Introduction

This paper establishes a finite-sample capacity characterization for covariate balancing in randomized experiments. Under independent uniform covariates, we consider centered prognostic functions consisting exactly of canonical main effects and pairwise interactions, with a common squared nonperiodic Sobolev restriction-norm budget pooled across components. Writing $n$ for the number of experimental units, $d$ for covariate dimension, and $s$ for component smoothness, the best achievable scaled excess variance of the unit-weight Horvitz–Thompson estimator for this class is of order $$\min \left\{1,{\left(\frac{\left(\genfrac{}{}{0ex}{}{d}{2}\right)}{n}\right)}^s\right\},n,d\ge 2,0<s\le 1.$$ One fair, outcome-independent assignment procedure attains this scale simultaneously across the stated smoothness range. The matching converse applies to every admissible covariate-dependent assignment law.

The statistical question is how much precision assignment can deliver when the outcome regression has many potentially relevant interactions. Pairwise structure associates each component with at most two covariates, while allowing the number of components to grow quadratically with ambient dimension. A pooled regularity budget bounds their total squared component norm and permits its allocation across coordinates and pairs to vary. Our characterization identifies $B=\left(\genfrac{}{}{0ex}{}{d}{2}\right)$, the coordinate-pair count, relative to sample size as the quantity governing worst-case excess variance. Below saturation, $B<n$, the rate is $(B/n{)}^s$; at and above saturation, it is constant order. For every fixed smoothness, vanishing minimax scaled excess is equivalent to $d=o(\sqrt{n})$, including along arbitrary integer dimension sequences.

The design problem fixes the estimator and specifies both the available information and the population comparison. An admissible design is a Borel assignment kernel using the complete original covariate array, with conditional treatment probability one-half for each unit and assignment independent of potential outcomes given covariates. Every original unit receives a treatment sign and enters the Horvitz–Thompson estimator at its original weight. Performance is evaluated by expected squared prognostic imbalance, averaging over covariate sampling and assignment. The supremum over prognostic functions lies outside that expectation, so each function is fixed across sample realizations. These choices define the minimax signing risk $R_s(n,d)$, the scaled population imbalance criterion in Definition 4, over the function and design classes in Definitions 2 and 3.

The causal interpretation follows from the excess-variance identity of Cytrynbaum (2026). For independent original units, admissible assignment, and finite potential-outcome second moments, scaled Horvitz–Thompson variance above the law-specific benchmark equals scaled expected squared imbalance of the conditional potential-outcome midpoint mean. Every prognostic function in our class is realized by a Gaussian outcome completion. Consequently, the minimax excess over all uniform-covariate outcome laws with square-integrable potential outcomes and the specified prognosis equals $R_s(n,d)$, as established in Lemma 2 and Theorem 3. The Gaussian subclass realizes the same capacity and provides a concrete calibration with scaled variance benchmark $4.01$.

The closest rate comparison is Cytrynbaum (2026). His correctly specified pairwise model admits a structured Gram–Schmidt Walk upper bound of order $(d^2\log n/n{)}^{s\wedge 1}$, while the full-dimensional smooth-class lower bound specializes to exponent $s$ in two dimensions. For our independent uniform sample and centered pooled order-two Sobolev class, Theorems 3 and 1 matches upper and lower bounds with explicit interaction-count dependence and a logarithm-free rate. The comparison includes the frequency-rich regime $B/n\to 0$ and the endpoint $s=1$. An existing random-matrix route supplies one-frequency saturation through Kogan et al. (2024); our characterization covers both that regime and the subcritical frequency-rich comparison at finite sample sizes.

The construction connects minimax experimental design (Kallus, 2018) to randomized balancing. In the subcritical regime, independent coordinate reflections and a common random shift supply ordered real spectral features. A finite boundary-rounding procedure protects an initial segment of these features while preserving coordinate means; the protected segment contracts as assignment coordinates reach their boundaries. The feature ordering is independent of smoothness. Combining its row-imbalance guarantees with the reflected nonperiodic Sobolev budget yields simultaneous attainment. In the saturated regime, the procedure uses independent fair signs. The complete assignment law and its guarantees are specified in Algorithm 1, Algorithm 2, and Theorem 1 under exact-real arithmetic conventions.

The converse uses a finite pair-cosine family with independent coefficient signs drawn before the covariate sample. Every realization satisfies the pooled Sobolev budget. A sample-size-dependent frequency cutoff supplies enough feature directions for a signing lower bound, including directions from pairs sharing an original coordinate. Averaging over this legal prior preserves the fixed-function population comparison and yields the full-design lower bound in Theorem 2. Together, the two comparisons have constants bounded uniformly over $0<s\le 1$. The uniform subclass also transfers the converse and the necessary dimension condition to the larger bounded-density outcome classes specified in Lemma 2.

The main capacity results are stated in Theorems 3 and 1.

The paper proceeds through related work, setup and assumptions, main results, the smoothness-independent assignment procedure, and discussion and extensions. The appendices develop causal variance transfer, spectral and rounding bounds, the independent-prior converse, and proof assembly, followed by proofs of the main results.

Related work

The closest econometric comparison is Cytrynbaum (2026). His excess-variance identity expresses the design-dependent part of Horvitz–Thompson variance as scaled expected squared prognostic imbalance, the criterion used in Definition 4. Writing $n$ for sample size, $d$ for covariate dimension, and $s$ for smoothness, his correctly specified pairwise model admits a structured Gram–Schmidt Walk (GSW) upper bound of order $(d^2\log n/n{)}^{s\wedge 1}$, supported by the pooled Matérn approximation calculation. His lower bound $n^{-2s/d}$ concerns a full-dimensional smooth class; its bivariate specialization has exponent $s$. For independent uniform covariates and the centered order-two pooled Sobolev class in Definition 2, the present characterization matches upper and lower bounds across sample size and dimension at order $\min \{1,(B/n{)}^s\}$, where $B=\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ is the number of coordinate pairs, throughout the parameter range $n,d\ge 2$ and $0<s\le 1$. The comparison therefore concerns the logarithmic factor and the explicit interaction-count dependence for this specified class; Proposition 1 supplies the class-transfer relation.

A precise comparison with random-matrix theory identifies an existing route to saturation. Applied to independent, centered, unit-variance cosine coordinates under uniform sampling, Kogan et al. (2024) implies the following spectral limit. Let $G$ denote the sample Gram matrix of the pair-product features using one cosine frequency per coordinate, normalized by $B$. As $n,d$ tend to infinity with $n/B\to \rho \in (0,1)$, where $\rho$ is the limiting sample-to-feature ratio, its smallest eigenvalue converges almost surely to $(1-\sqrt{\rho }{)}^2$. This positive spectral edge yields a constant-order signing lower bound for the normalized one-frequency submodel. The additional scope of Theorems 3 and 1 is the matched finite-sample characterization, including the frequency-rich regime $B/n\to 0$, together with attainment by the same assignment procedure for every smoothness in the stated range. The independent coefficient prior in Definition 9 supplies the growing frequency family for that subcritical comparison.

The criterion also belongs to the minimax experimental-design tradition. Kallus (2018) connects restrictions on conditional outcome functions to worst-case imbalance and optimal randomized designs, while Kallus (2020) studies mixed assignment distributions and randomization constraints. Here, Definitions 3 and 4 specifies fair, outcome-independent assignment laws and places the worst-function comparison outside the expectation over covariates and assignment. The lower bound in Theorem 2 applies to this entire admissible design class. Its role is to characterize the best achievable excess variance for the fixed Horvitz–Thompson criterion under the pooled smoothness restriction.

The attaining procedure connects this statistical objective to randomized balancing. The GSW design of Harshaw et al. (2024) combines assignment fairness with covariance control for augmented covariate features. The Edge-Walk method of Lovett et al. (2015) develops constrained random walks for partial coloring. The procedure in Algorithm 2 uses finite randomized boundary moves with an ordered family of balancing constraints. Its statistical application in Algorithm 1 uses a feature ordering independent of smoothness, allowing one assignment law to attain the comparison scale simultaneously across the stated Sobolev classes.

Chen et al. (2026) studies nonlinear feature balance through a normalized feature Gram matrix $K$. For $\varphi \in (0,1)$, the design has assignment covariance bounded by $\{\varphi I+(1-\varphi )K{\}}^{-1}$, and a fixed-schedule ridge approximation bound controls the actual Horvitz–Thompson variance. That comparison is conditional on the realized schedule and a chosen nonlinear feature map. The present result instead gives a population minimax capacity matched over an explicit pooled Sobolev class, with a single smoothness-independent construction and a converse for every admissible design.

Discrepancy-based integration provides a related use of randomized balancing with distinct sampling budgets and error criteria. Bansal et al. (2025) starts from $n^2$ independent uniform points and partitions them into $n$ sets of $n$ points. For each retained set, the mean squared integration error is bounded by ${\tilde{O}}_d({\sigma }_{\mathsf{SO}}(f{)}^2/n^2)$, where $f$ is the integrand and ${\sigma }_{\mathsf{SO}}(f)$ is its smoothed-out variation; the displayed order suppresses logarithmic factors. Chen et al. (2026) retains this quadratic input pool and develops bounds for weighted Fourier classes and low superposition dimension, with explicit coordinate-weight and dimension dependence. Ezeunala et al. (2026) obtains the smoothed-out-variation mean squared error bound while retaining $n$ points from $(1+\epsilon )n$ independent uniform inputs with high probability, for fixed $\epsilon \in (0,1)$; its constants and logarithmic factors depend on dimension and $\epsilon$.

Other thinning results emphasize different forms of approximation. The online procedure of Dwivedi et al. (2019) retains at least one point from every two consecutive uniform inputs and controls unnormalized counting discrepancy over axis-parallel rectangles. Kernel thinning in Dwivedi et al. (2024) compresses an input approximation into a smaller subset and bounds maximum mean discrepancy, equivalently worst-case integration error over the associated reproducing kernel Hilbert-space unit ball, under the stated regularity and input-radius conditions. In the present experiment, all $n$ original units receive treatment signs, and performance is measured by expected squared prognostic imbalance over a pooled nonperiodic Sobolev class. These sampling, function-class, and loss specifications make the comparison with integration results precise.

Finally, Sudijono et al. (2026) studies joint minimax choice of design and estimator for finite-population treatment-effect estimation with binary or bounded potential outcomes. For binary outcomes, Bernoulli randomization paired with a nonlinear estimator attains their minimax risk, for which they derive a second-order expansion. The present optimization fixes the Horvitz–Thompson estimator and varies covariate-dependent assignment under a smooth prognostic class. It characterizes the excess-variance gains available through assignment for that estimator and model.

Setup and assumptions

Throughout, $n,d\ge 2$ are integers and $0<s\le 1$. Unit indices $i$ range from $1$ to $n$, and coordinate indices $j$ and $\ell$ range from $1$ to $d$, with $j<\ell$ for coordinate pairs. A cube point is denoted by $x$. Integrals over cubes use Lebesgue measure, and torus integrals use normalized Haar probability. The notation $\Vert \cdot {\Vert }_2$ denotes the Euclidean norm for vectors and the $L^2$ norm for functions, as determined by the argument. Generic positive constants may change between occurrences; a subscript $s$ records permitted dependence on smoothness. For nonnegative quantities, $a{\lesssim }_{s}b$ means $a\le C(s)b$, and $a{\asymp }_{s}b$ means that both comparison inequalities hold.

Original-unit sampling

The full covariate sample $X=(X_1,\dots ,X_n)$ records the covariates of all original experimental units. Assignment takes values in the sign space ${\mathcal{Z}}_n=\{-1,1{\}}^n$. An assignment vector $Z=(Z_1,\dots ,Z_n)$ selects treatment through the indicators $A_i=(Z_i+1)/2$. A covariate-dependent assignment law $\pi$ is a Borel probability kernel from $[0,1{]}^{nd}$ to $\mathcal{P}({\mathcal{Z}}_n)$, the probability measures on the finite sign space. Thus the design can use the entire original covariate array when assigning every unit.

The sampling distribution fixes the population measure against which component centering, regularity, and expected imbalance are evaluated.

Independent uniform sampling is the exact uniform special case of the independent sampling and bounded-density condition in (Cytrynbaum, 2026). It supplies a common population distribution for the design criterion and the canonical component decomposition. Independence also separates the randomness of the original sample from the dependence among treatment assignments that the design creates.

Canonical components and regularity

The prognostic function $m$ describes the conditional mean of the potential-outcome midpoint. We work with its centered version, whose integral over the cube is zero. Under the product sampling distribution, the canonical main effects $g_j(m)$ and canonical pair effects $g_{j\ell }(m)$ are obtained by marginal integration. The following definition fixes their normalization and the interpretation of almost-everywhere equalities.

Coordinatewise centering assigns each pair effect to variation jointly associated with its two coordinates. Together with the zero integral of $m$, it fixes the intercept and identifies the components in the canonical ANOVA decomposition (Hoeffding, 1948; Sobol’, 1993). The specification restricts the prognostic function to the sum of these main and pair effects.

The exact order-two canonical ANOVA condition with zero intercept is the specification used here from (Cytrynbaum, 2026). It describes a population model in which main effects and pairwise interactions account for the prognostic function exactly. Canonical centering makes the allocation of regularity across components intrinsic to $m$.

Regularity is measured on the original cube by extension to Euclidean space. For a component dimension $p\in \{1,2\}$, the squared nonperiodic Sobolev restriction norm $N_{p,s}(g{)}^2$ is $$N_{p,s}(g{)}^2=\underset{\begin{array}{c}G\in L^1({\mathbb{R}}^p)\cap L^2({\mathbb{R}}^p)\\ G{|}_{[0,1{]}^p}=ga.e.\end{array}}{\inf }{\int }_{{\mathbb{R}}^p}{\left(1+\frac{\Vert \omega {\Vert }_2^2}{p}\right)}^s|\hat{G}(\omega ){|}^{2}d\omega .$$ Here the Euclidean Fourier transform $\hat{G}$, at angular frequency $\omega \in {\mathbb{R}}^p$, uses the convention $$\hat{G}(\omega )=(2\pi {)}^{-p/2}{\int }_{{\mathbb{R}}^p}{e}^{-\mathrm{i}{\omega }^{\mathsf{T}}u}G(u)du,$$ and the dimension-normalized weight is exactly $(1+\Vert \omega {\Vert }_2^2/p{)}^s$. The infimum runs over integrable, square-integrable extensions agreeing almost everywhere on the cube. We impose one common budget across all canonical components.

The pooled component Sobolev budget fixes the dimension-normalized restriction-norm convention used throughout this paper. Pooling bounds the total squared component norm while allowing its distribution across coordinates and pairs to vary. Consequently, the dimension dependence of the criterion reflects the number of available interaction directions under a common regularity budget.

These component conditions define the population class over which the design will be evaluated. Write ${\mathcal{H}}_{d,s}$ for this centered pooled Sobolev class.

The finite-valued Borel representatives permit evaluation at sampled covariates. Uniform sampling makes changes on Lebesgue null sets immaterial to the expected loss, so the class retains the usual almost-everywhere interpretation of Sobolev functions.

Assignment and signing loss

The assignment family combines covariate-dependent balancing with equal marginal treatment probabilities. Its first requirement concerns each unit’s assignment conditional on the complete covariate sample.

Conditional half-probability marginals are the fairness condition in (Cytrynbaum, 2026). Since signs take values in $\{-1,1\}$, each unit receives treatment with conditional probability $1/2$. The joint assignment distribution can nevertheless depend on all covariates and can correlate assignments across units to control prognostic imbalance.

The second requirement specifies the information available to the assignment law. Potential outcomes are generated before assignment, and the design draws signs using the covariate array.

Outcome-independent assignment is the information restriction in (Cytrynbaum, 2026). The sequential formulation expresses conditional independence for every pre-assignment outcome law: the design’s randomization uses covariates and its own randomness. In conjunction with conditional fairness, it supplies the assignment conditions needed for the causal variance identity below.

Let ${\mathcal{D}}_{n,d,s}$ denote the resulting admissible design class.

For a fixed admissible design and a fixed prognostic function, define the scaled signing loss ${\mathcal{L}}_{n,d,s}(\pi ,m)$ by $${\mathcal{L}}_{n,d,s}(\pi ,m)=\frac{4}{n}{\mathbb{E}}_{X,Z}\left[{\left(\sum _{i=1}^n{Z}_{i}m(X_i)\right)}^2\right].$$ The expectation integrates both the independent uniform sample and the conditional assignment law $\pi (X)$. The minimax signing risk $R_s(n,d)$ optimizes this population loss over admissible designs.

The order of optimization is substantive. A design specifies its assignment rule as a function of the covariate array; a fixed population function is then evaluated under that rule, with sampling and assignment averaged inside the loss. The supremum therefore ranges over functions fixed across sample realizations. This order measures how well a single covariate-based design controls population prognostic imbalance uniformly over ${\mathcal{H}}_{d,s}$.

To connect signing loss to treatment-effect estimation, fix the unit-weight Horvitz–Thompson estimator (Horvitz et al., 1952). In the next definition, $A(a)$ denotes the estimator as a function of a treatment vector $a$; the subscripted $A_i$ denotes unit $i$’s treatment indicator.

Conditional fairness gives each observed outcome its inverse assignment-probability weight. This estimator remains fixed throughout the design optimization in Definition 4.

Spectral assignment features

The assignment construction uses a reflected and commonly shifted representation of the original covariates. For an integer $p$, the period-two torus ${\mathbb{T}}_2^p=[0,2{)}^p$ has addition modulo two. Its coordinatewise fold $b_p$ is $$b_p(y{)}_h=\min \{y_h,2-y_h\},h=1,\dots ,p.$$ The reflected prognostic function $F=m\circ b_d$ is consequently a function on ${\mathbb{T}}_2^d$.

An independent fair bit array $(b_{ij})$, with $b_{ij}\in \{0,1\}$, selects the lifted covariates ${\tilde{X}}_i$ coordinatewise: $${\tilde{X}}_{ij}=\{\begin{array}{ll}{X}_{ij},& b_{ij}=0,\\ 2-X_{ij},& b_{ij}=1,\end{array}\text{interpreted modulo two}.$$ Let $T$ be a common Haar-uniform shift, independent of the covariates, reflections, and subsequent rounding randomness. The transformed covariate for original unit $i$ is $W_i=({\tilde{X}}_i+T)\mathrm{mod}2$. Each row used by the assignment procedure still has one entry for each original unit.

For an integer frequency $k\in {\mathbb{Z}}^d$, the torus character $e_k(y)=\exp (\mathrm{i}\pi k^{\mathsf{T}}y)$ gives the normalized Fourier coefficient $\hat{F}(k)$ through $$\hat{F}(k)={\int }_{{\mathbb{T}}_2^d}F(y)\overline{e_k(y)}dH(y),$$ where $H$ is normalized Haar probability. The frequency complexity $t(k)$ ranks frequencies by squared Euclidean magnitude divided by the number of active coordinates. The ordered real features $({\varphi }_h{)}_{h\ge 1}$ convert this ranking into the prescribed rows for assignment.

The support restriction matches the main and pair structure of the prognostic class. Ordering the real features before selecting rows gives a single prescribed balancing matrix for each sample size and dimension. Its independence from $s$ supplies the common feature input for the smoothness-independent assignment procedure.

Outcome laws and causal excess variance

The causal interpretation uses a single-unit law $P$ of covariates $U$ and potential outcomes $O(0),O(1)$. Define its average treatment effect and conditional prognostic function by $$\vartheta (P)={\mathbb{E}}_P[O(1)-O(0)],m_P(u)={\mathbb{E}}_P\left[\frac{O(1)+O(0)}{2}|U=u\right].$$ Let ${\mathcal{Q}}_{d,s}$ consist of these laws with uniform covariates, finite second moments for both potential outcomes, and $m_P=m$ almost everywhere for some $m\in {\mathcal{H}}_{d,s}$. Thus this outcome-law class fixes the prognostic restriction while allowing square-integrable conditional outcome distributions.

From $n$ independent copies of $P$, followed by assignment under $\pi (U_{1:n})$, the actual estimator is $${\hat{T}}_P=\frac{2}{n}\sum _{i=1}^n{Z}_i{O}_i\left(\frac{Z_i+1}{2}\right).$$ Under the admissibility conditions, its expectation is $\vartheta (P)$. The law-specific variance benchmark is $$\mathsf{V}(P)={\mathrm{Var}}_P\left({\mathbb{E}}_P[O(1)-O(0)\mid U]\right)+2{\mathbb{E}}_P{\mathrm{Var}}_P(O(1)\mid U)+2{\mathbb{E}}_P{\mathrm{Var}}_P(O(0)\mid U).$$ This definition supplies the benchmark directly in terms of the outcome law. It combines variation in the conditional treatment effect with the two conditional outcome variances.

Define the causal excess-variance criterion by $${\mathcal{V}}_n(\pi ,P)=n{\mathrm{Var}}_{P,\pi }({\hat{T}}_P)-\mathsf{V}(P).$$ The published excess-variance identity gives $${\mathcal{V}}_n(\pi ,P)=\frac{4}{n}{\mathbb{E}}_{P,\pi }\left[{\left(\sum _{i=1}^n{Z}_i{m}_P(U_i)\right)}^2\right]={\mathcal{L}}_{n,d,s}(\pi ,m_P)$$ for $P\in {\mathcal{Q}}_{d,s}$ and $\pi \in {\mathcal{D}}_{n,d,s}$ (Cytrynbaum, 2026). Hence the signing loss is the scaled actual Horvitz–Thompson variance above a benchmark specific to the outcome law.

A Gaussian causal completion supplies a concrete outcome law for every member of the prognostic class. Fix the mean-effect constant ${\tau }_0=0.2$ and heterogeneity amplitude $\delta =0.1$, and define the conditional treatment effect $\tau (x)$ by $$\tau (x)={\tau }_0+\delta \sqrt{2}\sin (2\pi x_1).$$ The Gaussian noise array $ϵ$ has law $\mathcal{N}(0,I_{2n})$, independently of $X$. The pre-assignment potential-outcome schedule $Y$ is generated by $$Y_i(0)=m(X_i)-\frac{\tau (X_i)}{2}+{ϵ}_i(0),Y_i(1)=m(X_i)+\frac{\tau (X_i)}{2}+{ϵ}_i(1).$$ Its conditional mean midpoint is $m(X_i)$, so its single-unit law belongs to ${\mathcal{Q}}_{d,s}$. The actual estimator $\hat{\theta }$, realized finite-population target ${\theta }_n$, and sampling-law target $\theta$ are $$\hat{\theta }=\frac{2}{n}\sum _{i=1}^n{Z}_i{Y}_i(A_i),{\theta }_n=\frac{1}{n}\sum _{i=1}^n(Y_i(1)-Y_i(0)),\theta =\mathbb{E}[{\theta }_n]={\tau }_0.$$ For this family, the law-specific benchmark is the constant $V^{\ast }={\delta }^2+4=4.01$. The published identity therefore expresses $n\mathrm{Var}(\hat{\theta })-V^{\ast }$ as the signing loss of $m$.

Because the Gaussian completion realizes every $m\in {\mathcal{H}}_{d,s}$, the prognostic image of ${\mathcal{Q}}_{d,s}$ is exactly ${\mathcal{H}}_{d,s}$. It follows from the same identity that $$\underset{\pi \in {\mathcal{D}}_{n,d,s}}{\inf }\underset{P\in {\mathcal{Q}}_{d,s}}{\sup }{\mathcal{V}}_n(\pi ,P)=R_s(n,d).$$ This equality makes the minimax signing criterion a causal excess-variance criterion over the full uniform-covariate outcome-law class. The following result records the relation between that class and its Gaussian subclass, the corresponding published admissibility conditions, and elementary lower and upper guarantees.

Proposition 1 distinguishes the outcome distributions from the prognostic functions that determine excess variance. The Gaussian family is a proper subclass of the admissible outcome laws, while realizing every prognostic function in the pooled Sobolev class. The admissibility equivalence aligns the original-unit assignment family with the published causal framework. Finally, the two-dimensional lower bound supplies an $n^{-s}$ benchmark, and independent fair assignment supplies the universal upper bound of $4$; these comparisons locate the design gains characterized by the main results.

Main results

The minimax criterion in Definition 4 admits a two-sided characterization in terms of the number of coordinate pairs relative to the sample size. For presentation, define the interaction count $B$ and the comparison scale $a_s(n,d)$ by $$B=\left(\genfrac{}{}{0ex}{}{d}{2}\right),a_s(n,d)=\min \{1,(B/n{)}^s\},n,d\ge 2,0<s\le 1.$$ The following result establishes the log-free capacity bound and the exact dimension condition for vanishing excess. The reflection lift and Gaussian completion appearing in the statement are specified in Definitions 8 and 7.

The comparison scale separates two regimes. In the subcritical regime $B<n$, minimax excess is of order $(B/n{)}^s$: the interaction count enters through its ratio to the number of experimental units. When $B\ge n$, the scale saturates at one, and the displayed lower and upper bounds are constants independent of $n$ and $d$. The assignment procedure ${\pi }_{n,d}^{\star }$ specified in Theorem 1 attains this scale in both regimes, using independent fair assignment in the saturated regime and spectral rounding in the subcritical regime.

At the endpoint $s=1$, the subcritical rate is $B/n$, with the same upper constant and a strictly positive lower constant. More generally, the lower constant $c_s$ is bounded below by $c_1>0$ throughout $0<s\le 1$. The two-sided comparison therefore describes the endpoint together with the fractional-smoothness cases, with a logarithm-free rate. For each fixed smoothness value, it also gives an exact criterion for vanishing minimax scaled excess along arbitrary integer dimension sequences: the number of coordinate pairs must be negligible relative to sample size, equivalently $d_n=o(\sqrt{n})$. This equivalence accommodates irregular as well as monotone dimension sequences.

Simultaneous attainment

For the spectral branch of the assignment procedure, the auxiliary randomization transforms the covariates of the original units as follows.

The reflections act separately across units and coordinates, whereas the Haar shift is shared by all units. The transformed covariates in Definition 8 supply the spectral rows used to generate assignment signs for the original sample.

The simultaneous guarantee separates the choice of assignment from the smoothness value used to assess precision. The procedure in Theorem 1 uses the same spectral rows and randomization for every $0<s\le 1$, while the resulting bound varies with the exponent $s$. Its finite completion property applies to every original sample and every allowed realization of the auxiliary seeds. Thus the displayed risk bounds describe a fully specified assignment law on the original units.

Log-free capacity and dimension frontier

The lower bound has a separate statistical role: it applies to every admissible assignment law through a single prior chosen independently of the covariate sample. The prior specification and its class-membership properties are developed further in Definition 9 and Lemma 6.

The normalized prognostic function $m_{\xi }$ in Definition 9 ranges over a finite family of pair-cosine functions. Its law, ${\nu }_{n,d,s}^{\star }$, depends on sample size, dimension, and smoothness, while remaining independent of both the covariate sample and the chosen assignment law. Every realization belongs to the centered pooled Sobolev class by Theorem 1. The contraction inequality used in the supporting converse analysis is Bartlett et al. (2002); its applications here use the absolute-value map with finite classes and finite-dimensional Euclidean unit-ball linear classes. The following result states the resulting prior-average bound over the full design class.

An independent-prior converse

The converse in Theorem 2 certifies the lower comparison scale $b_s(n,d)$ for every admissible assignment law, including laws that depend on the entire observed covariate sample. The frequency cutoff lets the prior supply the subcritical lower bound as well as the saturated bound. Because the guarantee concerns an average over legal prognostic functions drawn independently of the sample, each admissible design has worst-case loss at least as large as that average.

The next theorem isolates the prior-average lower bound that drives the converse for the entire admissible design class.

Causal transfer and prospective precision

The completion in Definition 7 retains $m$ as the conditional midpoint mean while fixing the treatment-effect function and the conditional noise variances. Its scaled variance benchmark is $4.01$, so a signing-loss bound translates directly into an actual excess-variance bound. This translation uses the excess-variance identity of Cytrynbaum (2026), under conditional fairness, outcome-independent assignment, and finite potential-outcome second moments.

The exact minimax equality has a causal interpretation. The published excess-variance identity makes the conditional midpoint mean the function that determines excess above the law-specific benchmark; the Gaussian completion realizes every function in the prognostic class. Accordingly, Theorem 1 equates the prognostic minimax criterion with the outcome-law minimax criterion over ${\mathcal{Q}}_{d,s}$, and gives the corresponding equality for the worst-case loss of the specified procedure. The same capacity is realized within the Gaussian subclass, where the benchmark is $4.01$.

For prospective precision calculations under the Gaussian completion, the sufficient sample-size bound guarantees scaled variance at most $4.01(1+\epsilon )$ uniformly over the prognostic class. The necessary bound applies when $0<\epsilon <c_s/4.01$. At $s=1$, $c_1\approx 3.36\times 10^{-7}$, so this displayed tolerance range is approximately $\epsilon <8.4\times 10^{-8}$; the ratio $3072/c_1$ exceeds nine billion. These constants are conservative finite-sample certificates of the interaction-count scaling rather than a claim of sharp numerical calibration.

The larger bounded-density class in Lemma 2 inherits the converse because its minimax scaled excess is at least $R_s(n,d)$. For fixed density bounds $0<\mathrm{lower}\le 1\le \mathrm{upper}$ and fixed smoothness, vanishing minimax scaled excess in that class therefore requires $d_n=o(\sqrt{n})$. This necessary dimension condition applies with the permitted intercept, the stated prognostic second-moment bound, and finite potential-outcome second moments.

A smoothness-independent assignment procedure

The simultaneous attainment in Theorem 1 is achieved by an assignment rule that prioritizes low-frequency covariate balance. The rule uses sample size, dimension, observed covariates, and auxiliary random seeds. The same rule serves every smoothness level $0<s\le 1$: smoothness enters the evaluation of its performance through the spectral budget.

We first describe the spectral construction used to supply the balancing rows. Work on the period-two torus ${\mathbb{T}}_2^d=[0,2{)}^d$, with addition modulo two and normalized Haar probability. Its coordinatewise fold onto the cube is $$b_d(y{)}_j=\min \{y_j,2-y_j\}.$$ For each original unit and coordinate, independently choose the lift $X_{ij}$ or $2-X_{ij}$, each with probability $1/2$. Add a common Haar-uniform shift $T\in {\mathbb{T}}_2^d$, independent of the sample, the reflection choices, and the rounding seeds. The resulting transformed covariates are $$W_i=(\text{lift of }{X}_i+T)\mathrm{mod}2.$$ These transformations retain the correspondence between each transformed covariate vector and its original experimental unit.

The reflected prognostic function is $F=m\circ b_d$. Writing $$e_k(y)=\exp (\mathrm{i}\pi k^{\mathsf{T}}y),\hat{F}(k)={\int }_{{\mathbb{T}}_2^d}F(y)\overline{e_k(y)}dy,$$ where integration uses normalized Haar probability, gives the Fourier representation underlying the spectral budget. Reflection and translation preserve the original prognostic values through $$F(W_i-T)=m(X_i).$$ The order-two structure in Assumption 2 directs attention to nonzero integer frequencies supported on one or two coordinates. Their frequency complexity is $$t(k)=\frac{\Vert k{\Vert }_2^2}{|\mathrm{supp}(k)|},\mathrm{supp}(k)=\{j:k_j\ne 0\}.$$ Take one representative of each pair $\{k,-k\}$, choosing the representative whose first nonzero entry is positive. Order these representatives by increasing complexity, with lexicographic tie breaking, and place the cosine row before the sine row for each representative. The corresponding real features are $$\sqrt{2}\cos (\pi k^{\mathsf{T}}{W}_i),\sqrt{2}\sin (\pi k^{\mathsf{T}}{W}_i).$$ The first $\lfloor n/4\rfloor$ rows of this ordered family form the input matrix for rounding. Thus reflection, the common shift, frequency ordering, and real features constitute a single construction shared across smoothness levels.

The assignment procedure protects an initial segment of these rows while enough fractional coordinates remain available. As coordinates reach their assignment boundaries, the protected segment becomes shorter. Low-frequency rows consequently receive protection for more of the procedure. Randomization between the two boundary directions preserves the conditional mean of each assignment coordinate, linking covariate balance to fair marginal assignment probabilities.

The independent prior used in the converse places random signs on a finite family of pair-cosine features. Its sample-dependent cutoff and smoothness-dependent normalization are specified next.

The pair-cosine features $V_{\alpha }(x)$ in Definition 9 are indexed by $\alpha$; the frequency cutoff $L$ gives the feature count $M$. The coefficient signs ${\xi }_{\alpha }$ are sampled independently of $X$, while the normalization constant $C_0$ and the factor $L^s\sqrt{M}$ determine the amplitude of the prior functions.

The following construction combines this assignment rule with the comparison scale and the independent prior used in the converse.

At saturation, $n\le \left(\genfrac{}{}{0ex}{}{d}{2}\right)$, the comparison scale in Algorithm 1 equals one, and independent fair signs attain the upper comparison established in Theorem 1. When the sample exceeds the coordinate-pair count, the same construction instead balances the ordered spectral features. Its assignment component ${\pi }_{n,d}^{\star }$, defined in Algorithm 1, is independent of $s$, whereas the comparison scale and the coefficient prior retain their smoothness dependence.

To implement the balancing branch, the fractional assignment starts at zero and moves within the active-coordinate subspace subject to the current row constraints. The boxed procedure specifies the ordering, boundary probabilities, and tie conventions that determine each move.

The protected rows in Algorithm 2 retain their current imbalances throughout each phase. Starting from zero therefore gives the earliest rows the strongest initial protection. The asymmetric boundary probabilities and the terminal coordinate probabilities preserve conditional means, yielding a fair sign for every original unit. Each boundary move fixes at least one previously active coordinate, so the construction requires at most $n$ boundary moves, followed by independent rounding of at most three remaining coordinates. This is a finite guarantee under the specified exact-real arithmetic conventions.

Every output coordinate remains attached to its original unit. Reflection and shifting provide balancing features, while all $n$ units enter the original Horvitz–Thompson estimator with their original weight $2/n$. Consequently, the assignment improvement applies directly to the original-sample criterion in Definition 4.

The quantitative link between ordered protection and prognostic imbalance is supplied by Lemma 5. Its row-imbalance bound combines with the reflected spectral budget and transfer in Lemmas 3 and 4. Frequency ordering allocates the strongest balance to the rows whose coefficients receive the greatest weight under smoothness. These bounds yield the simultaneous smoothness attainment stated in Theorem 1. The supporting bounds and measurable realization arguments accompany those auxiliary results in the appendix.

Discussion and extensions

The characterization in Theorem 1 gives an interaction-count benchmark for experimental sample size. Under the independent uniform sampling law of Assumption 1, the relevant comparison is between $B=\left(\genfrac{}{}{0ex}{}{d}{2}\right)$, the number of coordinate pairs, and $n$, the number of experimental units. For the centered prognostic class with the pooled Sobolev budget in Definition 2, minimax scaled excess variance is comparable to $\min \{1,(B/n{)}^s\}$. Thus the interaction count summarizes the dimension dependence of the worst-case precision requirement within this class.

Causal calibration and relative excess

To interpret the assignment guarantee in terms of causal precision, we first specify the potential outcomes, treatment, estimator, and targets used in the Gaussian calibration.

Concretely, for each $m\in {\mathcal{H}}_{d,s}$, the completion in Definition 7 fixes $$\tau (x)=0.2+0.1\sqrt{2}\sin (2\pi x_1)$$ and generates the pre-assignment schedule as $$Y_i(0)=m(X_i)-\frac{\tau (X_i)}{2}+{ϵ}_i(0),Y_i(1)=m(X_i)+\frac{\tau (X_i)}{2}+{ϵ}_i(1).$$ The $2n$ noise variables are independent standard Gaussians, independent of the covariates. After this schedule is generated, the admissible assignment law draws $Z$ using the covariates and satisfies Assumptions 4 and 5. With $A_i=(Z_i+1)/2$, the observed outcome is $Y_i(A_i)$, and the estimator and its two targets are $$\hat{\theta }=\frac{2}{n}\sum _{i=1}^n{Z}_i{Y}_i(A_i),{\theta }_n=\frac{1}{n}\sum _{i=1}^n(Y_i(1)-Y_i(0)),\theta =\mathbb{E}[{\theta }_n]=\mathrm{0.2.}$$ Fairness and outcome-independent assignment make the estimator conditionally unbiased for the realized schedule average ${\theta }_n$, and its sampling-law expectation is $\theta$.