chosen $\lambda$ minimises the mean validation error, with ties resolved in favour of the smaller $\lambda$. This is a predictive tuning rule, not a proof of denominator recovery. If it selects $\lambda =0$, the data provide no cross-validated reason to impose the ordered preference.

4.2. Choosing the Degree and Candidate Grid

An underestimated degree generally leaves $y^0$ outside ${\mathcal{V}}_{n_0}$ and can make a wrong denominator fit better. An overestimated degree enlarges every space, reduces residual gaps, and worsens conditioning. If d is unknown, it should be selected jointly with n using held-out prediction or an information criterion that penalises the dimension $d+1$. The ordered $n/N$ term does not replace that dimension penalty.

Domain information should set the initial upper limit N. A practical sequential rule is to enlarge $\{1,\dots ,N\}$ and retain a denominator claim only while (i) the selected n is stable, (ii) the observed objective gap does not collapse, and (iii) the scaled condition numbers and nontrivial angles remain acceptable. If $N<n_0$, recovery is impossible. An unnecessarily large N adds near-duplicate transformed grids, worsens multiple comparison, and may create poorly conditioned spaces.

4.3. A Data-Specific Stability Statement

Suppose $\widehat{n}$ minimises $J_{\lambda }(n;y)$ and an external perturbation radius $\eta <1$ is credible relative to ${\parallel y\parallel }_2$. Applying Lemma 1 with y as the reference shows that the selected model is unchanged throughout that response ball if

$$\underset{n\ne \widehat{n}}{\min }\{J_{\lambda }(n;y)-J_{\lambda }(\widehat{n};y)\}>2\Delta \left(\eta \right).$$

(16)

This certifies local stability of the computed decision. It does not certify that the assumed fractional-power family is scientifically correct.

5. Numerical Experiments

5.1. Protocol and Reproducibility

All random experiments use Python 3.12.14 with NumPy 2.3.5 and SciPy 1.17.0 default generator with master seed 20260904. Unless stated otherwise, the $x_i$ are regenerated independently in every trial from the log-uniform distribution on $[1,{10}^6]$; samples for different m are not nested. Conditional on the sampled $x_i$, independent Gaussian errors have standard deviation

$$\sigma ={\displaystyle \frac{\parallel y^0{\parallel }_2}{\sqrt{m}}}{10}^{-\mathrm{SNR}/20},$$

so that the displayed SNR is nominal. Each table entry is based on 500 trials. The candidate grid is $\mathcal{G}=\{1,\dots ,6\}$ unless varied explicitly. Five folds are newly randomised for each trial. The penalty grid is

$$\mathrm{\Lambda }=\{0,{10}^{-7},3·{10}^{-7},{10}^{-6},3·{10}^{-6},{10}^{-5},3·{10}^{-5},{10}^{-4},3·{10}^{-4},{10}^{-3},3·{10}^{-3},{10}^{-2}\}.$$

The supplementary script (Supplementary Materials) reproduces the tables and records the software-independent data source for the real example.

5.2. Baselines with Comparable Representation Capacity

The main experiment uses $d=1$ and $y^0=1+3x^{1/n_0}$ for $n_0\in \{2,3\}$. We compare:

• PR-CV: criterion (9), with $\lambda$ selected by the cross-validation rule above;
• PR-0: unpenalised residual selection on the same spaces;
• n-CV: direct five-fold selection of n on the same spaces;
• continuous: variable projection over $\nu \in [1,6]$, counted as correct when the rounded minimiser equals $n_0$;
• FD-CV: cross-validated selection of one power from $\{k/n:1\le k\le d,1\le n\le 6\}$, reduced to its denominator.

Thus, none of the baselines is handicapped by an integer-only dictionary.

The structured projection criterion is competitive with the methods having the same representational capacity, but Table 2 does not support a claim of universal superiority. Differences are small at high SNR. At 10 dB, $n_0=3$ is markedly harder than $n_0=2$, especially for $m=20$. The median cross-validated penalty is zero in every row; positive penalties account for the modest improvement over PR-0 in some difficult cases. This observation is consistent with Theorem 1: when the exact minimiser is unique, there is no theoretical lower requirement on $\lambda$.

Table 2. Exact denominator recovery proportions over 500 trials.

${\mathit{n}}_0$	m	SNR	PR-CV	PR-0	n-CV	Continuous	FD-CV
2	20	10	0.758	0.752	0.666	0.762	0.666
2	20	20	1.000	1.000	0.924	1.000	0.924
2	20	30	1.000	1.000	1.000	1.000	1.000
2	50	10	0.940	0.940	0.886	0.954	0.886
2	50	20	1.000	1.000	1.000	1.000	1.000
2	50	30	1.000	1.000	1.000	1.000	1.000
3	20	10	0.504	0.484	0.382	0.478	0.382
3	20	20	0.954	0.928	0.816	0.948	0.816
3	20	30	1.000	1.000	0.994	1.000	0.994
3	50	10	0.794	0.772	0.688	0.768	0.688
3	50	20	1.000	0.996	0.980	1.000	0.980
3	50	30	1.000	1.000	1.000	1.000	1.000

5.3. Degree Mis-Specification, Grid Size, Sampling, and Conditioning

For the sensitivity study, the noiseless law has $n_0=3$, true degree three, and coefficients scaled so that several terms remain visible on $[1,{10}^6]$:

$$y^0=0.2+{10}^{-2}{x}^{1/3}-7·{10}^{-5}{x}^{2/3}+5·{10}^{-7}x.$$

The nominal SNR is 40 dB. Table 3 changes one design choice at a time. Because the degree-sensitivity rows allow model mis-specification, the fourth column reports the signed residual contrast $c_d={\min }_{n\ne n_0}\{R_n\left(y^0\right)-R_{n_0}\left(y^0\right)\}$, rather than the margin g in Equation (5). When $y^0\in {\mathcal{V}}_{n_0}$, this contrast equals g; otherwise, it may be negative, indicating that a competing denominator fits the mis-specified family better than $n_0$. Condition numbers refer to column-scaled matrices.

Table 3. Sensitivity to the fitted degree and upper grid limit ($m=30$).

Factor	Setting	Recovery	Median Signed Contrast	Median Max. ${\mathit{\kappa }}_2$
Fitted degree	$d=2$	0.052	$-1.45·{10}^{-4}$	17.7
	$d=3$	0.972	$3.86·{10}^{-5}$	107.7
	$d=4$	0.556	$2.11·{10}^{-7}$	762.2
	$d=5$	0.630	$5.62·{10}^{-9}$	5865.8
Grid limit	$N=2$	0.000	not applicable	not applicable
	$N=3$	1.000	$6.08·{10}^{-4}$	62.5
	$N=6$	0.966	$3.96·{10}^{-5}$	107.6
	$N=10$	0.972	$3.97·{10}^{-5}$	285.1

Underfitting to $d=2$ changes the model and usually changes the selected denominator. Overfitting enlarges the spaces: the noiseless margin collapses by orders of magnitude and conditioning deteriorates. Choosing $N=2$ omits the truth. Here, $N=10$ does not reduce recovery relative to $N=6$, but it more than doubles the median maximum condition number. Thus, a large grid is not automatically harmful, but it is not cost-free and must be checked.

Table 4 reports the effects of the sampling distribution and sample size. Uniform sampling on a six-decade interval places most observations at large x and provides little information near the lower end.

Table 4. Effect of the sampling distribution and sample size ($d=3$, $N=6$).

Sampling	m	Recovery	Median Noiseless Margin	Median Max. ${\mathit{\kappa }}_2$
Uniform	8	0.060	$4.69·{10}^{-9}$	10,036.8
Uniform	12	0.062	$3.18·{10}^{-8}$	5247.6
Uniform	20	0.112	$9.56·{10}^{-8}$	3647.8
Uniform	50	0.244	$3.73·{10}^{-7}$	2190.4
Log-uniform	8	0.528	$2.12·{10}^{-6}$	433.9
Log-uniform	12	0.754	$9.06·{10}^{-6}$	194.1
Log-uniform	20	0.916	$2.81·{10}^{-5}$	124.5
Log-uniform	50	0.998	$4.99·{10}^{-5}$	99.6

Uniform sampling on a six-decade interval places most observations at large x and provides little information near the lower end. In this example, it produces both small margins and poor conditioning. Log-uniform sampling covers the scale more evenly; increasing m then improves the observed recovery. The comparison is specific to this design and should not be read as a universal optimal-design statement.

5.4. Real Data: Planetary Orbital Periods

The NASA/NSSDC Planetary Fact Sheet reports mean distance from the Sun and orbital period [13]. We use the eight planets, omit the Moon and Pluto, and scale distance and period by the corresponding Earth values. Table 5 lists the original, unscaled planetary data used. The physical reference law is Kepler’s $P\propto a^{3/2}$, which lies in the $d=3$, $n=2$ model through the $k=3$ term.

Table 5. Planetary data used in the real-data experiment.

Planet	Distance (${10}^6$ km)	Period (Days)	Planet	Distance (${10}^6$ km)	Period (Days)
Mercury	57.9	88.0	Jupiter	778.6	4331
Venus	108.2	224.7	Saturn	1433.5	10,747
Earth	149.6	365.2	Uranus	2872.5	30,589
Mars	227.9	687.0	Neptune	4495.1	59,800

Table 6 reports the normalised full-data residuals for the planetary data with $d=3$ and $n\in \{1,\dots ,6\}$.

Table 6. Normalised full-data residuals for the planetary data ($d=3$).

$\mathit{n}=1$	$\mathit{n}=2$	$\mathit{n}=3$	$\mathit{n}=4$	$\mathit{n}=5$	$\mathit{n}=6$
$5.55·{10}^{-6}$	$4.27·{10}^{-7}$	$1.05·{10}^{-4}$	$4.46·{10}^{-4}$	$8.71·{10}^{-4}$	$1.29·{10}^{-3}$

Four-fold cross-validation chooses $\lambda =0$ and the discrete estimator selects $n=2$; direct cross-validation over n also selects 2. The fitted coefficient of $a^{3/2}$ is $0.989$ after Earth normalisation, with smaller coefficients $0.068,-0.110,0.049$ on the lower powers. A continuous variable-projection fit returns $\nu =1.278$ because a cubic in $a^{1/\nu }$ can interpolate the short series in a different way. This is a useful distinction: the discrete grid encodes a rational-structure prior, while the continuous model optimises prediction over a larger family.

6. Discussion and Limitations

The theoretical and numerical results support a deliberately limited claim. The method is a transparent and inexpensive selector for a predeclared family of common-denominator models. Its guarantee is conditional on a finite-sample residual margin, a relative noise bound, and finite-grid identifiability. These conditions can fail even when the underlying functional representation is unique, because finite samples may make competing spaces nearly coincident.

No diagnostic can ensure correctness from the observed data alone. The residual curve, resampling frequencies, condition numbers, nontrivial angles, and stability under changes of d and N are warning devices. They should be reported together. Equation (16) gives a rigorous local stability statement only when an external error radius is credible.

The ordered penalty is not a conventional complexity penalty. All fixed-d models have the same number of coefficients. Its justified role is to prefer a smaller denominator among nearly tied or exactly aliased structured models. Cross-validation may reasonably choose zero, as it often did in our experiments. If the degree is selected at the same time, a separate dimension penalty or held-out criterion is necessary.

The simulations expose several limitations. Recovery can be poor at low SNR, for an incorrect degree, with a small sample, or when the sampling design compresses the transformed abscissae. A larger candidate grid may introduce near-duplicate spaces and numerical instability. The real-data experiment is physically interpretable but small and unusually clean; it does not establish performance for heteroscedastic, dependent, or pre-asymptotic measurements.

Finally, the framework assumes $x_i>0$, a fixed polynomial truncation, and a single common denominator. Logarithmic factors, multiple scaling regimes, unknown change points, and general Puiseux series with negative or mixed powers require an enlarged model class. In such settings, fractional polynomial, dictionary, or symbolic regression methods may be more appropriate.

7. Conclusions

Fractional-power denominator recovery can be organised as selection among a finite collection of linear subspaces. The useful part of this formulation is not that it removes identifiability difficulties, but that it makes them visible through projection residuals and numerical diagnostics.

The perturbation bound is first-order in the relative noise and explicitly requires a nonzero normalisation denominator. The recovery theorem shows when an ordered penalty can separate exact aliases without overwhelming genuine residual differences. It also makes clear that no lower penalty is needed in the generic unique-fit case and that the result is finite-sample, not asymptotic consistency.

Balanced experiments show competitive performance when the candidate family is well separated, and substantial degradation when degree, sampling, or conditioning erodes the residual margin. The planetary example demonstrates a setting in which the discrete rational structure has a direct scientific interpretation. Future work should derive sampling-design bounds for the signal-dependent margin and extend the model to jointly selected degrees, heteroscedastic errors, and multiple asymptotic regimes.