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Second-order Minimax Risk in Multi-arm Binary Randomization

Abstract

This paper studies design-based minimax risk for randomized experiments with a fixed number $K$ of treatment arms, $n$ labeled units, binary potential outcomes, and a prespecified nonzero zero-sum treatment contrast. The game, which optimizes jointly over the assignment law and an estimator clipped to the estimand’s own range against all complete labeled schedules, has the same value as an explicit finite game whose states are the response-type count vectors over the $2^{K}$ binary response types. For every fixed nonzero zero-sum contrast $c$ , the first-order minimax constant is $C_{0} (c) = L_{c}^{2} /4$ , where $L_{c}$ is the $ℓ_{1}$ norm of the contrast, and the contrast-weighted allocation $q_{a}^{⋆} \propto ∣ c_{a} ∣$ attains the finite-sample upper bound $C_{0} (c) / n$ with a projected Horvitz–Thompson rule. The same allocation, paired with an explicit clipped-shrinkage estimator, improves on that envelope by a positive multiple of $n^{- 4/3}$ for all sufficiently large $n$ , while an embedded two-arm Bayesian information argument gives a matching $O (n^{- 4/3})$ converse. Thus $4/3$ is the universal second-order exponent for every fixed nonzero zero-sum binary contrast. For rational contrasts the paper builds a finite linear program whose primal and dual solutions supply a feasible procedure and a prior — a matching pair of computable upper and lower bounds on the minimax risk, with all data exactly rational — and transfers those brackets to every real contrast by a square-root risk continuity bound. A three-arm example shows that at $n = 3$ , for the contrast $(1, - 1/2, - 1/2)$ under one fixed allocation, estimators using the full observed arm label and outcome attain a strictly smaller worst-case risk than estimators using only a scalar signed score.

Introduction

Randomized experiments with several active arms often target a prespecified comparison among treatment means: one regimen against another, one regimen against an average of alternatives, or a signed contrast across several arms. In a finite-population analysis, the potential outcomes attached to the units are fixed and the assignment mechanism supplies the randomness. This paper studies the corresponding design problem when each potential outcome is binary, the number of arms $K$ is fixed, and the target is a nonzero zero-sum contrast $c$ . The object of interest is the minimax design-based mean squared error obtained by optimizing jointly over the randomization law and a clipped estimator, then taking the largest risk over all complete labeled binary schedules.

The first contribution is an exact finite reduction. The unrestricted labeled-schedule minimax risk $ρ_{n} (c)$ , the worst-case risk over all labeled binary potential-outcome schedules, is equal to the finite response-type orbit value $G_{n} (c)$ in Definition 7. The reduction in Theorem 1 symmetrizes over unit labels, preserves the finite-population contrast target, and identifies a finite game whose states are response-type count vectors. For fixed $K$ , the response-type space has size $2^{K}$ , so the adversary ranges over the integer simplex of counts of binary response types and the statistician ranges over allocation-count distributions and orbit estimators.

The second contribution is the first-order minimax benchmark. Let $L_{c} = \sum_{a} ∣ c_{a} ∣$ , the contrast $ℓ_{1}$ norm, and let $S_{c} = {a : c_{a} \neq = 0}$ , the active support of the contrast. Theorem 3 shows that $n \to \infty lim n ρ_{n} (c) = C_{0} (c), C_{0} (c) = \frac{L _{c}^{2}}{4} .$ It also gives a finite-sample procedure attaining the upper envelope $C_{0} (c) / n$ : assign each unit independently among active arms with probabilities $q_{a}^{⋆} = ∣ c_{a} ∣/ L_{c}$ , and estimate the centered contrast by the projected contrast-weighted Horvitz–Thompson rule in Definition 14. Thus the first-order allocation depends on the signed contrast through absolute weights, matching the range-weighted logic familiar from finite-population sampling and Horvitz–Thompson estimation (Horvitz et al., 1952; Hansen et al., 1953; Särndal et al., 1992; Aronow et al., 2026).

The third contribution is the second-order rate. The improvement $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c)$ is positive at the $n^{- 4/3}$ scale for every fixed nonzero zero-sum contrast. Theorem 5 constructs an explicit clipped-shrinkage estimator under the same contrast-weighted allocation and proves a uniform risk bound of the form $C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}}$ for a positive contrast-dependent constant $κ_{c}$ . The converse side in Theorems 4 and 6 embeds the two-arm difficulty inside any fixed contrast and yields the upper bound $d_{n} (c) \leq 43 C_{0} (c) n^{- 4/3}$ for large $n$ . Together these results show that if a regularly varying normalization makes the improvement converge to a positive finite limit, then its index must be $4/3$ .

The analysis builds on the design-based causal-inference tradition initiated by Fisher, Neyman, and Rubin and developed in modern finite-population treatments (Fisher, 1935; Splawa-Neyman, 1990; Rubin, 1974; Rubin, 1978; Holland, 1986; Imbens et al., 2015; Imbens et al., 2009; Athey et al., 2017; Ding, 2024). It is also connected to optimum design, minimax decision theory, Bayesian information inequalities, and second-order minimax analysis (Smith, 1918; Kiefer, 1959; Kiefer, 1974; Atkinson et al., 2007; Wald, 1950; Le Cam, 1986; Ibragimov et al., 1981; Rao, 1945; Van Trees, 1968; Gill et al., 1995; Levit, 1981; Bickel, 1981; Pinsker, 1980; Johnstone et al., 1992). The nearest causal comparisons are sharp two-arm and bounded-outcome minimax analyses (Sudijono et al., 2026; Hull, 2026), together with recent minimax and optimal randomization work in experimental design (Kallus, 2018; Kallus, 2020; Bai, 2023; Harshaw et al., 2024; Basse et al., 2023; Dereziński et al., 2019; Hu et al., 2024; Kandiros et al., 2024; Kandiros et al., 2026). The present results give a fixed- $K$ binary finite-population counterpart: an exact orbit game, a design and estimator class as large as the estimand’s own range allows, and computable finite bounds on the minimax value.

Computable bounds are the fourth contribution. By a certificate we mean a finite object whose risk can be evaluated exactly and which therefore bounds $ρ_{n} (c)$ on one side: a feasible procedure bounds it above, a prior bounds it below. For rational contrasts, Definition 19 and Theorem 7 define a grid-action linear program over allocation-count orbits and estimator-grid actions. Its primal solution is an invariant randomized procedure, its dual response-count multipliers form a prior, and the gap between the two bounds is controlled by the grid mesh. Theorems 8 and 9 transfer these rational certificates to real contrasts through a Lipschitz bound for the square root of minimax risk. At the $n^{4/3}$ scale, the mesh can be chosen so that the certificate width is asymptotically negligible for the rational certificates, and for real contrasts after rational approximation and continuity transfer.

The three-arm diagnostic illustrates why the orbit experiment retains more structure than a scalar contrast score. For $c^{†} = (1, - 1/2, - 1/2)$ , Proposition 2 compares the full arm-label-and-outcome experiment with a signed-score reduction under the same contrast-weighted allocation. At $n = 3$ , a full-data clipped estimator attains a strictly smaller worst-case risk than any rule based on the scalar score. The support-specific results in Theorems 11 and 12 then separate two-active-arm contrasts, where the value is exactly the two-arm value scaled by $C_{0} (c)$ , from contrasts with at least three active arms, where the same order sandwich holds and the finite orbit programs supply the brackets.

The results apply directly to prespecified contrasts in multi-regimen binary-endpoint trials. In an ACTG 175-style four-regimen setting (Hammer et al., 1996), a contrast comparing one regimen with another falls under the support-two reduction, while a contrast comparing one regimen with an average of alternatives uses the support-at-least-three conclusions and the finite-program certificates. Section B records the scope of the mathematical results, the presentation-only notation, and the boundary of the formal claims.

Section 2 places the analysis in the literatures on design-based causal inference, finite-population sampling, optimum design, information inequalities, and minimax randomization. Section 3 defines the labeled schedule game, the response-type orbit likelihood, and the exact orbit reduction. Section 4 establishes the first-order constant, the clipped-shrinkage improvement, and the embedded two-arm converse. Section 5 develops rational grid certificates and the continuity transfer to real contrasts. Section 6 gives the three-arm diagnostic and the support-specific conclusions. Section 7 discusses design implications, limitations, and future work, and Sections A, B, and C collect the auxiliary lemmas, the verification record, and the proofs of the main results.

Randomization-based causal inference treats the assignment mechanism as the source of uncertainty and potential outcomes as fixed features of the finite population. This design-based perspective goes back to Fisher (1935), Splawa-Neyman (1990), and the potential-outcomes formulation of Rubin (1974); Rubin (1978); see also Holland (1986), Imbens et al. (2015), Imbens et al. (2009), Athey et al. (2017), and Ding (2024). Recent econometric treatments emphasize how the choice of randomization affects precision, robustness, and finite-sample behavior, including regression adjustment and covariate balance (Freedman, 2008; Lin, 2013; Dasgupta et al., 2015; Morgan et al., 2012; Li et al., 2018; Bai et al., 2024). The present paper works in the same finite-population tradition, with a fixed number of treatment arms $K$ , a fixed number of labeled units $n$ , binary potential outcomes, and a nonzero zero-sum treatment contrast $c$ . Its target is the exact design-estimator minimax risk and its first- and second-order behavior under the randomization distribution.

Finite-population sampling theory provides a parallel design-based language for choosing sampling laws and estimators under adversarial finite populations. Classical unequal-probability sampling and Horvitz–Thompson estimation (Horvitz et al., 1952; Hansen et al., 1953; Särndal et al., 1992) connect naturally to randomized experiments because treatment assignment can be read as a sampling design over exposure cells. Gabler (1990) and Aronow et al. (2026) study minimax ideas in finite-population sampling, with the latter giving a close comparison through finite-population minimax sampling under unbiasedness. The analysis here is aligned with that decision-theoretic finite-population viewpoint and specializes it to multi-arm binary randomization, where both the assignment law and the clipped estimator enter the minimax criterion.

The paper is also connected to optimum experimental design. The classical design literature studies allocation rules through criteria derived from information matrices or risk functionals (Smith, 1918; Kiefer, 1959; Kiefer, 1974; Atkinson et al., 2007). In causal experiments, minimax and optimality criteria have been used to evaluate and construct randomization designs under finite-population or model-assisted objectives. Kallus (2018) develops optimal a priori balance and Kallus (2020) studies optimal randomization designs; related modern work includes Basse et al. (2023), Harshaw et al. (2024), Kandiros et al. (2024); Kandiros et al. (2026), Eckles et al. (2014), Dereziński et al. (2019), and Hu et al. (2024). Relative to this literature, the contribution is an exact finite binary minimax analysis for treatment contrasts, including the orbit reduction and finite-program certificates used to evaluate sharp finite- $n$ quantities.

Information inequalities and local asymptotic arguments supply the lower-bound tools behind many minimax results. The decision-theoretic tradition begins with Wald (1950); asymptotic minimax and local experiment methods are developed in Le Cam (1986), Ibragimov et al. (1981), and related work. Bayesian Cramér–Rao and van Trees inequalities (Van Trees, 1968; Gill et al., 1995), together with classical information identities (Rao, 1945), provide lower bounds for estimation risk. Second-order minimax refinements have a long statistical lineage, including Levit (1981), and related shrinkage and nonparametric minimax phenomena appear in Bickel (1981), Pinsker (1980), Feldman (1991), and Johnstone et al. (1992). The results below use this tradition to identify the first-order minimax constant and to analyze the size of the second-order improvement in the finite-population binary randomization game.

The closest econometric comparisons are recent sharp analyses of two-arm designs. Sudijono et al. (2026) obtains a sharp two-arm binary expansion, and Hull (2026) studies bounded two-arm potential outcomes. Rosenman et al. (2026) is also related through shrinkage ideas for experimental estimation. The present paper places those two-arm themes inside a multi-arm contrast framework: it gives exact first-order risk for general fixed contrasts, embeds the two-arm converse where it is sharp, and constructs finite linear-program certificates for rational and real contrasts. A three-arm example then shows, at $n = 3$ and for the single contrast $c^{†} = (1, - 1/2, - 1/2)$ under one fixed allocation, that estimators using the full observed arm label and outcome achieve a strictly smaller worst-case risk than estimators using only a scalar signed score. That comparison isolates what the data reduction discards for this fixed three-arm support, fixed allocation, and sample size.

Setup and the finite orbit game

Throughout the paper, $n$ denotes the fixed number of labeled units, $K$ the fixed number of treatment arms, and asymptotic notation is taken with $K$ and the contrast fixed. Sums over arms run over the treatment-arm set and sums over response types run over the corresponding binary type space. Generic constants may depend on the fixed contrast and on $K$ , but not on $n$ .

The finite-population model starts with a labeled binary potential-outcome schedule. The treatment-arm set $A_{K}$ indexes the arms, and the response-type space records the vector of binary potential outcomes attached to each unit.

Definition 1 [synth_1] (Treatment-arm label set).

For each natural number $K$ , the treatment-arm set is the finite set of its $K$ arm labels: $A_{K} := {1, \dots, K} .$

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Definition 2 [synth_2] (Binary response types).

For a fixed number $K$ of treatment arms, let $A_{K}$ be the treatment-arm label set. The binary response-type space is $T_{K} := {0, 1}^{A_{K}} .$ Thus each $t \in T_{K}$ assigns one binary potential-outcome value $t_{a} \in {0, 1}$ to every arm $a \in A_{K}$ .

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This notation is the multi-arm version of the binary potential-outcome framework used in design-based causal inference (Splawa-Neyman, 1990; Rubin, 1974; Holland, 1986; Imbens et al., 2015). A complete schedule assigns one response type to each labeled unit; an assignment design then randomizes only the observed arm, while the schedule itself remains fixed.

Three objects enter the next definition and are fixed here once. The contrast $c$ is a nonzero zero-sum vector, that is $c \in R^{K} ∖ {0}$ with $\sum_{a \in A_{K}} c_{a} = 0$ (restated as Definition 10 below, where the rational case is also named); we write $L_{c} = a \in A_{K} \sum ∣ c_{a} ∣, I_{c} = [- \frac{L _{c}}{2}, \frac{L _{c}}{2}], τ_{c} (z) = \frac{1}{n} i = 1 \sum n a \in A_{K} \sum c_{a} t_{i, a}$ for its $ℓ_{1}$ norm, the interval it generates, and the finite-population estimand at the complete schedule $z = (t_{1}, \dots, t_{n})$ . Because $c$ sums to zero and each $t_{i, a}$ is $0$ or $1$ , the estimand always lies in $I_{c}$ . The infimum in the definition below runs over assignment laws $D \in Δ (A_{K}^{n})$ and over estimators $τ : (A_{K} \times {0, 1})^{n} \to I_{c} .$ Confining the estimator to $I_{c}$ costs nothing: projecting a real-valued rule onto $I_{c}$ never increases squared error at a target that lies in $I_{c}$ , so the value of the game is the same as it would be with arbitrary real-valued estimators. Finally, $M_{n, K}$ denotes the set of nonnegative integer vectors $(m_{t})_{t \in T_{K}}$ with $\sum_{t} m_{t} = n$ , and $R_{n, K}$ the set of nonnegative integer vectors $(r_{a})_{a \in A_{K}}$ with $\sum_{a} r_{a} = n$ ; these are finite sets of lattice points, and the shorter word “simplex” is occasionally used for them below.

Definition 3 [def:labeled-schedule-game] (Labeled-schedule risk

ρ_{n} (c)

The unrestricted labeled-schedule minimax risk for contrast $c$ is $ρ_{n} (c) = D \in Δ (A_{K}^{n}), τ in f z \in T_{K}^{n} max E_{D} [{τ ((A_{i}, Y_{i}^{obs})_{i = 1}^{n}) - τ_{c} (z)}^{2}] .$

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The risk $ρ_{n} (c)$ is the benchmark for joint design and estimation; by the previous paragraph nothing is lost by the clipping, so we drop the qualifier “unrestricted” from here on. It optimizes over an assignment law $D$ and a clipped estimator $τ$ , then evaluates the largest design-based squared error over complete binary schedules. The contrast $c$ is zero-sum, so the estimand is a treatment comparison rather than a level.

The next definitions pass from labeled schedules to response-type counts. This is the statistic that remains after averaging over relabelings of the units.

Definition 4 [synth_4] (Response-type count vector).

For a complete labeled schedule $z = (t_{1}, \dots, t_{n}) \in T_{K}^{n}$ , define $counts (z) = (counts (z)_{t})_{t \in T_{K}} \in M_{n, K}, counts (z)_{t} = # {i \in {1, \dots, n} : t_{i} = t} .$ Thus $counts (z)$ is the unit-permutation orbit of $z$ .

Definition 5 [synth_19] (Notation for response-type counts).

For a complete labeled response schedule $z = (t_{1}, \dots, t_{n}) \in T_{K}^{n}$ we write $m (z) = counts (z)$ for the response-type count vector of Definition 4, with coordinates $m_{t} (z) = # {i \in {1, \dots, n} : t_{i} = t}, t \in T_{K},$ so that $\sum_{t \in T_{K}} m_{t} (z) = n$ . A generic element of $M_{n, K}$ is written $m$ , and $m_{t}$ is its coordinate at $t$ .

The count vector $m$ belongs to the response-count simplex $M_{n, K}$ . It preserves the finite-population target because the contrast averages unit-level potential outcomes symmetrically across labels.

Given a response-count vector and an allocation-count vector, the observable data reduce to arm-specific success counts. The likelihood below counts all response-type-by-arm contingency tables compatible with those margins.

Definition 6 [def:orbit-likelihood] (Orbit likelihood

P_{m} (x ∣ r)

The orbit likelihood of arm-success counts $x$ given allocation counts $r$ and response-type counts $m$ is $P_{m} (x ∣ r) = \frac{\prod _{a \in A_{K}} r _{a} !}{n !} h \in H (m, r, x) \sum t \in T_{K} \prod \frac{m _{t} !}{\prod _{a \in A_{K}} h _{t, a} !} .$ Here $H (m, r, x)$ is the set of nonnegative integer tables $h = (h_{t, a})$ with response-type margins $m$ , allocation margins $r$ , and arm-success totals $x$ .

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The orbit likelihood $P_{m} (x ∣ r)$ is purely finite-population: it is induced by random assignment among labeled units with fixed response types. The allocation-count vector $r$ lies in the allocation-count simplex $R_{n, K}$ , and the observed-success vector $x$ records one count for each arm.

Definition 7 [def:finite-orbit-game] (Finite orbit game

G_{n} (c)

The finite response-type orbit minimax value is $G_{n} (c) = π \in Δ (R_{n, K}), δ in f m \in M_{n, K} max r \in R_{n, K} \sum π_{r} x \in \prod_{a \in A_{K}} {0, \dots, r_{a}} \sum P_{m} (x ∣ r) {δ (r, x) - τ_{c} (m)}^{2} .$ Here $M_{n, K}$ is the simplex of response-type count vectors and $R_{n, K}$ is the simplex of allocation-count vectors.

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The finite value $G_{n} (c)$ optimizes over an invariant design distribution $π$ on allocation-count orbits and an orbit estimator $δ$ . Its target $τ_{c} (m)$ is the count-level version of the labeled finite-population contrast.

For later reference, it is useful to name both the labeled and orbit risks explicitly.

Definition 8 [synth_3] (Design-based squared-error risk).

For a complete labeled response schedule $z \in T_{K}^{n}$ , an assignment design $D \in Δ (A_{K}^{n})$ , and an estimator $τ : (A_{K} \times {0, 1})^{n} \to [- L_{c} /2, L_{c} /2]$ , define $R_{n} (D, τ; z) = E_{D} [{τ ((A_{i}, Y_{i}^{obs})_{i = 1}^{n}) - τ_{c} (z)}^{2}],$ where $A \sim D$ , $Y_{i}^{obs} = t_{i, A_{i}}$ , and the expectation is only over the assignment randomization. If $p = (D, τ)$ is a labeled procedure, write $R_{n} (p; z)$ for the same quantity.

Definition 9 [synth_10] (Orbit-procedure risk).

For a contrast $c$ , an orbit procedure is a pair $q = (π, δ)$ with $π \in Δ (R_{n, K})$ and $δ (r, x) \in [- L_{c} /2, L_{c} /2]$ . Its risk at response-type count vector $m \in M_{n, K}$ is $R_{n} (q; m) = r \in R_{n, K} \sum π_{r} x \in \prod_{a \in A_{K}} {0, \dots, r_{a}} \sum P_{m} (x ∣ r) {δ (r, x) - τ_{c} (m)}^{2} .$ In the two-arm contrast $c = (1, - 1)$ , write $R_{(1, - 1)}^{orb} (q; m) = r \in R_{n, 2} \sum π_{r} x \in \prod_{a \in A_{2}} {0, \dots, r_{a}} \sum P_{m} (x ∣ r) {δ (r, x) - τ_{(1, - 1)} (m)}^{2} .$

Definition 10 [synth_20] (Nonzero zero-sum contrast).

For $K \geq 2$ , a zero-sum contrast is a vector $c = (c_{a})_{a \in A_{K}} \in R^{K}$ satisfying $\sum_{a \in A_{K}} c_{a} = 0$ . It is nonzero when $c \neq = 0$ . A rational nonzero zero-sum contrast is such a vector with $c \in Q^{K}$ .

Definition 11 [synth_15] (Compatible observation-success counts).

For an allocation-count vector $r \in R_{n, K}$ , an observed-success vector $x = (x_{a})_{a \in A_{K}}$ is compatible with $r$ when $x_{a} \in {0, \dots, r_{a}}$ for every arm $a$ . For $m \in M_{n, K}$ , the vector $x$ is compatible with $(m, r)$ when there exists a contingency table $h = (h_{t, a})_{t \in T_{K}, a \in A_{K}} \in Z_{+}^{T_{K} \times A_{K}}$ such that $\sum_{a} h_{t, a} = m_{t}$ for every $t$ , $\sum_{t} h_{t, a} = r_{a}$ for every $a$ , and $\sum_{t : t_{a} = 1} h_{t, a} = x_{a}$ for every $a$ ; equivalently, $H (m, r, x)$ is nonempty.

The compatibility condition spells out the feasible fibers in Definition 6. A table $h$ records how many units of each response type are assigned to each arm; its margins determine the allocation counts and the observed successes.

The exact reduction uses the unit-permutation orbit representation of the experiment. Invariance requires the design and estimator to ignore the arbitrary labels of the units.

Definition 12 [synth_14] (Invariant labeled procedure).

A labeled procedure is a pair $p = (D, τ)$ , where $D \in Δ (A_{K}^{n})$ is an assignment law and $τ : (A_{K} \times {0, 1})^{n} \to [- L_{c} /2, L_{c} /2]$ is an estimator. For $σ \in S_{n}$ , define $(σ A)_{i} = A_{σ^{- 1} (i)}$ , $(σ z)_{i} = z_{σ^{- 1} (i)}$ , and $σ ((A_{i}, Y_{i}^{obs})_{i = 1}^{n}) = ((A_{σ^{- 1} (i)}, Y_{σ^{- 1} (i)}^{obs}))_{i = 1}^{n}$ . The procedure $p$ is invariant when, for every $σ \in S_{n}$ , the law of $σ A$ under $D$ equals $D$ and $τ (σ o) = τ (o)$ for every observed-data vector $o \in (A_{K} \times {0, 1})^{n}$ .

The following theorem is the main reduction in this section. It establishes that symmetrizing over unit labels preserves the minimax value and identifies the resulting finite game exactly.

Theorem 1 [thm:exact-response-type-game] (Exact orbit game reduction).

Fix $n \in N$ , $K \geq 2$ , and a nonzero zero-sum contrast $c$ in the labeled-schedule and finite-orbit games of Definitions 3 and 7. Then:

(Lossless symmetrization.) For every labeled procedure $p$ , there are an invariant labeled procedure $\overset{p}{ˉ}$ and an orbit procedure $q$ such that $\overset{p}{ˉ}$ is the common-unit-permutation average of $p$ , and for every complete labeled schedule $z \in T_{K}^{n}$ , $R_{n} (\overset{p}{ˉ}; z) \leq \frac{1}{∣ S _{n} ∣} σ \in S_{n} \sum R_{n} (p; σ z) .$ The orbit procedure $q$ is dominated by this averaged invariant representative in the corresponding orbit risk.

(Exact invariant correspondence.) Invariant labeled procedures are in bijection with orbit procedures $(π, δ)$ . Under this bijection, each invariant labeled procedure and its orbit representative realize the same allocation-count and observation-count rule, and for every $z \in T_{K}^{n}$ , $R_{n} (q; counts (z)) = R_{n} (\overset{p}{ˉ}; z) .$

(Target preservation.) If $m = counts (z)$ , with $m_{t} = # {i : t_{i} = t}$ , then $τ_{c} (z) = \frac{1}{n} i \sum a \in A_{K} \sum c_{a} t_{i, a} = \frac{1}{n} t \sum m_{t} a \in A_{K} \sum c_{a} t_{a} = τ_{c} (m) .$

(Value equality.) The unrestricted labeled-schedule minimax risk equals the finite response-type orbit minimax value: $ρ_{n} (c) = G_{n} (c) .$

(Orbit saddle point.) There exist an orbit procedure $q^{⋆}$ and a prior $ν$ on $M_{n, K}$ such that, for every response-count vector $m$ and every orbit procedure $q$ , $R_{n} (q^{⋆}; m) \leq G_{n} (c) and G_{n} (c) \leq E_{ν} [R_{n} (q; m)] .$

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The intuition is that the labels of units carry no information about the contrast target once the complete schedule is fixed. Averaging a procedure over common relabelings cannot increase worst-case risk, and the averaged procedure depends on the assignment and outcomes only through allocation counts and observed-success counts. Theorem 1 therefore turns the unrestricted labeled problem into a finite decision problem over response-count vectors, while preserving both the target and the minimax value.

The final part of the setup records the two-arm benchmark and the fixed- $K$ extension. The two-arm bounded schedule domain is included because recent two-arm results provide an important comparison point for binary minimax analysis (Sudijono et al., 2026; Hull, 2026).

Definition 13 [synth_9] (Hull bounded-schedule domain and risk).

For bounds $L < U$ , let $Y_{n, L, U}^{Hull}$ be the set of two-arm bounded potential-outcome schedules $Y = (Y_{i} (1), Y_{i} (2))_{i = 1}^{n} with Y_{i} (a) \in [L, U]$ for every unit $i$ and arm $a \in {1, 2}$ . For a design $D$ , estimator $τ$ , and schedule $Y \in Y_{n, L, U}^{Hull}$ , define the Hull squared-error risk $R_{Hull} (D, τ; Y) = E_{D} {τ ((A_{i}, Y_{i}^{obs})_{i = 1}^{n}) - \frac{1}{n} i = 1 \sum n (Y_{i} (1) - Y_{i} (2))}^{2},$ where $Y_{i}^{obs} = Y_{i} (A_{i})$ and the expectation is over assignment from $D$ .

Theorem 2 [thm:multiarm-strict-extension] (Two-arm specialization and the fixed-

K

orbit game).

Throughout, $ρ_{n} (c)$ is the labeled-schedule minimax risk of Definition 3 and $G_{n} (c)$ the finite orbit-game value of Definition 7. The following statements hold.

(Two-arm orbit value.) For every $n > 0$ , with $c = (1, - 1)$ , $ρ_{n} (c) = G_{n} (c) and C_{0} (c) = 1.$

(Two-arm target.) For every $n > 0$ and every $m \in M_{n, 2}$ , with $p_{+} (m) = m_{(1, 0)}$ and $p_{-} (m) = m_{(0, 1)}$ , $τ_{c} (m) = \frac{p _{+} ( m ) - p _{-} ( m )}{n} .$

(Four two-arm response types and likelihood.) For every $n$ , every $m \in M_{n, 2}$ , every $r \in R_{n, 2}$ , and every compatible observed-success vector $x$ , let $t_{00} (a) = 0, t_{01} = (0, 1), t_{10} = (1, 0), t_{11} (a) = 1.$ These four response types are pairwise distinct and exhaust $T_{2}$ . Moreover, with the likelihood notation of Definition 6, $P_{m} (x ∣ r) = \frac{\prod _{a \in A_{2}} r _{a} !}{n !} h \in H (m, r, x) \sum t \in T_{2} \prod \frac{m _{t} !}{\prod _{a \in A_{2}} h _{t, a} !} .$

(Two-arm second-order scale.) For every $n \geq 8$ , $d_{n} (c) \leq 43 n^{- 4/3} .$ There are constants $κ > 0$ and $N$ such that, for every $n \geq N$ , $κ n^{- 4/3} \leq d_{n} (c) .$

(Exact multiarm orbit game.) For every $K \geq 3$ , every $n$ , and every contrast $c$ , $ρ_{n} (c) = G_{n} (c) and ∣ T_{K} ∣ = 2^{K} .$

(Hull risk-value inclusion.) For every $N, L, U$ , every finite design $D$ on two-arm assignments, and every measurable Hull estimator $τ$ , $Y sup R_{Hull} (D, τ; Y) \in {v : \exists D^{'}, τ^{'} with τ^{'} measurable and v = Y sup R_{Hull} (D^{'}, τ^{'}; Y)},$ where the suprema range over Hull schedules with parameters $N, L, U$ .

(Strict treatment-domain separation.) For every $K \geq 3$ , every $n > 0$ , and every $L < U$ , the Hull bounded-schedule domain with parameters $n, L, U$ is not a subset of the binary $K$ -arm schedule domain: $Y_{n, L, U}^{Hull} ⊈ T_{K}^{n} .$

⊢ Lean

Write $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c), C_{0} (c) = \frac{L _{c}^{2}}{4},$ for the improvement of the minimax risk on the first-order envelope; Section 4 is devoted to it. For $K = 2$ , Theorem 2 identifies the orbit value for $c = (1, - 1)$ , expresses the estimand through the effect-class counts $p_{+}$ and $p_{-}$ , reproduces the four-type likelihood, and places $d_{n} (c)$ on the $n^{- 4/3}$ scale. For every fixed $K \geq 3$ , it gives the exact multi-arm orbit equality and the binary type count $2^{K}$ , which supplies the finite state space used by the linear programs of Section 5.

Two clauses of that theorem fix the scope of the bounded-outcome comparison. They record that the worst-case Hull risk of a design–estimator pair is itself an attainable Hull risk value, and that the bounded real-valued two-arm schedule domain and the binary $K$ -arm schedule domain are distinct as sets. Both are elementary, and their role here is bookkeeping: they make precise that the two problems are posed over different schedule domains, so the route between them is an explicit construction. That is the route this paper takes, through the sign-pair embedding used in the converse of Section 4.

First-order risk and the second-order rate

The finite orbit reduction in Theorem 1 turns the unrestricted labeled-schedule risk into a finite response-type decision problem. This section first identifies the first-order benchmark in the original labeled game and then refines that benchmark by an explicit second-order construction. Throughout the section, the contrast $c$ is fixed, nonzero, and zero-sum on the treatment-arm set from Definition 3.

The first-order procedure allocates experimental effort in proportion to the absolute contrast weights and estimates the centered contrast by a clipped Horvitz–Thompson rule. The clipping interval is the natural range of the finite-population contrast target.

Definition 14 [def:first-order-procedure] (First-order rule

τ_{cHT}^{⋆}

Let $S_{c} = {a \in A_{K} : c_{a} \neq = 0}, L_{c} = a \in A_{K} \sum ∣ c_{a} ∣, q_{a}^{⋆} = \frac{∣ c _{a} ∣}{L _{c}} (a \in S_{c}) .$ Assign each $A_{i}$ independently on $S_{c}$ with $Pr (A_{i} = a) = q_{a}^{⋆}$ . The projected contrast-weighted Horvitz–Thompson rule is $τ_{cHT}^{⋆} = proj_{[- L_{c} /2, L_{c} /2]} [n^{- 1} i = 1 \sum n a \in S_{c} \sum c_{a} 1 {A_{i} = a} \frac{Y _{i}^{obs} - 1/2}{q _{a}^{⋆}}] .$

⊢ Lean

The active support $S_{c}$ selects the arms that enter the contrast, while the contrast norm $L_{c}$ fixes both the target range and the first-order scale. The assignment probabilities $q^{⋆}$ place larger sampling mass on arms with larger absolute contrast weights, matching the variance contribution of the centered Horvitz–Thompson summands.

Definition 15 [synth_5] (Contrast-weighted independent design).

Let $S_{c} = {a \in A_{K} : c_{a} \neq = 0}$ , $L_{c} = \sum_{a \in A_{K}} ∣ c_{a} ∣$ , and $q_{a}^{⋆} = {∣ c_{a} ∣/ L_{c}, 0, a \in S_{c}, a \in / S_{c} .$ The design $D^{⋆}$ is the product assignment law under which $A_{1}, \dots, A_{n}$ are independent and $Pr_{D^{⋆}} (A_{i} = a) = q_{a}^{⋆}$ for every unit $i$ and arm $a$ .

Definition 15 records the same allocation as a design object. This formulation will also be used for the shrinkage rule below, so that the first- and second-order procedures share the same contrast-weighted randomization scheme.

Theorem 3 [thm:first-order-saddle] (First-order minimax constant and attaining procedure).

Fix a natural number $K$ and a nonzero zero-sum contrast $c$ on the treatment-arm set $A_{K}$ , in the labeled-schedule game of Definition 3. Let $L_{c}$ , $q_{a}^{⋆}$ , and $τ_{cHT}^{⋆}$ be the contrast norm, contrast-weighted assignment probabilities, and projected contrast-weighted Horvitz–Thompson rule from Definition 14, and set $C_{0} (c) = \frac{L _{c}^{2}}{4} .$ Then the unrestricted labeled-schedule minimax risk satisfies $n \to \infty lim n ρ_{n} (c) = C_{0} (c) .$ Moreover, for every $n \geq 1$ , the independent contrast-weighted design together with $τ_{cHT}^{⋆}$ attains the finite-sample worst-case bound $z \in T_{K}^{n} max R_{n} (D^{⋆}, τ_{cHT}^{⋆}; z) \leq \frac{C _{0} ( c )}{n} .$

⊢ Lean

The theorem gives the first-order robustness constant and a finite-sample certificate attaining its envelope. The constant $C_{0} (c) = L_{c}^{2} /4$ depends on the contrast through its $ℓ_{1}$ norm, so the allocation benchmark is determined by the signed estimand rather than by a default equal-allocation rule. The bound is stated in the unrestricted labeled-schedule game of Definition 3, which places it on the same decision scale as the finite orbit value characterized in Theorem 1.

The converse input comes from embedding the sharp two-arm difficulty into the multi-arm contrast. This comparison links every fixed contrast to the two-arm contrast $(1, - 1)$ , and it also gives an exact specialization when the contrast has two active arms.

Theorem 4 [thm:embedded-two-arm-converse] (Embedded two-arm converse).

Fix integers $K$ and $n$ , and a contrast $c$ on $A_{K}$ , with labeled-schedule minimax risk $ρ_{n} (c)$ as in Definition 3. Suppose that

(Arm count.) $K \geq 2$ .

(Population size.) $n \geq 1$ .
(Contrast.) $c$ is nonzero and zero-sum.

Let $C_{0} (c) = \frac{L _{c}^{2}}{4}, ρ_{n, 2} = ρ_{n} ((1, - 1)), d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c),$ where $ρ_{n, 2}$ is the unrestricted two-arm minimax risk at population size $n$ . Then $C_{0} (c) ρ_{n, 2} \leq ρ_{n} (c) and ρ_{n} (c) \leq \frac{C _{0} ( c )}{n} .$ Moreover, if $n \geq 8$ , then $0 \leq d_{n} (c), d_{n} (c) \leq C_{0} (c) {n^{- 1} - ρ_{n, 2}}, d_{n} (c) \leq 43 C_{0} (c) n^{- 4/3} .$ Finally, writing $S_{c} = {a \in A_{K} : c_{a} \neq = 0}$ for the active support, if $∣ S_{c} ∣ = 2$ , then $ρ_{n} (c) = C_{0} (c) ρ_{n, 2} and d_{n} (c) = C_{0} (c) {n^{- 1} - ρ_{n, 2}} .$

⊢ Lean

The embedded comparison supplies the lower side of the second-order scale. It says that any improvement over the first-order envelope for a fixed multi-arm contrast is bounded by the corresponding two-arm improvement, up to the scale factor $C_{0} (c)$ . When the active support has size two, the comparison becomes an equality, so the two-arm subproblem fully determines the multi-arm risk for that contrast.

The second-order construction keeps the contrast-weighted design and modifies the estimator. The rule first normalizes the centered Horvitz–Thompson score to the interval scale $h_{c} = L_{c} /2$ , averages the normalized scores, and then applies a clipped shrinkage toward zero over a bandwidth of order $n^{- 1/3}$ .

Definition 16 [synth_16] (Regular variation of a normalizer).

A positive sequence $(a_{n})_{n \geq 1}$ is regularly varying with index $β$ when, for every $x > 0$ , $\frac{a _{⌊ x n ⌋}}{a _{n}} ⟶ x^{β} as n \to \infty.$

Definition 17 [synth_6] (Clipped-shrinkage estimator).

Put $h_{c} = L_{c} /2$ , and let $λ_{c} = min {\frac{\sum _{a \in A_{K}} c _{a} t _{a}}{h _{c}} : t \in T_{K}, a \in A_{K} \sum c_{a} t_{a} \neq = 0}$ be the smallest nonzero normalized response-type contrast magnitude. Under $D^{⋆}$ of Definition 15, define $W_{i} = a \in S_{c} \sum \frac{c _{a} 1 { A _{i} = a }}{q _{a}^{⋆}} (Y_{i}^{obs} - \frac{1}{2}), V_{i} = \frac{W _{i}}{h _{c}}, X_{n} = \frac{1}{n} i = 1 \sum n V_{i} .$ For $b_{n} = n^{- 1/3}, ε_{n} = \frac{λ _{c}}{4} n^{- 1/3}, g_{n} (x) = max {- b_{n}, min (x, b_{n})},$ define the clipped-shrinkage rule $τ_{n}^{sh} = proj_{[- h_{c}, h_{c}]} (h_{c} {X_{n} - ε_{n} g_{n} (X_{n})}) .$ The projection is never active for $n \geq 1$ , because $∣ X_{n} ∣ \leq 1$ , $0 \leq b_{n} \leq 1$ and $0 \leq ε_{n} \leq 1/4$ ; it is written explicitly only so that the rule takes values in the target interval by construction. This estimator is evaluated as a function of the realized assignment and observed outcomes.

The regular-variation definition in Definition 16 is used to state the uniqueness of the second-order normalizing rate. In Definition 17, the centered arm score $W_{i}$ , its normalized version $V_{i}$ , and their average $X_{n}$ are all computed under the same design as in Definition 15. The constants $b_{n}$ and $ε_{n}$ set the clipping width and shrinkage amplitude; their common $n^{- 1/3}$ scale produces an $n^{- 4/3}$ risk correction after averaging.

Theorem 5 [thm:universal-second-order-rate] (Universal second-order rate).

For every fixed $K \geq 2$ and every nonzero zero-sum contrast $c$ , let $λ_{c}$ , the smallest nonzero normalized response-type contrast magnitude, be $λ_{c} = min {\frac{\sum _{a \in A_{K}} c _{a} t _{a}}{L _{c} /2} : t \in T_{K}, a \in A_{K} \sum c_{a} t_{a} \neq = 0} .$ Define $C_{0} (c) = \frac{L _{c}^{2}}{4}, A_{c} = \frac{λ _{c}}{2} - \frac{λ _{c}}{16}, B_{c} = \frac{7 λ _{c}}{16}, κ_{c} = \frac{1}{2} min {A_{c}, B_{c}},$ and $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c), s_{n} = n^{4/3} .$ The following assertions hold:

(Spacing.) The constants satisfy $0 < λ_{c} \leq 1, 0 < κ_{c} .$

(First-order envelope.) For every $n$ , $ρ_{n} (c) \leq \frac{C _{0} ( c )}{n} .$

(Uniform shrinkage bound.) There is a finite integer $N_{c}$ such that, for every integer $m \geq N_{c}$ , $4 λ_{c} m exp {- m^{1/3} /8} \leq \frac{λ _{c}}{2} - \frac{λ _{c}}{16},$ and, for every $n \geq N_{c}$ , the explicit clipped-shrinkage procedure $τ_{n}^{sh}$ satisfies $z \in T_{K}^{n} sup R_{n} (D^{⋆}, τ_{n}^{sh}; z) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}} .$
(Second-order envelope.) The normalized second-order improvement satisfies $C_{0} (c) κ_{c} \leq n \to \infty lim inf n^{4/3} d_{n} (c) \leq n \to \infty lim sup n^{4/3} d_{n} (c) \leq 43 C_{0} (c) .$

(Regular-variation index.) For every positive sequence $a_{n}$ , every $β$ , and every $C > 0$ , if $a_{n}$ is regularly varying with index $β$ and $a_{n} d_{n} (c) \to C,$ then $β = 4/3$ .

⊢ Lean

Theorem 5 establishes the $n^{- 4/3}$ order of the improvement for every fixed nonzero zero-sum contrast. The positive spacing constant $λ_{c}$ is defined over the finite response-type space, so the shrinkage amount can be chosen directly from the contrast. The lower side of the displayed envelope is witnessed by the explicit clipped-shrinkage procedure of Definition 17, which guarantees an improvement of size $C_{0} (c) κ_{c} n^{- 4/3}$ ; the upper side follows from the embedded two-arm comparison in Theorem 4. The two sides pin the order of the improvement, not its constant. For contrasts with active support of size at least three, the same sandwich applies after the orbit reduction to the genuinely multi-arm response-type count game; support-two contrasts follow the exact two-arm reduction. The regular-variation clause characterizes $n^{4/3}$ as the unique regularly varying normalization that can yield a positive finite limit for the improvement.

It remains to record the two-arm lower bound used inside the embedded comparison. The result is stated for the unrestricted labeled-schedule minimax risk with the two-arm contrast $(1, - 1)$ , matching the notation of Theorem 4.

Theorem 6 [thm:coarse-two-arm-minimax-lower] (Coarse two-arm lower bound).

For every integer $n \geq 8$ , let $ρ_{n, 2} := ρ_{n} ((1, - 1))$ denote the unrestricted labeled-schedule minimax risk for the two-arm contrast $(1, - 1)$ . Then $\frac{( 1 - n ^{- 1/3} ) ^{2}}{n / ( 1 - n ^{- 2/3} /4 ) + 40 n ^{2/3}} \leq ρ_{n, 2}$ and $n^{- 1} - 43 n^{- 4/3} \leq ρ_{n, 2} .$

⊢ Lean

The bound supplies a fully internal rate input for the two-arm risk. Combined with Theorem 4, it transfers the $43 C_{0} (c) n^{- 4/3}$ upper bound on the second-order improvement to every fixed multi-arm contrast. Together with the shrinkage construction in Theorem 5, the section therefore pins down the second-order minimax scale while retaining the contrast-weighted design identified in Theorem 3.

Finite-program certificates for rational and real contrasts

The preceding section identifies $n^{4/3}$ as the scale on which the second-order improvement can have a nondegenerate limit. This section represents that scale by finite linear programs. The programs are finite and all their data are exactly rational, so a bracket at a given $(n, K)$ gives an exact rational certificate for that finite orbit game. The number of response-count rows is $(2 ^{K} - 1 n + 2 ^{K} - 1)$ , which grows quickly in both arguments. A finite bracket at a chosen $(n, K)$ compares a proposed asymptotic design against the exact finite orbit game and identifies which response-count states drive the lower bound. The construction replaces the continuum of estimator values by a finite rational grid, solves a primal linear program over allocation-count orbits and grid actions, and reads a matching lower bound from the dual response-count multipliers. The first version is the three-arm diagnostic contrast $c^{†}$ ; the general version covers rational contrasts and then transfers to real contrasts by continuity.

We begin with the three-arm program because it is the smallest setting in which the multi-arm response-type geometry is visible. The grid $Γ_{M}$ records candidate estimator values, the variable $w_{r, x, g}$ couples an allocation count and observation with a grid action $g$ , and the epigraph variable $u$ bounds the risk uniformly over response-count vectors.

Definition 18 [def:k3-rational-lp] (Three-arm LP value

Λ_{n, M}^{†}

For $K = 3$ , $c^{†} = (1, - 1/2, - 1/2)$ , and $Γ_{M} = {- 1 + j / M : 0 \leq j \leq 2 M},$ the three-arm rational grid-action LP value $Λ_{n, M}^{†}$ is the minimum of $u$ over rational variables $π_{r} \geq 0$ and $w_{r, x, g} \geq 0$ satisfying $r \in R_{n, 3} \sum π_{r} = 1,$ $g \in Γ_{M} \sum w_{r, x, g} = π_{r} for every r \in R_{n, 3} and x \in a \in A_{3} \prod {0, \dots, r_{a}},$ and $r, x, g \sum P_{m} (x ∣ r) w_{r, x, g} {g - τ_{c^{†}} (m)}^{2} \leq u for every m \in M_{n, 3} .$

⊢ Lean

The same construction applies to any rational nonzero zero-sum contrast. Its only additional bookkeeping is the contrast-dependent target range: the grid $Γ_{M, c}$ spans $[- L_{c} /2, L_{c} /2]$ , so each feasible barycenter remains in the admissible range for the orbit game of Definition 7. The resulting value $Λ_{n, M} (c)$ is the rational finite program used throughout the rest of the section.

Definition 19 [def:rational-contrast-grid-lp] (Rational grid-action value

Λ_{n, M} (c)

For $K \geq 2$ , $n, M \geq 1$ , and a rational nonzero zero-sum contrast $c \in Q^{K}$ , define the rational contrast grid-action value $Λ_{n, M} (c)$ as the optimal value of the following finite linear program. Let $h_{c} = L_{c} /2, Γ_{M, c} = {- h_{c} + j h_{c} / M : j = 0, \dots, 2 M} .$ The program minimizes $u$ over variables $π_{r} \geq 0$ and $w_{r, x, g} \geq 0$ , subject to $r \in R_{n, K} \sum π_{r} = 1,$ $g \in Γ_{M, c} \sum w_{r, x, g} = π_{r} for every (r, x),$ and $r \in R_{n, K} \sum x \sum g \in Γ_{M, c} \sum P_{m} (x ∣ r) w_{r, x, g} {g - τ_{c} (m)}^{2} \leq u for every m \in M_{n, K} .$

⊢ Lean

For later lower bounds, the dual variables are interpreted as a response-count prior. The next definition fixes the objective convention used to turn dual feasibility into a numerical certificate.

Definition 20 [synth_8] (Dual objective for the grid linear program).

For the grid-action linear program with primal constraints written in standard form $A ξ \leq b$ and objective $u$ , let $y$ denote the nonnegative vector of dual row multipliers satisfying the stationarity equations for the variables $(u, π, w)$ . Define $obj_{dual} (y) = - i \sum y_{i} b_{i} .$ In the grid program, the restriction of $y$ to the response-count risk rows is the response-count multiplier $ν$ , and dual feasibility gives a lower certificate for the corresponding primal epigraph value.

The three-arm diagnostic program has an exact rational certificate at every sample size and grid resolution. The statement also records the mesh size $M = n^{2}$ , which is fine enough for the grid error to vanish at the $n^{4/3}$ scale.

Proposition 1 [prop:k3-lp-certificate] (Three-arm LP certificate).

For the three-arm diagnostic contrast $c^{†} = (1, - 1/2, - 1/2)$ and every pair of positive integers $n$ and $M$ , let $ρ_{n}^{†}$ be the unrestricted three-arm minimax risk and let $Λ_{n, M}^{†}$ be the three-arm rational grid-action LP value from Definition 18. Then $ρ_{n}^{†} \leq Λ_{n, M}^{†} \leq ρ_{n}^{†} + \frac{2}{M} + \frac{1}{M ^{2}}, 0 \leq Λ_{n, M}^{†} \leq 1.$ At the resolution $M = n^{2}$ , there exist a rational allocation mass $π$ on allocation-count vectors, a joint design-action mass $w$ , a rational epigraph value $u$ , and rational response-count multipliers $ν$ on count vectors $m : T_{3} \to {0, \dots, n}$ with $\sum_{t \in T_{3}} m_{t} = n$ , such that there are a rational grid weight $w^{Q}$ and dual multipliers $y$ satisfying $w = w^{Q} after real embedding, (π, w^{Q}, u) is primal feasible, y is dual feasible, ν = y_{risk}, u = obj_{dual} (y),$ and $u = Λ_{n, n^{2}}^{†} .$ Moreover, every binary response type $t \in T_{3}$ is represented by some response-count vector $m$ with total population size $n$ and $m_{t} > 0$ , and the grid-discretization error at the certificate scale satisfies $k \to \infty lim k^{4/3} (\frac{2}{k ^{2}} + \frac{1}{k ^{4}}) = 0.$

⊢ Lean

Proposition 1 supplies a concrete finite certificate for the three-arm contrast. Its primal side gives a randomized invariant procedure whose risk is bounded by the grid value; its dual side gives response-count multipliers that certify a matching lower value up to the displayed mesh error. The representation clause records that every response type occurs in at least one feasible response-count vector, so the finite state space of the program covers the whole type space; which types carry positive dual mass is decided by the solution, not by this clause.

The lower endpoint in the general certificate is a Bayes risk computed at a fixed allocation count. The formula below is the posterior-variance form of that Bayes risk, expressed entirely with the orbit likelihood from Definition 6.

Definition 21 [synth_7] (Orbit Bayes risk at an allocation count).

For a response-count prior $ν \in Δ (M_{n, K})$ , a contrast $c$ , and an allocation count $r \in R_{n, K}$ , define $BayesRisk_{n, c} (r, ν) = δ_{r} in f m \in M_{n, K} \sum ν_{m} x \sum P_{m} (x ∣ r) {δ_{r} (x) - τ_{c} (m)}^{2},$ where $x \in \prod_{a \in A_{K}} {0, \dots, r_{a}}$ and $δ_{r} (x) \in [- L_{c} /2, L_{c} /2]$ . Equivalently, with $D_{r, x, c} (ν) = m \sum ν_{m} P_{m} (x ∣ r), N_{r, x, c} (ν) = m \sum ν_{m} P_{m} (x ∣ r) τ_{c} (m),$ $BayesRisk_{n, c} (r, ν) = m \sum ν_{m} τ_{c} (m)^{2} - x : D_{r, x, c} (ν) > 0 \sum \frac{N _{r, x, c} ( ν ) ^{2}}{D _{r, x, c} ( ν )} .$

The rational certificate theorem combines the primal grid solution, its barycenter estimator, and the dual Bayes-risk endpoint. The statement also gives the orbit counts, making the computational size of the certificate explicit.

Theorem 7 [thm:rational-contrast-grid-certificate-sandwich] (Rational grid certificate sandwich).

Fix natural numbers $K, n, M$ , a rational nonzero zero-sum contrast $c \in Q^{K}$ , and suppose $n \geq 1$ and $M \geq 1$ ; here $ρ_{n} (c)$ is the labeled-schedule minimax risk of Definition 3. In the rational grid-action program of Definition 19, there exist a rational allocation certificate $π^{n, M, c}$ , a joint design-action weight $w^{n, M, c}$ , a rational epigraph value $u$ , a rational response-count multiplier $ν_{n, M, c} : M_{n, K} \to Q$ , and an invariant estimator $δ_{n, M, c}$ on allocation and observation counts such that:

(Exact rational certificate.) There are rational grid weights $w^{Q}$ and rational dual multipliers $y$ such that $w^{n, M, c}$ is the real embedding of $w^{Q}$ , the tuple $(π^{n, M, c}, w^{Q}, u)$ is primal feasible for the grid program, $y$ is dual feasible, $ν_{n, M, c}$ is the risk component of $y$ , and $u$ is the dual objective value of $y$ .

(Barycenter estimator.) For every allocation count $r \in R_{n, K}$ and compatible observation $x$ , $δ_{n, M, c} (r, x) = ⎩ ⎨ ⎧ g \in Γ_{M, c} \sum g w_{r, x, g}^{n, M, c} / π_{r}^{n, M, c}, 0, π_{r}^{n, M, c} > 0, π_{r}^{n, M, c} = 0.$

(Optimal value.) The real value of $u$ is the grid LP value: $u = Λ_{n, M} (c) .$

(Certificate sandwich.) With $B_{n, c} (ν_{n, M, c}) = in f {v \in R : \exists r \in R_{n, K} with v = BayesRisk_{n, c} (r, ν_{n, M, c})},$ and with $U_{n, M} (c)$ denoting the unrestricted labeled-schedule risk of $δ_{n, M, c}$ under $π^{n, M, c}$ , the certificates satisfy $B_{n, c} (ν_{n, M, c}) \leq ρ_{n} (c) \leq U_{n, M} (c) \leq Λ_{n, M} (c) \leq B_{n, c} (ν_{n, M, c}) + \frac{C _{0} ( c )}{4 M ^{2}} .$

(Orbit counts.) The finite sets in the certificate have cardinalities $∣ M_{n, K} ∣ = (2 ^{K} - 1 n + 2 ^{K} - 1), ∣ R_{n, K} ∣ = (K - 1 n + K - 1),$ and $r \in R_{n, K} \sum ∣ {x : x is compatible with r} ∣ = (2 K - 1 n + 2 K - 1) .$

(Mesh-rate scaling.) For every positive sequence $a_{k}$ and every integer sequence $M_{k} \geq 1$ , if $\frac{a _{k}}{M _{k}^{2}} ⟶ 0,$ then $a_{k} \frac{C _{0} ( c )}{4 M _{k}^{2}} ⟶ 0.$

(Asymptotic certificates.) For every positive normalizing sequence $a_{k}$ and every grid-resolution sequence $M_{k} \geq 1$ satisfying $a_{k} / M_{k}^{2} \to 0$ , the rational certificates at mesh $M_{k}$ admit lower and upper endpoints $B_{k}$ and $U_{k}$ with $B_{k} \leq ρ_{k} (c) \leq U_{k}, a_{k} {U_{k} - B_{k}} ⟶ 0.$ Consequently, whenever $a_{k} d_{k} (c)$ has a finite limit $ℓ$ , the scaled lower and upper certificate improvements both converge to that limit: $a_{k} {\frac{C _{0} ( c )}{k} - U_{k}} ⟶ ℓ, a_{k} {\frac{C _{0} ( c )}{k} - B_{k}} ⟶ ℓ .$ At the second-order scale $a_{k} = k^{4/3}$ , the specialization $M_{k} = k$ satisfies the mesh condition and yields $k^{4/3} {U_{k} - B_{k}} ⟶ 0.$

⊢ Lean

Theorem 7 turns each rational contrast into two computable endpoints. The upper endpoint is obtained by replacing the randomized grid action with its conditional barycenter, which preserves the relevant risk inequality by convexity. The lower endpoint is the best allocation-count Bayes risk under the dual response-count multiplier. Strong duality for the finite rational program then leaves only the grid mesh gap, and that gap is $o (k^{- 4/3})$ when $M_{k} = k$ .

The transfer from rational to real contrasts uses a direct Lipschitz comparison for the square root of the minimax risk. The distance is the half- $ℓ_{1}$ contrast distance, so the comparison is expressed in the same scale as the target interval.

Theorem 8 [thm:contrast-risk-continuity] (Contrast risk continuity).

Let $K$ and $n$ be integers, and let $c, c^{'} \in R^{K}$ be nonzero zero-sum treatment contrasts for the labeled-schedule game in Definition 3. Suppose that $K \geq 2 and n \geq 1.$ Define the contrast distance $η (c, c^{'}) = \frac{1}{2} a \in A_{K} \sum ∣ c_{a} - c_{a}^{'} ∣.$ Then the unrestricted labeled-schedule minimax risk satisfies $ρ_{n} (c) - ρ_{n} (c^{'}) \leq η (c, c^{'}) .$

⊢ Lean

The square-root form of Theorem 8 matches the geometry of squared-error risk: changing the contrast shifts every target by at most $η (c, c^{'})$ , and the minimax risk changes accordingly in root-risk units. This gives a stable way to carry finite rational certificates to nearby real contrasts.

It remains to name the exact rational primal-dual certificate as a reusable object. The definition records the feasibility, dual stationarity, and zero-gap conditions that are invoked in the transfer theorem.

Definition 22 [synth_12] (Exact rational primal-dual grid certificate).

Fix rational nonzero zero-sum $c \in Q^{K}$ and integers $n, M \geq 1$ . An exact rational primal-dual grid certificate for the program defining $Λ_{n, M} (c)$ consists of rational values $(π_{r})_{r \in R_{n, K}}$ , rational values $(w_{r, x, g})$ , a rational epigraph value $u$ , and rational dual multipliers for the linear constraints, such that $π_{r} \geq 0$ , $w_{r, x, g} \geq 0$ , $\sum_{r} π_{r} = 1$ , $\sum_{g \in Γ_{M, c}} w_{r, x, g} = π_{r}$ for every $(r, x)$ , and $r \in R_{n, K} \sum x \sum g \in Γ_{M, c} \sum P_{m} (x ∣ r) w_{r, x, g} {g - τ_{c} (m)}^{2} \leq u for every m \in M_{n, K} .$ The dual multipliers are rational, nonnegative on inequality rows, satisfy the stationarity equations for the same rational linear program, and have dual objective equal to the primal objective $u$ . The common value is $Λ_{n, M} (c)$ .

The real-contrast result chooses rational approximants close enough that the continuity loss is dominated by the second-order scale. For an already rational contrast, the same statement specializes to a bracket with the $n^{- 2}$ mesh width.

Theorem 9 [thm:real-contrast-grid-certificate-transfer] (Real contrast certificate transfer).

Fix $K \geq 2$ and a nonzero zero-sum real contrast $c$ on $A_{K}$ . For real contrasts $c_{1}, c_{2}$ , write $η (c_{1}, c_{2}) = \frac{1}{2} a \in A_{K} \sum ∣ c_{1} (a) - c_{2} (a) ∣, C_{0} (c_{1}) = \frac{L _{c_{1}}^{2}}{4} .$ A tuple $(B, U, R^{-}, R^{+})$ is a transferred certificate for $(n, M, c, q)$ , where $q \in Q^{K} ∖ {0}$ is zero-sum, if there exist an invariant allocation-count distribution $π$ , a rational grid-action weight $w$ , a rational epigraph value $u$ , a rational response-count prior $ν$ , an invariant barycenter rule $δ$ , a projected upper procedure, and a schedule prior such that the exact rational primal-dual grid certificate and barycenter identities hold for $q, π, w, u, ν, δ$ , and $B = B_{n, q} (ν), U = U_{n, M} (q; π, δ), u = Λ_{n, M} (q),$ $B \leq ρ_{n} (q) \leq U \leq u \leq B + \frac{C _{0} ( q )}{4 M ^{2}},$ $R^{-} = {max (0, B - η (c, q))}^{2}, R^{+} = (U + η (c, q))^{2},$ $R^{-} \leq ρ_{n} (c) \leq R^{+},$ and $R^{+} - R^{-} \leq \frac{5 C _{0} ( q )}{4 M ^{2}} + 4 C_{0} (q) η (c, q) n^{- 1/2} + 3 η (c, q)^{2} .$ The projected upper procedure certifies worst-case risk at most $R^{+}$ , and the schedule prior certifies Bayes risk at least $R^{-}$ .

Then the following hold.

(Finite-grid transfer.) For every rational zero-sum contrast $q \in Q^{K} ∖ {0}$ and every pair of integers $n, M \geq 1$ , there exist $B, U, R^{-}, R^{+} \in R$ forming a transferred certificate for $(n, M, c, q)$ .

(Real-contrast approximation.) There exist rational zero-sum contrasts $c^{(n)} \in Q^{K} ∖ {0}$ and real sequences $B_{n}, U_{n}, R_{n}^{-}, R_{n}^{+}$ such that, for every arm $a \in A_{K}$ , $c^{(n)} (a) ⟶ c (a),$ $n^{5/6} η (c, c^{(n)}) ⟶ 0,$ for every $n \geq 1$ , $(B_{n}, U_{n}, R_{n}^{-}, R_{n}^{+})$ is a transferred certificate for $(n, n, c, c^{(n)})$ , and $n^{4/3} (R_{n}^{+} - R_{n}^{-}) ⟶ 0.$

(Exact rational case.) If a rational zero-sum contrast $q \in Q^{K} ∖ {0}$ embeds coefficientwise as $c$ , then for every $n \geq 1$ there exist $B, U, R^{-}, R^{+} \in R$ forming a transferred certificate for $(n, n, c, q)$ and satisfying $U - B \leq \frac{C _{0} ( c )}{4 n ^{2}} .$

⊢ Lean

Theorem 9 gives the certificate construction for every real contrast. The rational program supplies the inner bracket for the approximating contrast $q$ , while Theorem 8 expands the bracket just enough to cover the target contrast $c$ . The displayed bound separates the mesh contribution, the contrast-approximation contribution, and the quadratic approximation term, so choosing $M = n$ and $n^{5/6} η (c, c^{(n)}) \to 0$ produces a bracket whose width is negligible at the $n^{4/3}$ scale.

We close the section by recording the specialized three-arm sandwich. It packages the diagnostic program of Definition 18 in the same lower-upper form as the general rational theorem and gives the certificate comparison used in the three-arm analysis that follows.

Theorem 10 [thm:k3-grid-certificate-sandwich] (Three-arm grid sandwich).

For integers $n \geq 1$ and $M \geq 1$ , in the three-arm rational grid program of Definition 18, there exist a rational allocation certificate $π^{n, M}$ , a joint design–action weight $w^{n, M}$ , a rational epigraph value $u$ , a rational response-count multiplier $ν_{n, M}$ , and an estimator $δ_{n, M}$ such that:

(Exact certificate.) The tuple $(π^{n, M}, w^{n, M}, u, ν_{n, M})$ is represented by a rational grid weight whose real embedding is $w^{n, M}$ , a primal feasible point of the rational grid LP, and a dual feasible point whose risk component is $ν_{n, M}$ and whose dual objective is $u$ .

(Barycenter rule.) For every allocation count $r \in R_{n, 3}$ and compatible observation $x$ , the estimator is $δ_{n, M} (r, x) = ⎩ ⎨ ⎧ g \in Γ_{M} \sum g \frac{w _{r, x, g}^{n, M}}{π _{r}^{n, M}}, 0, π_{r}^{n, M} > 0, π_{r}^{n, M} = 0.$

(Grid value.) The real value of $u$ is $Λ_{n, M}^{†}$ .

Define $B_{n} (ν_{n, M}) = r \in R_{n, 3} in f BayesRisk_{n, c^{†}} (r, ν_{n, M})$ and $U_{n, M} = m \in M_{n, 3} sup r \in R_{n, 3} \sum π_{r}^{n, M} x \sum P_{m} (x ∣ r) {δ_{n, M} (r, x) - τ_{c^{†}} (m)}^{2} .$ Then $B_{n} (ν_{n, M}) \leq ρ_{n}^{†} \leq U_{n, M} \leq Λ_{n, M}^{†} \leq B_{n} (ν_{n, M}) + \frac{1}{4 M ^{2}} .$ Moreover, for every positive normalizing sequence $a_{k}$ and every grid-resolution sequence $M_{k}$ with $M_{k} \geq 1$ for all $k$ , if $\frac{a _{k}}{M _{k}^{2}} ⟶ 0,$ then $a_{k} \frac{1}{4 M _{k}^{2}} ⟶ 0.$ Finally, for every positive normalizing sequence $a_{k}$ , every grid-resolution sequence $M_{k}$ , every pair of real sequences $L_{k}, U_{k}$ , and every real number $C$ , if for every $k \geq 1$ , $M_{k} \geq 1, L_{k} \leq ρ_{k}^{†} \leq U_{k}, U_{k} - L_{k} \leq \frac{1}{4 M _{k}^{2}},$ and if $\frac{a _{k}}{M _{k}^{2}} ⟶ 0, a_{k} d_{k} (c^{†}) ⟶ C,$ then $a_{k} {\frac{C _{0} ( c ^{†} )}{k} - L_{k}} ⟶ C and a_{k} {\frac{C _{0} ( c ^{†} )}{k} - U_{k}} ⟶ C .$

⊢ Lean

Theorem 10 specializes the rational programs to $c^{†}$ , where the mesh gap is $1/ (4 M^{2})$ . Thus any independently computed bracket with that width inherits the same second-order limit for the scaled improvement. The next section uses this certificate form to diagnose the three-arm geometry and to compare support-specific conclusions at the $n^{4/3}$ scale.

Three-arm diagnostics and support-specific conclusions

The finite certificates of Theorem 10 give a computable way to audit three-arm risks. We first use them to separate the full labeled experiment from a natural scalar signed-score reduction. For the diagnostic contrast $c^{†} = (1, - 1/2, - 1/2)$ , the scalar statistic records the sign-adjusted observed outcome, which retains the one-dimensional target direction and nothing else. The full experiment keeps a richer summary of the assignment and the outcomes: the rule exhibited below reads the number of units sent to the positively weighted arm, the signed outcome total within that arm, and the signed outcome total within the negatively weighted group. That summary already suffices, under the same independent allocation rule, to attain a strictly smaller worst-case risk than any rule based on the scalar score. The comparison is therefore between the scalar score and this richer assignment–outcome summary, which the exhibited rule reads while aggregating the two negatively weighted arms.

Proposition 2 [prop:k3-scalar-score-not-minimax-preserving] (Scalar score separation).

For $K = 3$ , let $c^{†} = (1, - 1/2, - 1/2)$ , and let $D^{⋆}$ be the independent design assigning each unit with probabilities $q^{⋆} = (1/2, 1/4, 1/4)$ . For any $n \geq 1$ , any schedule $z \in T_{3}^{n}$ , and any unit $i$ , define $Z_{i} = sign (c_{A_{i}}^{†}) (2 Y_{i}^{obs} - 1), μ_{i} = t_{i, 1} - \frac{t _{i, 2} + t _{i, 3}}{2} .$ Then the following statements hold.

(Score moments.) For every $n \geq 1$ , every $z \in T_{3}^{n}$ , and every unit $i$ , $E_{D^{⋆}} (Z_{i} ∣ z) = μ_{i}, Var_{D^{⋆}} (Z_{i} ∣ z) = 1 - μ_{i}^{2} .$

(Average-score risk.) For every $n \geq 1$ and every $z \in T_{3}^{n}$ , $E_{D^{⋆}} {\frac{1}{n} i \sum Z_{i} - τ_{c^{†}} (z)}^{2} z = \frac{1}{n} - \frac{1}{n ^{2}} i \sum μ_{i}^{2} .$

(Scalar minimax value.) For the scalar experiment at $n = 3$ that retains $(Z_{1}, Z_{2}, Z_{3})$ and lets each $μ_{i}$ range over the five feasible values, its minimax value is $V_{3}^{Z} = 1 - \frac{3}{2} .$

(Full-data certificate.) There exists a clipped full-data estimator using the observed arm labels and outcomes under $D^{⋆}$ whose worst-case risk over $z \in T_{3}^{3}$ is $\frac{511653}{4000000} .$ Moreover, $\frac{511653}{4000000} < \frac{18213}{136000} < 1 - \frac{3}{2} .$

(Scalar moment envelope.) For every $n$ , every real $M$ , and every feasible three-arm scalar total $M = ∣ \sum_{i} μ_{i} ∣$ , the minimum attainable value of $\sum_{i} μ_{i}^{2}$ among scalar mean vectors with $∣ \sum_{i} μ_{i} ∣ = M$ is ${M /2, (3 M - n) /2, M \leq n /2, M > n /2.$

Thus the scalar signed-score experiment has exact value $V_{3}^{Z}$ at $n = 3$ , while the full observed arm-label-and-outcome experiment under the same independent $q^{⋆}$ design attains the smaller certified worst-case risk displayed above.

⊢ Lean

The diagnostic has two consequences for the rest of the section. First, scalar moment calculations remain useful for locating difficult schedules, because the displayed envelope identifies how the average-score risk depends on the feasible means. Second, the minimax audit is conducted in the full labeled experiment: Proposition 2 gives an explicit $n = 3$ comparison in which arm labels strictly lower the worst-case risk.

We now package the certificate objects used to compare rational finite programs and real contrasts. The cluster terminology names the primal weights, barycenter estimator, and dual response-count prior that move together in the finite orbit calculation.

Definition 23 [synth_18] (Exact rational grid-certificate cluster).

Fix a rational nonzero zero-sum contrast $c \in Q^{K}$ and integers $n, M \geq 1$ . An exact rational grid-certificate cluster at $(n, M, c)$ consists of an exact rational primal-dual grid certificate for the grid-action program of Definition 19, in the sense of Definition 22 — rational primal weights $(π^{n, M, c}, w^{n, M, c})$ and the rational epigraph value $Λ_{n, M} (c)$ — together with the rational nonnegative response-count multiplier $ν_{n, M, c} \in Δ (M_{n, K})$ read off its dual solution. Its derived objects are the barycenter estimator $δ_{n, M, c}$ and the upper risk certificate $U_{n, M} (c)$ , both built from $w^{n, M, c}$ , and the posterior-mean lower Bayes-risk certificate $B_{n, c} (ν_{n, M, c})$ , built from $ν_{n, M, c}$ .

For real contrasts, the transfer argument uses the same half- $ℓ_{1}$ metric as Theorem 8. We record the notation before stating the support-three-and-larger result.

Definition 24 [synth_11] (Contrast distance).

For two real contrasts $c_{1}, c_{2}$ on $A_{K}$ , define $dist (c_{1}, c_{2}) = η (c_{1}, c_{2}) = \frac{L _{c_{1} - c_{2}}}{2} = \frac{1}{2} a \in A_{K} \sum ∣ c_{1} (a) - c_{2} (a) ∣.$ Thus $dist (c, c^{(n)}) = η (c, c^{(n)})$ .

The next result collects the resolved second-order scale and the finite-certificate approximations for contrasts with at least three active arms. It uses the improvement $d_{n} (c) = C_{0} (c) / n - ρ_{n} (c)$ and the normalized quantity $x_{n} (c) = n^{4/3} d_{n} (c)$ , matching the rate established in Theorem 5 and the lower comparison in Theorem 6.

Theorem 11 [thm:second-order-rate-and-certificate-frontier] (Second-order certificate frontier).

Fix $K \in N$ and a contrast $c$ on $A_{K}$ as in Definition 3. Suppose that

(Arm count.) $2 \leq K$ .

(Active support.) With $S_{c} = {a \in A_{K} : c_{a} \neq = 0}$ , one has $∣ S_{c} ∣ \geq 3$ .

Define $x_{n} (c) = n^{4/3} d_{n} (c), d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c) .$ Then $0 < n \to \infty lim inf x_{n} (c) \leq n \to \infty lim sup x_{n} (c) \leq 43 C_{0} (c) .$ Moreover, $x_{n} (c)$ has at least one subsequential limit, and its subsequential-limit set ${a \in R : there is a strictly increasing ϕ : N \to N with x_{ϕ (n)} (c) \to a}$ is compact.

For every positive sequence $a_{n}$ , every $β \in R$ , and every $C > 0$ , if $\frac{a _{⌊ t n ⌋}}{a _{n}} \to t^{β} for every t > 0$ and $a_{n} d_{n} (c) \to C,$ then $β = 4/3$ . In addition, $\exists a_{n} > 0, \exists C > 0 such that \frac{a _{⌊ t n ⌋}}{a _{n}} \to t^{4/3} for every t > 0 and a_{n} d_{n} (c) \to C ⟺ x_{n} (c) > 0 eventually and \frac{x _{⌊ t n ⌋} ( c )}{x _{n} ( c )} \to 1 for every t > 0.$

For every rational contrast $q$ on $A_{K}$ whose real-valued contrast is $c$ , there exist sequences $G_{n}$ , $P_{n}$ , and $L_{n}$ such that $G_{n}$ is an exact rational grid-certificate cluster for $q$ , and $G_{n} - x_{n} (c) \to 0, P_{n} - G_{n} \to 0, L_{n} - G_{n} \to 0.$

There also exist rational contrasts $c^{(n)}$ , certificate values $B_{n}, U_{n}$ , and endpoints $R_{n}^{-}, R_{n}^{+}$ such that, for every arm $a$ , $c_{a}^{(n)} \to c_{a},$ and $n^{5/6} dist (c, c^{(n)}) \to 0.$ For every $n > 0$ , there are an exact rational primal-dual certificate, a grid barycenter estimator, and a rational prior for $c^{(n)}$ such that $B_{n} = the lower certificate from the prior, U_{n} = the upper certificate from the grid barycenter,$ $R_{n}^{-} = (max {0, B_{n} - dist (c, c^{(n)})})^{2}, R_{n}^{+} = (U_{n} + dist (c, c^{(n)}))^{2},$ and $R_{n}^{-} \leq ρ_{n} (c) \leq R_{n}^{+} .$ The transferred normalized endpoint improvements are asymptotically coincident with $x_{n} (c)$ : $n^{4/3} (\frac{C _{0} ( c )}{n} - R_{n}^{-}) - x_{n} (c) \to 0, n^{4/3} (\frac{C _{0} ( c )}{n} - R_{n}^{+}) - x_{n} (c) \to 0.$

Finally, for the three-arm contrast $c^{†} = (1, - 1/2, - 1/2)$ , there exists an estimator rule $τ$ in the $K = 3, n = 3$ labeled game such that $z \in T_{3}^{3} sup R_{3} (D^{⋆}, τ; z) < V_{3}^{Z},$ where $D^{⋆}$ is the product design using the $q^{⋆}$ allocation rule for $c^{†}$ and $V_{3}^{Z}$ is the three-observation scalar sign-score minimax value.

⊢ Lean

The theorem gives the $n^{- 4/3}$ scale an operational interpretation. The bounded positive liminf and finite limsup locate the improvement between two constants on that scale, while the regular-variation clause says that any convergent power normalization must use exponent $4/3$ . The finite-program clauses connect the asymptotic quantity to exactly solvable rational programs in principle: rational clusters approximate the normalized improvement, and rational approximants transfer the same bracketing statement to real contrasts through the distance defined in the setup. The three-arm scalar diagnostic is stated separately as a finite $n = 3$ illustration and is then used as the running example for the bracket construction.

It remains to isolate the support-two case. When the contrast has exactly one positive and one negative active arm, the multi-arm problem embeds the two-arm labeled experiment by assigning mass only to those active arms and rescaling the target by the half-range $h_{c}$ .

Definition 25 [synth_17] (Sign-pair embedding and lift).

Assume $∣ S_{c} ∣ = 2$ , and write $S_{c} = {a_{+}, a_{-}}$ with $c_{a_{+}} = h_{c} = L_{c} /2$ and $c_{a_{-}} = - h_{c}$ . The sign-schedule embedding $ι_{c} : T_{2}^{n} \to T_{K}^{n}$ sends a two-arm schedule $s = ((s_{i, +}, s_{i, -}))_{i = 1}^{n}$ to the $K$ -arm schedule $z = ι_{c} (s)$ with $z_{i, a_{+}} = s_{i, +}$ , $z_{i, a_{-}} = s_{i, -}$ , and fixed zero coordinates on arms outside ${a_{+}, a_{-}}$ . The sign-pair lift of a two-arm labeled procedure $p_{2} = (D_{2}, τ_{2})$ is the $K$ -arm procedure that maps the two labels $(+, -)$ to $(a_{+}, a_{-})$ , assigns no mass to other arms, and reports $h_{c} τ_{2}$ after translating the observed active-arm data back to two-arm notation. Under this embedding, $τ_{c} (ι_{c} (s)) = h_{c} τ_{(1, - 1)} (s)$ , and pushed-forward priors on $T_{2}^{n}$ become priors on $T_{K}^{n}$ .

The concluding theorem combines the support-two exact value, the order-attaining shrinkage rule, and the certificate statements used for finite audits. In applications with several randomized arms, such as multi-arm binary endpoint analyses in the style of ACTG 175 (Hammer et al., 1996), these clauses identify which support geometry determines the minimax comparison.

Theorem 12 [thm:attainment-and-k3-certified-converse] (Two-arm value transfer and finite brackets).

Fix $K \geq 2$ and a nonzero zero-sum contrast $c$ on $A_{K}$ . Here $ρ_{n} (c)$ is the labeled-schedule minimax risk of Definition 3 and $G_{n} (c)$ the finite orbit-game value of Definition 7. The following statements hold.

(Two-active-arm value and saddle transfer.) For every $n \geq 1$ , if $∣ S_{c} ∣ = 2$ , then $ρ_{n} (c) = C_{0} (c) ρ_{n, 2} .$ Moreover, there exist a two-arm orbit procedure $q_{2}$ , a prior $ν_{2}$ on $M_{n, 2}$ , a two-arm labeled procedure $p_{2}$ , a prior $prior_{2}$ on $T_{2}^{n}$ , a $K$ -arm labeled procedure $p_{K}$ , and a prior $prior_{K}$ on $T_{K}^{n}$ such that $R_{(1, - 1)}^{orb} (q_{2}; m) \leq G_{n} ((1, - 1)) for every m \in M_{n, 2},$ and, for every two-arm orbit procedure $q_{2}^{'}$ , $G_{n} ((1, - 1)) \leq E_{ν_{2}} [R_{(1, - 1)}^{orb} (q_{2}^{'}; m)] .$ The labeled procedure $p_{2}$ is the invariant labeled procedure induced by $q_{2}$ , and the labeled prior satisfies $prior_{2} (z) = \frac{ν _{2} ( counts ( z ))}{∣ { z ^{'} \in T _{2}^{n} : counts ( z ^{'} ) = counts ( z )} ∣} .$ The $K$ -arm procedure $p_{K}$ is the sign-pair lift of $p_{2}$ along $c$ , and $prior_{K}$ is the pushforward of $prior_{2}$ under the sign-schedule embedding. These objects satisfy $R_{n} (p_{2}; z) \leq ρ_{n, 2} for every z \in T_{2}^{n},$ $ρ_{n, 2} \leq E_{prior_{2}} [R_{n} (p_{2}^{'}; z)] for every two-arm labeled procedure p_{2}^{'},$ $R_{n} (p_{K}; z) \leq ρ_{n} (c) for every z \in T_{K}^{n},$ and $ρ_{n} (c) \leq E_{prior_{K}} [R_{n} (p_{K}^{'}; z)] for every K -arm labeled procedure p_{K}^{'} .$
(Second-order scale.) The normalized improvement $d_{n} (c) = C_{0} (c) / n - ρ_{n} (c)$ satisfies $C_{0} (c) κ_{c} \leq n \to \infty lim inf n^{4/3} d_{n} (c) \leq n \to \infty lim sup n^{4/3} d_{n} (c) \leq 43 C_{0} (c) .$

(Shrinkage attainment.) There is an integer $N$ such that, for every $n \geq N$ , the explicit clipped-shrinkage procedure satisfies $z \in T_{K}^{n} sup R_{n} (D^{⋆}, τ_{n}^{sh}; z) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}} .$

(Real-contrast transfer certificates.) For every rational nonzero zero-sum contrast $q$ , every $n \geq 1$ , and every $M \geq 1$ , there exist real numbers $B, U, R^{-}, R^{+}$ satisfying the real-contrast transfer-certificate conditions of Theorem 9 for $(K, n, M, c, q)$ . There also exist rational contrasts $c^{(n)}$ and real sequences $B_{n}, U_{n}, R_{n}^{-}, R_{n}^{+}$ such that $c_{a}^{(n)} \to c_{a} for every a \in A_{K},$ $n^{5/6} η (c, c^{(n)}) \to 0,$ the tuple $(B_{n}, U_{n}, R_{n}^{-}, R_{n}^{+})$ satisfies the real-contrast transfer-certificate conditions of Theorem 9 for $(K, n, n, c, c^{(n)})$ whenever $n \geq 1$ , and $n^{4/3} (R_{n}^{+} - R_{n}^{-}) \to 0.$ If a rational contrast $q$ represents $c$ exactly, then for every $n \geq 1$ there exist real numbers $B, U, R^{-}, R^{+}$ satisfying the same transfer-certificate conditions for $(K, n, n, c, q)$ and $U - B \leq \frac{C _{0} ( c )}{4 n ^{2}} .$

(Three-arm rational certificates.) For every $n \geq 1$ and $M \geq 1$ , there exist a three-arm grid allocation certificate $π$ , a grid weight $w$ , a rational epigraph value $u$ , a rational response-count prior $ν$ , and a barycenter estimator $δ$ satisfying the exact primal-dual certificate and barycenter conditions of Theorem 10 for $c^{†}$ . These objects satisfy $B_{n} (ν) \leq ρ_{n}^{†} \leq U_{n, M} \leq B_{n} (ν) + \frac{1}{4 M ^{2}} .$

⊢ Lean

The support-two clause is an exact reduction: the sign-pair embedding in Definition 25 transports both the upper procedure and the least favorable prior, so the $K$ -arm value is the two-arm value scaled by $C_{0} (c)$ . For larger supports, the shrinkage and certificate clauses give a common conclusion on the same second-order scale. The explicit procedure attains a $C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}}$ bound eventually, while the rational and real-certificate brackets provide finite lower and upper checks around the unrestricted minimax risk. The specialized three-arm certificate closes the loop with Proposition 2: the same orbit-program machinery that audits $ρ_{n}^{†}$ also records where the scalar signed-score reduction separates from the full observed-data experiment.

Discussion, limitations, and future work

Design implications for multi-arm binary experiments

The results give minimax benchmarks and finite-program tools for prespecified finite-population contrasts in multi-arm binary randomization. For any fixed nonzero zero-sum contrast, Theorem 3 identifies the first-order minimax constant and shows that the contrast-weighted assignment rule, paired with the projected contrast-weighted Horvitz–Thompson estimator, attains the first-order bound. Theorem 5 then shows that the same design can be paired with a clipped shrinkage estimator to improve the worst-case risk on the $n^{- 4/3}$ scale. The model fixes a finite population and fixed $K$ , encodes each unit by binary potential outcomes for the arms, uses zero-sum contrasts to target treatment differences, clips estimators to the feasible contrast range, and lets the adversary range over all labeled binary schedules; the active support records which arms enter the contrast. The finite orbit reduction in Theorem 1 and the finite LP constructions in Theorems 7 and 9 turn the minimax problem into finite optimization problems that give exact rational certificates and transferred finite bounds at fixed $n$ .

A concrete setting is ACTG 175 (Hammer et al., 1996), which randomized HIV-1 infected patients among four regimens — zidovudine monotherapy, didanosine monotherapy, zidovudine–didanosine, and zidovudine–zalcitabine — and whose primary endpoint was a binary composite event: a decline of at least $50%$ in CD4 cell count, progression to AIDS, or death. Taking $K = 4$ , $Y_{i} (a) \in {0, 1}$ for the indicator of that event under regimen $a$ , and the enrolled cohort as the finite population, the model of Section 3 applies verbatim once a contrast $c$ is fixed in the protocol, before outcomes are seen. With the contrast fixed, its active support determines which of the support-two or support-at-least-three conclusions applies. A comparison that places mass on one regimen against one other regimen falls under the support-two reduction in Theorem 12. A contrast comparing one regimen against an average of two or more alternatives uses the multi-arm conclusions of Theorems 11 and 12, with finite lower and upper certificates available through the rational or real-contrast programs.

The three-arm diagnostic case clarifies why the full observed-data experiment matters. Proposition 2 separates the signed-score reduction from the orbit experiment for the contrast $c^{†} = (1, - 1/2, - 1/2)$ , while Theorem 10 supplies matching finite-program certificates for the corresponding minimax risk. Together these statements say that collapsing observations to a scalar sign score can change the minimax value, and that the response-type orbit game retains the information needed for finite- $n$ certification.

The logical spine is short, and it is worth stating once. Everything rests on the exact orbit equality of Theorem 1. Given that equality, Theorem 3 fixes the first-order constant, Theorem 5 supplies the attaining shrinkage rule, and Theorems 4 and 6 supply the matching converse; Theorems 7, 8, and 9 then make the finite value computable, first for rational contrasts and then for all real ones. The remaining statements — Theorem 11, Theorem 12, Theorem 10, and Proposition 1 — assemble those ingredients for specific supports and for the three-arm example, and are read as consequences rather than as new mechanisms.

For design practice, the results suggest three complementary checks. First, Definition 14 and Theorem 3 identify the contrast-weighted randomization probabilities as the first-order benchmark. Second, Theorem 5 gives an explicit estimator achieving a uniform second-order improvement over that benchmark. Third, Theorems 7, 9, and 10 provide computable finite-sample brackets around the minimax risk, so the asymptotic design criterion can be checked against the exact finite orbit game.

Limitations and future work

Four limitations bound the reach of what is proved here, and the ACTG 175 reading above carries two more: the realized trial used a fixed equal-allocation randomization rather than the contrast-weighted $q^{⋆}$ , and its published analyses use time-to-event information that the binary composite endpoint discards. The potential outcomes are binary, so the results do not cover bounded or continuous responses. The number of arms $K$ and the contrast $c$ are fixed before the asymptotics, so nothing here speaks to regimes in which $K$ grows with $n$ . The second-order result is an order statement: the exponent $4/3$ is pinned down, but the constant in front of $n^{- 4/3}$ is bracketed rather than identified, and the scaled improvement $n^{4/3} d_{n} (c)$ is not shown to converge. And the finite linear programs are established as objects, with their primal–dual structure proved; the paper reports no solved instance beyond the three-arm example at $n = 3$ , and the number of response-count rows grows as $(2 ^{K} - 1 n + 2 ^{K} - 1)$ , so the size of a program at useful $(n, K)$ is an open practical question.

Beyond those, the theory leaves two analytic programs that would sharpen the present finite-bracket and rate conclusions. The first concerns the limiting constant and the local structure of the hardest response-type configurations. The second concerns attainment: the form of procedures and priors that realize, or approximate, the limiting second-order value. Both questions are naturally tied to the orbit representation and to the contrast-weighted scores used in Theorem 5.

Remark 1 [def:boundary-layer-handle] (Boundary-layer problem).

A natural next question is whether the exact orbit game admits a boundary-layer limit obtained by enumerating binding response-type and allocation faces, expanding the discrete risk by a vector Stein identity within each stratum, using the $c^{†}$ LP sequence to choose the normalizer, and passing the primal and dual games to a stratified diffusion-control or spectral value problem with face and corner compatibility conditions.

⊢ Lean

The boundary-layer program would convert the finite orbit certificates into an analytic limiting problem. Its role is to identify the faces that bind at the second-order scale and to determine whether the normalized finite- $n$ values converge to a sharp constant. The proposed Stein expansion and stratified limiting game parallel classical links between minimax estimation and differential-operator problems (Levit, 1986; Johnstone et al., 1992), adapted here to the discrete response-type and allocation faces generated by multi-arm binary randomization.

Remark 2 [def:attainment-handle] (Attainment program).

A natural direction for future work is to start from the vector of centered arm scores induced by $q^{⋆}$ , derive an orbit-saddle posterior mean or feedback correction from the limiting value function, lift that correction to finite $n$ , and discretize the squared ground state or dual occupation measure into a least-favorable prior on response-type counts.

⊢ Lean

The attainment program would connect the constructive shrinkage estimator in Theorem 5 with the finite dual certificates in Theorems 7 and 10. It asks for a posterior-mean or feedback correction whose finite- $n$ version matches the orbit game, together with a least-favorable prior on response-type counts. Related shrinkage ideas appear in recent work on randomized experiments (Rosenman et al., 2026); the orbit formulation here gives a finite-population route for studying their minimax constants and support-specific behavior.

Appendices

Proofs and auxiliary lemmas

This appendix collects the auxiliary statements used by the paper’s reduction, first-order, finite-program, continuity, and two-arm lower-bound arguments. The organizing principle is the finite orbit representation from Definition 7: permutation symmetry turns the original labeled experiment into a finite response-count game, while the two-arm lower bound uses a mixture over labeled binary schedules whose induced experiment depends on the assignment only through a one-dimensional score, so that the remaining assignment randomness can be averaged out without changing the Bayes risk. The scalar construction below supplies that interface before the Bayesian information inequality is applied to the resulting one-dimensional family.

The first auxiliary object is the canonical schedule law for the two-arm reduced scalar experiment. It lifts a distribution on effect-class triples to a distribution on complete response schedules, while recording the scalar observation induced by any assignment vector.

Definition 26 [synth_13] (Canonical two-arm scalar schedule kernel).

For a triple $ϑ = (p_{+}, p_{-}, r_{0})$ with $p_{+} + p_{-} + r_{0} = n$ , the canonical schedule kernel $K_{ϑ}$ is the probability law on complete two-arm schedules obtained by choosing uniformly an ordered partition $(P, N, R)$ of ${1, \dots, n}$ with sizes $(p_{+}, p_{-}, r_{0})$ , assigning response type $(1, 0)$ on $P$ and $(0, 1)$ on $N$ , and, independently for $i \in R$ , drawing $H_{i} \sim Bernoulli (1/2)$ and assigning type $(H_{i}, H_{i})$ . Equivalently, $K_{ϑ} (z) = 1 {m_{(1, 0)} (z) = p_{+}, m_{(0, 1)} (z) = p_{-}, m_{(0, 0)} (z) + m_{(1, 1)} (z) = r_{0}} \frac{p _{+} ! p _{-} ! r _{0} !}{n !} 2^{- r_{0}} .$ For a schedule $z$ and assignment vector $A \in A_{2}^{n}$ , define the transformed observed-score vector $S (z, A) \in {0, 1}^{n}$ by $S_{i} (z, A) = Y_{i}^{obs}$ when $A_{i} = 1$ and $S_{i} (z, A) = 1 - Y_{i}^{obs}$ when $A_{i} = 2$ , and define the scalar observation $X (z, A) = i = 1 \sum n S_{i} (z, A) .$ Under $K_{ϑ}$ , conditionally on $ϑ$ , $X (z, A) = p_{+} + B$ with $B \sim Binomial (r_{0}, 1/2)$ , whose probability mass is $binom_{1/2} (r_{0}, k) = (k r _{0}) 2^{- r_{0}}, k = 0, \dots, r_{0} .$

The construction in Definition 26 is useful because it keeps the prior on complete schedules inside the original design problem while exposing a scalar sufficient statistic for the two-arm comparison. The next result states the exact mixture identity and the associated Rao–Blackwell comparison.

Lemma 1 [lem:two-arm-scalar-prior-schedule-kernel] (Scalar two-arm lift).

Fix $n \geq 0$ , and let $ν$ be any probability distribution on triples $ϑ = (p_{+}, p_{-}, r_{0})$ of nonnegative integers with $p_{+} + p_{-} + r_{0} = n$ . There exists a probability distribution $ν$ on complete two-arm schedules $z \in T_{2}^{n}$ such that:

(Canonical mixture.) For every schedule $z$ , $ν (z) = ϑ \sum ν_{ϑ} K_{ϑ} (z),$ where $K_{ϑ}$ is the canonical schedule kernel for the triple $ϑ$ .

(Scalar kernel.) For every triple $ϑ = (p_{+}, p_{-}, r_{0})$ , every assignment vector $A \in A_{2}^{n}$ , and every $x \in {0, \dots, n}$ , $z \in T_{2}^{n} \sum 1 {X (z, A) = x} K_{ϑ} (z) = k = 0 \sum r_{0} 1 {x = p_{+} + k} binom_{1/2} (r_{0}, k) .$ Moreover, for every $A \in A_{2}^{n}$ , $z \in T_{2}^{n} \sum K_{ϑ} (z) = 1,$ and, for every binary vector $s$ with $\sum_{i} s_{i} = x$ , $z \in T_{2}^{n} \sum 1 {X (z, A) = x, S (z, A) = s} K_{ϑ} (z) = \frac{\sum _{z \in T_{2}^{n}} 1 { X ( z , A ) = x } K _{ϑ} ( z )}{( x n )} .$

(Rao–Blackwell domination.) For every assignment law $D$ on $A_{2}^{n}$ and every estimator $τ$ , there is a function $f : N \to R$ such that $ϑ = (p_{+}, p_{-}, r_{0}) \sum ν_{ϑ} k = 0 \sum r_{0} binom_{1/2} (r_{0}, k) {f (p_{+} + k) - \frac{p _{+} - p _{-}}{n}}^{2} \leq E_{ν} [R_{n} (D, τ; z)] .$

(Bayes-risk identity.) For every assignment law $D$ on $A_{2}^{n}$ , $τ in f E_{ν} [R_{n} (D, τ; z)] = f : N \to R in f ϑ = (p_{+}, p_{-}, r_{0}) \sum ν_{ϑ} k = 0 \sum r_{0} binom_{1/2} (r_{0}, k) {f (p_{+} + k) - \frac{p _{+} - p _{-}}{n}}^{2} .$

⊢ Lean

Proof of Lemma 1.

Fix a triple $ϑ = (p_{+}, p_{-}, r_{0})$ . For a complete two-arm schedule $z$ , write $P (z) = {i : z_{i, 1} = 1, z_{i, 2} = 0}, N (z) = {i : z_{i, 1} = 0, z_{i, 2} = 1}, R (z) = {i : z_{i, 1} = z_{i, 2}} .$ Let $Ω_{ϑ} = \frac{n !}{p _{+} ! p _{-} ! r _{0} !} 2^{r_{0}} = (p _{+} n) (p _{-} n - p _{+}) 2^{r_{0}},$ where the second equality uses $p_{+} + p_{-} + r_{0} = n$ . Define $K_{ϑ} (z) = {Ω_{ϑ}^{- 1}, 0, ∣ P (z) ∣ = p_{+}, ∣ N (z) ∣ = p_{-}, ∣ R (z) ∣ = r_{0}, otherwise.$ When $n = 0$ , the only triple is $(0, 0, 0)$ , $Ω_{ϑ} = 1$ , and all sums below are the corresponding one-point or empty sums.

Define the lifted prior by $ν (z) = ϑ \sum ν_{ϑ} K_{ϑ} (z), z \in T_{2}^{n} .$ Each summand is nonnegative. For fixed $ϑ$ , there are $(p _{+} n) (p _{-} n - p _{+})$ choices of the sets $P, N$ , after which $R$ is forced, and the $r_{0}$ equal-potential-outcome units have $2^{r_{0}}$ binary choices. Thus exactly $Ω_{ϑ}$ schedules have positive $K_{ϑ}$ -mass, each with mass $Ω_{ϑ}^{- 1}$ , so $z \in T_{2}^{n} \sum K_{ϑ} (z) = 1.$ It follows that $z \in T_{2}^{n} \sum ν (z) = ϑ \sum ν_{ϑ} z \in T_{2}^{n} \sum K_{ϑ} (z) = ϑ \sum ν_{ϑ} = 1,$ and the displayed definition is exactly the canonical mixture formula.
For $A \in A_{2}^{n}$ , define the transformed score vector and scalar count by $S_{i} (z, A) = {z_{i, 1}, 1 - z_{i, 2}, A_{i} = 1, A_{i} = 2, X (z, A) = i = 1 \sum n S_{i} (z, A) .$ Fix a binary vector $s \in {0, 1}^{n}$ , and put $x_{s} = \sum_{i} s_{i}$ . A schedule in the $ϑ$ -fiber can produce $S (z, A) = s$ exactly by choosing $P (z)$ as a $p_{+}$ -element subset of ${i : s_{i} = 1}$ and $N (z)$ as a $p_{-}$ -element subset of ${i : s_{i} = 0}$ ; after these choices the schedule is determined. Hence $z \in T_{2}^{n} \sum 1 {S (z, A) = s} K_{ϑ} (z) = \frac{( p _{+} x _{s} ) ( p _{-} n - x _{s} )}{( p _{+} n ) ( p _{-} n - p _{+} ) 2 ^{r_{0}}} .$ If $p_{+} \leq x_{s} \leq p_{+} + r_{0}$ , the factorial identity $(x _{s} n) (p _{+} x _{s}) (p _{-} n - x _{s}) = (p _{+} n) (p _{-} n - p _{+}) (x _{s} - p _{+} r _{0})$ gives $z \in T_{2}^{n} \sum 1 {S (z, A) = s} K_{ϑ} (z) = \frac{binom _{1/2} ( r _{0} , x _{s} - p _{+} )}{( x _{s} n )} .$ If $x_{s} < p_{+}$ or $p_{+} + r_{0} < x_{s}$ , the same mass is $0$ , since respectively $(p _{+} x _{s}) = 0$ or $(p _{-} n - x _{s}) = 0$ . Summing over the $(x n)$ vectors $s$ with $x_{s} = x$ yields $z \in T_{2}^{n} \sum 1 {X (z, A) = x} K_{ϑ} (z) = k = 0 \sum r_{0} 1 {x = p_{+} + k} binom_{1/2} (r_{0}, k) .$ Summing this identity over $x = 0, \dots, n$ gives $z \in T_{2}^{n} \sum K_{ϑ} (z) = k = 0 \sum r_{0} binom_{1/2} (r_{0}, k) = 1.$ Finally, if $\sum_{i} s_{i} = x$ , then the event ${S (z, A) = s}$ already implies ${X (z, A) = x}$ , and the preceding display gives $z \in T_{2}^{n} \sum 1 {X (z, A) = x, S (z, A) = s} K_{ϑ} (z) = \frac{\sum _{z \in T_{2}^{n}} 1 { X ( z , A ) = x } K _{ϑ} ( z )}{( x n )} .$ These identities are independent of $A$ .
Fix an assignment law $D$ and an estimator $τ$ . For a binary score vector $s$ , let $y (A, s)$ be the observed-outcome vector obtained by inverting the preceding score map: $y_{i} (A, s) = {s_{i}, 1 - s_{i}, A_{i} = 1, A_{i} = 2.$ Set $T (A, s) = τ (A, y (A, s)), ψ (ϑ) = {(p_{+} - p_{-}) / n, 0, n > 0, n = 0.$ Throughout the displays below, the expression $(p_{+} - p_{-}) / n$ is read with this same total convention when $n = 0$ . The risk under the lifted schedule prior is the prior risk in the induced score experiment: $E_{ν} [R_{n} (D, τ; z)] = ϑ \sum ν_{ϑ} A \sum D (A) z \sum K_{ϑ} (z) {T (A, S (z, A)) - ψ (ϑ)}^{2} .$ For $x = 0, \dots, n$ , define $β_{ϑ} (x) = k = 0 \sum r_{0} 1 {x = p_{+} + k} binom_{1/2} (r_{0}, k), ω (A, s) = \frac{D ( A )}{( \sum _{i} s _{i} n )} .$ The scalar-kernel calculation gives the factorization $D (A) z \sum 1 {S (z, A) = s} K_{ϑ} (z) = ω (A, s) β_{ϑ} (i \sum s_{i}) .$ Moreover, for each $x$ , $A \sum s : \sum_{i} s_{i} = x \sum ω (A, s) = A \sum D (A) \frac{( x n )}{( x n )} = 1.$ Since the displayed weights form a probability distribution on each fiber ${(A, s) : \sum_{i} s_{i} = x}$ , define the scalar average $g (x) = A \sum s : \sum_{i} s_{i} = x \sum ω (A, s) T (A, s), x = 0, \dots, n .$ For each fixed $ϑ$ and $x$ , Jensen’s inequality on this finite fiber gives ${g (x) - ψ (ϑ)}^{2} \leq A \sum s : \sum_{i} s_{i} = x \sum ω (A, s) {T (A, s) - ψ (ϑ)}^{2} .$ Multiplying by $ν_{ϑ} β_{ϑ} (x)$ , summing over $ϑ$ and $x$ , and using the preceding factorization gives $ϑ \sum ν_{ϑ} x = 0 \sum n β_{ϑ} (x) {g (x) - ψ (ϑ)}^{2} \leq E_{ν} [R_{n} (D, τ; z)] .$ Extend $g$ to $f : N \to R$ by setting $f (x) = g (x)$ for $0 \leq x \leq n$ and $f (x) = 0$ for $x > n$ . Expanding $β_{ϑ}$ gives $ϑ = (p_{+}, p_{-}, r_{0}) \sum ν_{ϑ} k = 0 \sum r_{0} binom_{1/2} (r_{0}, k) {f (p_{+} + k) - \frac{p _{+} - p _{-}}{n}}^{2} \leq E_{ν} [R_{n} (D, τ; z)] .$
It remains to identify the two infima. Let $clip (u) = max {- 1, min {1, u}} .$ For every $ϑ$ , the target $ψ (ϑ)$ lies in $[- 1, 1]$ : this is immediate when $n = 0$ , and for $n > 0$ the identities $p_{+} + p_{-} + r_{0} = n$ and nonnegativity give $0 \leq p_{+}, p_{-} \leq n$ , so $- n \leq p_{+} - p_{-} \leq n$ , and division by the positive number $n$ gives $- 1 \leq (p_{+} - p_{-}) / n \leq 1$ . Therefore, for every scalar rule $f : N \to R$ , ${clip (f (p_{+} + k)) - ψ (ϑ)}^{2} \leq {f (p_{+} + k) - ψ (ϑ)}^{2} .$ The full-data estimator $τ_{f} (A, y) = clip (f (i = 1 \sum n {y_{i}, 1 - y_{i}, A_{i} = 1, A_{i} = 2))$ therefore has lifted Bayes risk at most the scalar risk of $f$ . Taking the infimum over $f$ gives $τ in f E_{ν} [R_{n} (D, τ; z)] \leq f : N \to R in f ϑ = (p_{+}, p_{-}, r_{0}) \sum ν_{ϑ} k = 0 \sum r_{0} binom_{1/2} (r_{0}, k) {f (p_{+} + k) - \frac{p _{+} - p _{-}}{n}}^{2} .$ Conversely, the Rao–Blackwell domination from the previous step says that every full-data estimator has a scalar rule whose scalar risk is no larger than its lifted Bayes risk. Taking the infimum over $τ$ gives the reverse inequality. Both risk classes are nonempty and bounded below by $0$ , since they contain the zero rule and all losses are squared losses, so the two infimum comparisons combine to the claimed identity.

∎

Lemma 1 provides the paper-local Rao–Blackwell step behind the coarse two-arm converse in Theorem 6. For every prior on effect-class triples, it constructs a complete-schedule prior, identifies the binomial scalar experiment, and equates the induced Bayes risk with the best scalar decision rule. In this form the lower-bound calculation can be carried out on $X$ while remaining a lower bound for the original assignment problem.

The final auxiliary ingredient is a Bayesian information inequality in the spirit of the van Trees bound (Van Trees, 1968; Gill et al., 1995). The statement is written for an observation-dependent target $g (θ, x)$ , because the two-arm lower bound applies the inequality after conditioning and then evaluates the derivative of the induced conditional mean. The regularity clauses impose the smoothness, integrability, and boundary cancellation needed for the integration-by-parts argument underlying the information bound.

Lemma 2 [lem:bayesian-information-inequality] (Bayesian information inequality).

Let $Θ = (ℓ, u)$ be a scalar parameter interval with $ℓ < u$ , let $(Ω, F)$ be a measurable observation space, and let $μ$ be a $σ$ -finite dominating measure. Let $w, d w : R \to R$ , let $p_{θ} (x)$ , $d p_{θ} (x)$ , $g (θ, x)$ , and $d g (θ, x)$ be real-valued measurable fields, and let $T : Ω \to R$ . Define $J (w) = \int_{ℓ}^{u} {d w (θ)^{2} / w (θ), 0, w (θ) > 0, w (θ) \leq 0, d θ$ and $I (θ) = \int {d p_{θ} (x)^{2} / p_{θ} (x), 0, p_{θ} (x) > 0, p_{θ} (x) \leq 0, d μ (x) .$ Assume the following regularity conditions.

(Prior.) The density $w$ is nonnegative, satisfies $\int_{ℓ}^{u} w (θ) d θ = 1$ , is $C^{1}$ , has compact topological support contained in $(ℓ, u)$ , is positive on the interior of that support, and satisfies $d w (θ) = w^{'} (θ)$ for every $θ$ .

(Likelihood.) For every $θ \in (ℓ, u)$ , $p_{θ} (x) \geq 0$ for $μ$ -almost every $x$ , $\int p_{θ} (x) d μ (x) = 1$ , the fields $p_{θ}$ and $d p_{θ}$ are $μ$ -integrable, and $\frac{d}{d t}_{t = θ} \int p_{t} (x) d μ (x) = \int d p_{θ} (x) d μ (x) = 0.$ For $μ$ -almost every $x$ , $θ \mapsto p_{θ} (x)$ is absolutely continuous on every compact subinterval of $(ℓ, u)$ , and $d p_{θ} (x) = \partial_{θ} p_{θ} (x)$ for Lebesgue-almost every $θ \in (ℓ, u)$ .

(Target.) For $μ$ -almost every $x$ , $θ \mapsto g (θ, x)$ is absolutely continuous on every compact subinterval of $(ℓ, u)$ , and $d g (θ, x) = \partial_{θ} g (θ, x)$ for Lebesgue-almost every $θ \in (ℓ, u)$ .

(Pointwise derivatives.) For $(θ, x)$ -almost every point under Lebesgue measure on $(ℓ, u)$ times $μ$ , the derivatives $\partial_{θ} p_{θ} (x) = d p_{θ} (x)$ and $\partial_{θ} g (θ, x) = d g (θ, x)$ exist in the ordinary pointwise sense.

(Information and joint integrability.) The prior information $J (w)$ is finite, $I (θ)$ is finite for every $θ \in (ℓ, u)$ , and $0 < J (w) + \int_{ℓ}^{u} I (θ) w (θ) d θ .$ Moreover, all joint fields entering the risk, target derivative, squared prior score, squared likelihood score, score cross-product, squared total score, derivative-balance identity, and error-score identity are integrable with respect to $w (θ) p_{θ} (x) d θ d μ (x)$ on $[ℓ, u] \times Ω$ .

(Boundary product.) For $μ$ -almost every $x$ , the map $θ \mapsto w (θ) p_{θ} (x) {T (x) - g (θ, x)}$ is absolutely continuous on $[ℓ, u]$ , and its endpoint values vanish: $w (ℓ) p_{ℓ} (x) {T (x) - g (ℓ, x)} = 0, w (u) p_{u} (x) {T (x) - g (u, x)} = 0.$

Then $\int_{ℓ}^{u} \int {T (x) - g (θ, x)}^{2} p_{θ} (x) w (θ) d μ (x) d θ \geq \frac{( \int _{ℓ}^{u} \int d g ( θ , x ) p _{θ} ( x ) w ( θ ) d μ ( x ) d θ ) ^{2}}{J ( w ) + \int _{ℓ}^{u} I ( θ ) w ( θ ) d θ} .$

⊢ Lean

Proof of Lemma 2.

Let $λ_{ℓ, u}$ denote Lebesgue measure restricted to $[ℓ, u]$ , and write $ϖ (θ, x) = w (θ) p_{θ} (x), e (θ, x) = T (x) - g (θ, x)$ for $(θ, x) \in [ℓ, u] \times Ω$ . Since $ℓ < u$ , the interval integral over $[ℓ, u]$ is the integral with respect to $λ_{ℓ, u}$ . The stated joint integrability hypotheses justify Fubini and the following identifications: $\int_{ℓ}^{u} \int e (θ, x)^{2} p_{θ} (x) w (θ) d μ (x) d θ = \iint e (θ, x)^{2} ϖ (θ, x) d (λ_{ℓ, u} \otimes μ) (θ, x),$ and $\int_{ℓ}^{u} \int d g (θ, x) p_{θ} (x) w (θ) d μ (x) d θ = \iint d g (θ, x) ϖ (θ, x) d (λ_{ℓ, u} \otimes μ) (θ, x) .$ The same change of measure rewrites $J (w)$ and the prior-averaged Fisher information in the guarded-score notation introduced below.
Define the guarded prior, likelihood, and joint scores on their full domains by $s_{w} (θ) = {d w (θ) / w (θ), 0, w (θ) > 0, w (θ) \leq 0,$ $s_{p} (θ, x) = {d p_{θ} (x) / p_{θ} (x), 0, p_{θ} (x) > 0, p_{θ} (x) \leq 0,$ and $s (θ, x) = {{d w (θ) p_{θ} (x) + w (θ) d p_{θ} (x)} / ϖ (θ, x), 0, ϖ (θ, x) > 0, ϖ (θ, x) \leq 0.$ For $μ$ -almost every $x$ , the section $θ \mapsto p_{θ} (x)$ is absolutely continuous, hence continuous on compact subintervals of $(ℓ, u)$ . Intersecting the full-measure nonnegativity sets over rational $θ$ ’s and passing to limits gives $p_{θ} (x) \geq 0 for every θ \in (ℓ, u)$ for $μ$ -almost every $x$ . Since the endpoints have $λ_{ℓ, u}$ -measure zero, $ϖ (θ, x) \geq 0$ for $(λ_{ℓ, u} \otimes μ)$ -almost every $(θ, x)$ .
The zero-density cases are controlled by local-minimum arguments. Because $w \geq 0$ and $d w = w^{'}$ , whenever $w (θ) = 0$ one has $d w (θ) = 0$ . Similarly, at $(λ_{ℓ, u} \otimes μ)$ -almost every $(θ, x)$ , if $p_{θ} (x) = 0$ , then $θ$ is a local minimum of the nonnegative section $t \mapsto p_{t} (x)$ , and therefore $d p_{θ} (x) = 0$ . Consequently the guarded likelihood score satisfies $s_{p} (θ, x) p_{θ} (x) = d p_{θ} (x)$ for almost every relevant point. The identity is immediate on the positive-density set and follows from the displayed zero-derivative fact on the zero-density set. The same case analysis will also be used below for the guarded total score: on the set $w (θ) p_{θ} (x) > 0$ it is the quotient defining the logarithmic derivative of $w (θ) p_{θ} (x)$ , while if $w (θ) p_{θ} (x) = 0$ , the preceding zero-derivative facts make the corresponding numerator vanish.
Define the derivative-balance field $B (θ, x) = {d w (θ) p_{θ} (x) + w (θ) d p_{θ} (x)} e (θ, x) - ϖ (θ, x) d g (θ, x) .$ For $μ$ -almost every $x$ , the map $θ ⟼ w (θ) p_{θ} (x) e (θ, x)$ is absolutely continuous on $[ℓ, u]$ and has endpoint values equal to zero. At $λ_{ℓ, u}$ -almost every $θ$ , the product rule gives its derivative as ${d w (θ) p_{θ} (x) + w (θ) d p_{θ} (x)} e (θ, x) + w (θ) p_{θ} (x) (- d g (θ, x)),$ because $T (x)$ is constant in $θ$ and $\partial_{θ} e (θ, x) = - d g (θ, x)$ . This displayed derivative is exactly $B (θ, x)$ . Thus the fundamental theorem for absolutely continuous functions gives $\int_{ℓ}^{u} B (θ, x) d θ = w (u) p_{u} (x) e (u, x) - w (ℓ) p_{ℓ} (x) e (ℓ, x) = 0.$ Fubini and the assumed integrability of $B$ yield $\iint B (θ, x) d (λ_{ℓ, u} \otimes μ) (θ, x) = 0.$ Using the guarded total-score case analysis from the previous step, $e (θ, x) s (θ, x) ϖ (θ, x) = e (θ, x) {d w (θ) p_{θ} (x) + w (θ) d p_{θ} (x)}$ almost everywhere: on $w (θ) p_{θ} (x) > 0$ this is the defining quotient multiplied by $w (θ) p_{θ} (x)$ , and on $w (θ) p_{θ} (x) = 0$ the numerator vanishes by the zero-density derivative facts for $w$ and $p_{θ}$ . Hence $B (θ, x) = e (θ, x) s (θ, x) ϖ (θ, x) - d g (θ, x) ϖ (θ, x)$ almost everywhere. Therefore $\iint e (θ, x) s (θ, x) ϖ (θ, x) d (λ_{ℓ, u} \otimes μ) = \iint d g (θ, x) ϖ (θ, x) d (λ_{ℓ, u} \otimes μ) .$
For $λ_{ℓ, u}$ -almost every $θ$ , the point lies in $(ℓ, u)$ , and the likelihood normalization and differentiation-under-the-integral assumptions give $\int d p_{θ} (x) d μ (x) = 0.$ Combining this with $s_{p} (θ, x) p_{θ} (x) = d p_{θ} (x)$ gives the conditional centering identity $\int s_{p} (θ, x) p_{θ} (x) d μ (x) = 0.$ Now expand the squared joint score. If $w (θ) > 0$ and $p_{θ} (x) > 0$ , then $s (θ, x) = s_{w} (θ) + s_{p} (θ, x)$ . If $w (θ) \leq 0$ , the prior nonnegativity hypothesis gives $w (θ) = 0$ , so every term below contains the factor $w (θ)$ or the guarded joint density and is zero. If $w (θ) > 0$ but $p_{θ} (x) \leq 0$ , the almost-everywhere likelihood nonnegativity established above gives $p_{θ} (x) = 0$ , and again every term below vanishes by the guarded conventions. Hence, almost everywhere, $s (θ, x)^{2} ϖ (θ, x) = w (θ) p_{θ} (x) s_{w} (θ)^{2} + w (θ) p_{θ} (x) s_{p} (θ, x)^{2} + 2 w (θ) p_{θ} (x) s_{w} (θ) s_{p} (θ, x) .$ Integrating, the cross term vanishes by the centering identity: $\iint w (θ) p_{θ} (x) s_{w} (θ) s_{p} (θ, x) d (λ_{ℓ, u} \otimes μ) = 0.$ The prior term evaluates to $\iint w (θ) p_{θ} (x) s_{w} (θ)^{2} d (λ_{ℓ, u} \otimes μ) = \int_{ℓ}^{u} {d w (θ)^{2} / w (θ), 0, w (θ) > 0, w (θ) \leq 0, d θ = J (w),$ using $\int p_{θ} d μ = 1$ for almost every interior $θ$ . The likelihood term evaluates to $\iint w (θ) p_{θ} (x) s_{p} (θ, x)^{2} d (λ_{ℓ, u} \otimes μ) = \int_{ℓ}^{u} I (θ) w (θ) d θ .$ Thus $\iint s (θ, x)^{2} ϖ (θ, x) d (λ_{ℓ, u} \otimes μ) = J (w) + \int_{ℓ}^{u} I (θ) w (θ) d θ .$
Finally, apply the weighted $L^{2}$ inequality with weight $ϖ$ , functions $e$ and $s$ . Its proof is the standard quadratic argument: for every $α \in R$ , $0 \leq \iint {e (θ, x) - α s (θ, x)}^{2} ϖ (θ, x) d (λ_{ℓ, u} \otimes μ),$ and expanding the square, using the assumed integrability of the two square terms and the cross term, gives a nonnegative quadratic in $α$ . Its discriminant is nonpositive, so $(\iint e (θ, x) s (θ, x) ϖ (θ, x) d (λ_{ℓ, u} \otimes μ))^{2} \leq (\iint e (θ, x)^{2} ϖ (θ, x) d (λ_{ℓ, u} \otimes μ)) (\iint s (θ, x)^{2} ϖ (θ, x) d (λ_{ℓ, u} \otimes μ)) .$ Substituting the error–sensitivity identity and the information decomposition obtained above gives $(\iint d g (θ, x) ϖ (θ, x) d (λ_{ℓ, u} \otimes μ))^{2} \leq (\iint e (θ, x)^{2} ϖ (θ, x) d (λ_{ℓ, u} \otimes μ)) (J (w) + \int_{ℓ}^{u} I (θ) w (θ) d θ) .$ The denominator is strictly positive by assumption, so division yields the claimed inequality, and the identifications made at the outset translate it back to the statement’s iterated-integral notation.

∎

Lemma 2 is the reusable information step for the two-arm Bayes lower bound. The numerator measures the averaged derivative of the target, and the denominator combines the prior information with the likelihood information, matching the standard Bayesian Cramér–Rao structure associated with Van Trees (1968). In the application to Theorem 6, Lemma 1 first reduces the complete-schedule prior to the scalar experiment, and Lemma 2 then lower-bounds the scalar Bayes risk used in the converse.

Verification note

This appendix records what is machine-checked, in what environment, and what is not.

Every displayed definition, proposition, lemma and theorem in the paper falls into exactly one of two classes. A matched environment renders a Lean declaration in reader-facing mathematics: the displayed statement unfolds the declaration’s structures and composite predicates into ordinary notation, with the same hypotheses and the same conclusion, and the proof reproduced in Section C follows the machine-checked proof of that declaration. The rendering is faithful to the declaration’s content, not a transcription of Lean source. A presentation-only environment introduces notation or packages several matched objects for the reader; it carries no separate mathematical content and is not itself checked. The two lists below are exhaustive.

Environment.

The development is written in Lean 4 against the toolchain leanprover/lean4:v4.33.0, on top of Mathlib and the project’s own Causalean library. The results reported here were checked at commit 715add8e9475f5193526bce0c26baef559a47b68 of the source repository. The source of this development contains no sorry and no axiom declaration, and an axiom audit of all $26$ declarations listed below reports exactly the three standard Lean axioms — propext, Classical.choice and Quot.sound — and nothing else. In particular, the paper contains no statement that is conditional on an unproved input, and no result is imported from a published source as a black box.

Matched environments.

Each entry gives the paper environment, its Lean declaration, and the file that contains it.

Definition 1: ().
Definition 2: ().
Definition 3: ().
Definition 6: ().
Definition 7: ().
Theorem 1: ().
Theorem 2: ().
Definition 14: ().
Theorem 3: ().
Theorem 4: ().
Theorem 5: ().
Theorem 6: ().
Definition 18: ().
Definition 19: ().
Proposition 1: ().
Theorem 7: ().
Theorem 8: ().
Theorem 9: ().
Theorem 10: ().
Proposition 2: ().
Theorem 11: ().
Theorem 12: ().
Remark 1: ().
Remark 2: ().
Lemma 1: ().
Lemma 2: ().

Presentation-only environments.

These introduce notation used by the matched statements and are not separately checked.

Definition 4: Response-type count vector.
Definition 5: Notation for response-type counts.
Definition 8: Design-based squared-error risk.
Definition 9: Orbit-procedure risk.
Definition 10: Nonzero zero-sum contrast.
Definition 11: Compatible observation-success counts.
Definition 12: Invariant labeled procedure.
Definition 13: Hull bounded-schedule domain and risk.
Definition 15: Contrast-weighted independent design.
Definition 16: Regular variation of a normalizer.
Definition 17: Clipped-shrinkage estimator.
Definition 20: Dual objective for the grid linear program.
Definition 21: Orbit Bayes risk at an allocation count.
Definition 22: Exact rational primal-dual grid certificate.
Definition 23: Exact rational grid-certificate cluster.
Definition 24: Contrast distance.
Definition 25: Sign-pair embedding and lift.
Definition 26: Canonical two-arm scalar schedule kernel.

What is not checked.

The bibliographic comparisons in Section 2, including those to Sudijono et al. (2026) and Hull (2026), are positioning claims about the published literature and are not part of the formal development; no theorem in this paper takes a published result as a hypothesis. The applied reading of the ACTG 175 design in Section 7 is likewise interpretive. The numerical values reported for the three-arm example — the table $T_{F}$ and the exact rational risk $511653/4000000$ — are checked, but the search that produced the table is not: only the risk it attains is claimed.

Proofs of the main results

Proof of Theorem 1.

Write $S_{n}$ for the group of unit permutations. The proof packages five finite-sum facts.

Fix a labeled procedure $p = (D, τ)$ . For an assignment $A$ and observed vector $y$ , define $den (A) = σ \in S_{n} \sum D (σ A), num (A, y) = σ \in S_{n} \sum D (σ A) τ (σ A, σ y),$ where $(σ A)_{i} = A_{σ^{- 1} (i)}$ and $(σ y)_{i} = y_{σ^{- 1} (i)}$ . Let $\overset{ˉ}{D} (A) = \frac{1}{∣ S _{n} ∣} σ \in S_{n} \sum D (σ A),$ and let $\overset{ˉ}{τ} (A, y) = ⎩ ⎨ ⎧ clip_{[- L_{c} /2, L_{c} /2]} (num (A, y) / den (A)), 0, den (A) > 0, den (A) = 0.$ Since $L_{c} = \sum_{a} ∣ c_{a} ∣ \geq 0$ , the second value lies in the estimator interval. Reindexing the finite sum by left composition with any fixed unit permutation gives $\overset{ˉ}{D} (σ_{0} A) = \overset{ˉ}{D} (A), \overset{ˉ}{τ} (σ_{0} A, σ_{0} y) = \overset{ˉ}{τ} (A, y),$ so $\overset{p}{ˉ} = (\overset{ˉ}{D}, \overset{ˉ}{τ})$ is invariant.

For fixed $A$ with $den (A) > 0$ , the weighted mean $num (A, y) / den (A)$ is a convex combination of values in $[- L_{c} /2, L_{c} /2]$ , hence the clipping is inactive. Therefore, for every real target value $θ$ , $den (A) {\overset{ˉ}{τ} (A, y) - θ}^{2} \leq σ \in S_{n} \sum D (σ A) {τ (σ A, σ y) - θ}^{2}$ by Jensen’s inequality for the convex map $t \mapsto (t - θ)^{2}$ , applied to the weights $D (σ A)$ . If $den (A) = 0$ , the left side is zero and the right side is nonnegative, so the same inequality holds.

Apply this with $y = Y^{obs} (z, A)$ and $θ = τ_{c} (z)$ , then sum over $A$ . The identities $τ_{c} (σ z) = τ_{c} (z), Y^{obs} (σ z, σ A) = σ Y^{obs} (z, A)$ convert the right-hand side into the permutation average of labeled risks: $R_{n} (\overset{p}{ˉ}; z) \leq \frac{1}{∣ S _{n} ∣} σ \in S_{n} \sum R_{n} (p; σ z) .$ Let $q$ be the orbit procedure obtained by collapsing the invariant procedure $\overset{p}{ˉ}$ to allocation and observation counts. For this collapsed procedure, the labeled and orbit descriptions put the same total probability on each allocation-count and observation-count fiber, so summing the squared loss over those fibers gives $R_{n} (q; counts (z)) = R_{n} (\overset{p}{ˉ}; z) .$ Hence $q$ is dominated by the averaged invariant representative.
An invariant labeled procedure is determined exactly by allocation-count and observation-count orbits. For an assignment $A \in A_{K}^{n}$ and an observed binary vector $y$ , define $alloc (A)_{a} = # {i : A_{i} = a}, succ (A, y)_{a} = # {i : A_{i} = a, y_{i} = 1} .$ Thus $alloc (A) \in R_{n, K}$ , and, when $alloc (A) = r$ , the vector $succ (A, y)$ is an element of $\prod_{a \in A_{K}} {0, \dots, r_{a}}$ . Write $O_{alloc} (r) = {A^{'} \in A_{K}^{n} : alloc (A^{'}) = r} .$ For an orbit procedure $q = (π, δ)$ , inflate it to a labeled procedure by $D_{q} (A) = \frac{π _{alloc (A)}}{∣ O _{alloc} ( alloc ( A )) ∣}, τ_{q} (A, y) = δ (alloc (A), succ (A, y)) .$ Conversely, if $p = (D, τ)$ is invariant, choose one representative $A_{r}^{\circ} \in O_{alloc} (r)$ for each allocation count $r$ , and one representative pair $(A_{r, x}^{\circ}, y_{r, x}^{\circ})$ with $alloc (A_{r, x}^{\circ}) = r$ and $succ (A_{r, x}^{\circ}, y_{r, x}^{\circ}) = x$ for each observation count $x \in \prod_{a \in A_{K}} {0, \dots, r_{a}}$ . Set $π_{p} (r) = ∣ O_{alloc} (r) ∣ D (A_{r}^{\circ}), δ_{p} (r, x) = τ (A_{r, x}^{\circ}, y_{r, x}^{\circ}) .$ These definitions are independent of the representatives: equal allocation counts are exactly equality up to a unit permutation, and equal allocation and observation counts are exactly equality up to a simultaneous unit permutation of assignment and observed outcomes.

The two constructions are inverse to each other. Under either construction, $D (A) = \frac{π _{alloc (A)}}{∣ O _{alloc} ( alloc ( A )) ∣}, τ (A, y) = δ (alloc (A), succ (A, y)) .$ It remains to identify the regrouped labeled risk with the orbit risk. For a labeled schedule $z$ , put $m_{z} = counts (z)$ and define the observed labeled fiber $L_{z} (r, x) = {A \in A_{K}^{n} : alloc (A) = r, succ (A, Y^{obs} (z, A)) = x} .$ For a table $h \in H (m_{z}, r, x)$ , the number of assignments in $L_{z} (r, x)$ having $h_{t, a}$ units of response type $t$ assigned to arm $a$ is $t \in T_{K} \prod \frac{m _{z, t} !}{\prod _{a \in A_{K}} h _{t, a} !} .$ Summing over the feasible contingency-table fiber and dividing by the allocation-orbit cardinality $∣ O_{alloc} (r) ∣ = \frac{n !}{\prod _{a \in A_{K}} r _{a} !}$ gives $\frac{∣ L _{z} ( r , x ) ∣}{∣ O _{alloc} ( r ) ∣} = \frac{\prod _{a \in A_{K}} r _{a} !}{n !} h \in H (m_{z}, r, x) \sum t \in T_{K} \prod \frac{m _{z, t} !}{\prod _{a \in A_{K}} h _{t, a} !} = P_{m_{z}} (x ∣ r),$ where the last equality is the likelihood formula in Definition 6. The target term is also constant on the response-count orbit, $τ_{c} (z) = \frac{1}{n} i \sum a \in A_{K} \sum c_{a} t_{i, a} = \frac{1}{n} t \in T_{K} \sum m_{z, t} a \in A_{K} \sum c_{a} t_{a} = τ_{c} (m_{z}) .$ Therefore, regrouping first by allocation counts and then by observed fibers yields $R_{n} (p; z) = r \sum A \in O_{alloc} (r) \sum \frac{π _{r}}{∣ O _{alloc} ( r ) ∣} {δ (r, succ (A, Y^{obs} (z, A))) - τ_{c} (m_{z})}^{2} = r \sum π_{r} x \sum \frac{∣ L _{z} ( r , x ) ∣}{∣ O _{alloc} ( r ) ∣} {δ (r, x) - τ_{c} (m_{z})}^{2} = r \sum π_{r} x \sum P_{m_{z}} (x ∣ r) {δ (r, x) - τ_{c} (m_{z})}^{2} = R_{n} (q; m_{z}) .$ This proves the stated bijection and the exact equality of allocation-count, observation-count, and risk rules.
For every schedule $z$ , the count vector satisfies $counts (z)_{t} = # {i : t_{i} = t} .$ Thus the finite sum over labeled units may be regrouped by its response-type fibers: $i \sum a \sum c_{a} t_{i, a} = t \sum # {i : t_{i} = t} a \sum c_{a} t_{a} .$ Here and in the displayed target formula, $1/ n$ denotes the real inverse $(n : R)^{- 1}$ , with the convention $0^{- 1} = 0$ . Scaling the regrouped identity by this factor gives $τ_{c} (z) = \frac{1}{n} i \sum a \sum c_{a} t_{i, a} = \frac{1}{n} t \sum m_{t} a \sum c_{a} t_{a} = τ_{c} (m), m = counts (z) .$
The arm-count condition $K \geq 2$ supplies nonempty assignment and orbit spaces, while the zero clipped estimator supplies nonempty procedure classes. Risks are nonnegative, so both minimax values in Definitions 3 and 7 are formed over bounded-below finite risk ranges.

First fix an orbit procedure $q$ , and inflate it to its invariant labeled representative $p_{q}$ . By the risk identity above, for every labeled schedule $z$ , $R_{n} (p_{q}; z) = R_{n} (q; counts (z)) \leq m max R_{n} (q; m) .$ Taking the worst case over $z$ and then the infimum over $q$ gives $ρ_{n} (c) \leq G_{n} (c) .$ Conversely, fix a labeled procedure $p$ . The lossless symmetrization step gives an orbit procedure $q_{p}$ such that, for every $z$ , $R_{n} (q_{p}; counts (z)) \leq \frac{1}{∣ S _{n} ∣} σ \in S_{n} \sum R_{n} (p; σ z) \leq z^{'} max R_{n} (p; z^{'}) .$ Every response-count vector is $counts (z)$ for some schedule $z$ , so $m max R_{n} (q_{p}; m) \leq z^{'} max R_{n} (p; z^{'}) .$ Taking the infimum over $p$ gives $G_{n} (c) \leq ρ_{n} (c)$ . Hence $ρ_{n} (c) = G_{n} (c) .$
Put the orbit game in finite squared-loss form with state space $M_{n, K}$ , design space $R_{n, K}$ , observation space $X (r) = a \in A_{K} \prod {0, \dots, r_{a}},$ likelihood coefficient $L_{orb} (m, r, x) = P_{m} (x ∣ r),$ target $τ_{c} (m)$ , and action interval $[- L_{c} /2, L_{c} /2]$ . The interval is nonempty because $L_{c} \geq 0$ , and $L_{orb} (m, r, x) = P_{m} (x ∣ r) \geq 0$ follows term by term from the factorial formula in Definition 6.

The state set $M_{n, K}$ is finite, the action set is the compact interval $[- L_{c} /2, L_{c} /2]$ , and squared loss is bounded and convex in the action, so the game has a saddle point (Wald, 1950). Applied to these data it gives an orbit procedure $q_{sad}$ and a probability law $ν$ on $M_{n, K}$ such that $R_{n} (q_{sad}; m) \leq q in f m^{'} max R_{n} (q; m^{'}) for every m,$ and $q in f m^{'} max R_{n} (q; m^{'}) \leq E_{ν} [R_{n} (q; m)] for every orbit procedure q .$ By Definition 7, the middle value is $G_{n} (c)$ , giving $R_{n} (q_{sad}; m) \leq G_{n} (c), G_{n} (c) \leq E_{ν} [R_{n} (q; m)] .$

∎

Proof of Theorem 2.

For $K = 2$ and $c = (1, - 1)$ , the arm-count condition required in Theorem 1 is immediate. Its value-equality conclusion, applied to this two-arm contrast, gives $ρ_{n} (c) = G_{n} (c)$ for every $n > 0$ .
For the same contrast, $L_{c} = ∣1∣ + ∣ - 1∣ = 2, C_{0} (c) = \frac{L _{c}^{2}}{4} = 1.$ This proves the asserted first-order constant.
Fix $n > 0$ and $m \in M_{n, 2}$ . The four two-arm response types are $t_{00} = (0, 0), t_{01} = (0, 1), t_{10} = (1, 0), t_{11} = (1, 1) .$ They are pairwise distinct and every $t \in T_{2}$ is exactly one of these four types: the pair $(t_{1}, t_{2})$ is one of $(0, 0), (0, 1), (1, 0), (1, 1) .$ Thus the orbit-target sum may be evaluated over this four-element list: $τ_{c} (m) = \frac{1}{n} t \in {t_{00}, t_{01}, t_{10}, t_{11}} \sum m_{t} a \in A_{2} \sum c_{a} t_{a} .$ For $c = (1, - 1)$ , their contrast scores are $a \sum c_{a} t_{00, a} = 0, a \sum c_{a} t_{01, a} = - 1, a \sum c_{a} t_{10, a} = 1, a \sum c_{a} t_{11, a} = 0.$ Therefore $τ_{c} (m) = \frac{m _{t_{10}} - m _{t_{01}}}{n} = \frac{p _{+} ( m ) - p _{-} ( m )}{n},$ which is the claimed two-arm target formula.
The four functions above differ by evaluating them at one of the two arms: $t_{00}$ differs from $t_{01}$ at the second arm, from $t_{10}$ at the first arm, and from $t_{11}$ at either arm; $t_{01}$ and $t_{10}$ differ at the first arm; $t_{01}$ and $t_{11}$ differ at the first arm; and $t_{10}$ and $t_{11}$ differ at the second arm. Hence they are pairwise distinct.
If $t \in T_{2}$ , then the pair $(t_{1}, t_{2})$ is one of $(0, 0), (0, 1), (1, 0), (1, 1),$ so $t$ is respectively $t_{00}, t_{01}, t_{10}$ , or $t_{11}$ . Thus these four types exhaust $T_{2}$ .
With these four response types kept as the elements of $T_{2}$ , the likelihood identity is exactly the specialization of Definition 6 to $K = 2$ : $P_{m} (x ∣ r) = \frac{\prod _{a \in A_{2}} r _{a} !}{n !} h \in H (m, r, x) \sum t \in T_{2} \prod \frac{m _{t} !}{\prod _{a \in A_{2}} h _{t, a} !} .$ Here the fiber $H (m, r, x)$ imposes the response-type margins $m$ , allocation margins $r$ , and arm-success totals $x$ , as in Definition 6.
Let $n \geq 8$ . Applying Theorem 4 with $K = 2$ and $c = (1, - 1)$ gives $d_{n} (c) \leq 43 C_{0} (c) n^{- 4/3} .$ Using $C_{0} (c) = 1$ from the calculation above yields $d_{n} (c) \leq 43 n^{- 4/3} .$
For the lower second-order bound, apply Theorem 5 with $K = 2$ and $c = (1, - 1)$ . Let $κ = κ_{c} > 0$ , and choose $N_{⋆}$ large enough that the shrinkage procedure in that result satisfies, for every $n \geq N_{⋆}$ , $z \in T_{2}^{n} sup R_{n} (D^{⋆}, τ_{n}^{sh}; z) \leq C_{0} (c) {n^{- 1} - κ n^{- 4/3}} .$ Since $ρ_{n} (c)$ is the infimum of the same worst-case risk over all admissible procedures, $ρ_{n} (c) \leq C_{0} (c) {n^{- 1} - κ n^{- 4/3}} .$ With $C_{0} (c) = 1$ and $d_{n} (c) = n^{- 1} - ρ_{n} (c)$ , this rearranges to $κ n^{- 4/3} \leq d_{n} (c)$ for every $n \geq N_{⋆}$ .
Now fix $K \geq 3$ , $n$ , and a contrast $c$ . Since $K \geq 3$ implies $K \geq 2$ , Theorem 1 applies and gives $ρ_{n} (c) = G_{n} (c) .$ The response-type set is the set of binary functions on $A_{K}$ . Because $∣ A_{K} ∣ = K$ , its cardinality is $∣ T_{K} ∣ = ∣ {0, 1}^{A_{K}} ∣ = 2^{K} .$
Fix $N, L, U$ , a finite design $D$ on two-arm assignments, and a measurable Hull estimator $τ$ . Set $v = Y sup R_{Hull} (D, τ; Y),$ where $Y$ ranges over Hull schedules with parameters $N, L, U$ . Choosing $D^{'} = D, τ^{'} = τ$ preserves measurability and gives $v = Y sup R_{Hull} (D^{'}, τ^{'}; Y) .$ Thus $v$ belongs to the displayed risk-value class.
Finally fix $K \geq 3$ , $n > 0$ , and $L < U$ . First choose a number $v_{sep} \in [L, U], v_{sep} \neq = 0, v_{sep} \neq = 1.$ The choice is by endpoint cases. If $L = 0$ and $U = 1$ , take $v_{sep} = 1/2$ . If $L = 0$ and $U \neq = 1$ , then the subcase $U = 0$ contradicts $L < U$ , and in the remaining subcase take $v_{sep} = U$ . If $L \neq = 0$ and $L = 1$ , then the subcases $U = 0$ and $U = 1$ contradict $L < U$ , and in the remaining subcase take $v_{sep} = U$ . If $L \neq = 0$ and $L \neq = 1$ , take $v_{sep} = L$ . In every surviving case this gives the displayed membership and exclusions.

Define the Hull schedule $Y_{i} (a) = v_{sep}, i = 1, \dots, n, a \in A_{2} .$ This schedule lies in $Y_{n, L, U}^{Hull}$ . Since $n > 0$ , choose one unit $i_{0}$ . If this embedded Hull schedule belonged to the binary $K$ -arm schedule domain, then at the first arm its value would be the value of a binary potential outcome, hence either $0$ or $1$ . But the same coordinate equals $v_{sep}$ , contradicting $v_{sep} \in / {0, 1}$ . Therefore $Y_{n, L, U}^{Hull} ⊈ T_{K}^{n} .$

∎

Proof of Theorem 3.

We first prove the limiting assertion. The hypotheses on $c$ imply $K \geq 2$ : if $K = 0$ , there are no arm coefficients and the contrast is the zero function; if $K = 1$ , the zero-sum identity forces the unique coefficient to be zero. Both alternatives contradict the assumed nonzero contrast.

For $n \geq 1$ , set $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c) .$ With $K \geq 2$ , $n \geq 1$ , and $c$ nonzero and zero-sum, Theorem 4 applies. In particular, for every $n \geq 8$ , $0 \leq d_{n} (c), d_{n} (c) \leq 43 C_{0} (c) n^{- 4/3} .$ Since $C_{0} (c) = L_{c}^{2} /4 \geq 0$ , for $n \geq 8$ we have $n ρ_{n} (c) - C_{0} (c) = n {\frac{C _{0} ( c )}{n} - d_{n} (c)} - C_{0} (c) = - n d_{n} (c),$ and therefore $∣ n ρ_{n} (c) - C_{0} (c) ∣ = n d_{n} (c) \leq 43 C_{0} (c) n^{- 1/3} .$ The upper bound tends to $0$ , so the squeeze theorem gives $n \to \infty lim n ρ_{n} (c) = C_{0} (c) .$
We next prove the finite-sample risk bound. Fix $n \geq 1$ and a labeled schedule $z \in T_{K}^{n}$ . Since $c$ is nonzero, $L_{c} = a \in A_{K} \sum ∣ c_{a} ∣ > 0.$ Thus $q_{a}^{⋆} = ∣ c_{a} ∣/ L_{c}$ is a probability vector, and $q_{a}^{⋆} > 0$ exactly on the active support $S_{c} = {a : c_{a} \neq = 0}$ .

For each unit $i$ and arm $a$ , define the centered one-unit score on all of $A_{K}$ by $ψ_{i} (a; z) = ⎩ ⎨ ⎧ \frac{c _{a} ( t _{i, a} - 1/2 )}{q _{a}^{⋆}}, 0, q_{a}^{⋆} > 0, q_{a}^{⋆} = 0.$ Under the independent $q^{⋆}$ design in Definition 14, the unprojected estimator is $τ_{raw} (A, z) = \frac{1}{n} i = 1 \sum n ψ_{i} (A_{i}; z) .$ For each $i$ , $E {ψ_{i} (A_{i}; z)} = a \in S_{c} \sum q_{a}^{⋆} \frac{c _{a} ( t _{i, a} - 1/2 )}{q _{a}^{⋆}} = a \in A_{K} \sum c_{a} (t_{i, a} - 1/2) = a \in A_{K} \sum c_{a} t_{i, a},$ where the last equality uses $\sum_{a} c_{a} = 0$ . Hence $E {τ_{raw} (A, z)} = \frac{1}{n} i = 1 \sum n a \in A_{K} \sum c_{a} t_{i, a} = τ_{c} (z) .$

Also, since $t_{i, a} \in {0, 1}$ , $(t_{i, a} - 1/2)^{2} = \frac{1}{4},$ and therefore $E {ψ_{i} (A_{i}; z)^{2}} = \frac{1}{4} a \in S_{c} \sum \frac{c _{a}^{2}}{q _{a}^{⋆}} = \frac{1}{4} a \in S_{c} \sum L_{c} ∣ c_{a} ∣ = \frac{L _{c}^{2}}{4} = C_{0} (c) .$ Thus $Var {ψ_{i} (A_{i}; z)} = C_{0} (c) - {E ψ_{i} (A_{i}; z)}^{2} \leq C_{0} (c) .$ Independence across units gives $MSE {τ_{raw}; z} = Var {τ_{raw} (A, z)} = \frac{1}{n ^{2}} i = 1 \sum n Var {ψ_{i} (A_{i}; z)} \leq \frac{C _{0} ( c )}{n} .$

It remains to pass from the raw rule to the projected rule. For any two zero-sum contrasts $c_{1}, c_{2}$ , put $dist (c_{1}, c_{2}) = \frac{1}{2} a \in A_{K} \sum ∣ (c_{1})_{a} - (c_{2})_{a} ∣.$ For any binary response type, splitting the arms according to $t_{a} = 1$ and $t_{a} = 0$ , the zero-sum identity for $c_{1} - c_{2}$ yields $a \in A_{K} \sum {(c_{1})_{a} - (c_{2})_{a}} t_{a} \leq dist (c_{1}, c_{2}) .$ Averaging over the $n$ units gives $∣ τ_{c_{1}} (z) - τ_{c_{2}} (z) ∣ \leq dist (c_{1}, c_{2}) .$ Taking $c_{1} = c$ and $c_{2} = - c$ , we have $τ_{- c} (z) = - τ_{c} (z), dist (c, - c) = L_{c},$ so $∣ τ_{c} (z) ∣ \leq \frac{L _{c}}{2}, that is, τ_{c} (z) \in [- L_{c} /2, L_{c} /2] .$

For every $y \in [- L_{c} /2, L_{c} /2]$ and every real $x$ , ${proj_{[- L_{c} /2, L_{c} /2]} (x) - y}^{2} \leq (x - y)^{2} .$ Indeed, if $x$ lies in the interval there is equality; if $x < - L_{c} /2$ , then $0 \leq y + L_{c} /2 \leq y - x$ ; and if $x > L_{c} /2$ , then $0 \leq L_{c} /2 - y \leq x - y$ . Squaring gives the displayed inequality in the two boundary cases. Applying it with $y = τ_{c} (z)$ shows, pointwise in the assignment, ${τ_{cHT}^{⋆} (A, Y^{obs}) - τ_{c} (z)}^{2} \leq {τ_{raw} (A, z) - τ_{c} (z)}^{2} .$ Consequently, $R_{n} (D^{⋆}, τ_{cHT}^{⋆}; z) \leq \frac{C _{0} ( c )}{n} .$ Since $z$ was arbitrary, $z \in T_{K}^{n} max R_{n} (D^{⋆}, τ_{cHT}^{⋆}; z) \leq \frac{C _{0} ( c )}{n} .$
The two preceding steps are exactly the two components of the asserted conjunction: the first gives the asymptotic value of the unrestricted labeled-schedule minimax risk, and the second gives the finite-sample worst-case bound for the independent contrast-weighted design together with the projected contrast-weighted Horvitz–Thompson rule.

∎

Proof of Theorem 4.

Write $R_{n} (D, τ; z) = E_{D} [{τ ((A_{i}, Y_{i}^{obs})_{i = 1}^{n}) - τ_{c} (z)}^{2}]$ for the schedulewise risk appearing in Definition 3. For the two-arm contrast $(1, - 1)$ , write $τ_{2} (y) = \frac{1}{n} i = 1 \sum n (y_{i, +} - y_{i, -}), R_{n, 2} (D_{2}, υ; y) = E_{D_{2}} [{υ ((B_{i}, V_{i})_{i = 1}^{n}) - τ_{2} (y)}^{2}],$ for $y \in T_{2}^{n}$ , with $y_{i, +}, y_{i, -} \in {0, 1}$ .

Define the sign groups and scale $S_{c}^{+} = {a \in A_{K} : c_{a} > 0}, S_{c}^{-} = {a \in A_{K} : c_{a} < 0}, S_{c}^{0} = {a \in A_{K} : c_{a} = 0}, h_{c} = \frac{L _{c}}{2} .$ Since $c$ is nonzero and zero-sum, both $S_{c}^{+}$ and $S_{c}^{-}$ are nonempty, and $a \in S_{c}^{+} \sum c_{a} = - a \in S_{c}^{-} \sum c_{a} = \frac{1}{2} a \in A_{K} \sum ∣ c_{a} ∣ = h_{c}, h_{c}^{2} = C_{0} (c) .$ For $y \in T_{2}^{n}$ , embed it into a $K$ -arm schedule $E_{c} y \in T_{K}^{n}$ by $(E_{c} y)_{i, a} = ⎩ ⎨ ⎧ y_{i, +}, y_{i, -}, 0, a \in S_{c}^{+}, a \in S_{c}^{-}, a \in S_{c}^{0} .$ Then $τ_{c} (E_{c} y) = \frac{1}{n} i = 1 \sum n ⎩ ⎨ ⎧ y_{i, +} a \in S_{c}^{+} \sum c_{a} + y_{i, -} a \in S_{c}^{-} \sum c_{a} ⎭ ⎬ ⎫ = h_{c} τ_{2} (y) .$

Fix an arbitrary $K$ -arm procedure $(D, τ)$ . Let $Ω = {0, 1}^{{1, \dots, n}}$ , put the product law $L$ on $(A, ω) \in A_{K}^{n} \times Ω$ by drawing $A \sim D$ and independent fair bits $ω_{i}$ , and define the induced two-arm assignment $σ_{c, i} (A, ω) = ⎩ ⎨ ⎧ +, -, +, -, A_{i} \in S_{c}^{+}, A_{i} \in S_{c}^{-}, A_{i} \in S_{c}^{0} and ω_{i} = 1, A_{i} \in S_{c}^{0} and ω_{i} = 0.$ For $v \in {0, 1}^{n}$ , define the inactive-zero observation vector $v_{i} (A, v) = {v_{i}, 0, c_{A_{i}} \neq = 0, c_{A_{i}} = 0.$ Let $D_{2}^{\circ}$ be the pushforward of $L$ under $σ_{c}$ , so that $D_{2}^{\circ} {b} = L {(A, ω) : σ_{c} (A, ω) = b}, b \in {+, -}^{n} .$ All conditional means below use the following totalization convention: on a cell with $D_{2}^{\circ} {b} = 0$ , the conditional mean is set to $0$ ; this value is immaterial in risks because the cell has zero $D_{2}^{\circ}$ -mass. Define the induced two-arm estimator, on every $b \in {+, -}^{n}$ and $v \in {0, 1}^{n}$ , by $υ^{\circ} (b, v) = {E_{L} [h_{c}^{- 1} τ (A, v (A, v)) σ_{c} (A, ω) = b], 0, D_{2}^{\circ} {b} > 0, D_{2}^{\circ} {b} = 0.$ Because $τ$ is clipped to $[- h_{c}, h_{c}]$ , this conditional mean lies in $[- 1, 1]$ . For each $y \in T_{2}^{n}$ , expanding the risk under the pushed-forward law and applying conditional Jensen’s inequality gives $R_{n, 2} (D_{2}^{\circ}, υ^{\circ}; y) = E_{D_{2}^{\circ}} [{υ^{\circ} (B, (y_{i, B_{i}})_{i = 1}^{n}) - τ_{2} (y)}^{2}] \leq E_{L} [{h_{c}^{- 1} τ (A, v (A, (y_{i, σ_{c, i} (A, ω)})_{i = 1}^{n})) - τ_{2} (y)}^{2}] = h_{c}^{- 2} R_{n} (D, τ; E_{c} y) .$ The last equality follows from the observation identity $v_{i} (A, (y_{j, σ_{c, j} (A, ω)})_{j = 1}^{n}) = (E_{c} y)_{i, A_{i}}, i = 1, \dots, n,$ and the target identity $τ_{c} (E_{c} y) = h_{c} τ_{2} (y)$ . Thus the construction supplies the statewise comparison $R_{n, 2} (D_{2}^{\circ}, υ^{\circ}; y) \leq h_{c}^{- 2} R_{n} (D, τ; E_{c} y), y \in T_{2}^{n} .$ Taking the supremum over $y$ , and using $E_{c} y \in T_{K}^{n}$ , yields the induced-procedure worst-case bound $y \in T_{2}^{n} sup R_{n, 2} (D_{2}^{\circ}, υ^{\circ}; y) \leq h_{c}^{- 2} z \in T_{K}^{n} sup R_{n} (D, τ; z) .$ Since $ρ_{n, 2}$ is the infimum of the left-hand worst-case risk over all two-arm procedures, $ρ_{n, 2} \leq h_{c}^{- 2} z \in T_{K}^{n} sup R_{n} (D, τ; z) .$ Multiplying by $h_{c}^{2} = C_{0} (c)$ and taking the infimum over the arbitrary $K$ -arm procedure yields $C_{0} (c) ρ_{n, 2} \leq ρ_{n} (c) .$
The identity $C_{0} (c) = L_{c}^{2} /4$ gives $C_{0} (c) \geq 0$ . Consider the product design and projected estimator in Definition 14. For every schedule $z \in T_{K}^{n}$ , introduce the auxiliary totalized weights $\overset{q}{ˉ}_{a}^{⋆} = \frac{∣ c _{a} ∣}{L _{c}}, a \in A_{K} .$ These agree with the probabilities $q_{a}^{⋆}$ from Definition 14 on $S_{c}$ , and $\overset{q}{ˉ}_{a}^{⋆} = 0$ exactly when $c_{a} = 0$ . For $a \in A_{K}$ , define $ψ_{i} (a; z) = ⎩ ⎨ ⎧ \frac{c _{a} { z _{i, a} - 1/2 }}{q ˉ _{a}^{⋆}}, 0, \overset{q}{ˉ}_{a}^{⋆} > 0, \overset{q}{ˉ}_{a}^{⋆} = 0.$ Under the independent assignment law in Definition 14, equivalently with totalized weights $\overset{q}{ˉ}^{⋆}$ on $A_{K}$ , $E {ψ_{i} (A_{i}; z)} = a \in S_{c} \sum \overset{q}{ˉ}_{a}^{⋆} \frac{c _{a} ( z _{i, a} - 1/2 )}{q ˉ _{a}^{⋆}} = a \sum c_{a} (z_{i, a} - 1/2) = a \sum c_{a} z_{i, a},$ because $\sum_{a} c_{a} = 0$ . Also, $E {ψ_{i} (A_{i}; z)^{2}} = a \in S_{c} \sum \overset{q}{ˉ}_{a}^{⋆} \frac{c _{a}^{2} ( z _{i, a} - 1/2 ) ^{2}}{( q ˉ _{a}^{⋆} ) ^{2}} = \frac{1}{4} a \in S_{c} \sum \frac{c _{a}^{2}}{q ˉ _{a}^{⋆}} = \frac{1}{4} a \in S_{c} \sum L_{c} ∣ c_{a} ∣ = C_{0} (c) .$ Thus, for $\overline{ψ} (A; z) = \frac{1}{n} i = 1 \sum n ψ_{i} (A_{i}; z),$ the product structure gives $E {\overline{ψ} (A; z)} = τ_{c} (z), Var {\overline{ψ} (A; z)} = \frac{1}{n ^{2}} i = 1 \sum n Var {ψ_{i} (A_{i}; z)} \leq \frac{C _{0} ( c )}{n} .$ For each unit, $- \frac{L _{c}}{2} \leq a \sum c_{a} z_{i, a} \leq \frac{L _{c}}{2},$ so $τ_{c} (z) \in [- L_{c} /2, L_{c} /2]$ . Projection onto this interval cannot increase squared distance from $τ_{c} (z)$ , hence $R_{n} (D^{⋆}, τ_{cHT}^{⋆}; z) \leq Var {\overline{ψ} (A; z)} \leq \frac{C _{0} ( c )}{n} .$ Taking the maximum over $z$ and then the infimum over procedures gives $ρ_{n} (c) \leq \frac{C _{0} ( c )}{n} .$
Suppose $∣ S_{c} ∣ = 2$ . Let $a_{c}^{+}$ and $a_{c}^{-}$ be the unique active arms with positive and negative coefficients. The preceding sign-sum identities give $c_{a_{c}^{+}} = h_{c}, c_{a_{c}^{-}} = - h_{c}, c_{a} = 0 (a \in / {a_{c}^{+}, a_{c}^{-}}) .$ For any two-arm procedure $(D_{2}, υ)$ , lift a two-arm assignment $b \in {+, -}^{n}$ to the $K$ -arm assignment $A^{b}$ given by $A_{i}^{b} = {a_{c}^{+}, a_{c}^{-}, b_{i} = +, b_{i} = - .$ Let $D_{2}^{↑}$ be the pushforward of $D_{2}$ under $b \mapsto A^{b}$ . For a general $K$ -arm assignment $A$ , define the associated two-arm assignment $b_{i}^{A} = {+, -, A_{i} = a_{c}^{+}, A_{i} \neq = a_{c}^{+},$ and define the lifted estimator on its full domain by $τ^{↑} (A, v) = h_{c} υ (b^{A}, v), A \in A_{K}^{n}, v \in {0, 1}^{n} .$ For a $K$ -arm schedule $z$ , define the active two-arm schedule $z_{i, +}^{↓} = z_{i, a_{c}^{+}}, z_{i, -}^{↓} = z_{i, a_{c}^{-}} .$ The coefficient identities above give $τ_{c} (z) = h_{c} τ_{2} (z^{↓}) .$ Moreover $b^{A^{b}} = b$ and the lifted observations satisfy $(z^{↓})_{i, b_{i}} = z_{i, A_{i}^{b}}, i = 1, \dots, n .$ Therefore, for every $z \in T_{K}^{n}$ , the lifted procedure has the statewise risk identity $R_{n} (D_{2}^{↑}, τ^{↑}; z) = h_{c}^{2} R_{n, 2} (D_{2}, υ; z^{↓}) .$ It follows that $z \in T_{K}^{n} sup R_{n} (D_{2}^{↑}, τ^{↑}; z) \leq h_{c}^{2} y \in T_{2}^{n} sup R_{n, 2} (D_{2}, υ; y) .$ Taking the infimum over two-arm procedures and using $h_{c}^{2} > 0$ to pull the fixed scale through the infimum gives the minimax comparison $ρ_{n} (c) \leq h_{c}^{2} ρ_{n, 2} = C_{0} (c) ρ_{n, 2} .$ Together with the lower bound obtained above, $ρ_{n} (c) = C_{0} (c) ρ_{n, 2} whenever ∣ S_{c} ∣ = 2.$
Now assume $n \geq 8$ . From the upper bound, $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c) \geq 0.$ From the lower bound, $d_{n} (c) = C_{0} (c) n^{- 1} - ρ_{n} (c) \leq C_{0} (c) n^{- 1} - C_{0} (c) ρ_{n, 2} = C_{0} (c) {n^{- 1} - ρ_{n, 2}} .$ Finally, Theorem 6 gives $n^{- 1} - 43 n^{- 4/3} \leq ρ_{n, 2},$ and therefore $n^{- 1} - ρ_{n, 2} \leq 43 n^{- 4/3} .$ Since $C_{0} (c) \geq 0$ , $d_{n} (c) \leq C_{0} (c) {n^{- 1} - ρ_{n, 2}} \leq 43 C_{0} (c) n^{- 4/3} .$
In the support-two case, the equality established above gives $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c) = C_{0} (c) n^{- 1} - C_{0} (c) ρ_{n, 2} = C_{0} (c) {n^{- 1} - ρ_{n, 2}} .$ This proves both asserted identities for $∣ S_{c} ∣ = 2$ .

∎

Proof of Theorem 5.

Fix $K \geq 2$ and a nonzero zero-sum contrast $c$ . We prove the five asserted conclusions in the order displayed.

Let $σ_{c} (t) = \frac{\sum _{a \in A_{K}} c _{a} t _{a}}{L _{c} /2}, t \in T_{K} .$ Since $c \neq = 0$ , choose $a_{0}$ with $c_{a_{0}} \neq = 0$ . For the response type $t^{a_{0}}$ with $t_{a_{0}}^{a_{0}} = 1$ and $t_{a}^{a_{0}} = 0$ for $a \neq = a_{0}$ , the numerator is $c_{a_{0}}$ , so the finite set of nonzero values $∣ σ_{c} (t) ∣$ is nonempty and has a positive minimum. This is exactly $λ_{c}$ , hence $0 < λ_{c}$ . Also, zero-summability gives $c_{a_{0}} = - a \neq = a_{0} \sum c_{a}, ∣ c_{a_{0}} ∣ \leq a \neq = a_{0} \sum ∣ c_{a} ∣,$ and therefore $2∣ c_{a_{0}} ∣ \leq L_{c}$ . Thus one admissible normalized nonzero value is at most one, and the minimum satisfies $λ_{c} \leq 1$ . Finally, $A_{c} = \frac{λ _{c}}{2} - \frac{λ _{c}}{16} > 0, B_{c} = \frac{7 λ _{c}}{16} > 0,$ because $0 < λ_{c} \leq 1$ . Hence $κ_{c} = \frac{1}{2} min {A_{c}, B_{c}} > 0.$
The first-order rule $τ_{cHT}^{⋆}$ from Definition 14 is controlled first before projection. For the degenerate case $n = 0$ , there is a single empty schedule, its target is $τ_{c} (\emptyset) = 0,$ and the centered contrast-weighted sum is the empty sum, hence equals $0$ . The projection fixes $0$ , so the first-order procedure has risk $R_{0} (D^{⋆}, τ_{cHT}^{⋆}; \emptyset) = 0.$ Since squared-error risks are nonnegative, this gives $ρ_{0} (c) = 0$ . The reciprocal estimate now begins with positive population size, so assume $n \geq 1$ . For $z = (t_{i})_{i = 1}^{n}$ , define, for an assignment vector $A \in A_{K}^{n}$ , $ξ_{i} (A_{i}, z) = a \in S_{c} \sum c_{a} 1 {A_{i} = a} \frac{t _{i, a} - 1/2}{q _{a}^{⋆}}, τ_{n} (A, z) = \frac{1}{n} i = 1 \sum n ξ_{i} (A_{i}, z) .$ Under the independent $q^{⋆}$ -design, inverse-probability cancellation and $\sum_{a} c_{a} = 0$ give $E {ξ_{i} (A_{i}, z) ∣ z} = a \in A_{K} \sum c_{a} t_{i, a}, E {ξ_{i} (A_{i}, z)^{2} ∣ z} = C_{0} (c),$ and hence $E {τ_{n} (A, z) ∣ z} = τ_{c} (z), Var {τ_{n} (A, z) ∣ z} \leq \frac{C _{0} ( c )}{n} .$ The target lies in the projection interval: for each response type $t$ , the positive and negative coefficient masses are both $L_{c} /2$ , so $- \frac{L _{c}}{2} \leq a \in A_{K} \sum c_{a} t_{a} \leq \frac{L _{c}}{2}, - \frac{L _{c}}{2} \leq τ_{c} (z) \leq \frac{L _{c}}{2} .$ For every $y \in [- L_{c} /2, L_{c} /2]$ , interval projection satisfies $proj_{[- L_{c} /2, L_{c} /2]} (x) - y \leq ∣ x - y ∣,$ because the projection fixes $x$ inside the interval and otherwise moves $x$ to the nearer endpoint between $x$ and $y$ . Therefore ${proj_{[- L_{c} /2, L_{c} /2]} (τ_{n} (A, z)) - τ_{c} (z)}^{2} \leq {τ_{n} (A, z) - τ_{c} (z)}^{2} .$ Taking expectations gives $R_{n} (D^{⋆}, τ_{cHT}^{⋆}; z) \leq \frac{C _{0} ( c )}{n} for every z \in T_{K}^{n},$ and taking the infimum over procedures gives $ρ_{n} (c) \leq C_{0} (c) / n$ for $n \geq 1$ . Together with the zero-unit branch, this proves the first-order envelope for every $n$ .
Since $j_{tail} exp {- j_{tail}^{1/3} /8} ⟶ 0 (j_{tail} \to \infty),$ multiplying by $4 λ_{c}$ gives $4 λ_{c} j_{tail} exp {- j_{tail}^{1/3} /8} ⟶ 0.$ Because $A_{c} > 0$ , there is $N_{0}$ such that, for all integers $j_{tail} \geq N_{0}$ , $4 λ_{c} j_{tail} exp {- j_{tail}^{1/3} /8} \leq A_{c} .$ Set $N_{c} = max {N_{0}, 1}$ . Then the stated tail side condition holds for every integer threshold variable at least $N_{c}$ .

For the risk bound, fix $n \geq N_{c}$ and a schedule $z = (t_{i})_{i = 1}^{n}$ . Under the independent $q^{⋆}$ -design, define $V_{i} = \frac{1}{L _{c} /2} a \in S_{c} \sum c_{a} 1 {A_{i} = a} \frac{t _{i, a} - 1/2}{q _{a}^{⋆}}, X_{n} = \frac{1}{n} i = 1 \sum n V_{i},$ $Θ_{n} (z) = \frac{1}{n} i = 1 \sum n σ_{c} (t_{i}), Q_{n} (z) = i = 1 \sum n σ_{c} (t_{i})^{2} .$ Then $E (X_{n} ∣ z) = Θ_{n} (z), Var (X_{n} ∣ z) = \frac{1}{n} - \frac{Q _{n} ( z )}{n ^{2}},$ and the definition of $λ_{c}$ gives $Q_{n} (z) \geq λ_{c} i = 1 \sum n ∣ σ_{c} (t_{i}) ∣ \geq λ_{c} n ∣ Θ_{n} (z) ∣.$ For the tail estimate, write $ζ_{i} (A) = V_{i} (A) - σ_{c} (t_{i}), X_{n} (A) - Θ_{n} (z) = \frac{1}{n} i = 1 \sum n ζ_{i} (A) .$ The product design makes the variables $ζ_{i}$ independent conditional on $z$ . Also $V_{i} (A) \in [- 1, 1]$ and $E (V_{i} ∣ z) = σ_{c} (t_{i})$ , so $ζ_{i}$ has conditional mean zero and lies in the interval $[- 1 - σ_{c} (t_{i}), 1 - σ_{c} (t_{i})]$ , whose length is $2$ . Hoeffding’s lemma therefore gives, for every real $α_{mgf}$ , $E [exp {α_{mgf} ζ_{i}} ∣ z] \leq exp {α_{mgf}^{2} /2} .$ Multiplying the moment-generating-function bounds across the independent units yields $E [exp {α_{mgf} i = 1 \sum n ζ_{i}} ∣ z] \leq exp {n α_{mgf}^{2} /2} .$ Thus Chernoff’s bound, with $α_{mgf} = ω_{tail}$ for $ω_{tail} > 0$ and with the endpoint $ω_{tail} = 0$ immediate, gives $Pr {i = 1 \sum n ζ_{i} \geq n ω_{tail} ∣ z} \leq exp {- n ω_{tail}^{2} /2} (ω_{tail} \geq 0) .$ Applying the same argument to $- \sum_{i} ζ_{i}$ and taking the union of the two one-sided events gives, for every $ω_{tail} \geq 0$ , $Pr {∣ X_{n} - Θ_{n} (z) ∣ \geq ω_{tail} ∣ z} \leq 2 exp {- n ω_{tail}^{2} /2} .$

Put $b_{n} = n^{- 1/3}, ε_{n} = \frac{λ _{c}}{4} b_{n}, g_{n} (x) = max {- b_{n}, min {x, b_{n}}} .$ The clipped-shrinkage estimator is $τ_{n}^{sh} = \frac{L _{c}}{2} {X_{n} - ε_{n} g_{n} (X_{n})},$ followed by projection onto $[- L_{c} /2, L_{c} /2]$ ; that projection fixes the displayed value. Indeed, fix a realized assignment and write $x = X_{n}$ . The bounds $∣ x ∣ \leq 1, 0 \leq b_{n} \leq 1, 0 \leq ε_{n} \leq \frac{1}{4} \leq 1$ imply $∣ x - ε_{n} g_{n} (x) ∣ \leq 1.$ To see this, use the three regions in the definition of $g_{n}$ . If $x < - b_{n}$ , then $g_{n} (x) = - b_{n}$ and $- 1 \leq x \leq x + ε_{n} b_{n} \leq x + b_{n} < 0 \leq 1.$ If $b_{n} < x$ , then $g_{n} (x) = b_{n}$ and $- 1 \leq 0 < x - b_{n} \leq x - ε_{n} b_{n} \leq x \leq 1.$ In the remaining case, $- b_{n} \leq x \leq b_{n}$ , so $g_{n} (x) = x$ and $∣ x - ε_{n} g_{n} (x) ∣ = (1 - ε_{n}) ∣ x ∣ \leq ∣ x ∣ \leq 1.$ Thus, for every realized assignment, $- \frac{L _{c}}{2} \leq \frac{L _{c}}{2} {X_{n} - ε_{n} g_{n} (X_{n})} \leq \frac{L _{c}}{2},$ because $L_{c} /2 \geq 0$ , and the final interval projection is inactive.

Let $C_{n} (z) = Cov {X_{n}, g_{n} (X_{n}) ∣ z} .$ Since $g_{n}$ is monotone, $C_{n} (z) \geq 0,$ and $E [{X_{n} - ε_{n} g_{n} (X_{n}) - Θ_{n} (z)}^{2} ∣ z] \leq Var (X_{n} ∣ z) - 2 ε_{n} C_{n} (z) + ε_{n}^{2} b_{n}^{2} .$

If $∣ Θ_{n} (z) ∣ \leq b_{n} /2$ , define the far-tail event $E_{tail} (z) = {A \in A_{K}^{n} : ∣ X_{n} (A) - Θ_{n} (z) ∣ \geq b_{n} /2} .$ The tail bound gives $Pr {E_{tail} (z) ∣ z} \leq 2 exp {- n^{1/3} /8} .$ On $E_{tail} (z)^{c}$ , the local condition gives $∣ X_{n} (A) ∣ \leq ∣ X_{n} (A) - Θ_{n} (z) ∣ + ∣ Θ_{n} (z) ∣ \leq b_{n},$ so $g_{n} (X_{n} (A)) = X_{n} (A)$ . Since $∣ X_{n} (A) ∣ \leq 1$ and $b_{n} \leq 1$ , clipping also gives the uniform perturbation bound $∣ g_{n} (X_{n} (A)) - X_{n} (A) ∣ \leq 1.$ Combining the two displays, $∣ g_{n} (X_{n} (A)) - X_{n} (A) ∣ \leq 1 {A \in E_{tail} (z)} .$ Moreover $∣ X_{n} (A) - Θ_{n} (z) ∣ \leq ∣ X_{n} (A) ∣ + ∣ Θ_{n} (z) ∣ \leq 2$ . Since $E (X_{n} ∣ z) = Θ_{n} (z)$ , $C_{n} (z) = Var (X_{n} ∣ z) + E [(X_{n} - Θ_{n} (z)) {g_{n} (X_{n}) - X_{n}} ∣ z],$ and therefore $C_{n} (z) \geq Var (X_{n} ∣ z) - E [∣ X_{n} - Θ_{n} (z) ∣ ∣ g_{n} (X_{n}) - X_{n} ∣ ∣ z] \geq Var (X_{n} ∣ z) - 2 Pr {E_{tail} (z) ∣ z} \geq Var (X_{n} ∣ z) - 4 exp {- n^{1/3} /8} .$ Since $0 \leq ε_{n} \leq 1/4$ , $1 - 2 ε_{n} \geq 0, Q_{n} (z) \geq 0.$ Substituting $Var (X_{n} ∣ z) = n^{- 1} - Q_{n} (z) / n^{2}$ and the covariance lower bound into the preceding MSE inequality gives $E [{X_{n} - ε_{n} g_{n} (X_{n}) - Θ_{n} (z)}^{2} ∣ z] \leq (1 - 2 ε_{n}) {\frac{1}{n} - \frac{Q _{n} ( z )}{n ^{2}}} + 8 ε_{n} exp {- n^{1/3} /8} + ε_{n}^{2} b_{n}^{2} \leq \frac{1}{n} - \frac{2 ε _{n}}{n} + 8 ε_{n} exp {- n^{1/3} /8} + ε_{n}^{2} b_{n}^{2},$ where the last line drops the nonpositive term $- (1 - 2 ε_{n}) Q_{n} (z) / n^{2}$ . Using $\frac{b _{n}}{n} = n^{- 4/3}, ε_{n} = \frac{λ _{c}}{4} b_{n}, ε_{n}^{2} b_{n}^{2} = \frac{λ _{c}}{16} n^{- 4/3},$ we obtain $E [{X_{n} - ε_{n} g_{n} (X_{n}) - Θ_{n} (z)}^{2} ∣ z] \leq \frac{1}{n} - A_{c} n^{- 4/3} + 2 λ_{c} b_{n} exp {- n^{1/3} /8} .$ The choice of $N_{c}$ implies $2 λ_{c} b_{n} exp {- n^{1/3} /8} \leq \frac{A _{c}}{2} n^{- 4/3},$ so the local case gives $E [{X_{n} - ε_{n} g_{n} (X_{n}) - Θ_{n} (z)}^{2} ∣ z] \leq \frac{1}{n} - \frac{A _{c}}{2} n^{- 4/3} \leq \frac{1}{n} - κ_{c} n^{- 4/3} .$

If $∣ Θ_{n} (z) ∣ > b_{n} /2$ , then the lower bound on $Q_{n} (z)$ yields $\frac{Q _{n} ( z )}{n ^{2}} \geq \frac{λ _{c}}{2} n^{- 4/3} .$ Using $C_{n} (z) \geq 0$ and the same identity for $ε_{n}^{2} b_{n}^{2}$ , $E [{X_{n} - ε_{n} g_{n} (X_{n}) - Θ_{n} (z)}^{2} ∣ z] \leq \frac{1}{n} - \frac{7 λ _{c}}{16} n^{- 4/3} \leq \frac{1}{n} - κ_{c} n^{- 4/3} .$ The two cases cover every schedule. Multiplying by $(L_{c} /2)^{2} = C_{0} (c)$ gives, uniformly in $z$ , $R_{n} (D^{⋆}, τ_{n}^{sh}; z) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}} .$ Taking the supremum over schedules proves the asserted clipped-shrinkage risk bound for all $n \geq N_{c}$ .
Since $ρ_{n} (c)$ is the minimax infimum, the shrinkage bound implies, for all $n \geq N_{c}$ , $ρ_{n} (c) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}} .$ Thus $d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c) \geq C_{0} (c) κ_{c} n^{- 4/3},$ and therefore $n^{4/3} d_{n} (c) \geq C_{0} (c) κ_{c} (n \geq N_{c}) .$ On the other hand, Theorem 4 gives, for all $n \geq 8$ , $d_{n} (c) \leq 43 C_{0} (c) n^{- 4/3},$ so $n^{4/3} d_{n} (c) \leq 43 C_{0} (c) (n \geq 8) .$ The eventual lower and upper bounds imply $C_{0} (c) κ_{c} \leq n \to \infty lim inf n^{4/3} d_{n} (c) \leq n \to \infty lim sup n^{4/3} d_{n} (c) \leq 43 C_{0} (c) .$
Let $a_{n}$ be a positive regularly varying sequence with index $β$ , and suppose $a_{n} d_{n} (c) ⟶ C with C > 0.$ Define the auxiliary positive sequence $υ_{n} = {1, a_{n} n^{- 4/3}, n = 0, n \geq 1.$ For $n \geq 1$ , $υ_{n} n^{4/3} d_{n} (c) = a_{n} d_{n} (c) .$ The bounds from the previous step and the convergence to $C$ imply that, eventually, $\frac{C}{2 \cdot 43 C _{0} ( c )} \leq υ_{n} \leq \frac{2 C}{C _{0} ( c ) κ _{c}} .$ Regular variation at the scale factor $2$ gives $\frac{a _{2 n}}{a _{n}} ⟶ 2^{β},$ and hence $\frac{υ _{2 n}}{υ _{n}} = \frac{a _{2 n}}{a _{n}} 2^{- 4/3} ⟶ 2^{β - 4/3} .$ A positive sequence that is eventually bounded above and away from zero can have a dyadic ratio limit only equal to $1$ : indeed, for the dyadic logarithms $χ_{k} = lo g υ_{2^{k}},$ the increments $χ_{k + 1} - χ_{k}$ converge to the logarithm of the dyadic ratio, while the eventual boundedness of $υ_{2^{k}}$ makes $χ_{k}$ eventually bounded; Cesaro averaging of $\frac{χ _{J} - χ _{0}}{J} = \frac{1}{J} k = 0 \sum J - 1 (χ_{k + 1} - χ_{k})$ therefore forces that logarithm to be $0$ . Applying this to $υ_{n}$ gives $2^{β - 4/3} = 1.$ Since the map $r \mapsto 2^{r}$ is strictly increasing, $β = \frac{4}{3} .$

∎

Proof of Theorem 6.

Fix $n \geq 8$ and put $α := n^{- 1/3} .$

Since $n > 0$ , also $α > 0$ , and the real-power identities give $α^{3} = n^{- 1}, n^{- 2/3} = α^{2}, n^{2/3} = n α .$ Moreover $α \leq 1/2$ . Indeed, $n^{- 1} \leq 8^{- 1}$ . If $1/2 < α$ , then $\frac{1}{4} < α^{2}$ and, multiplying by the positive inequality $1/2 < α$ , $\frac{1}{8} < α^{3} = n^{- 1},$ contradicting $n^{- 1} \leq 8^{- 1}$ . The same identities also give $\frac{1}{α ^{2}} = n^{2/3} .$
We first prove the sharper fractional lower bound in the auxiliary parameter $α$ . Let $w_{α} (θ) = \frac{15}{8 α} (1 - \frac{4 θ ^{2}}{α ^{2}})^{2} 1 {∣ θ ∣ \leq α /2} .$ This is a probability density on $(- 1/2, 1/2)$ , is positive on its support interior, and vanishes at the support endpoints. Direct differentiation on $∣ θ ∣ < α /2$ gives $w_{α}^{'} (θ) = - \frac{30 θ}{α ^{3}} (1 - \frac{4 θ ^{2}}{α ^{2}}),$ so $J (w_{α}) = \int \frac{w _{α}^{'} ( θ ) ^{2}}{w _{α} ( θ )} d θ = \int_{- α /2}^{α /2} \frac{480 θ ^{2}}{α ^{5}} d θ = \frac{40}{α ^{2}} .$ For fixed $θ$ with $∣ θ ∣ \leq α /2$ , draw independent two-arm response types $r_{i} = (r_{i, 1}, r_{i, 2}) \in {0, 1}^{2}$ with $θ Pr {(1, 0)} = \frac{α + θ}{2}, θ Pr {(0, 1)} = \frac{α - θ}{2}, θ Pr {(0, 0)} = θ Pr {(1, 1)} = \frac{1 - α}{2} .$ These masses are nonnegative and sum to one because $0 < α \leq 1/2$ and $∣ θ ∣ \leq α /2$ . Let $σ_{i} := r_{i, 1}, X (σ) := i = 1 \sum n 1 {σ_{i} = 1}, \overset{ˉ}{R} (σ) := \frac{1}{n} i = 1 \sum n (21 {σ_{i} = 1} - 1) .$ Then the marginal law of $σ$ is the product Bernoulli law $p_{θ} (σ) = i = 1 \prod n (\frac{1 + θ}{2})^{1 {σ_{i} = 1}} (\frac{1 - θ}{2})^{1 {σ_{i} = 0}},$ with Fisher information $I_{n} (θ) = \frac{n}{1 - θ ^{2}} \leq \frac{n}{1 - α ^{2} /4} .$ The finite-population contrast target for a response vector $r$ is $τ (r) = \frac{1}{n} i = 1 \sum n (r_{i, 1} - r_{i, 2}) .$ Conditional on $σ_{i} = 1$ , $E_{θ} (r_{i, 1} - r_{i, 2} ∣ σ_{i} = 1) = \frac{α + θ}{1 + θ},$ and conditional on $σ_{i} = 0$ , $E_{θ} (r_{i, 1} - r_{i, 2} ∣ σ_{i} = 0) = - \frac{α - θ}{1 - θ} .$ Thus the posterior target is $γ_{α} (θ, σ) := E_{θ} (τ (r) ∣ σ) = θ + ω_{α} (θ) {\overset{ˉ}{R} (σ) - θ}, ω_{α} (θ) := \frac{α - θ ^{2}}{1 - θ ^{2}} .$ For $∣ θ ∣ \leq α /2$ , $0 \leq ω_{α} (θ) \leq α .$ At fixed $σ$ , $\partial_{θ} γ_{α} (θ, σ) = 1 - ω_{α} (θ) + \frac{2 θ ( α - 1 )}{( 1 - θ ^{2} ) ^{2}} {\overset{ˉ}{R} (σ) - θ} .$ Since $E_{θ} \overset{ˉ}{R} (σ) = θ$ , $E_{θ} {\partial_{θ} γ_{α} (θ, σ)} = 1 - ω_{α} (θ) \geq 1 - α .$ For any scalar estimator $t : {0, \dots, n} \to R$ , let $t^{\circ}$ be its clipping to $[- 1, 1]$ . Since $τ (r) \in [- 1, 1]$ , ${t^{\circ} (X (σ)) - τ (r)}^{2} \leq {t (X (σ)) - τ (r)}^{2} .$ The conditional square completion gives, for each $θ$ and $σ$ , $p_{θ} (σ) {t^{\circ} (X (σ)) - γ_{α} (θ, σ)}^{2} \leq r \sum θ Pr (r) 1 {r_{i, 1} = σ_{i} \forall i} {t^{\circ} (X (σ)) - τ (r)}^{2} .$ Applying Lemma 2 to the finite observation space ${0, 1}^{n}$ , the density $w_{α}$ , likelihood $p_{θ}$ , target $γ_{α}$ , and estimator $t^{\circ} (X (σ))$ , with the regularity hypotheses discharged by the displayed compact support and finite product likelihood, yields $\int w_{α} (θ) σ \sum p_{θ} (σ) {t^{\circ} (X (σ)) - γ_{α} (θ, σ)}^{2} d θ \geq \frac{( 1 - α ) ^{2}}{40/ α ^{2} + n / ( 1 - α ^{2} /4 )} .$ Combining this estimate with the square-completion and clipping inequalities shows that every scalar estimator has Bayes risk at least $\frac{( 1 - α ) ^{2}}{40/ α ^{2} + n / ( 1 - α ^{2} /4 )} .$ Let $ν_{α}$ be the induced distribution on triples $ϑ = (p_{+}, p_{-}, r_{0}),$ where $p_{+}$ counts $(1, 0)$ units, $p_{-}$ counts $(0, 1)$ units, and $r_{0}$ counts equal-potential-outcome units. Applied to $ν_{α}$ , Lemma 1 gives a complete-schedule prior whose Rao–Blackwell scalar risk is dominated by every design-estimator Bayes risk. Hence, by Definition 3, $\frac{( 1 - α ) ^{2}}{40/ α ^{2} + n / ( 1 - α ^{2} /4 )} \leq ρ_{n, 2} .$ Using $n^{- 2/3} = α^{2}$ , $α = n^{- 1/3}$ , and $1/ α^{2} = n^{2/3}$ , this is exactly $\frac{( 1 - n ^{- 1/3} ) ^{2}}{n / ( 1 - n ^{- 2/3} /4 ) + 40 n ^{2/3}} \leq ρ_{n, 2} .$
It remains to convert the fractional bound to the advertised coarse form. Since $0 < α \leq 1/2$ , $0 < 1 - \frac{α ^{2}}{4}$ and $(1 - \frac{α ^{2}}{4})^{- 1} \leq 1 + α .$ Therefore $40 α + (1 - \frac{α ^{2}}{4})^{- 1} \leq 1 + 41 α .$ We claim $1 - 43 α \leq \frac{( 1 - α ) ^{2}}{40 α + ( 1 - α ^{2} /4 ) ^{- 1}} .$ The denominator is positive. If $1 - 43 α \geq 0$ , then $(1 - 43 α) {40 α + (1 - \frac{α ^{2}}{4})^{- 1}} \leq (1 - 43 α) (1 + 41 α) \leq (1 - α)^{2} .$ If $1 - 43 α < 0$ , then the left-hand side after multiplication by the positive denominator is at most $0$ , while $(1 - α)^{2} \geq 0$ . This proves the claim in both cases.

Finally, $n^{- 4/3} = n^{- 1} α$ and $\frac{n}{1 - α ^{2} /4} + 40 n α = n {40 α + (1 - \frac{α ^{2}}{4})^{- 1}} .$ Thus $n^{- 1} - 43 n^{- 4/3} = n^{- 1} (1 - 43 α) \leq n^{- 1} \frac{( 1 - α ) ^{2}}{40 α + ( 1 - α ^{2} /4 ) ^{- 1}} = \frac{( 1 - α ) ^{2}}{n / ( 1 - α ^{2} /4 ) + 40 n α} .$ The last expression is the fractional lower bound proved in the preceding step, so $n^{- 1} - 43 n^{- 4/3} \leq ρ_{n, 2} .$ Together with the first displayed bound, this proves both conclusions.

∎

Proof of Proposition 1.

Let $n, M \geq 1$ . We write $ρ_{n}^{†} = ρ_{n} (c^{†})$ and use the identification of $Λ_{n, M}^{†}$ with the three-arm specialization of the rational grid value from Definition 18.

Applying Theorem 7 with $K = 3$ , contrast $c^{†}$ , and mesh $M$ gives certificates $π^{n, M, c^{†}}$ , $w^{n, M, c^{†}}$ , $u$ , $ν_{n, M, c^{†}}$ , and a barycenter estimator $δ_{n, M, c^{†}}$ satisfying $B_{n, c^{†}} (ν_{n, M, c^{†}}) \leq ρ_{n}^{†} \leq U_{n, M} (c^{†}) \leq Λ_{n, M}^{†} \leq B_{n, c^{†}} (ν_{n, M, c^{†}}) + \frac{C _{0} ( c ^{†} )}{4 M ^{2}} .$ For $c^{†} = (1, - 1/2, - 1/2)$ , $L_{c^{†}} = ∣1∣ + ∣ - 1/2∣ + ∣ - 1/2∣ = 2, C_{0} (c^{†}) = \frac{L _{c^{†}}^{2}}{4} = 1.$ The displayed sandwich first gives $ρ_{n}^{†} \leq Λ_{n, M}^{†} .$ It also gives $Λ_{n, M}^{†} \leq B_{n, c^{†}} (ν_{n, M, c^{†}}) + \frac{1}{4 M ^{2}} \leq ρ_{n}^{†} + \frac{1}{4 M ^{2}} .$ Since $M \geq 1$ , $\frac{1}{4 M ^{2}} \leq \frac{2}{M} + \frac{1}{M ^{2}},$ and hence $Λ_{n, M}^{†} \leq ρ_{n}^{†} + \frac{2}{M} + \frac{1}{M ^{2}} .$
The unrestricted minimax risk is an infimum of suprema of squared-error risks, so $0 \leq ρ_{n}^{†} .$ Combining this with $ρ_{n}^{†} \leq Λ_{n, M}^{†}$ yields $0 \leq Λ_{n, M}^{†} .$
It remains to record the elementary feasible point giving the upper bound $1$ . Let $r^{⋆} = (n, 0, 0) \in R_{n, 3}$ , let $0 \in Γ_{M}$ be the grid action corresponding to the index $j = M$ , and define $π_{r} = 1 {r = r^{⋆}}, w_{r, x, g} = 1 {r = r^{⋆}, g = 0}, u_{⋆} = 1.$ The normalization constraints in Definition 18 are then immediate. For a response-count vector $m \in M_{n, 3}$ , set $α (ζ) = ζ_{1} - \frac{1}{2} ζ_{2} - \frac{1}{2} ζ_{3}, ζ \in T_{3} .$ Because each $ζ_{a} \in {0, 1}$ , one has $- 1 \leq α (ζ) \leq 1$ . Therefore $τ_{c^{†}} (m) = \frac{1}{n} ζ \in T_{3} \sum m_{ζ} α (ζ) \in [- 1, 1], τ_{c^{†}} (m)^{2} \leq 1.$ Using the orbit likelihood of Definition 6, whose finite probabilities in $x$ are nonnegative and sum to $1$ , the risk row for the displayed feasible point is $r, x, g \sum P_{m} (x ∣ r) w_{r, x, g} {g - τ_{c^{†}} (m)}^{2} = x \sum P_{m} (x ∣ r^{⋆}) τ_{c^{†}} (m)^{2} = τ_{c^{†}} (m)^{2} \leq 1.$ Thus $u_{⋆} = 1$ is feasible, and the minimizing value satisfies $Λ_{n, M}^{†} \leq 1.$
Apply Theorem 7 again, now with mesh $M = n^{2}$ . Since $n \geq 1$ , also $n^{2} \geq 1$ . The exact rational certificate clause gives a rational allocation mass $π$ , real joint design-action mass $w$ , rational epigraph value $u$ , and rational response-count multipliers $ν$ , packaged as an exact primal-dual certificate. Unpacking that certificate supplies rational grid weights $w^{Q}$ and dual multipliers $y$ such that $w = w^{Q} after real embedding, (π, w^{Q}, u) is primal feasible, y is dual feasible,$ $ν = y_{risk}, u = obj_{dual} (y) .$ The optimal-value clause gives $u = Λ_{n, n^{2}}^{†} .$
Fix $t \in T_{3}$ . Choose any labeling of the $n$ units by ${1, \dots, n}$ , and let $z^{(t)} \in T_{3}^{n}$ be the constant labeled schedule $z_{i}^{(t)} = t, i = 1, \dots, n,$ and let $m^{(t)} \in M_{n, 3}$ be its response-count vector, $m_{ζ}^{(t)} = # {i : z_{i}^{(t)} = ζ}, ζ \in T_{3} .$ Then $m_{t}^{(t)} = n$ . Since $n \geq 1$ , $m_{t}^{(t)} > 0,$ so the response type $t$ is represented by a response-count vector with total population size $n$ .
Finally, work at the second-order scale $k^{4/3}$ . The asserted limit follows from the following direct calculation. For all $k \geq 1$ , $k^{4/3} (\frac{2}{k ^{2}} + \frac{1}{k ^{4}}) = 2 k^{- 2/3} + k^{- 8/3} .$ Both powers on the right tend to zero as $k \to \infty$ , because $2/3 > 0$ and $8/3 > 0$ . Hence $k \to \infty lim k^{4/3} (\frac{2}{k ^{2}} + \frac{1}{k ^{4}}) = 0.$

∎

Proof of Theorem 7.

Since a zero-sum contrast on $K = 0$ or $K = 1$ is identically zero, the hypotheses imply $K \geq 2$ . The rational grid program of Definition 19 is a finite rational linear program. A feasible point is obtained by placing all allocation mass on one allocation orbit and all grid-action mass at $0$ , with an epigraph coordinate large enough to dominate the finitely many risk rows. The risk rows also force every feasible epigraph coordinate to be nonnegative. Finite-dimensional rational LP strong duality therefore gives rational primal and dual optimizers. Decoding the primal optimizer gives $π^{n, M, c}, w^{Q}, u \in Q,$ and decoding the dual optimizer gives multipliers $y$ , whose risk-row component is $ν_{n, M, c} (m) = y_{m}^{risk}, m \in M_{n, K} .$ The decoded certificate satisfies $w_{r, x, g}^{n, M, c} = w_{r, x, g}^{Q}, u = y^{norm, -} - y^{norm, +}, u = Λ_{n, M} (c),$ with primal feasibility and dual feasibility in the grid program. In particular, $y_{m}^{risk} \geq 0 for all m, m \in M_{n, K} \sum y_{m}^{risk} = 1,$ so $ν_{n, M, c}$ is a rational prior on response-count vectors.
Define the estimator, for $r \in R_{n, K}$ and compatible $x$ , by $δ_{n, M, c} (r, x) = ⎩ ⎨ ⎧ g \in Γ_{M, c} \sum g w_{r, x, g}^{n, M, c} / π_{r}^{n, M, c}, 0, π_{r}^{n, M, c} > 0, π_{r}^{n, M, c} = 0.$ This is exactly the barycenter condition in the certificate statement; the zero convention applies precisely on allocation orbits with zero certificate mass.
For a rational prior $ν$ on $M_{n, K}$ , set $D_{r, x, c} (ν) = m \in M_{n, K} \sum ν_{m} P_{m} (x ∣ r), N_{r, x, c} (ν) = m \in M_{n, K} \sum ν_{m} P_{m} (x ∣ r) τ_{c} (m),$ and use the posterior-mean convention $\overset{τ}{ˉ}_{r, x, c} (ν) = {N_{r, x, c} (ν) / D_{r, x, c} (ν), 0, D_{r, x, c} (ν) > 0, D_{r, x, c} (ν) = 0.$ Then the Bayes risk at allocation orbit $r$ is $BayesRisk_{n, c} (r, ν) = m \sum ν_{m} τ_{c} (m)^{2} - x \sum {N_{r, x, c} (ν)^{2} / D_{r, x, c} (ν), 0, D_{r, x, c} (ν) > 0, D_{r, x, c} (ν) = 0.$ Completing the square gives, for every invariant orbit procedure $(π, δ)$ , $m \sum ν_{m} R_{n} ((π, δ); m) \geq r \sum π_{r} BayesRisk_{n, c} (r, ν) \geq B_{n, c} (ν) .$ Thus $B_{n, c} (ν)$ is a Bayes lower bound for the finite orbit game. By the value equality in Theorem 1, $B_{n, c} (ν) \leq ρ_{n} (c) .$ Applying this to $ν = ν_{n, M, c}$ gives the stated lower certificate inequality.
The barycenter rule $δ_{n, M, c}$ , together with $π^{n, M, c}$ , is an admissible orbit procedure. Through the invariant correspondence and value equality in Theorem 1, $ρ_{n} (c) \leq m \in M_{n, K} sup r \sum π_{r}^{n, M, c} x \sum P_{m} (x ∣ r) {δ_{n, M, c} (r, x) - τ_{c} (m)}^{2} = U_{n, M} (c) .$ This proves $ρ_{n} (c) \leq U_{n, M} (c)$ .
For $π_{r}^{n, M, c} > 0$ , let $κ_{r, x}^{n, M, c} (g) = w_{r, x, g}^{n, M, c} / π_{r}^{n, M, c}, g \in Γ_{M, c} .$ Primal feasibility gives $\sum_{g} κ_{r, x}^{n, M, c} (g) = 1$ , and the barycenter definition gives $δ_{n, M, c} (r, x) = g \sum g κ_{r, x}^{n, M, c} (g) .$ Hence, for every $m, r, x$ with $π_{r}^{n, M, c} > 0$ , ${δ_{n, M, c} (r, x) - τ_{c} (m)}^{2} \leq g \sum κ_{r, x}^{n, M, c} (g) {g - τ_{c} (m)}^{2} .$ When $π_{r}^{n, M, c} = 0$ , the corresponding contribution is multiplied by zero. Multiplying by $π_{r}^{n, M, c} P_{m} (x ∣ r)$ , summing over $(r, x)$ , and using the primal risk row gives $U_{n, M} (c) \leq u = Λ_{n, M} (c) .$
It remains to compare $Λ_{n, M} (c)$ with the lower certificate. Fix an allocation orbit $r$ . Since each $τ_{c} (m)$ lies in $[- h_{c}, h_{c}]$ , the posterior mean $\overset{τ}{ˉ}_{r, x, c} (ν_{n, M, c})$ also lies in this interval when $D_{r, x, c} (ν_{n, M, c}) > 0$ , and the convention value $0$ lies there when $D_{r, x, c} (ν_{n, M, c}) = 0$ . The grid $Γ_{M, c}$ has mesh $h_{c} / M$ , so choose $γ_{r, x}^{⋆} \in Γ_{M, c}$ with $γ_{r, x}^{⋆} - \overset{τ}{ˉ}_{r, x, c} (ν_{n, M, c}) \leq \frac{h _{c}}{2 M} .$ Write the relevant dual occupancy difference and nonnegativity multipliers as $α_{r, x} = y_{r, x}^{occ, +} - y_{r, x}^{occ, -}, ζ_{r} = y_{r}^{π} \geq 0, ω_{r, x}^{⋆} = y_{r, x, γ_{r, x}^{⋆}}^{w} \geq 0.$ For the fixed allocation orbit $r$ and the selected grid points $γ_{r, x}^{⋆}$ , dual feasibility gives the stationarity rows $y^{norm, +} - y^{norm, -} - x \sum α_{r, x} - ζ_{r} = 0$ and, for every compatible observation $x$ , $α_{r, x} - ω_{r, x}^{⋆} + m \sum ν_{n, M, c} (m) P_{m} (x ∣ r) {γ_{r, x}^{⋆} - τ_{c} (m)}^{2} = 0.$ Since the dual objective equals $u = y^{norm, -} - y^{norm, +}$ , these rows imply $u = - x \sum α_{r, x} - ζ_{r} \leq x \sum (- α_{r, x}) \leq x \sum m \sum ν_{n, M, c} (m) P_{m} (x ∣ r) {γ_{r, x}^{⋆} - τ_{c} (m)}^{2} = m \sum ν_{n, M, c} (m) x \sum P_{m} (x ∣ r) {γ_{r, x}^{⋆} - τ_{c} (m)}^{2} .$ The posterior square decomposition gives $m \sum ν_{n, M, c} (m) x \sum P_{m} (x ∣ r) {γ_{r, x}^{⋆} - τ_{c} (m)}^{2} = BayesRisk_{n, c} (r, ν_{n, M, c}) + x \sum D_{r, x, c} (ν_{n, M, c}) {γ_{r, x}^{⋆} - \overset{τ}{ˉ}_{r, x, c} (ν_{n, M, c})}^{2} .$ Indeed, for each $x$ with positive predictive mass this is completion of the square around the posterior mean $\overset{τ}{ˉ}_{r, x, c} (ν_{n, M, c})$ ; when the predictive mass is zero, all terms carrying $P_{m} (x ∣ r) ν_{n, M, c} (m)$ vanish and the displayed convention contributes zero. Moreover $x \sum D_{r, x, c} (ν_{n, M, c}) = m \sum ν_{n, M, c} (m) x \sum P_{m} (x ∣ r) = 1.$ Using the grid approximation bound for every $x$ , $x \sum D_{r, x, c} (ν_{n, M, c}) {γ_{r, x}^{⋆} - \overset{τ}{ˉ}_{r, x, c} (ν_{n, M, c})}^{2} \leq \frac{h _{c}^{2}}{4 M ^{2}} = \frac{C _{0} ( c )}{4 M ^{2}} .$ Thus, for every allocation orbit $r$ , $u \leq BayesRisk_{n, c} (r, ν_{n, M, c}) + \frac{C _{0} ( c )}{4 M ^{2}} .$ The allocation-orbit set is nonempty, so taking the infimum over $r$ and using $u = Λ_{n, M} (c)$ gives $Λ_{n, M} (c) \leq B_{n, c} (ν_{n, M, c}) + \frac{C _{0} ( c )}{4 M ^{2}} .$ Together with the preceding steps, this proves the full certificate sandwich.
The orbit counts are stars-and-bars counts. Response-count vectors are nonnegative counts over $2^{K}$ binary response types with total $n$ , so $∣ M_{n, K} ∣ = (2 ^{K} - 1 n + 2 ^{K} - 1) .$ Allocation-count vectors are nonnegative counts over $K$ arms with total $n$ , so $∣ R_{n, K} ∣ = (K - 1 n + K - 1) .$ Finally, a pair $(r, x)$ , with $x$ compatible with $r$ , is equivalently a nonnegative count vector on arm-outcome cells $A_{K} \times {0, 1}$ with total $n$ . There are $2 K$ such cells, hence $r \in R_{n, K} \sum ∣ {x : x is compatible with r} ∣ = (2 K - 1 n + 2 K - 1) .$
Let $(a_{k})$ be any positive sequence and let $M_{k} \geq 1$ . If $\frac{a _{k}}{M _{k}^{2}} ⟶ 0,$ then multiplication by the fixed constant $C_{0} (c) /4$ gives $a_{k} \frac{C _{0} ( c )}{4 M _{k}^{2}} = \frac{C _{0} ( c )}{4} \frac{a _{k}}{M _{k}^{2}} ⟶ 0.$ For certificate sequences at mesh $M_{k}$ , the finite-sample sandwich gives, for all positive $k$ , $B_{k} \leq ρ_{k} (c) \leq U_{k}, 0 \leq U_{k} - B_{k} \leq \frac{C _{0} ( c )}{4 M _{k}^{2}} .$ Finite initial values do not affect limits, and the preceding display therefore implies $a_{k} (U_{k} - B_{k}) ⟶ 0.$ Writing $d_{k} (c) = C_{0} (c) / k - ρ_{k} (c)$ , if $a_{k} d_{k} (c) \to ℓ$ , then $a_{k} {\frac{C _{0} ( c )}{k} - U_{k}} = a_{k} d_{k} (c) - a_{k} {U_{k} - ρ_{k} (c)} ⟶ ℓ$ and $a_{k} {\frac{C _{0} ( c )}{k} - B_{k}} = a_{k} d_{k} (c) + a_{k} {ρ_{k} (c) - B_{k}} ⟶ ℓ,$ because both error terms are squeezed by $a_{k} (U_{k} - B_{k})$ . At the second-order scale $a_{k} = k^{4/3}$ and with $M_{k} = k$ , $\frac{k ^{4/3}}{k ^{2}} = k^{- 2/3} ⟶ 0,$ so $k^{4/3} (U_{k} - B_{k}) ⟶ 0.$

∎

Proof of Theorem 8.

First prove the one-sided estimate $ρ_{n} (c) \leq ρ_{n} (c^{'}) + η (c, c^{'}) .$ Put $ξ_{a} = c_{a} - c_{a}^{'}, η (c, c^{'}) = \frac{1}{2} a \in A_{K} \sum ∣ ξ_{a} ∣.$ For a binary response type $t$ , let $I (t) = {a \in A_{K} : t_{a} = 1} .$ Since both contrasts have zero sum, $\sum_{a} ξ_{a} = 0$ , and therefore $a \in I (t) \sum ξ_{a} = - a \in / I (t) \sum ξ_{a} .$ Thus $2 a \in I (t) \sum ξ_{a} \leq a \in I (t) \sum ∣ ξ_{a} ∣ + a \in / I (t) \sum ∣ ξ_{a} ∣ = a \in A_{K} \sum ∣ ξ_{a} ∣,$ so $a \in A_{K} \sum ξ_{a} t_{a} \leq η (c, c^{'}) .$ For any labeled schedule $z \in T_{K}^{n}$ , using $n \geq 1$ , $∣ τ_{c} (z) - τ_{c^{'}} (z) ∣ = \frac{1}{n} i = 1 \sum n a \in A_{K} \sum ξ_{a} z_{i, a} \leq \frac{1}{n} i = 1 \sum n a \in A_{K} \sum ξ_{a} z_{i, a} \leq η (c, c^{'}) .$ Also set $L_{c} = a \in A_{K} \sum ∣ c_{a} ∣, I_{c} = [- \frac{L _{c}}{2}, \frac{L _{c}}{2}] .$ Applying the preceding target bound to the pair $c, - c$ gives $∣2 τ_{c} (z) ∣ = ∣ τ_{c} (z) - τ_{- c} (z) ∣ \leq L_{c},$ hence $τ_{c} (z) \in I_{c}$ .

Fix any procedure $(D, τ^{'})$ for contrast $c^{'}$ . Define the transferred procedure for $c$ by keeping the same assignment law and projecting the estimator output onto $I_{c}$ : $τ^{c} (A, Y) = proj_{I_{c}} {τ^{'} (A, Y)} .$ For $y \in I_{c}$ , projection onto the closed interval $I_{c}$ satisfies $proj_{I_{c}} (s) - y^{2} \leq ∣ s - y ∣^{2} (s \in R),$ by the three cases $s < - L_{c} /2$ , $s \in I_{c}$ , and $s > L_{c} /2$ . For fixed $z$ and assignment vector $A$ , define $e_{z} (A) = τ^{'} (A, Y^{obs} (z, A)) - τ_{c^{'}} (z) .$ Using $τ_{c} (z) \in I_{c}$ and the target bound above, $τ^{c} (A, Y^{obs} (z, A)) - τ_{c} (z) \leq τ^{'} (A, Y^{obs} (z, A)) - τ_{c} (z) \leq ∣ e_{z} (A) ∣ + ∣ τ_{c^{'}} (z) - τ_{c} (z) ∣ \leq ∣ e_{z} (A) ∣ + η (c, c^{'}) .$ For each target contrast $\overset{c}{ˉ} \in {c, c^{'}}$ and any estimator $τ$ , write its squared-error risk at the fixed schedule $z$ as $R_{n}^{\overset{c}{ˉ}} (D, τ; z) = E_{D} [{τ (A, Y^{obs} (z, A)) - τ_{\overset{c}{ˉ}} (z)}^{2}] .$ Taking $D$ -expectations gives $R_{n}^{c} (D, τ^{c}; z) \leq E_{D} [(∣ e_{z} (A) ∣ + η (c, c^{'}))^{2}] .$ Since $η (c, c^{'}) \geq 0$ , Cauchy–Schwarz yields $E_{D} ∣ e_{z} (A) ∣ \leq E_{D} e_{z} (A)^{2},$ and therefore $E_{D} [(∣ e_{z} (A) ∣ + η (c, c^{'}))^{2}] \leq E_{D} e_{z} (A)^{2} + η (c, c^{'}) .$ Here $E_{D} e_{z} (A)^{2} = R_{n}^{c^{'}} (D, τ^{'}; z)$ , so $R_{n}^{c} (D, τ^{c}; z) \leq R_{n}^{c^{'}} (D, τ^{'}; z) + η (c, c^{'}) .$ It remains to pass from this pointwise comparison to the two minimax values. For each $\overset{c}{ˉ} \in {c, c^{'}}$ , set $Ψ_{\overset{c}{ˉ}} (D, τ) = z \in T_{K}^{n} max R_{n}^{\overset{c}{ˉ}} (D, τ; z),$ so Definition 3 says $ρ_{n} (\overset{c}{ˉ}) = D, τ in f Ψ_{\overset{c}{ˉ}} (D, τ) .$ The schedule set is finite and nonempty, and $\cdot$ is increasing on $[0, \infty)$ , hence $z \in T_{K}^{n} max R_{n}^{\overset{c}{ˉ}} (D, τ; z) = Ψ_{\overset{c}{ˉ}} (D, τ) .$ We also have $ρ_{n} (\overset{c}{ˉ}) = D, τ in f Ψ_{\overset{c}{ˉ}} (D, τ) .$ Indeed, every $Ψ_{\overset{c}{ˉ}} (D, τ)$ is at least $ρ_{n} (\overset{c}{ˉ})$ , giving the $\geq$ inequality. Conversely, the procedure class is nonempty, and for every $ε > 0$ there is a procedure with $Ψ_{\overset{c}{ˉ}} (D, τ) < ρ_{n} (\overset{c}{ˉ}) + ε;$ therefore the infimum of the square roots is at most $ρ_{n} (\overset{c}{ˉ}) + ε$ , and continuity at $ρ_{n} (\overset{c}{ˉ})$ gives the reverse inequality as $ε ↓ 0$ .

Taking the maximum over $z$ in the previous pointwise display gives $Ψ_{c} (D, τ^{c}) \leq Ψ_{c^{'}} (D, τ^{'}) + η (c, c^{'}) .$ Since $(D, τ^{c})$ is one admissible procedure for contrast $c$ , taking the infimum over all procedures $(D, τ^{'})$ for $c^{'}$ gives $ρ_{n} (c) \leq ρ_{n} (c^{'}) + η (c, c^{'}) .$
The contrast distance is symmetric: $η (c^{'}, c) = \frac{1}{2} a \in A_{K} \sum ∣ c_{a}^{'} - c_{a} ∣ = \frac{1}{2} a \in A_{K} \sum ∣ c_{a} - c_{a}^{'} ∣ = η (c, c^{'}) .$ Applying the one-sided estimate to the ordered pair $(c^{'}, c)$ gives $ρ_{n} (c^{'}) \leq ρ_{n} (c) + η (c^{'}, c) = ρ_{n} (c) + η (c, c^{'}),$ hence $- η (c, c^{'}) \leq ρ_{n} (c) - ρ_{n} (c^{'}) .$
Applying the same one-sided estimate to the ordered pair $(c, c^{'})$ gives $ρ_{n} (c) - ρ_{n} (c^{'}) \leq η (c, c^{'}) .$ Together with the previous displayed lower bound, this is exactly $ρ_{n} (c) - ρ_{n} (c^{'}) \leq η (c, c^{'}) .$

∎

Proof of Theorem 9.

Fix a rational zero-sum contrast $q \in Q^{K} ∖ {0}$ and integers $n, M \geq 1$ . Apply Theorem 7 to $q$ . It gives an invariant allocation certificate $π$ , a rational grid weight $w$ , a rational epigraph value $u$ , a rational response-count multiplier $ν$ , and a barycenter rule $δ$ , with $B := B_{n, q} (ν), U := U_{n, M} (q; π, δ), u = Λ_{n, M} (q),$ and $B \leq ρ_{n} (q) \leq U \leq u \leq B + \frac{C _{0} ( q )}{4 M ^{2}} .$ The multiplier $ν$ is a response-count probability prior. Indeed, by Theorem 7, $ν$ is the risk component of a dual-feasible multiplier for the grid-action program of Definition 19. In that program the response-count constraints are the inequality rows $r, x, g \sum P_{m} (x ∣ r) w_{r, x, g} {g - τ_{q} (m)}^{2} - u \leq 0, m \in M_{n, K},$ so dual feasibility gives $ν (m) \geq 0$ for every $m$ . The epigraph variable $u$ has objective coefficient $1$ , coefficient $- 1$ in each of those rows, and no other occurrence, so its dual stationarity equation is $1 - m \in M_{n, K} \sum ν (m) = 0.$ Thus $ν (m) \geq 0 for every m \in M_{n, K}, m \in M_{n, K} \sum ν (m) = 1.$ With $ν$ now a probability prior, the lower certificate is an infimum of Bayes risks over the finite allocation-count set, hence $B \geq 0$ ; the displayed sandwich gives $U \geq 0$ . Put $η_{q} := η (c, q), R^{-} := {max (0, B - η_{q})}^{2}, R^{+} := (U + η_{q})^{2} .$

The pointwise transfer estimate used for both certificate directions is the following. If a procedure for one contrast is clipped to the natural interval of another contrast, then for every schedule $z$ , $R_{n} (D, τ^{c}; z; c) \leq R_{n} (D, τ^{q}; z; q) + η_{q} .$ Indeed, for each assignment $A$ , $∣ τ_{c} (z) - τ_{q} (z) ∣ \leq η_{q}$ because the target difference is an average of binary response-type scores and $η_{q} = \frac{1}{2} a \in A_{K} \sum ∣ c (a) - q (a) ∣.$ Projection onto $[- L_{c} /2, L_{c} /2]$ can only reduce squared distance to $τ_{c} (z)$ , and Cauchy–Schwarz gives $E_{D} (∣ E_{A} ∣ + η_{q})^{2} \leq {E_{D} E_{A}^{2} + η_{q}}^{2}$ for the original residual $E_{A} = τ^{q} (A, Y^{obs}) - τ_{q} (z)$ .

The upper procedure keeps the invariant allocation law induced by $π$ and uses $τ^{c} (A, Y^{obs}) = proj_{[- L_{c} /2, L_{c} /2]} {δ (r (A), x (A, Y^{obs}))} .$ The rational certificate bounds the $q$ -risk of the barycenter rule by $U$ , so the transfer estimate yields worst-case $c$ -risk at most $R^{+}$ , and therefore $ρ_{n} (c) \leq R^{+} .$

For the lower endpoint, spread this response-count prior uniformly over each labeled schedule orbit, and write the resulting expectation as $E_{ν}^{sch} [φ] := z \in T_{K}^{n} \sum \frac{ν ( m ( z ))}{∣ { z ^{'} : m ( z ^{'} ) = m ( z )} ∣} φ (z) .$ The denominator is positive because each response-count vector is represented by at least one labeled schedule, and the displayed nonnegativity and mass identity make $E_{ν}^{sch}$ a probability expectation. We first record the all-procedure lower bound supplied by this lifted prior. Let $p = (D_{p}, τ_{p})$ be any labeled procedure for contrast $q$ , and average $p$ over the unit-permutation group $S_{n}$ . By the lossless symmetrization and exact invariant correspondence in Theorem 1, the averaged procedure is invariant and corresponds to an orbit procedure $o$ . If $α_{o}$ is its allocation-count law, define its orbit risk at response-count vector $m$ by $Risk_{n, q}^{orb} (o; m) := r \in R_{n, K} \sum α_{o} (r) x \sum P_{m} (x ∣ r) {τ_{o} (r, x) - τ_{q} (m)}^{2} .$ For a fixed allocation count $r$ , the conditional Bayes risk under $ν$ is bounded below by $B = B_{n, q} (ν)$ ; on zero prior-predictive fibers the corresponding summand is zero, and on positive prior-predictive fibers the posterior-mean residual is the conditional minimum. Hence, for every orbit procedure, $B \leq m \sum ν (m) Risk_{n, q}^{orb} (o; m) .$ Because the lifted prior is uniform within each response-count orbit, $m \sum ν (m) Risk_{n, q}^{orb} (o; m) = E_{ν}^{sch} [Risk_{n, q}^{orb} (o; m (z))] .$ Lossless symmetrization gives, for every labeled schedule $z$ , $Risk_{n, q}^{orb} (o; m (z)) \leq \frac{1}{∣ S _{n} ∣} σ \in S_{n} \sum R_{n} (D_{p}, τ_{p}; σ z; q) .$ The lifted prior is invariant under unit permutations, so averaging the last display over $z$ yields $B \leq E_{ν}^{sch} [R_{n} (D_{p}, τ_{p}; z; q)]$ for every labeled $q$ -procedure $p$ .

Now let $r = (D_{r}, τ_{r})$ be any procedure for the real contrast $c$ , and let $r^{q}$ be the procedure obtained by clipping its action to $[- L_{q} /2, L_{q} /2]$ . Applying the pointwise transfer estimate in the direction from $c$ to $q$ , $R_{n} (r^{q}; z; q) \leq R_{n} (D_{r}, τ_{r}; z; c) + η_{q} .$ Set $V_{c} (r) := E_{ν}^{sch} [R_{n} (D_{r}, τ_{r}; z; c)] .$ Squaring the preceding inequality, integrating with respect to $E_{ν}^{sch}$ , and using Cauchy–Schwarz in the finite schedule space gives $E_{ν}^{sch} [R_{n} (r^{q}; z; q)] \leq V_{c} (r) + 2 η_{q} V_{c} (r) + η_{q}^{2} = {V_{c} (r) + η_{q}}^{2} .$ Combining this with the all-procedure $q$ -lower bound applied to $r^{q}$ gives $B \leq {V_{c} (r) + η_{q}}^{2} .$ If $B \leq 0$ , then $B = 0$ and $R^{-} = 0 \leq V_{c} (r)$ . If $B > 0$ , the displayed inequality gives $B \leq V_{c} (r) + η_{q}$ . When $B \leq η_{q}$ this again gives $R^{-} = 0 \leq V_{c} (r)$ , and otherwise $R^{-} = (B - η_{q})^{2} \leq V_{c} (r) .$ Thus the lifted schedule prior has Bayes risk at least $R^{-}$ against every real-contrast procedure. Since prior risk is bounded above by worst-case risk for each procedure, taking the minimax value gives $R^{-} \leq ρ_{n} (c) .$

It remains to verify the stated width. From the rational sandwich, $U - B \leq \frac{C _{0} ( q )}{4 M ^{2}} .$ Also Theorem 4 gives $ρ_{n} (q) \leq \frac{C _{0} ( q )}{n},$ so $U \leq \frac{C _{0} ( q )}{n} + \frac{C _{0} ( q )}{4 M ^{2}} .$ Consequently, $U \leq C_{0} (q) n^{- 1/2} + \frac{C _{0} ( q )}{2 M},$ because the square of the right-hand side contains the displayed upper bound plus a nonnegative cross term. Finally, $B - {max (0, B - η_{q})}^{2} \leq 2 η_{q} B + η_{q}^{2},$ and $B \leq U$ , whence $R^{+} - R^{-} \leq \frac{C _{0} ( q )}{4 M ^{2}} + 4 η_{q} U + 2 η_{q}^{2} .$ Substituting the bound on $U$ and using $\frac{2 C _{0} ( q ) η _{q}}{M} \leq \frac{C _{0} ( q )}{M ^{2}} + η_{q}^{2}$ gives $R^{+} - R^{-} \leq \frac{5 C _{0} ( q )}{4 M ^{2}} + 4 C_{0} (q) η_{q} n^{- 1/2} + 3 η_{q}^{2} .$ Thus $B, U, R^{-}, R^{+}$ form a transferred certificate for $(n, M, c, q)$ , proving the finite-grid transfer assertion.
We next construct the rational approximating sequence. Since $c$ is nonzero, $L_{c} > 0$ . For $s \in N$ , define the approximation radius $rad_{s} = {L_{c} /4, min {L_{c} /4, s^{- 3}}, s = 0, s \geq 1.$ Then $0 < rad_{s} < L_{c} /2$ . Choose a fixed arm $a_{\circ}$ . By density of $Q$ in $R$ , choose rational numbers $c_{s} (a)$ for $a \neq = a_{\circ}$ with $∣ c (a) - c_{s} (a) ∣ < \frac{rad _{s}}{2 K},$ and set $c_{s} (a_{\circ}) = - a \neq = a_{\circ} \sum c_{s} (a) .$ Because $c$ is zero-sum, $c (a_{\circ}) = - a \neq = a_{\circ} \sum c (a),$ and hence $∣ c (a_{\circ}) - c_{s} (a_{\circ}) ∣ \leq a \neq = a_{\circ} \sum ∣ c (a) - c_{s} (a) ∣.$ Therefore $a \in A_{K} \sum ∣ c (a) - c_{s} (a) ∣ \leq rad_{s} .$ Since $rad_{s} < L_{c} /2$ , this rational zero-sum vector is nonzero; otherwise $L_{c} \leq rad_{s}$ . Denote the resulting rational contrast by $c^{(s)}$ . Then $η (c, c^{(s)}) \leq rad_{s},$ and for $s \geq 1$ , $η (c, c^{(s)}) \leq s^{- 3} .$ It follows that $η (c, c^{(s)}) \to 0, s^{5/6} η (c, c^{(s)}) \leq s^{5/6} s^{- 3} = s^{- 13/6} \to 0.$ For each arm $a$ , $∣ c^{(s)} (a) - c (a) ∣ \leq b \in A_{K} \sum ∣ c^{(s)} (b) - c (b) ∣ = 2 η (c, c^{(s)}),$ so $c^{(s)} (a) \to c (a)$ .
For each $s \geq 1$ , apply the fixed construction from [it:real-contrast-finite-grid-transfer] with $q = c^{(s)}$ , $n = s$ , and $M = s$ , obtaining $B_{s}, U_{s}, R_{s}^{-}, R_{s}^{+}$ . The construction gives a transferred certificate for every $s \geq 1$ . Write $η_{s} := η (c, c^{(s)}), C_{s} := C_{0} (c^{(s)}) .$ Because $c^{(s)} (a) \to c (a)$ for every arm and the arm set is finite, $L_{c^{(s)}} = a \in A_{K} \sum ∣ c^{(s)} (a) ∣ ⟶ a \in A_{K} \sum ∣ c (a) ∣ = L_{c},$ and hence $C_{s} = \frac{L _{c^{(s)}}^{2}}{4} ⟶ C_{0} (c) .$ The certificate width bound gives, for $s \geq 1$ , $0 \leq s^{4/3} (R_{s}^{+} - R_{s}^{-}) \leq s^{4/3} {\frac{5 C _{s}}{4 s ^{2}} + 4 C_{s} η_{s} s^{- 1/2} + 3 η_{s}^{2}} .$ The three terms on the right converge to zero: $s^{4/3} \frac{5 C _{s}}{4 s ^{2}} = \frac{5 C _{s}}{4} s^{- 2/3} \to 0,$ $s^{4/3} 4 C_{s} η_{s} s^{- 1/2} = 4 C_{s} {s^{5/6} η_{s}} \to 0,$ and, using $η_{s} \leq s^{- 3}$ , $s^{4/3} 3 η_{s}^{2} \leq 3 s^{4/3} s^{- 6} = 3 s^{- 14/3} \to 0.$ By squeezing, $s^{4/3} (R_{s}^{+} - R_{s}^{-}) \to 0.$ This proves the real-contrast approximation assertion.
Finally suppose a rational zero-sum contrast $q$ embeds coefficientwise as $c$ . For any $n \geq 1$ , apply the fixed construction with $M = n$ . The rational sandwich gives $U - B \leq \frac{C _{0} ( q )}{4 n ^{2}} .$ Since the real embedding of $q$ is $c$ , $C_{0} (q) = C_{0} (c)$ , and hence $U - B \leq \frac{C _{0} ( c )}{4 n ^{2}} .$ Together with the transferred certificate constructed in [it:real-contrast-finite-grid-transfer], this proves the exact rational case.

∎

Proof of Theorem 10.

Fix $n \geq 1$ and $M \geq 1$ .

Let $c_{Q}^{†} = (1, - 1/2, - 1/2) \in Q^{3}, c^{†} = (1, - 1/2, - 1/2) \in R^{3} .$ This contrast is nonzero and zero-sum, and $L_{c^{†}} = ∣1∣ + ∣ - 1/2∣ + ∣ - 1/2∣ = 2, C_{0} (c^{†}) = (\frac{L _{c^{†}}}{2})^{2} = 1.$ Hence the real embedding of $c_{Q}^{†}$ has $C_{0} = 1$ , and the grid in Definition 19 becomes $Γ_{M, c^{†}} = {- 1 + j / M : 0 \leq j \leq 2 M} = Γ_{M} .$ Therefore, after specializing Theorem 7 to $K = 3$ and $c = c_{Q}^{†}$ , the rational contrast grid program of Definition 19 is the three-arm rational grid program of Definition 18, with $Λ_{n, M} (c^{†}) = Λ_{n, M}^{†}, ρ_{n} (c^{†}) = ρ_{n}^{†} .$
The specialized certificate theorem supplies a rational allocation certificate $π^{n, M}$ , a joint design–action weight $w^{n, M}$ , a rational epigraph value $u$ , a rational response-count multiplier $ν_{n, M}$ , and an estimator $δ_{n, M}$ . Its exact-certificate conclusion gives the rational grid weight whose real embedding is $w^{n, M}$ , primal feasibility, dual feasibility, the identification of the risk component with $ν_{n, M}$ , and dual objective value $u$ . Its barycenter conclusion gives, for every $r \in R_{n, 3}$ and compatible $x$ , $δ_{n, M} (r, x) = ⎩ ⎨ ⎧ g \in Γ_{M} \sum g \frac{w _{r, x, g}^{n, M}}{π _{r}^{n, M}}, 0, π_{r}^{n, M} > 0, π_{r}^{n, M} = 0.$ The second branch fixes the estimator at allocation counts that carry no certificate mass; those counts are never charged by the certificate, so the choice is immaterial.

The same specialization gives $u = Λ_{n, M}^{†} .$ For a response-count multiplier $ν$ , the specialized lower endpoint is $B_{n, c^{†}} (ν) = r \in R_{n, 3} in f BayesRisk_{n, c^{†}} (r, ν) = B_{n} (ν),$ and the specialized upper endpoint is precisely $U_{n, M} = m \in M_{n, 3} sup r \in R_{n, 3} \sum π_{r}^{n, M} x \sum P_{m} (x ∣ r) {δ_{n, M} (r, x) - τ_{c^{†}} (m)}^{2} .$ Thus the certificate sandwich in Theorem 7, together with $C_{0} (c^{†}) = 1$ , yields $B_{n} (ν_{n, M}) \leq ρ_{n}^{†} \leq U_{n, M} \leq Λ_{n, M}^{†} \leq B_{n} (ν_{n, M}) + \frac{1}{4 M ^{2}} .$
Let $a_{k} > 0$ be any positive normalizing sequence and let $M_{k} \geq 1$ for all $k$ . If $\frac{a _{k}}{M _{k}^{2}} ⟶ 0,$ then $a_{k} \frac{1}{4 M _{k}^{2}} = \frac{1}{4} \frac{a _{k}}{M _{k}^{2}} ⟶ 0.$ This is the mesh-rate conclusion at $C_{0} (c^{†}) = 1$ .
Now let $a_{k} > 0$ , $M_{k}$ , $L_{k}$ , $U_{k}$ , and $C$ satisfy the hypotheses of the final assertion. All inequalities below are for $k \geq 1$ , which is an eventual range for limits along $k \to \infty$ . From $L_{k} \leq ρ_{k}^{†} \leq U_{k}, U_{k} - L_{k} \leq \frac{1}{4 M _{k}^{2}},$ we have $0 \leq a_{k} (ρ_{k}^{†} - L_{k}) \leq a_{k} (U_{k} - L_{k}) \leq a_{k} \frac{1}{4 M _{k}^{2}},$ and also $0 \leq a_{k} (U_{k} - ρ_{k}^{†}) \leq a_{k} (U_{k} - L_{k}) \leq a_{k} \frac{1}{4 M _{k}^{2}} .$ The mesh assumption and the preceding step give $a_{k} (ρ_{k}^{†} - L_{k}) ⟶ 0, a_{k} (U_{k} - ρ_{k}^{†}) ⟶ 0.$ Finally, $d_{k} (c^{†}) = \frac{C _{0} ( c ^{†} )}{k} - ρ_{k}^{†} .$ Therefore $a_{k} {\frac{C _{0} ( c ^{†} )}{k} - L_{k}} = a_{k} d_{k} (c^{†}) + a_{k} (ρ_{k}^{†} - L_{k}),$ and $a_{k} {\frac{C _{0} ( c ^{†} )}{k} - U_{k}} = a_{k} d_{k} (c^{†}) - a_{k} (U_{k} - ρ_{k}^{†}) .$ Since $a_{k} d_{k} (c^{†}) \to C$ and both residual terms converge to $0$ , the two displayed sequences converge to $C$ .

∎

Proof of Proposition 2.

For $c^{†} = (1, - 1/2, - 1/2)$ , the contrast-weighted one-unit law is $Pr (A_{i} = 1) = \frac{∣1∣}{∣1∣ + ∣ - 1/2∣ + ∣ - 1/2∣} = \frac{1}{2}, Pr (A_{i} = 2) = Pr (A_{i} = 3) = \frac{1}{4} .$ Since $c^{†}$ is positive on arm $1$ and negative on arms $2, 3$ , $Z_{i} = ⎩ ⎨ ⎧ 2 t_{i, 1} - 1, 1 - 2 t_{i, 2}, 1 - 2 t_{i, 3}, A_{i} = 1, A_{i} = 2, A_{i} = 3.$ Thus, conditionally on $z$ , $E_{D^{⋆}} (Z_{i} ∣ z) = \frac{1}{2} (2 t_{i, 1} - 1) + \frac{1}{4} (1 - 2 t_{i, 2}) + \frac{1}{4} (1 - 2 t_{i, 3}) = t_{i, 1} - \frac{t _{i, 2} + t _{i, 3}}{2} = μ_{i} .$ This proves the asserted conditional mean for every $n \geq 1$ , schedule $z$ , and unit $i$ .
The score is always sign-valued, so $Z_{i}^{2} = 1$ . Combining this with the preceding mean identity gives $Var_{D^{⋆}} (Z_{i} ∣ z) = E_{D^{⋆}} (Z_{i}^{2} ∣ z) - {E_{D^{⋆}} (Z_{i} ∣ z)}^{2} = 1 - μ_{i}^{2} .$
Write $\overline{Z} = \frac{1}{n} i \sum Z_{i} .$ The assignment law is a product law, so the $Z_{i}$ ’s are conditionally independent given $z$ . Moreover $τ_{c^{†}} (z) = \frac{1}{n} i \sum (t_{i, 1} - \frac{t _{i, 2} + t _{i, 3}}{2}) = \frac{1}{n} i \sum μ_{i},$ and therefore $\overline{Z}$ is conditionally unbiased for $τ_{c^{†}} (z)$ . Hence, for $n \geq 1$ , $E_{D^{⋆}} [{\overline{Z} - τ_{c^{†}} (z)}^{2} z] = Var_{D^{⋆}} (\overline{Z} ∣ z) = \frac{1}{n ^{2}} i \sum Var_{D^{⋆}} (Z_{i} ∣ z) = \frac{1}{n ^{2}} i \sum (1 - μ_{i}^{2}) = \frac{1}{n} - \frac{1}{n ^{2}} i \sum μ_{i}^{2} .$
For the scalar experiment at $n = 3$ , let $L_{Z} = {- 1, - 1/2, 0, 1/2, 1}$ be the feasible set of scalar means. Given $μ = (μ_{1}, μ_{2}, μ_{3}) \in L_{Z}^{3}$ , the retained observation has independent coordinates with $μ Pr (Z_{i} = 1) = \frac{1 + μ _{i}}{2}, μ Pr (Z_{i} = - 1) = \frac{1 - μ _{i}}{2},$ and the scalar target is $M_{μ} /3$ , where $M_{μ} = i = 1 \sum 3 μ_{i}, Q_{μ} = i = 1 \sum 3 μ_{i}^{2} .$ Set $α = \frac{3 - 3}{6}, θ_{Z} (Z_{1}, Z_{2}, Z_{3}) = α i = 1 \sum 3 Z_{i} .$ Then $E_{μ} [{θ_{Z} - \frac{M _{μ}}{3}}^{2}] = α^{2} (3 - Q_{μ}) + (α - \frac{1}{3})^{2} M_{μ}^{2} = \frac{2 - 3}{18} {9 - 3 Q_{μ} + M_{μ}^{2}} .$ By Cauchy’s inequality, $M_{μ}^{2} \leq 3 Q_{μ}$ , and hence $E_{μ} [{θ_{Z} - \frac{M _{μ}}{3}}^{2}] \leq \frac{2 - 3}{2} = 1 - \frac{3}{2} .$ Taking the supremum over $μ \in L_{Z}^{3}$ and then the infimum over scalar rules gives $V_{3}^{Z} \leq 1 - \frac{3}{2} .$
For the reverse inequality, put the following weights on the five homogeneous states $λ_{j} = - 1 + \frac{j}{2}, j = 0, 1, 2, 3, 4 :$ $ϖ_{0} = ϖ_{4} = \frac{- 11 + 7 3}{12}, ϖ_{1} = ϖ_{3} = \frac{8 - 4 3}{3}, ϖ_{2} = \frac{- 5 + 3 3}{2} .$ They are nonnegative and satisfy $\sum_{j = 0}^{4} ϖ_{j} = 1$ . For any scalar rule $θ$ , its Bayes risk under this prior is $B (θ) = j = 0 \sum 4 ϖ_{j} E_{λ_{j}} [(θ (Z) - λ_{j})^{2}],$ where $E_{λ_{j}}$ denotes the product law with all three scalar means equal to $λ_{j}$ . If $N_{+} = # {i : Z_{i} = 1}$ , then for $b = 0, 1, 2, 3$ the prior predictive mass of the event ${N_{+} = b}$ and its target numerator are $P_{b} = (b 3) j = 0 \sum 4 ϖ_{j} (\frac{1 + λ _{j}}{2})^{b} (\frac{1 - λ _{j}}{2})^{3 - b}, H_{b} = (b 3) j = 0 \sum 4 ϖ_{j} λ_{j} (\frac{1 + λ _{j}}{2})^{b} (\frac{1 - λ _{j}}{2})^{3 - b} .$ A direct expansion gives $b 0123 P_{b} (- 1 + 33) /16 (9 - 33) /16 (9 - 33) /16 (- 1 + 33) /16 H_{b} (6 - 53) /16 (- 6 + 33) /16 (6 - 33) /16 (- 6 + 53) /16 H_{b} / P_{b} - 3 α - α α 3 α .$ Thus the posterior mean is exactly $α (2 N_{+} - 3) = θ_{Z} (Z)$ , and completing the square yields $B (θ) = B (θ_{Z}) + Z \in {- 1, 1}^{3} \sum prior Pr (Z) {θ (Z) - θ_{Z} (Z)}^{2} \geq B (θ_{Z}) .$ For a homogeneous level $λ_{j}$ , $E_{λ_{j}} [(θ_{Z} (Z) - λ_{j})^{2}] = α^{2} 3 (1 - λ_{j}^{2}) + (3 α λ_{j} - λ_{j})^{2},$ and summing this identity with the weights $ϖ_{j}$ gives $B (θ_{Z}) = 1 - \frac{3}{2} .$ Every supremum risk dominates this Bayes risk, so $1 - \frac{3}{2} \leq V_{3}^{Z} .$ Together with the preceding item, this proves $V_{3}^{Z} = 1 - 3 /2$ .
The full-data rule is defined from the observed arm labels and outcomes by $r_{1} = i \sum 1 {A_{i} = 1}, s_{1} = i : A_{i} = 1 \sum (2 Y_{i}^{obs} - 1), s_{-} = i : A_{i} \in {2, 3} \sum (1 - 2 Y_{i}^{obs}) .$ The rule is given by a lookup table. Let $T_{F} (r_{1}, s_{1}, s_{-})$ be the integer in the table $(0, 0, - 3) (0, 0, 1) (1, - 1, - 2) (1, - 1, 2) (1, 1, 0) (2, - 2, - 1) (2, 0, - 1) (2, 2, - 1) (3, - 3, 0) (3, 1, 0) - 623 183 - 638 148199 - 648 - 167 210 - 660 167 (0, 0, - 1) (0, 0, 3) (1, - 1, 0) (1, 1, - 2) (1, 1, 2) (2, - 2, 1) (2, 0, 1) (2, 2, 1) (3, - 1, 0) (3, 3, 0) - 183 623 - 199 - 148 638 - 210 167648 - 167 660.$ The displayed triples are exactly the triples $(r_{1}, s_{1}, s_{-})$ that assignments and binary outcomes can generate, so the rule is defined wherever it is used; set $T_{F} = 0$ elsewhere. Define $τ_{F} (A, Y^{obs}) = proj_{[- 1, 1]} (\frac{T _{F} ( r _{1} , s _{1} , s _{-} )}{1000}) .$ All displayed values lie in $[- 1000, 1000]$ , so the projection leaves the table value divided by $1000$ unchanged. For any schedule $z$ , its risk under $D^{⋆}$ is the finite rational sum $R_{F} (z) = a \in A_{3}^{3} \sum i = 1 \prod 3 q_{a_{i}}^{⋆} {\frac{T _{F} ( r _{1} ( a ) , s _{1} ( a , z ) , s _{-} ( a , z ))}{1000} - τ_{c^{†}} (z)}^{2} .$ Identifying an assignment $a \in A_{3}^{3}$ with its ordered triple gives a $27$ -term rational sum. To certify the uniform bound, write each schedule as $z = (t^{(1)}, t^{(2)}, t^{(3)}), t^{(j)} \in {0, 1}^{3},$ and order the eight possible response types as $O_{3} = (000, 001, 010, 011, 100, 101, 110, 111) .$ For $t^{(1)}, t^{(2)}, t^{(3)} \in O_{3}$ , define the cleared slack $Ξ (t^{(1)}, t^{(2)}, t^{(3)}) = 144000000 {\frac{511653}{4000000} - R_{F} (t^{(1)}, t^{(2)}, t^{(3)})} .$ Substituting the $27$ -term expression for $R_{F}$ and clearing denominators makes $Ξ$ an integer-valued function of the triple, and the check is a finite one: there are $8^{3} = 512$ triples, and $Ξ \geq 0$ at every one of them. The evaluation is an exact rational computation: after substituting the displayed formula for $R_{F}$ , the $512$ integer values of $Ξ$ are all nonnegative. The bound is attained: for instance $Ξ (000, 111, 111) = 0,$ so no smaller constant than $511653/4000000$ works for this estimator. Since $Ξ \geq 0$ on all of $({0, 1}^{3})^{3}$ , $R_{F} (z) \leq \frac{511653}{4000000} for every z \in T_{3}^{3} .$ For the schedule with response types $(0, 0, 0), (1, 1, 1), (1, 1, 1)$ , the target is $0$ , and the same finite sum reduces to $R_{F} (z) = \frac{1}{8} (\frac{183}{1000})^{2} + \frac{1}{8} (\frac{638}{1000})^{2} + \frac{1}{4} (\frac{199}{1000})^{2} + \frac{1}{4} (\frac{167}{1000})^{2} + \frac{1}{8} (\frac{648}{1000})^{2} + \frac{1}{8} (\frac{167}{1000})^{2} = \frac{511653}{4000000} .$ Therefore the worst-case risk of this full-data rule is exactly $511653/4000000$ .
The rational comparison in the full-data certificate is $\frac{511653}{4000000} < \frac{18213}{136000} .$

Also, $3 < \frac{117787}{68000},$ because squaring both positive sides gives $3 < (117787/68000)^{2}$ . Hence $1 - \frac{3}{2} > 1 - \frac{1}{2} \frac{117787}{68000} = \frac{18213}{136000} .$ Thus $\frac{511653}{4000000} < \frac{18213}{136000} < 1 - \frac{3}{2} .$
It remains to compute the scalar moment envelope. Let $L_{Z} = {- 1, - 1/2, 0, 1/2, 1}, M = i \sum μ_{i}, Q_{μ} = i \sum μ_{i}^{2},$ with each $μ_{i} \in L_{Z}$ . Pointwise on $L_{Z}$ , $\frac{∣ μ _{i} ∣}{2} \leq μ_{i}^{2}, \frac{3∣ μ _{i} ∣ - 1}{2} \leq μ_{i}^{2} .$ Since $\sum_{i} ∣ μ_{i} ∣ \geq M$ , summing these two inequalities gives $Q_{μ} \geq \frac{M}{2}, Q_{μ} \geq \frac{3 M - n}{2} .$ Consequently $Q_{μ} \geq {M /2, (3 M - n) /2, M \leq n /2, M > n /2.$ For attainment, feasibility implies $M = k /2$ for some integer $0 \leq k \leq 2 n$ . If $k \leq n$ , take $k$ coordinates equal to $1/2$ and the remaining coordinates equal to $0$ . Then $i \sum μ_{i} = \frac{k}{2} = M, i \sum μ_{i}^{2} = \frac{k}{4} = \frac{M}{2} .$ If $k > n$ , write $ℓ = 2 n - k$ , so $0 \leq ℓ < n$ , take $ℓ$ coordinates equal to $1/2$ , and take the remaining $n - ℓ$ coordinates equal to $1$ . Then $i \sum μ_{i} = n - \frac{ℓ}{2} = \frac{k}{2} = M,$ and $i \sum μ_{i}^{2} = n - \frac{3 ℓ}{4} = \frac{3 M - n}{2} .$ For $n = 0$ , feasibility forces $M = 0$ , and the empty vector gives value $0$ , matching the first branch. Taking the infimum over feasible scalar mean vectors yields $min {Q_{μ} : μ \in L_{Z}^{n}, i \sum μ_{i} = M} = {M /2, (3 M - n) /2, M \leq n /2, M > n /2.$ Equivalently, in the notation of the statement, this is the asserted formula for the minimum attainable $\sum_{i} μ_{i}^{2}$ .
Combining the score identities, the average-score risk formula, the exact scalar minimax value, the full-data witness with its rational separation, and the scalar moment envelope proves all asserted clauses.

∎

Proof of Theorem 11.

Let $s_{n} = n^{4/3}, x_{n} = x_{n} (c) = s_{n} d_{n} (c), d_{n} (c) = \frac{C _{0} ( c )}{n} - ρ_{n} (c),$ Since $C_{0} (c) = L_{c}^{2} /4$ and the contrast is nonzero, $C_{0} (c) > 0$ .

Eventual two-sided bounds. Theorem 5 supplies $κ_{c} > 0$ , the liminf–limsup chain $C_{0} (c) κ_{c} \leq n \to \infty lim inf x_{n} \leq n \to \infty lim sup x_{n} \leq 43 C_{0} (c),$ and, for all sufficiently large $n$ , a clipped-shrinkage rule satisfying $z \in T_{K}^{n} sup R_{n} (D^{⋆}, τ_{n}^{sh}; z) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}} .$ By the definition of $ρ_{n} (c)$ in Definition 3, $ρ_{n} (c) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}},$ and hence $d_{n} (c) \geq C_{0} (c) κ_{c} n^{- 4/3}, x_{n} = s_{n} d_{n} (c) \geq C_{0} (c) κ_{c},$ for all sufficiently large $n$ , using $s_{n} n^{- 4/3} = 1$ . On the other side, Theorem 4 gives, for every $n \geq 8$ , $d_{n} (c) \leq 43 C_{0} (c) n^{- 4/3}, x_{n} \leq 43 C_{0} (c) .$ Thus, eventually, $x_{n} \in [C_{0} (c) κ_{c}, 43 C_{0} (c)], 0 < x_{n} .$ The displayed liminf–limsup inequalities above give the first assertion of the theorem.
Subsequential limits. Put $L_{c} = {y \in R : there is a strictly increasing ϕ : N \to N with x_{ϕ (n)} \to y} .$ The eventual containment from the previous step places every tail of $(x_{n})$ in the compact interval $I_{c} = [C_{0} (c) κ_{c}, 43 C_{0} (c)] .$ Bolzano–Weierstrass applied to a tail subsequence gives $L_{c} \neq = \emptyset$ . Also $L_{c} \subseteq I_{c}$ . To see that $L_{c}$ is closed, take $y_{j} \in L_{c}$ with $y_{j} \to y$ . For each $j$ , choose an index $n_{j} > n_{j - 1}$ such that $∣ x_{n_{j}} - y_{j} ∣ < \frac{1}{j},$ possible because $y_{j}$ is a subsequential limit. Then $x_{n_{j}} \to y$ , so $y \in L_{c}$ . Hence $L_{c}$ is a closed subset of the compact interval $I_{c}$ , and is compact.
Regular-variation index and the slowly varying criterion. The index assertion is exactly the regular-variation conclusion in Theorem 5: if $a_{n} > 0$ is regularly varying with index $β$ , $C > 0$ , and $a_{n} d_{n} (c) \to C,$ then $β = \frac{4}{3} .$

It remains to identify when a normalizer of index $4/3$ exists. Suppose first that $a_{n} > 0$ , $a_{n}$ is regularly varying with index $4/3$ , $C > 0$ , and $a_{n} d_{n} (c) \to C$ . For $t > 0$ , set $p_{n} = a_{n} d_{n} (c) .$ Since $⌊ t n ⌋ \to \infty$ , $\frac{p _{⌊ t n ⌋}}{p _{n}} \to 1.$ Moreover, $\frac{s _{⌊ t n ⌋}}{s _{n}} \to t^{4/3}, \frac{a _{⌊ t n ⌋}}{a _{n}} \to t^{4/3} .$ For all sufficiently large $n$ , the denominators are nonzero, and $\frac{x _{⌊ t n ⌋}}{x _{n}} = \frac{s _{⌊ t n ⌋} d _{⌊ t n ⌋} ( c )}{s _{n} d _{n} ( c )} = \frac{s _{⌊ t n ⌋}}{s _{n}} \frac{p _{⌊ t n ⌋}}{p _{n}} (\frac{a _{⌊ t n ⌋}}{a _{n}})^{- 1} ⟶ t^{4/3} \cdot 1 \cdot t^{- 4/3} = 1.$ Together with the eventual positivity established above, this is the asserted eventual positivity and slow variation of $x_{n}$ .

Conversely, suppose $x_{n} > 0$ eventually and $\frac{x _{⌊ t n ⌋}}{x _{n}} \to 1 (t > 0) .$ Choose an integer $N_{+} \geq 1$ such that $x_{n} > 0$ for all $n \geq N_{+}$ , and define a positive sequence $b_{n}$ on all of $N$ by assigning arbitrary positive values for $n < N_{+}$ and setting $b_{n} = x_{n} (n \geq N_{+}) .$ Since $⌊ t n ⌋ \to \infty$ , the finite initial assignments are invisible in the ratio, and hence $\frac{b _{⌊ t n ⌋}}{b _{n}} \to 1,$ so $b_{n}$ is regularly varying with index $0$ . Now define a positive sequence $a_{n}$ on all of $N$ by assigning arbitrary positive values for $n < N_{+}$ and setting $a_{n} = \frac{s _{n}}{b _{n}} (n \geq N_{+}) .$ This totalization is positive because $N_{+} \geq 1$ , so $s_{n} = n^{4/3} > 0$ on the displayed tail and $b_{n} > 0$ everywhere. Since finite initial values do not affect regular variation, and since $s_{n}$ is regularly varying with index $4/3$ , the quotient $a_{n} = s_{n} / b_{n}$ on the tail is regularly varying with index $4/3$ . For all $n \geq N_{+}$ , $a_{n} d_{n} (c) = \frac{s _{n}}{x _{n}} d_{n} (c) = 1.$ Thus the desired normalizer exists, with $C = 1$ .
Exact rational grid-certificate cluster. Let $q$ be a rational contrast whose real embedding is $c$ . For each $n \geq 1$ , apply Theorem 7 with mesh $M = n$ . This gives rational certificate data $π_{n}, w_{n}, u_{n}, ν_{n}, δ_{n}$ such that the exact primal-dual certificate and barycenter identities hold, and such that, writing $B_{n}^{q} = B_{n, q} (ν_{n}), U_{n}^{q} = U_{n, n} (q; π_{n}, δ_{n}), T_{n}^{q} = u_{n} = Λ_{n, n} (q),$ one has $B_{n}^{q} \leq ρ_{n} (c) \leq U_{n}^{q} \leq T_{n}^{q} \leq B_{n}^{q} + \frac{C _{0} ( c )}{4 n ^{2}} .$ Define, with arbitrary zero-th values, $G_{n} = s_{n} {\frac{C _{0} ( c )}{n} - T_{n}^{q}}, P_{n} = s_{n} {\frac{C _{0} ( c )}{n} - U_{n}^{q}}, L_{n} = s_{n} {\frac{C _{0} ( c )}{n} - B_{n}^{q}} .$ The same certificate data make $G_{n}$ an exact rational grid-certificate cluster for $q$ . The sandwich width satisfies $0 \leq s_{n} (T_{n}^{q} - B_{n}^{q}) \leq s_{n} \frac{C _{0} ( c )}{4 n ^{2}} = \frac{C _{0} ( c )}{4} n^{- 2/3} ⟶ 0.$ Since $B_{n}^{q} \leq ρ_{n} (c) \leq T_{n}^{q},$ we get $∣ G_{n} - x_{n} ∣ \leq s_{n} (T_{n}^{q} - B_{n}^{q}) \to 0.$ Similarly, because $U_{n}^{q} \leq T_{n}^{q}$ and $B_{n}^{q} \leq U_{n}^{q}$ , $∣ P_{n} - G_{n} ∣ = s_{n} (T_{n}^{q} - U_{n}^{q}) \leq s_{n} (T_{n}^{q} - B_{n}^{q}) \to 0,$ and $∣ L_{n} - G_{n} ∣ = s_{n} (T_{n}^{q} - B_{n}^{q}) \to 0.$ These are precisely $G_{n} - x_{n} (c) \to 0, P_{n} - G_{n} \to 0, L_{n} - G_{n} \to 0.$
Transferred certificates for real contrasts. Apply Theorem 9 to the real contrast $c$ . It gives rational contrasts $c^{(n)}$ , certificate values $B_{n}, U_{n}$ , and transferred endpoints $R_{n}^{-}, R_{n}^{+}$ such that, for every arm $a$ , $c_{a}^{(n)} \to c_{a}, n^{5/6} η (c, c^{(n)}) \to 0,$ where $η (c, c^{(n)}) = \frac{1}{2} a \in A_{K} \sum ∣ c_{a} - c_{a}^{(n)} ∣.$ For every $n \geq 1$ , the transferred certificate supplies an exact rational primal-dual certificate, a grid barycenter estimator, and a rational prior for $c^{(n)}$ , with $B_{n} = the lower certificate from the prior, U_{n} = the upper certificate from the grid barycenter,$ and $R_{n}^{-} = (max {0, B_{n} - η (c, c^{(n)})})^{2}, R_{n}^{+} = (U_{n} + η (c, c^{(n)}))^{2} .$ The same transferred certificate gives $R_{n}^{-} \leq ρ_{n} (c) \leq R_{n}^{+}$ and, at mesh $M = n$ , $s_{n} (R_{n}^{+} - R_{n}^{-}) \to 0.$ Therefore $0 \leq s_{n} {\frac{C _{0} ( c )}{n} - R_{n}^{-}} - x_{n} (c) = s_{n} {ρ_{n} (c) - R_{n}^{-}} \leq s_{n} (R_{n}^{+} - R_{n}^{-}) \to 0,$ and $0 \leq x_{n} (c) - s_{n} {\frac{C _{0} ( c )}{n} - R_{n}^{+}} = s_{n} {R_{n}^{+} - ρ_{n} (c)} \leq s_{n} (R_{n}^{+} - R_{n}^{-}) \to 0.$ Thus $n^{4/3} (\frac{C _{0} ( c )}{n} - R_{n}^{-}) - x_{n} (c) \to 0, n^{4/3} (\frac{C _{0} ( c )}{n} - R_{n}^{+}) - x_{n} (c) \to 0.$
The three-arm scalar-score separation. For $K = 3$ and $c^{†} = (1, - 1/2, - 1/2)$ , Proposition 2 supplies a full observed arm-label-and-outcome estimator $τ$ under the product $q^{⋆}$ design $D^{⋆}$ whose worst-case risk over $T_{3}^{3}$ is $\frac{511653}{4000000} .$ The same result identifies the three-observation scalar sign-score minimax value as $V_{3}^{Z} = 1 - \frac{3}{2}$ and gives the strict numerical chain $\frac{511653}{4000000} < \frac{18213}{136000} < 1 - \frac{3}{2} .$ Consequently $z \in T_{3}^{3} sup R_{3} (D^{⋆}, τ; z) < V_{3}^{Z},$ as claimed.

∎

Proof of Theorem 12.

Suppose first that $n \geq 1$ and $∣ S_{c} ∣ = 2$ . Choose the two active arms $a_{+} (c)$ and $a_{-} (c)$ so that $c_{a_{+} (c)} = α_{c}, c_{a_{-} (c)} = - α_{c}, α_{c} = \frac{L _{c}}{2},$ and hence $α_{c}^{2} = C_{0} (c) .$ For $z_{K} = (t_{i})_{i = 1}^{n} \in T_{K}^{n}$ , define the active two-arm schedule $act_{c} (z_{K})_{i} = (t_{i, a_{+} (c)}, t_{i, a_{-} (c)}) \in T_{2} .$ For $z_{2} = (s_{i})_{i = 1}^{n} \in T_{2}^{n}$ , define the sign-schedule embedding $E_{c} (z_{2}) \in T_{K}^{n}$ by $E_{c} (z_{2})_{i, a_{+} (c)} = s_{i, 1}, E_{c} (z_{2})_{i, a_{-} (c)} = s_{i, 2}, E_{c} (z_{2})_{i, a} = 0 (a \in / S_{c}) .$ Then $τ_{c} (z_{K}) = α_{c} τ_{(1, - 1)} {act_{c} (z_{K})}, τ_{c} {E_{c} (z_{2})} = α_{c} τ_{(1, - 1)} (z_{2}) .$ By Theorem 4, $ρ_{n} (c) = C_{0} (c) ρ_{n, 2} .$
Apply Theorem 1 with $K = 2$ and contrast $(1, - 1)$ . It gives a two-arm orbit procedure $q_{2}$ and a prior $ν_{2}$ on $M_{n, 2}$ such that $R_{(1, - 1)}^{orb} (q_{2}; m) \leq G_{n} ((1, - 1)) (m \in M_{n, 2}),$ and, for every two-arm orbit procedure $q_{2}^{'}$ , $G_{n} ((1, - 1)) \leq E_{ν_{2}} [R_{(1, - 1)}^{orb} (q_{2}^{'}; m)] .$ The same result identifies $G_{n} ((1, - 1)) = ρ_{n, 2}$ . Let $p_{2}$ be the invariant labeled procedure induced by $q_{2}$ . For $m \in M_{n, 2}$ , set $Orb_{2} (m) = {z_{2} \in T_{2}^{n} : counts (z_{2}) = m},$ and define the labeled prior $prior_{2} (z_{2}) = \frac{ν _{2} ( counts ( z _{2} ))}{∣ Orb _{2} ( counts ( z _{2} )) ∣} .$ Since $z_{2} \in Orb_{2} (counts (z_{2}))$ , the denominator is positive for every schedule in the displayed formula. The invariant correspondence gives $R_{n}^{(1, - 1)} (p_{2}; z_{2}) = R_{(1, - 1)}^{orb} (q_{2}; counts (z_{2})),$ so $R_{n}^{(1, - 1)} (p_{2}; z_{2}) \leq ρ_{n, 2} (z_{2} \in T_{2}^{n}) .$ For any two-arm labeled procedure $p_{2}^{'}$ , symmetrizing $p_{2}^{'}$ over unit permutations and passing through the invariant-orbit correspondence yields an orbit procedure whose orbit risk, averaged under $ν_{2}$ , is bounded above by the $prior_{2}$ -Bayes risk of $p_{2}^{'}$ . Therefore $ρ_{n, 2} \leq E_{prior_{2}} [R_{n}^{(1, - 1)} (p_{2}^{'}; z_{2})] .$
Let $p_{K}$ be the sign-pair lift of $p_{2}$ , and let $prior_{K} = (E_{c})_{#} prior_{2}, E_{prior_{K}} F = E_{prior_{2}} [F {E_{c} (z_{2})}] .$ The lifted experiment observes exactly the two active coordinates and multiplies the two-arm estimate by $α_{c}$ . Hence, for every $z_{K} \in T_{K}^{n}$ , $R_{n}^{c} (p_{K}; z_{K}) = α_{c}^{2} R_{n}^{(1, - 1)} (p_{2}; act_{c} (z_{K})) \leq C_{0} (c) ρ_{n, 2} = ρ_{n} (c) .$ Now fix an arbitrary $K$ -arm labeled procedure $p_{K}^{'} = (D_{K}^{'}, τ_{K}^{'})$ . Define the induced two-arm procedure $p_{2 \leftarrow K}^{'}$ by the following Rao–Blackwellization. Let $D_{K}^{'}$ be the product experiment in which $a \in A_{K}^{n}$ is drawn from $D_{K}^{'}$ and, independently, $ξ_{i} \in {0, 1}$ is fair for each unit. Define the sign-group assignment map $ϕ_{c} : A_{K}^{n} \times {0, 1}^{n} \to A_{2}^{n}$ by $ϕ_{c} (a, ξ)_{i} = ⎩ ⎨ ⎧ 1, 2, 1, 2, c_{a_{i}} > 0, c_{a_{i}} < 0, c_{a_{i}} = 0 and ξ_{i} = 1, c_{a_{i}} = 0 and ξ_{i} = 0.$ Thus the fair bits are used exactly on zero-coefficient assignments. Define the inactive-zero pseudo-observation by $y_{c} (a, y)_{i} = {y_{i}, 0, c_{a_{i}} \neq = 0, c_{a_{i}} = 0,$ where $0$ is the binary response value assigned to inactive coordinates. For a two-arm observed vector $y$ , set $ζ_{c}^{'} (a, ξ; y) = α_{c}^{- 1} τ_{K}^{'} {a, y_{c} (a, y)} .$ Here $α_{c} = L_{c} /2$ and $α_{c} > 0$ , while the $K$ -arm estimator is clipped to $[- L_{c} /2, L_{c} /2] = [- α_{c}, α_{c}]$ . Hence $ζ_{c}^{'} (a, ξ; y) \in [- 1, 1] .$ The induced two-arm procedure $p_{2 \leftarrow K}^{'}$ has assignment law $(ϕ_{c})_{#} D_{K}^{'}$ and estimator $τ_{2 \leftarrow K}^{'} (b, y) = {E_{D_{K}^{'}} [ζ_{c}^{'} (a, ξ; y) ∣ ϕ_{c} (a, ξ) = b], 0, P_{D_{K}^{'}} {ϕ_{c} = b} > 0, P_{D_{K}^{'}} {ϕ_{c} = b} = 0.$ The displayed interval containment makes this a legitimate two-arm procedure. For a fixed $z_{2} \in T_{2}^{n}$ , write $Y_{z_{2}} (b)$ for the observed outcome vector under two-arm assignment $b$ . Conditional Jensen, applied on each fiber of $ϕ_{c}$ , gives $R_{n}^{(1, - 1)} (p_{2 \leftarrow K}^{'}; z_{2}) \leq E_{D_{K}^{'}} [{ζ_{c}^{'} (a, ξ; Y_{z_{2}} (ϕ_{c} (a, ξ))) - τ_{(1, - 1)} (z_{2})}^{2}] .$ For every $(a, ξ)$ , the sign-schedule embedding and inactive-zero convention give $y_{c} {a, Y_{z_{2}} (ϕ_{c} (a, ξ))} = Y_{E_{c} (z_{2})} (a) .$ Indeed, a positive-coefficient assigned arm reads the first two-arm coordinate, a negative-coefficient assigned arm reads the second, and an inactive assigned arm has binary response value $0$ on both sides: $E_{c} (z_{2})$ fixes inactive coordinates at $0$ , while $y_{c}$ replaces inactive observations by $0$ . Also, $τ_{c} {E_{c} (z_{2})} = \frac{1}{n} i = 1 \sum n (a : c_{a} > 0 \sum c_{a} (z_{2})_{i, 1} + a : c_{a} < 0 \sum c_{a} (z_{2})_{i, 2}) = \frac{1}{n} i = 1 \sum n (α_{c} (z_{2})_{i, 1} - α_{c} (z_{2})_{i, 2}) = α_{c} τ_{(1, - 1)} (z_{2}),$ where the middle equality uses $\sum_{a : c_{a} > 0} c_{a} = α_{c}$ and $\sum_{a : c_{a} < 0} c_{a} = - α_{c}$ . Substituting these two identities into the preceding Jensen bound yields $R_{n}^{(1, - 1)} (p_{2 \leftarrow K}^{'}; z_{2}) \leq E_{D_{K}^{'}} [α_{c}^{- 2} {τ_{K}^{'} (a, Y_{E_{c} (z_{2})} (a)) - τ_{c} {E_{c} (z_{2})}}^{2}] = α_{c}^{- 2} R_{n}^{c} (p_{K}^{'}; E_{c} (z_{2})),$ since $D_{K}^{'}$ has $D_{K}^{'}$ as its $a$ -marginal. Integrating this inequality under $prior_{2}$ and using the two-arm Bayes lower bound from the preceding step gives $ρ_{n, 2} \leq E_{prior_{2}} [R_{n}^{(1, - 1)} (p_{2 \leftarrow K}^{'}; z_{2})] \leq α_{c}^{- 2} E_{prior_{K}} [R_{n}^{c} (p_{K}^{'}; z_{K})] .$ Multiplying by $α_{c}^{2} = C_{0} (c)$ and using $ρ_{n} (c) = C_{0} (c) ρ_{n, 2}$ yields $ρ_{n} (c) \leq E_{prior_{K}} [R_{n}^{c} (p_{K}^{'}; z_{K})] .$ Together with the definitions of $p_{2}, prior_{2}, p_{K}, prior_{K}$ , this proves the two-active-arm value and saddle-transfer assertion.
For arbitrary active support, Theorem 5 supplies $C_{0} (c) κ_{c} \leq n \to \infty lim inf n^{4/3} d_{n} (c) \leq n \to \infty lim sup n^{4/3} d_{n} (c) \leq 43 C_{0} (c) .$ It also supplies an integer $N$ such that, for every $n \geq N$ , the clipped-shrinkage procedure satisfies the contrast- $c$ risk bound $z \in T_{K}^{n} sup R_{n}^{c} (D^{⋆}, τ_{n}^{sh}; z) \leq C_{0} (c) {n^{- 1} - κ_{c} n^{- 4/3}} .$ These are exactly the second-order scale and shrinkage-attainment assertions.
By Theorem 9, for every rational nonzero zero-sum contrast $q$ and every $n, M \geq 1$ , there exist real numbers $B, U, R^{-}, R^{+}$ forming a transferred certificate for $(n, M, c, q)$ . The same result supplies rational contrasts $c^{(n)}$ and real sequences $B_{n}, U_{n}, R_{n}^{-}, R_{n}^{+}$ such that $c_{a}^{(n)} \to c_{a} (a \in A_{K}), n^{5/6} η (c, c^{(n)}) \to 0,$ for every $n \geq 1$ , the tuple $(B_{n}, U_{n}, R_{n}^{-}, R_{n}^{+})$ is a transferred certificate for $(n, n, c, c^{(n)})$ , and $n^{4/3} (R_{n}^{+} - R_{n}^{-}) \to 0.$ Finally, if $q$ represents $c$ coefficientwise, the exact rational part of the same result gives, for every $n \geq 1$ , a transferred certificate for $(n, n, c, q)$ satisfying $U - B \leq \frac{C _{0} ( c )}{4 n ^{2}} .$ This proves all real-contrast transfer certificate assertions.
Apply Theorem 10 with the given $n, M \geq 1$ , and write the objects it supplies as $π, w, u, ν, δ$ . For these, $B_{n} (ν)$ is the lower certificate formed from the response-count prior $ν$ , and $U_{n, M}$ is the upper certificate formed from $π$ and $δ$ . The cited result gives the exact primal-dual certificate and barycenter conditions for $c^{†}$ , and its returned inequalities include $B_{n} (ν) \leq ρ_{n}^{†}, ρ_{n}^{†} \leq U_{n, M}, U_{n, M} \leq B_{n} (ν) + \frac{1}{4 M ^{2}} .$ These are exactly the displayed sandwich in the three-arm rational certificate assertion.

∎

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Formal layer — Lean correspondence

Every object below is formalized in Lean 4. Click a verified row to view its statement; the rest link to the Lean source.

Setups (4)

Setup S-1 — A fixed finite population of $n$ labeled units with $K\ge2$ treatment arms $\mathcal A_K=\{1,\ldots,K\}$ and fixed binary potential outcomes. Nature fixes a complete labeled response schedule $z=(t_1,\ldots,t_n)\in\mathcal T_K^n$ , $\mathcal T_K=\{0,1\}^{\mathcal A_K}$ , and $Y_i(a)=t_{i,a}$ . The experimenter chooses, before the schedule is revealed, an arbitrary assignment law $\mathcal D\in\Delta(\mathcal A_K^n)$ — dependence across units and zero allocation probabilities permitted — and observes $Y_i^{\mathrm{obs}}=Y_i(A_i)$ . Assignment is the only source of randomness. For a fixed nonzero zero-sum contrast $c\in\mathbb R^K$ with $\sum_a c_a=0$ , support $S_c=\{a:c_a\ne0\}$ and $\ell_1$ norm $L_c=\sum_a|c_a|$ , the target is $\tau_c(z)=n^{-1}\sum_i\sum_a c_at_{i,a}\in[-L_c/2,L_c/2]$ . An estimator $\widehat\tau$ is an arbitrary, possibly biased map of $(A_i,Y_i^{\mathrm{obs}})_{i=1}^n$ clipped to $[-L_c/2,L_c/2]$ ; the loss is design-based squared error $R_n(\mathcal D,\widehat\tau;z)=\mathbb E_{\mathcal D}[(\widehat\tau-\tau_c(z))^2]$ and the value is $\rho_n(c)=\inf_{\mathcal D,\widehat\tau}\max_{z}R_n$ . The first-order constant is $C_0(c)=L_c^2/4$ , the contrast-weighted allocation is $q_a^\star=|c_a|/L_c$ on $S_c$ , and the second-order sequence is $d_n(c)=C_0(c)/n-\rho_n(c)$ . No exposure-positivity, overlap, rank, or inverse condition is imposed on the unrestricted game; only the displayed independent upper procedures use $q_a^\star>0$ on active arms. — This is EXACTLY the abstraction of the Causalean design-based substrate: FiniteDesign Ω is a pmf on a finite assignment space, and here Ω := Fin n → Fin K. Reused verbatim: FiniteDesign (DesignCore.lean:46), FiniteDesign.E/Var/Cov and the whole linearity/Var_linear_comb algebra, FiniteDesign.mse/bias/mse_eq_var_add_bias_sq/var_le_mse (Risk.lean:42-85) for R_n, and Causalean.Stat.worstCaseRisk for max_z together with Causalean.Stat.minimaxValue and le_minimaxValue, minimaxValue_le_worstCaseRisk_of_nonneg, minimaxValue_le_minimaxValue, minimaxValue_subtype_eq_sInf_setOf (Stat/Minimax/MinimaxValue.lean:50,59,136,145,198) for the inf-over-(design,estimator) layer of rho_n — checked with lean_hover_info/source read: worstCaseRisk (risk : E -> Theta -> R) (e : E) = iSup_theta risk e theta and minimaxValue risk = iInf_e worstCaseRisk risk e quantify over an ARBITRARY estimator index type E, so instantiating E := Procedure K n c (design paired with clipped estimator) and Theta := Schedule K n is an exact fit for the unrestricted game. EXPLICITLY NOT REUSED (corrected in round 3, previously claimed here in error): Causalean.Experimentation.DesignBased.{worstRisk, le_worstRisk, IsMinimaxOn, exists_isMinimaxOn} (Optimality/Minimax.lean:40-66, Optimality.lean:43,75). Two independent signature misfits, confirmed by reading the source: (i) their risk criterion has type Y -> FiniteDesign Omega -> alpha, i.e. it is indexed by the DESIGN ALONE, so it cannot express R_n(D, tauHat; z) which is jointly indexed by the design and the estimator; (ii) exists_isMinimaxOn additionally requires (hfin : D.Finite) (hne : D.Nonempty) on a DesignFamily Omega = Set (FiniteDesign Omega) and merely inherits exists_isOptimalOn's finite-Finset.min' argument, whereas this paper's game ranges over the UNCOUNTABLE set of all assignment laws on A_K^n paired with all clipped estimators. It therefore cannot deliver either the minimax-procedure attainment or the saddle/least-favourable-prior clause of thm:exact-response-type-game. Those two clauses are delivered instead by the planned compact-game construction in Helpers/FiniteGame.lean (attainable upper-risk set U — nonempty, compact, convex, upper-comprehensive; Euclidean projection of (V - eps) * 1 onto U; normalized separating vector giving mu^*), exactly as the note's own proof_tex argues, and the whole Optimality/Optimality.Minimax pair is dropped from this S-block's modules. prodDesign (Product.lean:47) + FiniteDesign.Var_prod_linear_comb (ProductVariance.lean:90) give the independent q^\star design and its exact variance decomposition for free. Search trace: npm run search -- "finite population design based minimax worst case risk squared error" --cluster experimentation; read ../doc/API.md §18 (lines 10264-10420) and §10z (10519-10696); grepped Causalean/Experimentation/DesignBased/ for worstRisk|mse|prodDesign|minimax; lean_local_search prodDesign / Var_prod_linear_comb / minimaxRisk. No bypass: the paper's world is literally the finite design-based world, so scaffolding over Mathlib alone would be an unjustified bypass. NOT reused: Exposure.*/htTotal/htMean/htEffect/EdgeVarianceBound.*/IndepSummandsCLT.*/DepGraphCLT.*/LocalDependenceCLT.*/GaussianCDF.*/SteinMethod.* from the brief's reuse list — bypass-justified: this paper has no exposure mapping, no interference/dependency graph, no CLT/Wald leg, and its estimator is a clipped (hence biased) shrinkage rule rather than an HT/Hajek estimator, so the HT-unbiasedness and asymptotic-normality layers are at the wrong abstraction. E_compound_tower/Var_compound_eq_tower/chebyshev/var_htMean_le likewise unused: bypass-justified (no two-stage design; the tail step needs a sub-Gaussian, not a Chebyshev, bound).
Setup S-2 — The unit-permutation orbit (symmetrized) form of the same experiment. $\mathfrak S_n$ acts simultaneously on schedules, assignments, and observed outcome vectors. Schedule orbits are the response-type count vectors $m=(m_t)_{t\in\mathcal T_K}\in\mathcal M_{n,K}=\{m\in\mathbb Z_+^{\mathcal T_K}:\sum_t m_t=n\}$ , with $\tau_c(m)=n^{-1}\sum_t m_t\sum_a c_at_a$ . Assignment orbits are the allocation counts $r\in\mathcal R_{n,K}=\{r\in\mathbb Z_+^{\mathcal A_K}:\sum_a r_a=n\}$ , so an invariant design is exactly a mixture $\pi\in\Delta(\mathcal R_{n,K})$ of uniform assignments conditional on $r$ . Observed-data orbits are the pairs $(r,x)$ with arm-success counts $x_a=\#\{i:A_i=a,\;Y_i^{\mathrm{obs}}=1\}$ , and an invariant estimator is a map $\delta(r,x)\in[-L_c/2,L_c/2]$ . Writing $h=(h_{t,a})$ for the response-type-by-arm contingency table and $\mathcal H(m,r,x)$ for its feasible fiber, the orbit likelihood is $P_m(x\mid r)=\bigl(\prod_a r_a!/n!\bigr)\sum_{h\in\mathcal H(m,r,x)}\prod_t m_t!/\prod_a h_{t,a}!$ , and the finite $2^K$ -response-type game value is $G_n(c)=\inf_{\pi,\delta}\max_{m\in\mathcal M_{n,K}}\sum_r\pi_r\sum_x P_m(x\mid r)\{\delta(r,x)-\tau_c(m)\}^2$ . All state, action, and observation sets are finite; zero allocation counts and the all-zero and all-one response types are retained. — Genuinely new: no Causalean or Mathlib module carries a response-type orbit experiment, a contingency-table fiber, or an orbit likelihood. Search trace: npm run search -- "response type orbit likelihood contingency table fiber permutation symmetrization finite population" --cluster experimentation and --type def -> nothing; grepped Causalean/Experimentation/ and Causalean/Stat/Minimax/ for orbit|contingency|responseType|countVec|symmetriz -> nothing; lean_local_search orbitLik / responseType / allocationCount -> empty. Mathlib-side check: lean_loogle "Nat.multinomial" -> Nat.multinomial in Mathlib.Data.Nat.Choose.Multinomial (reused for the fiber count), Mathlib.Data.Sym.Card (Sym.card_sym_eq_multichoose, reused for the stars-and-bars cardinalities |M_{n,K}| = C(n+2^K-1, 2^K-1), |R_{n,K}| = C(n+K-1,K-1)); lean_leansearch "minimax theorem concave-convex compact convex sets" -> Mathlib.Order.SaddlePoint (IsSaddlePointOn, isSaddlePointOn_value, iSup2_iInf2_le_iInf2_iSup2) reused for the saddle statement itself, but Mathlib carries no Sion/von-Neumann existence theorem, so the paper's own projection-and-separation existence argument is built inline in Helpers/FiniteGame.lean. The orbit-count simplexes are encoded as Finsets over Fintype.piFinset (fun _ => Finset.range (n+1)) filtered by the sum constraint; pi and delta live in a compact convex subset of a finite-dimensional Euclidean space so Mathlib's convex-projection layer (Mathlib.Analysis.InnerProductSpace.Convex) supplies the separation step.
Setup S-3 — The rational grid-action linear program attached to the orbit experiment. For a rational nonzero zero-sum contrast $c\in\mathbb Q^K$ put $h_c=L_c/2$ and let the estimator action grid be $\Gamma_{M,c}=\{-h_c+jh_c/M:j=0,\ldots,2M\}$ (mesh $h_c/M$ , containing both endpoints and $0$ ); for the three-arm diagnostic contrast $c^\dagger=(1,-1/2,-1/2)$ , $h_{c^\dagger}=1$ and $\Gamma_{M,c^\dagger}=\Gamma_M=\{-1+j/M\}$ . The program minimizes the epigraph variable $u\ge0$ over $\pi_r\ge0$ and joint design-action weights $w_{r,x,g}\ge0$ subject to $\sum_r\pi_r=1$ , $\sum_g w_{r,x,g}=\pi_r$ for every $(r,x)$ , and $\sum_{r,x,g}P_m(x\mid r)w_{r,x,g}\{g-\tau_c(m)\}^2\le u$ for every $m\in\mathcal M_{n,K}$ . Its value is $\Lambda_{n,M}(c)$ , and $\Lambda_{n,M}^\dagger=\Lambda_{n,M}(c^\dagger)$ with $\rho_n^\dagger=\rho_n(c^\dagger)$ . Every index set is finite and, for rational $c$ , every objective and constraint coefficient is rational. — ROUND-10 CORRECTION (FLAG-unbounded-LP) — READ FIRST; IT SUPERSEDES EVERY SENTENCE IN THIS WORLD BLOCK AND IN THE LP NODES THAT TREATS u <= h_c^2 (OR u <= 1 AT c_dagger) AS A CONSEQUENCE OF FEASIBILITY, OR THAT CLAIMS THE LP'S FEASIBLE SET IS A COMPACT POLYTOPE WITH AN OPTIMUM SUPPLIED BY IsCompact.exists_isMinOn. THE DEFECT: in the encoded program the epigraph variable u appears ONLY through the m-indexed risk inequalities sum_{r,x,g} P_m(x|r) w_{r,x,g} {g - tau_c(m)}^2 <= u, which are monotone in u; hence (pi, w, u) feasible implies (pi, w, u') feasible for every u' >= u, and the feasible set is UNBOUNDED ABOVE in u. The explicit feasible point (point-mass pi, all action mass on g = 0, u = h_c^2) therefore bounds the OPTIMAL VALUE — Lambda_{n,M}(c) <= h_c^2 — and NOT an arbitrary feasible point. Only the lower side is a feasibility consequence: 0 <= u at every feasible point, by gridLP_u_nonneg (fix any m_0 in the nonempty M_{n,K}; its risk row bounds u below by a nonnegative combination of the w's, which are nonnegative by their retained rows) — this is the round-9 argument and it is unchanged. WHAT THIS WORLD NOW RECORDS, primary route: NO real-compactness claim is made anywhere. gridLPValue K n M c : R is the sInf of the real feasible objective values, bounded below by 0, and its ATTAINMENT is algebraic, from the pair returned by the landed 0-sorry CausalSmith.RationalLP.exists_rational_optimal_primal_dual, through two named lemmas that Helpers/RationalLPBridge.lean must prove: gridLPValue_le_rat — the returned rational optimizer, coerced coordinatewise into R, is a real feasible point with objective (gridLPValueRat : R), so csInf_le gives gridLPValue <= (gridLPValueRat : R); and gridLP_dual_bound_real — the returned rational nonnegative multipliers yStar bound EVERY REAL feasible point, since for real feasible x, dot c x = - sum_j (sum_i y_i * A i j) * x_j = - sum_i y_i * (dot (A i) x) >= - sum_i y_i * b_i = dualObjective yStar = objective xStar, using only DualFeasible's nonnegativity and stationarity conjuncts and PrimalFeasible's row inequalities. Together: gridLPValue = (gridLPValueRat : R), with the infimum attained at the coerced rational optimizer. This is strictly stronger than the withdrawn compactness route, which would have left the real-versus-rational optimum gap open (the landed theorem's optimality clause quantifies over RATIONAL feasible points only). RECORDED FALLBACK, topological: restrict to the objective sublevel S = {feasible and u <= h_c^2} and prove the restriction preserves the infimum. S is closed and bounded — pi in the unit simplex, 0 <= w_{r,x,g} <= pi_r <= 1 from the occupancy equalities plus nonnegativity, and 0 <= u <= h_c^2 BY THE RESTRICTION rather than by feasibility — hence compact, so IsCompact.exists_isMinOn applies ON S; and gridLPValue_eq_sInf_sublevel holds because the explicit feasible point lies in S with objective h_c^2, so every discarded feasible point has objective u > h_c^2 and cannot lower the infimum. IsCompact.exists_isMinOn (Mathlib/Topology/Order/Compact.lean, re-confirmed by lean_local_search) is thus retained as a Mathlib reuse with scope RESTRICTED TO S. BINDING PROHIBITION FOR F2/F3: no declaration may state or use u <= h_c^2 (or u <= 1 at c_dagger) as a consequence of GridLPFeasible, and no compactness or IsCompact.exists_isMinOn step may be taken on the full feasible set. EVERYTHING ELSE IN THIS BLOCK STANDS: the round-9 sign-row-free row set and its hbdd/hne discharges (the hne witness is unchanged — only the false global reading of its u value is withdrawn), dualRiskRow_nonneg / dualRiskRow_sum_eq_one as the only permitted source of nu in Delta(M_{n,K}), the deleted RationalLPExactCertificate gate, the empty external_substrate_assumptions, and the demotion of Causalean.PartialID.ConicProgram to an optional real-valued analogue (note that its strong_duality_primal_attained carries its own constraint-qualification binders and was never, and is still not, a substitute for the algebraic route above). SUPERSEDED ROUND-9 TEXT FOLLOWS, retained for the audit trail and still in force on the dual normalization. ROUND-9 CORRECTION (FLAG-DUAL-NORMALIZATION) — IT SUPERSEDES THE ROUND-8 ENCODING AND EVERY 'nu ... normalized by the epigraph stationarity' SENTENCE IN THIS BLOCK AND IN THE LP NODES. THE WITHDRAWN CLAIM: round 8 said the core's response-count prior nu_{n,M,c} in Delta(M_{n,K}) is the restriction of the returned yStar to the risk rows, 'normalized by the epigraph stationarity sum_m nu_m = 1, which is the j = u coordinate of DualFeasible'. WITH THE ROUND-8 ROW SET THAT IS FALSE. Recompute it against the signature read at source (Program.DualFeasible y : (forall i, 0 <= y i) and (forall j, sum_i y i * A i j = -c j), RationalLP.lean:715): the objective is minimize u, so c u = 1 and c j = 0 at every other variable; each risk row sum_{r,x,g} P_m(x|r) w_{r,x,g} {g - tau_c(m)}^2 - u <= 0 has A m u = -1; the round-8 row set ALSO contained an explicit u >= 0 row, encoded as -u <= 0, whose u-coefficient is likewise -1; the sum_r pi_r = 1 and sum_g w_{r,x,g} = pi_r equality pairs and the pi/w nonnegativity rows all have u-coefficient 0. Stationarity at j = u therefore reads -(sum_m y_m) - y_(u>=0) = -1, i.e. sum_m y_m + y_(u>=0) = 1, so the risk-row restriction is a SUB-probability vector sum_m nu_m = 1 - y_(u>=0) <= 1, and nothing in the theorem's conclusion forces y_(u>=0) = 0. The signature alone does not deliver nu in Delta(M_{n,K}). THE FIX, PRIMARY ROUTE, adopted by this plan and to be implemented in Helpers/RationalLPBridge.lean: ENCODE THE PROGRAM WITHOUT THE REDUNDANT u >= 0 ROW. The row Fintype is then exactly (i) sum_r pi_r = 1 as two inequalities, (ii) each sum_g w_{r,x,g} = pi_r as two inequalities, one pair per (r,x), (iii) one nonnegativity row -pi_r <= 0 per r and -w_{r,x,g} <= 0 per (r,x,g), and (iv) one risk row per m in M_{n,K} — NO row constrains the sign of u. The risk rows are then the ONLY rows with a nonzero u-coefficient, each equal to -1, so stationarity at j = u gives sum_m y_m = 1 EXACTLY, and together with DualFeasible's nonnegativity conjunct nu_m := y_m IS a probability vector on M_{n,K}. nu in Delta(M_{n,K}) is thereby DERIVED from the landed conclusion, by an explicit named lemma, and not read off from an under-determined stationarity identity. RE-DISCHARGING hbdd WITHOUT THAT ROW (this is the step the deleted row used to do, and it must not be skipped): M_{n,K} is nonempty for every n — take the count vector placing all n units on a single response type — so fix any m_0 in it; its risk row gives sum_{r,x,g} P_{m_0}(x|r) w_{r,x,g} {g - tau_c(m_0)}^2 <= u for every primal-feasible point; every coefficient P_{m_0}(x|r) {g - tau_c(m_0)}^2 is nonnegative and every w_{r,x,g} is nonnegative by its own RETAINED nonnegativity row, so the left side is >= 0 and hence 0 <= u = objective. hbdd therefore holds with l = 0, proved FROM THE RISK ROWS. hne is unchanged — the explicit feasible point (point-mass pi on any allocation orbit, all action mass on g = 0, u = h_c^2) satisfies every retained row — and both hypotheses remain proved build-inline and unconditionally on def:rational-contrast-grid-lp. FAITHFULNESS OF THE DELETION: the core's symbol table declares u in R_+, and the core's own program text lists only pi_r >= 0 and w_{r,x,g} >= 0 among its sign constraints. The deletion is faithful because 0 <= u is DERIVED on the reduced feasible set by exactly the argument just given (emitted as the lemma gridLP_u_nonneg), so the reduced and round-8 programs have the SAME feasible objective set and the SAME value; gridLPValue / gridLPValueRat are unchanged, and the core's Lambda_{n,M}(c) in Q_+ becomes a proved nonnegativity rather than an imposed constraint. The Prop GridLPFeasible must be restated to drop u >= 0 from its conjuncts so that the transport lemma GridLPFeasible ... <-> (gridProgram ...).PrimalFeasible (encode ...) stays a genuine iff. FALLBACK ROUTE, recorded in case F3 prefers to keep the sign row: keep it and prove COMPLEMENTARITY plus STRICT POSITIVITY of the optimal epigraph value. Complementarity is derivable from the returned conclusion alone — from objective xStar = dualObjective yStar and stationarity, sum_i y_i (b_i - dot (A i) xStar) = sum_i y_i b_i + dot c xStar = -dualObjective yStar + objective xStar = 0, and every summand is nonnegative by DualFeasible.1 and PrimalFeasible, hence each is zero — so y_(u>=0) > 0 would force its slack 0 - (-u^*) = u^* to vanish, i.e. Lambda_{n,M}(c) = 0. The route therefore closes if and only if one additionally PROVES Lambda_{n,M}(c) > 0, which is true (any two response-count vectors with distinct targets cannot both have zero risk under one procedure) but is extra work the primary route avoids entirely. BINDING PROHIBITION FOR F2/F3: nu in Delta(M_{n,K}) may NOT be asserted as a by-product of applying exists_rational_optimal_primal_dual. It must come from a named lemma in Helpers/RationalLPBridge.lean — dualRiskRow_nonneg and dualRiskRow_sum_eq_one, the latter proved from the j = u stationarity coordinate OF THE ROW SET THAT HAS NO SIGN ROW ON u (or, on the fallback route, from complementarity plus Lambda > 0). Everything else in the round-8 discharge stands unchanged: the gate RationalLPExactCertificate is still deleted, external_substrate_assumptions is still empty, no declaration takes an (hLP : ...) binder, all six certificate consumers are still unconditional users of the landed 0-sorry CausalSmith.RationalLP.exists_rational_optimal_primal_dual, and infrastructure_needed[I-1] stays dropped — this round corrects HOW nu is obtained from that theorem, not WHETHER the theorem discharges the gate. SUPERSEDED ROUND-8 TEXT FOLLOWS, retained for the audit trail; every sentence in it about the u >= 0 row (as the source of hbdd, or as harmless to the dual) and about normalizing nu by stationarity is withdrawn by the paragraph above, while its signature record, its index confirmations and its gate-deletion record remain accurate and in force. ROUND-8 DISCHARGE (F2.5 fix-locus triage, after verified substrate landing) — READ FIRST. IT SUPERSEDES EVERY RationalLPExactCertificate PARAGRAPH BELOW AND EVERY GATE PARAGRAPH IN THE LP NODES. The exact-rational LP certificate theory is NO LONGER external debt and is NO LONGER a gate: it has LANDED in this repo as CausalSmith/Mathlib/Optimization/RationalLP.lean (module CausalSmith.Mathlib.Optimization.RationalLP, namespace CausalSmith.RationalLP), 0-sorry (grepped the file for sorry|admit|native_decide -> 0 matches) and present in the declaration index (lean_local_search exists_rational_optimal_primal_dual -> CausalSmith.RationalLP.exists_rational_optimal_primal_dual; lean_local_search exists_rational_primal_optimizer -> CausalSmith.RationalLP.exists_rational_primal_optimizer). SIGNATURES READ AT SOURCE, not from an index: RationalLP.Program (iota) (n) bundles A : iota -> Fin n -> Q, b : iota -> Q, c : Fin n -> Q (RationalLP.lean:234-240) with Program.PrimalFeasible x : forall i, dot (A i) x <= b i (245), Program.objective x = dot c x (249), Program.DualFeasible y : (forall i, 0 <= y i) and (forall j, sum_i y i * A i j = -c j) (715), Program.dualObjective y = -(sum_i y i * b i) (720); theorem exists_rational_optimal_primal_dual (P : Program iota n) (hne : exists x, P.PrimalFeasible x) (hbdd : exists l : Q, forall x, P.PrimalFeasible x -> l <= P.objective x) : exists (xStar : Fin n -> Q) (yStar : iota -> Q), P.PrimalFeasible xStar and P.DualFeasible yStar and P.objective xStar = P.dualObjective yStar and forall x, P.PrimalFeasible x -> P.objective xStar <= P.objective x (728-734), with [Fintype iota] [DecidableEq iota]; theorem exists_rational_primal_optimizer has the same two hypotheses and the primal half of the conclusion (374-378). The module docstring fixes the standard form as: minimize c . x subject to A i . x <= b i, variables UNRESTRICTED IN SIGN, with equalities and nonnegativity represented by pairs of inequalities — which is exactly how this world's program presents sum_r pi_r = 1, sum_g w_{r,x,g} = pi_r, pi_r >= 0, w_{r,x,g} >= 0, u >= 0 and the m-indexed risk rows. CONSEQUENCES FOR THIS PLAN, all applied in this round: (1) the tracked external object ext:rational-lp-exact-certificate is DELETED — external_substrate_assumptions is now empty, no RationalLPExactCertificate Prop is declared in Basic.lean, no declaration anywhere takes an (hLP : ...) binder, no _of_rationalLPExactCertificate declaration is emitted, and every node's external_hyps / external_hyps_scope / external_declares / external_depends_on field is removed; (2) all six former certificate consumers (thm:rational-contrast-grid-certificate-sandwich, prop:k3-lp-certificate, thm:k3-grid-certificate-sandwich, thm:real-contrast-grid-certificate-transfer, thm:second-order-rate-and-certificate-frontier, thm:attainment-and-k3-certified-converse) are planned as UNCONDITIONAL users of the landed theorem, with their exact-rational existence clauses now PROVED; (3) infrastructure_needed[I-1] is dropped. THE ONLY REMAINING WORK, and it is build-inline (medium), is the ENCODING BRIDGE, planned as the new helper Helpers/RationalLPBridge.lean: index the grid program's variables (u, (pi_r)_r, (w_{r,x,g})_{r,x,g}) by a Fintype and transport to Fin N through Fintype.equivFin; index its rows (one sum_r pi_r = 1 equality as two inequalities, one sum_g w = pi_r equality per (r,x) as two, one nonnegativity row per variable, one risk row per m in M_{n,K}) by a Fintype; read off A, b, c from orbitLik (Q-valued by construction) and tauC (Q-valued for a rational contrast); then discharge hne with the explicit feasible point (pi a point mass on any allocation orbit, all action mass on g = 0, u = h_c^2) and hbdd with l = 0 (the program's own u >= 0 row), BOTH of which are already proved build-inline and unconditionally on def:rational-contrast-grid-lp. Applying exists_rational_optimal_primal_dual to that Program then yields, in one step, the rational primal optimizer, the rational nonnegative dual multipliers, equality of primal and dual objective values, and optimality — i.e. precisely the core's Lambda_{n,M}(c) in Q_+ plus exact-rational primal and dual certificates. The dual response-count multiplier nu_{n,M,c} in Delta(M_{n,K}) is then the restriction of yStar to the risk rows, normalized by the epigraph stationarity sum_m nu_m = 1 (which is the j = u coordinate of DualFeasible), so the core's nu is obtained from the landed conclusion rather than assumed. Causalean.PartialID.ConicProgram.{weak_duality, dualValue_le_primalValue, strong_duality_zero_gap, strong_duality_primal_attained, farkas} is DEMOTED to optional: it remains the analogous REAL cone-program layer F3 may consult for the real-valued dual identity Lambda_{n,M}(c) = max_nu B^grid_{n,c,M}(nu), but it is no longer load-bearing for anything, since the rational route now delivers optimality and zero gap directly. The absence traces recorded below (npm run search -- "Fourier Motzkin ...", the Causalean grep, the Mathlib linarith simplex-oracle finding) remain accurate as of the round they were run and are exactly what motivated building the module; they are retained for the audit trail and are NO LONGER a reason to gate anything. SUPERSEDED ROUND-3/5/7 TEXT FOLLOWS. Analogous-module hit: Causalean.PartialID.ConicProgram (Causalean/PO/ID/Partial/LP/ConicDuality.lean) is the repo's finite-dimensional cone-program layer and supplies weak_duality, dualValue_le_primalValue, farkas, strong_duality_primal_attained and strong_duality_zero_gap over a real inner-product space with a proper cone — this is the reference F3 should adapt for the LP dual identity Lambda_{n,M}(c) = max_nu B^grid_{n,c,M}(nu). Gap to bridge: ConicProgram is a real cone program, so it gives attainment and zero duality gap but NOT rationality of the optimum. Search trace: npm run search -- "linear program strong duality rational polyhedron Fourier Motzkin elimination certificate"; grepped Causalean/ for LinearProgram|Farkas|Fourier|Motzkin|rationalPolyhedron; lean_local_search farkas -> only Causalean.PartialID.ConicProgram.farkas; lean_local_search minimaxRisk / bayesRisk. Mathlib-side check: lean_local_search bayesRisk -> ProbabilityTheory.bayesRisk/minimaxRisk in Mathlib/Probability/Decision/Risk/* — bypass-justified, those are kernel/measure-theoretic decision problems while this game is a purely finite real-valued one over Finsets, and Mathlib carries no rational-polyhedron / Fourier-Motzkin layer at all. ROUND-3 CORRECTION — the previous plan weakened the core's exact-rational certificate claim to rational coefficients + real attainment + zero duality gap and filed the remainder as non-blocking debt; that is NOT a sound completion of the node and is withdrawn. The missing theory is now SIZED AND ROUTED as a single named external substrate gate, the Prop def RationalLPExactCertificate (declared in Basic.lean beside the cited-gate Props, listed as infrastructure_needed[I-1], effort large): for a finite linear program presented over Q — finitely many variables, finitely many rational-coefficient linear equalities and inequalities, a rational linear objective, feasible and bounded below — there exist a RATIONAL feasible primal point and RATIONAL nonnegative dual multipliers attaining a common objective value; consequently the optimal value lies in Q and both are exact certificates. This is the classical rational-LP / rational-polyhedron fact (Fourier-Motzkin elimination over Q with rational back-substitution and recorded nonnegative rational multipliers; Schrijver, Theory of Linear and Integer Programming, Sec. 7.1) invoked verbatim by the notes' own elimination paragraphs. ROUND-5 CORRECTION (FLAG-hidden-external-gate). Two changes, both aimed at the reviewer's finding that the gate existed only as an F2 signature binder. (i) IT IS NOW A TRACKED PLAN OBJECT. The gate has the plan id ext:rational-lp-exact-certificate and a first-class record in the plan's top-level external_substrate_assumptions[] — statement, Lean name, declaring file, kind, effort, buildability tag, infrastructure_needed id, discharge condition, search-trace pointer, and the EXPLICIT list of carrier nodes with the single clause each one carries — and every carrier node repeats it in a machine-readable external_hyps field with an external_hyps_scope naming that clause. The binder can therefore no longer appear in Lean while appearing nowhere in the plan. It cannot go into a node's hyps array: hyps is checked by the deterministic plan gate as a subset of the CORE's depends_on (P4, tools/src/formalization/plan/plan_gate.ts:161), and ext:rational-lp-exact-certificate is not a core node, so external_hyps is the tracked non-core equivalent and hyps stays core-only by construction rather than by omission. ROUND-7 CORRECTION (F2.5 fix-locus triage) SUPERSEDES ITEM (ii) BELOW. The scaffolder found hLP bound in emitted Lean on nodes whose plan record showed no hypothesis, so the round-5 narrowing to two carriers is WITHDRAWN: the gate is threaded into ALL SIX certificate consumers (thm:rational-contrast-grid-certificate-sandwich, prop:k3-lp-certificate, thm:k3-grid-certificate-sandwich, thm:real-contrast-grid-certificate-transfer, thm:second-order-rate-and-certificate-frontier, thm:attainment-and-k3-certified-converse), each with a non-empty external_hyps, a carried_by entry in external_substrate_assumptions[0] naming its clause and declaration, and a _of_rationalLPExactCertificate-suffixed declaration. The mathematics of item (ii) is KEPT, not discarded: it is exactly what justifies giving each of the four downstream consumers a gate-free _of_certificates companion alongside its gated headline, since their proofs use only optimality/feasibility of the handed-in objects. The binding contract F2 must follow — a declaration may bind hLP only if its node's external_hyps lists the gate, must be named for it, and must name the single clause in its docstring — is in external_substrate_assumptions[0].threading_rule, together with the reason the id cannot be placed in hyps (P4 subset-of-core-depends_on, P1 node-key equality). SUPERSEDED ROUND-5 TEXT FOLLOWS, retained for the audit trail. (ii) ITS BLAST RADIUS IS CUT FROM SIX CONSUMERS TO TWO. The core states its sandwich clauses in let (pi, w, Lambda) be such a primal certificate ... let nu be the response-count multiplier in such a dual certificate form — UNIVERSALLY QUANTIFIED over handed-in certificate objects — and the sandwich proofs (barycentre/conditional-variance identity, posterior-mean square completion, mesh bound, scale transfer) never use rationality at any step. So thm:k3-grid-certificate-sandwich, thm:real-contrast-grid-certificate-transfer, thm:second-order-rate-and-certificate-frontier and thm:attainment-and-k3-certified-converse take those objects as explicit universally quantified arguments carrying their own optimality/feasibility hypotheses and are proved with NO gate binder at all; their hypothesis class is shown nonempty by the inline REAL attainment proved unconditionally on def:rational-contrast-grid-lp, so the quantified form is not vacuous. Only the two EXISTENCE assertions carry the gate: thm:rational-contrast-grid-certificate-sandwich's opening clause "admits exact rational optimal primal and dual certificates" and prop:k3-lp-certificate's clause "choosing M = n^2 yields exact-rational primal and dual certificates". Each is emitted as a SEPARATELY NAMED corollary suffixed _of_rationalLPExactCertificate, so the conditionality is visible in the declaration name itself, in its docstring, in external_hyps, and in the tracked record — never as an unannounced premise on an otherwise unconditional statement. It is tagged SUBSTRATE-GATE and NOT crux: it is a classical, named, externally established theorem, not this paper's own content, and building it here would mean contributing a whole rational-polyhedral / FM-elimination layer (elimination over a symbolic variable count, vertex/back-substitution theory, dual-multiplier extraction) upstream — out of scope for a bounded dedicated build in this artifact. It cannot make any consumer vacuous: it constrains certificates of an independently constructed, provably feasible and bounded program. Causalean.PartialID.ConicProgram.{strong_duality_primal_attained, strong_duality_zero_gap} is retained ONLY as the real-valued attainment/zero-gap analogue F3 adapts for the real dual identity; it is explicitly NOT a discharge of the rational-certificate gate, since ConicProgram is a structure over real inner-product spaces (ConicDuality.lean:46-49) and says nothing about Q. Search trace for the gate's absence: npm run search -- "Fourier Motzkin elimination rational polyhedron linear program rational optimal vertex certificate" -> nothing; grepped Causalean/ for Fourier|Motzkin|rationalPolyhedron|LinearProgram|basicFeasibleSolution -> 4 incidental hits, all Walsh/Fourier-basis or sieve-regression prose (ML/Core/Hypothesis.lean:48, ML/Linear.lean:18, DesignBased/LocalDependenceVariance.lean:16, DesignBased/Designs/BernoulliMoments.lean:15), no LP or polyhedron layer. Mathlib-side check: lean_leansearch "linear programming duality over the rationals rational optimal solution of a rational polyhedron" -> the only rational-LP artefacts in Mathlib are Mathlib.Tactic.Linarith.SimplexAlgorithm.{findPositiveVector, extractSolution} and Linarith.CertificateOracle.simplexAlgorithm{Dense,Sparse} (Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm*), which are META-LEVEL MetaM-valued oracles returning Array Rat for linarith's own proof search — not a proved mathematical statement about rational LPs and not usable as one; the remaining hits (ProperCone.mem_dual, PointedCone.mem_dual) are ordered-ring cone duality with no rationality or optimum-attainment content.
Setup S-4 — The two-arm reduced scalar experiment. For $K=2$ and $c=(1,-1)$ a binary schedule is summarized by the effect-class counts $p_+=m_{(1,0)}$ , $p_-=m_{(0,1)}$ , and $r_0=m_{(0,0)}+m_{(1,1)}$ with $p_++p_-+r_0=n$ ; the target is $(p_+-p_-)/n$ and the two-arm unrestricted minimax risk is written $\rho_{n,2}=\rho_n((1,-1))$ . Writing $S_i=Y_i^{\mathrm{obs}}$ on arm $1$ and $S_i=1-Y_i^{\mathrm{obs}}$ on arm $2$ , the reduced observation is $X_2=\sum_iS_i$ ; for any prior on $(p_+,p_-,r_0)$ lifted to complete labeled schedules by choosing a uniform ordered partition of sizes $(p_+,p_-,r_0)$ , assigning types $(1,0)$ and $(0,1)$ on the first two blocks and an independent fair bit $H_i$ giving type $(H_i,H_i)$ on the third, one has $X_2=p_++\operatorname{Binomial}(r_0,1/2)$ conditionally on $(p_+,p_-,r_0)$ , the assignment $A$ is ancillary, and $S$ is conditionally uniform given $X_2$ with a kernel free of the parameter and of $A$ . This world carries both the paper's own internal two-arm constructions — that generic lifted prior, the smooth mixture $w_a(\theta)=a^{-1}w_0(\theta/a)$ with $w_0(u)=\tfrac{15}8(1-4u^2)^2\mathbf 1\{|u|\le1/2\}$ and $a=n^{-1/3}$ , and the resulting coarse converse $\rho_{n,2}\ge n^{-1}-43n^{-4/3}$ for $n\ge8$ — and, separately, the citation interface in which $C_A=1.617\ldots$ denotes the scalar Airy ground-state constant of the published expansion $\rho_{n,2}=n^{-1}-C_An^{-4/3}+o(n^{-4/3})$ . The aggregation to $(p_+,p_-,r_0)$ is an identity for the target only; the four counts $m_{(0,0)},m_{(0,1)},m_{(1,0)},m_{(1,1)}$ remain separately indexed in the orbit likelihood. — ROUND-9 CORRECTION (FLAG-PARTIAL-TYPEFIT) — READ FIRST; IT SUPERSEDES THE ROUND-8 RE-CONFIRMATION AND EVERY 'REUSES van_trees_inequality' SENTENCE BELOW. This world's analytic-ingredient record is corrected on one point and unchanged on the rest. WITHDRAWN: the claim that the scalar core of the two-arm converse REUSES Causalean.Stat.Limit.van_trees_inequality. That decl's regularity binders are everywhere-HasDerivAt on the prior, the target and the posterior mean (VanTreesInequality.lean:83, 88, 92-93), strictly stronger than the absolute-continuity / a.e.-derivative regularity this world's constructions advertise, so it cannot discharge the scalar core at the stated hypotheses. IN FORCE INSTEAD: the whole information inequality used by this world — general observation-dependent target AND parameter-only special case — is PROVED LOCALLY in Helpers/BayesInformation.lean at AC regularity, and thm:coarse-two-arm-minimax-lower applies its general form (its target g(theta, A, S) = theta + b(theta)(Rbar - theta) depends on the data through Rbar). The Causalean theorem is retained only as a stricter-DQM reference/subcase; Causalean.Stat.Limit.VanTreesInequality is therefore dropped from this world's modules and from the two nodes it carried, and kept only on thm:first-order-saddle as an optional corroborating route whose stronger binders that node's polynomial construction happens to satisfy. UNCHANGED AND RE-CONFIRMED: AbsolutelyContinuousOnInterval.integral_mul_deriv_eq_deriv_mul with AbsolutelyContinuousOnInterval.intervalIntegrable_deriv are the integration-by-parts pair for the local proof (both re-confirmed present), the everywhere-HasDerivAt intervalIntegral.integral_mul_deriv_eq_deriv_mul stays WITHDRAWN, the Binomial(r_0,1/2) finite-pmf encoding is unchanged, and the two status: "cited" Sudijono nodes carried by this world stay SECONDARY-ONLY gates with no delivered consumer. SUPERSEDED ROUND-6/7/8 TEXT FOLLOWS, retained for the audit trail. ROUND-8 RE-CONFIRMATION (F2.5 directive). This world's analytic-ingredient record is CORRECT AS WRITTEN and is re-verified, not changed: the scalar core of the two-arm converse REUSES Causalean.Stat.Limit.van_trees_inequality (module Causalean.Stat.Limit.VanTreesInequality, namespace Causalean.Stat.Limit at VanTreesInequality.lean:44, theorem van_trees_inequality at line 76 — re-read at source this round), and only the genuinely observation-dependent numerator extension uses AbsolutelyContinuousOnInterval.integral_mul_deriv_eq_deriv_mul together with AbsolutelyContinuousOnInterval.intervalIntegrable_deriv (both re-confirmed present this round), with the everywhere-HasDerivAt intervalIntegral.integral_mul_deriv_eq_deriv_mul remaining withdrawn. The two status: "cited" Sudijono nodes carried by this world stay SECONDARY-ONLY with no delivered consumer, exactly as the revised core states; round 8 does not touch them. ROUND-4 CORRECTION (revised core). This world previously existed ONLY to carry the status:"cited" Airy/scalar-attainment gate Props. The revised core adds two internally proved nodes that live here — lem:two-arm-scalar-prior-schedule-kernel (now stated for an ARBITRARY prior nu on effect-class triples, with no least-favourability and no published optimizer assumed) and thm:coarse-two-arm-minimax-lower (the smooth-mixture Bayesian-information converse rho_{n,2} >= n^{-1} - 43 n^{-4/3} for n >= 8) — so the world text now states the reduced experiment, the ordered-partition-plus-fair-bit lift, and the smooth prior as first-class objects of the paper, with the Airy constant demoted to a separately flagged citation interface. Consequently the two cited Sudijono gates are no longer load-bearing anywhere: the revised core gives lem:published-two-arm-minimax-airy and lem:published-two-arm-scalar-attainment-scope NO delivered consumer (consumer: "No delivered theorem consumes this node."), so they stay gate:true, gate_class:"cited" and are threaded into nothing. Encoding is unchanged: Binomial(r_0,1/2) stays the finite pmf 2^{-r_0} * C(r_0,k) over Finset.range (r_0+1), not PMF.binomial, to stay inside the finite-sum layer. Search trace: npm run search -- "Airy constant second order minimax binomial finite population two arm"; grepped Causalean/ and CausalSmith/ for Airy|airyGroundState|C_A; lean_local_search Airy -> empty. Mathlib-side check: lean_loogle "Real.airy" / lean_leansearch "Airy function ground state largest negative zero of derivative" -> Mathlib has no Airy function, which is one reason lem:published-two-arm-minimax-airy and lem:published-two-arm-scalar-attainment-scope stay cited gates rather than being reconstructed. ROUND-6 CORRECTION (FLAG-missed-reuse) — the round-4 zero-hit claim recorded here for the two newly added internal nodes was WRONG and is WITHDRAWN. Causalean.Stat.Limit.van_trees_inequality (Causalean/Stat/Limit/VanTreesInequality.lean:76, module Causalean.Stat.Limit.VanTreesInequality, imported from Causalean.lean:538, present in doc/library_index.json) already PROVES the single-parameter van Trees / Bayesian Cramer-Rao assembly. ROOT CAUSE of the miss: the round-4 grep pattern vanTrees|VanTrees|fisherInfo|FisherInfo|cramerRao|CramerRao|informationInequality was camelCase-only and therefore could not match the snake_case declaration van_trees_inequality or the prose Fisher information; lean_local_search vanTrees was likewise a name-prefix miss (and lean_local_search "Causalean.Stat.Limit.van_trees_inequality" still returns empty, so the local-search index is stale here and MUST NOT be used to assert absence). CORRECTED TRACE (re-run, snake_case and prose included): grep -r "van_trees|Cramer|Cramér|fisher|Fisher" ../Causalean --include=*.lean -> 9 files, of which Causalean/Stat/Limit/VanTreesInequality.lean is the hit that matters (the others are CLT/quantile/KL-expansion/ML-Fisher-consistency modules, none an information inequality); signature confirmed by lean_hover_info at VanTreesInequality.lean:76:9, elaborating clean with no diagnostics. Same-repo check: CausalSmith/Stat/STAT_NeymanRegretMinimax_Research/Helpers/VanTrees.lean carries only the Prop gate defs LocalNeighborhoodRiskInputs and BayesCramerRaoPredictable (external assumptions, NOT proofs), so it is not a reuse source. CONSEQUENCE for this world: lem:bayesian-information-inequality REUSES that theorem for its scalar core and stays define-local ONLY for the strictly more general observation-dependent target g(theta, x), which the Causalean statement does not cover (its target is the parameter-only psi : R -> R); thm:coarse-two-arm-minimax-lower inherits the reuse through that helper. Mathlib-side check (unchanged and still correct): lean_leansearch "Cramer-Rao lower bound Fisher information unbiased estimator variance" -> only ProbabilityTheory.variance* lemmas; Mathlib itself has no Cramer-Rao/van Trees/Fisher-information layer, which is why the Causalean theorem is the right substrate; lean_local_search bayesRisk -> ProbabilityTheory.bayesRisk and the Mathlib/Probability/Decision/Risk/* family, which remain bypass-justified (kernel- and ENNReal-valued). The Mathlib ingredients the extension still needs DO exist and are reused. ROUND-7 CORRECTION (F2.5 directive, propagated into this world block because it is where the extension's analytic ingredients are recorded): the integration-by-parts lemma is AbsolutelyContinuousOnInterval.integral_mul_deriv_eq_deriv_mul (Mathlib/MeasureTheory/Integral/IntervalIntegral/AbsolutelyContinuousFun.lean:257, module Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun), with AbsolutelyContinuousOnInterval.intervalIntegrable_deriv (Mathlib/MeasureTheory/Integral/IntervalIntegral/DerivIntegrable.lean:160, module Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable) for the integrability side condition and the predicate itself from Mathlib.MeasureTheory.Function.AbsolutelyContinuous — NOT intervalIntegral.integral_mul_deriv_eq_deriv_mul, which is WITHDRAWN here because it demands HasDerivAt everywhere on the interval while this world's extension advertises only absolute continuity in theta and an a.e. derivative. MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg (Mathlib/MeasureTheory/Integral/Bochner/Basic.lean) at p = q = 2 remains the integral Cauchy-Schwarz. (The Causalean proof internally uses the everywhere-differentiable intervalIntegral form under its own binders; that is unchanged and untouched, since Causalean/Stat/Limit/VanTreesInequality.lean imports Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts itself.)

Definitions (8)

Definition P-1 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rhoN $\rho_n(c)=\inf_{\mathcal D\in\Delta(\mathcal A_K^n),\,\widehat\tau}\max_{z\in\mathcal T_K^n}\mathbb E_{\mathcal D}\left[\left\{\widehat\tau((A_i,Y_i^{\mathrm{obs}})_{i=1}^n)-\tau_c(z)\right\}^2\right].$
Definition P-2 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.orbitLik $P_m(x\mid r)=\frac{\prod_{a\in\mathcal A_K}r_a!}{n!}\sum_{h\in\mathcal H(m,r,x)}\prod_{t\in\mathcal T_K}\frac{m_t!}{\prod_{a\in\mathcal A_K}h_{t,a}!}.$
Definition P-3 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.orbitGameValue $G_n(c)=\inf_{\pi\in\Delta(\mathcal R_{n,K}),\,\delta}\max_{m\in\mathcal M_{n,K}}\sum_{r\in\mathcal R_{n,K}}\pi_r\sum_{x\in\prod_{a\in\mathcal A_K}\{0,\ldots,r_a\}}P_m(x\mid r)\{\delta(r,x)-\tau_c(m)\}^2.$
Definition P-4 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrastWeightedProcedure Assign each $A_i$ independently on $S_c$ with $\Pr(A_i=a)=q_a^\star$ , and report $\widehat\tau_{\mathrm{cHT}}^\star=\operatorname{proj}_{[-L_c/2,L_c/2]}\!\left[n^{-1}\sum_{i=1}^n\sum_{a\in S_c}c_a\mathbf 1\{A_i=a\}(Y_i^{\mathrm{obs}}-1/2)/q_a^\star\right]$ .
Definition P-5 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPValueK3 For $K=3$ and $c=c^\dagger$ , minimize $u$ over rational variables $\pi_r\ge0$ and $w_{r,x,g}\ge0$ , subject to $\sum_r\pi_r=1$ , $\sum_{g\in\Gamma_M}w_{r,x,g}=\pi_r$ for every $(r,x)$ , and $\sum_{r,x,g}P_m(x\mid r)w_{r,x,g}\{g-\tau_{c^\dagger}(m)\}^2\le u$ for every $m\in\mathcal M_{n,3}$ ; its value is $\Lambda_{n,M}^\dagger$ .
Definition P-6 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.activeFaceBoundaryLayerHandle Starting from the exact orbit game, enumerate binding response-type and allocation faces, expand the discrete risk by a vector Stein identity within each stratum, use the certified $c^\dagger$ LP sequence to select the normalizer, and pass the primal and dual games to a stratified diffusion-control or spectral value problem with face and corner compatibility conditions.
Definition P-7 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrastScoreFeedbackHandle Begin with the vector of centered arm scores generated by $q^\star$ ; derive an orbit-saddle posterior mean or feedback correction from the limiting value function, lift that correction to finite $n$ , and discretize the squared ground state or dual occupation measure into a least-favorable prior on response-type counts.
Definition P-8 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPValue Fix $K\geq2$ , $n,M\geq1$ , and a rational nonzero zero-sum contrast $c\in\mathbb Q^K$ . Put $h_c=L_c/2$ and $\Gamma_{M,c}=\{-h_c+jh_c/M:j=0,\ldots,2M\}.$ Minimize $u$ over $\pi_r\geq0$ and $w_{r,x,g}\geq0$ , subject to $\sum_{r\in\mathcal R_{n,K}}\pi_r=1, \qquad \sum_{g\in\Gamma_{M,c}}w_{r,x,g}=\pi_r \quad\text{for every }(r,x),$ and $\sum_{r\in\mathcal R_{n,K}}\sum_x\sum_{g\in\Gamma_{M,c}} P_m(x\mid r)w_{r,x,g}\{g-\tau_c(m)\}^2\leq u \quad\text{for every }m\in\mathcal M_{n,K}.$ The value of this finite grid-action linear program is $\Lambda_{n,M}(c)$ .

Lemmas (2)

Lemma L-1 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScalarPriorScheduleKernel For $K=2$ and $c=(1,-1)$ , let $\nu$ be any probability distribution on triples $\vartheta=(p_+,p_-,r_0)$ of nonnegative integers satisfying $p_++p_-+r_0=n$ . Conditional on $\vartheta$ , choose uniformly an ordered partition $(P,N,R)$ of $\{1,\ldots,n\}$ with sizes $(p_+,p_-,r_0)$ , assign response type $(1,0)$ on $P$ and $(0,1)$ on $N$ , and, independently over $i\in R$ , draw $H_i\sim\operatorname{Bernoulli}(1/2)$ and assign type $(H_i,H_i)$ . For every assignment law $\mathcal D$ , including laws dependent across units and chosen independently of the schedule draw, put $S_i=Y_i^{\mathrm{obs}}$ on arm $1$ , put $S_i=1-Y_i^{\mathrm{obs}}$ on arm $2$ , and put $X=\sum_iS_i$ . Conditional on $\vartheta$ , $X=p_++\operatorname{Binomial}(r_0,1/2);$ the assignment $A$ is ancillary; and, conditional on $X=x$ , $S$ is uniform over the binary vectors of sum $x$ , with a kernel independent of $\vartheta$ and $A$ . Consequently every full-data estimator can be Rao--Blackwellized first over $A$ and then over $S$ conditional on $X$ , without increasing squared Bayes risk. The Bayes risk of the lifted complete-schedule prior therefore equals $\inf_f\sum_\vartheta\nu_\vartheta \mathbb E_\vartheta\!\left[ \left\{f\!\left(p_++\operatorname{Binomial}(r_0,1/2)\right) -\frac{p_+-p_-}{n}\right\}^2\right].$ These conclusions include every boundary case in which one or more of $p_+$ , $p_-$ , and $r_0$ is zero; no least-favorability or published optimizer is assumed.
Lemma L-2 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.bayesianInformationInequality Let $\Theta=(\ell,u)\subseteq\mathbb R$ , let $w$ be a continuously differentiable probability density with compact support in $\Theta$ , positive on the interior of its support, and put $J(w)=\int_{\{w>0\}}\frac{\{w'(\theta)\}^2}{w(\theta)}\,d\theta<\infty.$ Let $P_\theta$ be dominated by a $\sigma$ -finite measure with density $p_\theta(x)$ . Assume that for almost every $x$ , $p_\theta(x)$ is absolutely continuous in $\theta$ , that differentiation under the integral is valid, that $\int \partial_\theta p_\theta(x)\,dx=0, \qquad I(\theta)=\int_{\{p_\theta>0\}} \frac{\{\partial_\theta p_\theta(x)\}^2}{p_\theta(x)}\,dx<\infty,$ and that $0<J(w)+\int I(\theta)w(\theta)\,d\theta<\infty$ . Let $g(\theta,x)$ be absolutely continuous in $\theta$ , and let $T(x)$ be any estimator. Assume that all quantities below are integrable, that the product $w(\theta)p_\theta(x)\{T(x)-g(\theta,x)\}$ is absolutely continuous in $\theta$ , and that its boundary values vanish. Then $\mathbb E\{T(X)-g(\theta,X)\}^2 \geq \frac{\left[\mathbb E\{\partial_\theta g(\theta,X)\}\right]^2} {J(w)+\mathbb E\{I(\theta)\}},$ where the expectation is under $w(\theta)p_\theta(x)\,d\theta\,dx$ . The statement allows the target $g$ to depend on the observation and includes the ordinary Bayesian Cram\'er--Rao inequality as the special case $g(\theta,x)=g(\theta)$ .

Theorems (14)

Theorem T-1 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.exact_response_type_game For every fixed $n$ , $K\ge2$ , and contrast $c$ , begin with nature choosing a complete labeled schedule $z\in\mathcal T_K^n$ , an arbitrary assignment law $\mathcal D$ , and an arbitrary estimator $\widehat\tau$ . Averaging a procedure over common permutations of unit labels is lossless, and the symmetrized risk is constant on each unit-permutation orbit of schedules. Consequently, with $m_t=\#\{i:t_i=t\}$ , $\tau_c(z)=\tau_c(m)$ , and the labeled minimax risk equals the explicit orbit game: $\rho_n(c)=G_n(c)$ . A symmetrized design is exactly a mixture $\pi\in\Delta(\mathcal R_{n,K})$ of uniform assignments conditional on $r$ , and a nonrandomized invariant estimator is $\delta(r,x)$ . The compact convex game admits a minimax procedure and a least-favorable prior on $\mathcal M_{n,K}$ forming a saddle point.
Theorem T-2 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.multiarm_strict_extension The exact response-type game is a strict treatment-domain extension of the unrestricted binary decision problem from two arms to every fixed $K\geq3$ ; it is not a setwise extension of Hull's bounded-outcome model. When $K=2$ and $c=(1,-1)$ , the orbit experiment retains the four distinct counts $m_{(0,0)},m_{(0,1)},m_{(1,0)},m_{(1,1)}$ in $P_m(x\mid r)$ . Although $\tau_c(m)=\frac{m_{(1,0)}-m_{(0,1)}}n=\frac{p_+-p_-}{n},$ no aggregation by individual-effect class is asserted to identify the likelihood kernel. The exact orbit equality and the internal sign-group and Bayesian-information arguments give $\rho_n((1,-1))=G_n((1,-1)), \qquad C_0((1,-1))=1, \qquad d_n((1,-1))=\Theta(n^{-4/3}),$ with the explicit coarse ceiling $d_n((1,-1))\leq43n^{-4/3}$ for $n\geq8$ , but no exact second-order constant or source-specific optimizer is delivered. Hull's theorem permits unrestricted designs and measurable estimators for two bounded-outcome arms. By contrast, for every fixed $K\geq3$ and nonzero zero-sum $c$ , the present theorem assumes binary response schedules and proves equality of the unrestricted labeled-schedule risk and the explicit joint $2^K$ -response-type orbit game, including arbitrary assignment dependence and possibly biased estimators. Thus the strict extension is on the number-of-arms axis within the binary model. This statement asserts neither a bounded-outcome $K\geq3$ extension nor that the multi-arm game is scalar-nonrepresentable.
Theorem T-3 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.first_order_saddle Under fixed $K$ , $\lim_{n\to\infty}n\rho_n(c)=C_0(c)=\frac{1}{4}\left(\sum_{a\in\mathcal A_K}|c_a|\right)^2.$ Independent assignment supported on $S_c$ with $q_a^\star=|c_a|/L_c$ and the projected rule $\widehat\tau_{\mathrm{cHT}}^\star$ satisfies $\max_{z\in\mathcal T_K^n}R_n(\mathcal D,\widehat\tau_{\mathrm{cHT}}^\star;z)\le C_0(c)/n$ . Because every target $\tau_c(z)$ lies in $[-L_c/2,L_c/2]$ , projection onto this interval cannot increase pointwise squared loss, so $\widehat\tau_{\mathrm{cHT}}^\star$ belongs to the same minimax estimator action space. Conversely, every sequence of randomized designs and possibly biased estimators has asymptotic worst-case labeled-schedule risk at least $C_0(c)/n+o(n^{-1})$ .
Theorem T-4 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3_lp_certificate For $K=3$ , $c=c^\dagger=(1,-1/2,-1/2)$ , and every $n,M\ge1$ , the grid-action program defining $\Lambda_{n,M}^\dagger$ is a finite linear program with rational coefficients and $\rho_n^\dagger\le\Lambda_{n,M}^\dagger\le\rho_n^\dagger+\frac{2}{M}+\frac{1}{M^2}.$ Consequently, choosing $M=n^2$ yields exact-rational primal and dual certificates whose discretization error is $o(n^{-4/3})$ , so the sequence $n^{4/3}\{n^{-1}-\Lambda_{n,n^2}^\dagger\}$ is a valid early diagnostic for, but not an assumption of, the $4/3$ exponent. The LP retains the all-zero and all-one response types and every response-type count vector in $\mathcal M_{n,3}$ .
Theorem T-5 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3_scalar_score_not_minimax_preserving For $K=3$ , $c=c^\dagger=(1,-1/2,-1/2)$ , and independent assignment with $q^\star=(1/2,1/4,1/4)$ , define $Z_i=\operatorname{sign}(c_{A_i}^\dagger)(2Y_i^{\mathrm{obs}}-1), \qquad \mu_i=t_{i,1}-\frac{t_{i,2}+t_{i,3}}{2}.$ Then $Z_i\in\{-1,1\}$ , $\mathbb E(Z_i\mid t_i)=\mu_i, \qquad \operatorname{Var}(Z_i\mid t_i)=1-\mu_i^2,$ and $\mathbb E\!\left[\left\{\frac1n\sum_iZ_i-\tau_{c^\dagger}(z)\right\}^2\middle|z\right] =\frac1n-\frac1{n^2}\sum_i\mu_i^2.$ The exact scalar minimax value at $n=3$ , when only $(Z_1,Z_2,Z_3)$ is retained and each $\mu_i$ ranges over $\{-1,-1/2,0,1/2,1\}$ , is $V_3^Z=1-\frac{\sqrt3}{2}.$ Nevertheless, an explicit deterministic rule based on the full observed arm labels and outcomes under the same independent $q^\star$ design has maximum risk $\frac{511653}{4000000}<\frac{18213}{136000}<V_3^Z.$ Thus the map from the full data to $(Z_i)_{i=1}^3$ is not minimax-preserving, even at the fixed $q^\star$ design, and cannot itself be an exact scalar reduction of the unrestricted three-arm orbit game. In addition, if $M_\mu=\sum_i\mu_i$ and $Q_\mu=\sum_i\mu_i^2$ , then over feasible half-integer $M_\mu$ , $\min Q_\mu= \begin{cases} |M_\mu|/2,&|M_\mu|\leq n/2,\\ (3|M_\mu|-n)/2,&n/2\leq|M_\mu|\leq n. \end{cases}$ Hence the coefficient $\tfrac12|x|$ is the correct local wedge for the scalar relaxation, but this proposition does not establish that the discarded arm-label information is negligible at any second-order scale.
Theorem T-6 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.embedded_two_arm_converse For every integer $n\geq1$ , every $K\geq2$ , and every nonzero zero-sum contrast $c$ , write $\rho_{n,2}=\rho_n((1,-1))$ for the unrestricted two-arm minimax risk with the same population size. Then $\rho_n(c)\geq C_0(c)\rho_{n,2}.$ If $|S_c|=2$ , this inequality is an equality. The contrast-weighted first-order procedure gives $\rho_n(c)\leq C_0(c)/n$ , and the internal Bayesian-information construction gives, for every $n\geq8$ , $\rho_{n,2}\geq n^{-1}-43n^{-4/3}.$ Consequently, for every $n\geq8$ , $0\leq d_n(c)\leq C_0(c)\{n^{-1}-\rho_{n,2}\} \leq43C_0(c)n^{-4/3}.$ When $|S_c|=2$ , the middle inequality is an equality, so $d_n(c)=C_0(c)\{n^{-1}-\rho_{n,2}\}$ . The finite-sample sign-group embedding permits arbitrary dependent assignment designs and arbitrary possibly biased estimators; no published two-arm optimizer, least-favorable prior, or Airy expansion is used.
Theorem T-7 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.universal_second_order_rate For every fixed $K\geq2$ and every nonzero zero-sum contrast $c$ , put $h_c=\frac{L_c}{2}=\sqrt{C_0(c)}, \qquad \lambda_c=\min\left\{ \left|h_c^{-1}\sum_{a\in\mathcal A_K}c_at_a\right|: t\in\mathcal T_K,\ \sum_{a\in\mathcal A_K}c_at_a\neq0 \right\}.$ Then $0<\lambda_c\leq1$ . Under independent assignment on $S_c$ with $\Pr(A_i=a)=q_a^\star$ , define $W_i=\sum_{a\in S_c}\frac{c_a\mathbf 1\{A_i=a\}}{q_a^\star} \left(Y_i^{\mathrm{obs}}-\frac12\right), \qquad V_i=\frac{W_i}{h_c}, \qquad X_n=\frac1n\sum_{i=1}^nV_i.$ Let $b_n=n^{-1/3},\qquad \varepsilon_n=\frac{\sqrt{\lambda_c}}4n^{-1/3},\qquad g_n(x)=\max\{-b_n,\min(x,b_n)\},$ and report the explicit clipped-shrinkage rule $\widehat\tau_n^{\mathrm{sh}}=h_c\{X_n-\varepsilon_ng_n(X_n)\}.$ This rule is a map from the realized assignment and observed outcomes into $[-L_c/2,L_c/2]$ . Define $A_c=\frac{\sqrt{\lambda_c}}2-\frac{\lambda_c}{16}, \qquad B_c=\frac{7\lambda_c}{16}, \qquad \kappa_c=\frac12\min\{A_c,B_c\}>0,$ and choose a finite integer $N_c$ such that $4\sqrt{\lambda_c}\,m\exp(-m^{1/3}/8)\leq A_c \qquad\text{for every integer }m\geq N_c.$ Then, for every $n\geq N_c$ , $\max_{z\in\mathcal T_K^n} R_n(\mathcal D^\star,\widehat\tau_n^{\mathrm{sh}};z) \leq C_0(c)\{n^{-1}-\kappa_cn^{-4/3}\},$ where $\mathcal D^\star$ is the displayed independent $q^\star$ design. Consequently, $0<C_0(c)\kappa_c \leq\liminf_{n\to\infty}n^{4/3}d_n(c) \leq\limsup_{n\to\infty}n^{4/3}d_n(c) \leq43C_0(c),$ so $d_n(c)=\Theta(n^{-4/3})$ . Moreover, if a positive normalizer $a_n$ is regularly varying with index $\beta$ , meaning $a_{\lfloor xn\rfloor}/a_n\to x^\beta$ for every $x>0$ , and if $a_nd_n(c)$ converges to a finite positive limit, then necessarily $\beta=4/3$ . The upper procedure and the matching-order converse are both derived internally and do not use a published Airy constant or optimizer.
Theorem T-8 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rational_contrast_grid_certificate_sandwich Fix $K\geq2$ and a rational nonzero zero-sum contrast $c\in\mathbb Q^K$ , and put $h_c=L_c/2$ . For every $n,M\geq1$ , the program defining $\Lambda_{n,M}(c)$ has rational coefficients and admits exact rational optimal primal and dual certificates. Let $(\pi^{n,M,c},w^{n,M,c},\Lambda_{n,M}(c))$ be such a primal certificate and let $\nu_{n,M,c}\in\Delta(\mathcal M_{n,K})$ be the response-count multiplier in such a dual certificate. For $\pi_r^{n,M,c}>0$ , define $\kappa_{r,x}^{n,M,c}(g)=\frac{w_{r,x,g}^{n,M,c}}{\pi_r^{n,M,c}}, \qquad \delta_{n,M,c}(r,x)=\sum_{g\in\Gamma_{M,c}}g\kappa_{r,x}^{n,M,c}(g),$ and put $\delta_{n,M,c}(r,x)=0$ when $\pi_r^{n,M,c}=0$ . Define $U_{n,M}(c)=\max_{m\in\mathcal M_{n,K}} \sum_{r\in\mathcal R_{n,K}}\pi_r^{n,M,c}\sum_xP_m(x\mid r) \{\delta_{n,M,c}(r,x)-\tau_c(m)\}^2.$ For a rational prior $\nu\in\Delta(\mathcal M_{n,K})$ , put $D_{r,x,c}(\nu)=\sum_m\nu_mP_m(x\mid r), \qquad N_{r,x,c}(\nu)=\sum_m\nu_mP_m(x\mid r)\tau_c(m),$ and define $B_{n,c}(\nu)=\min_{r\in\mathcal R_{n,K}} \left[ \sum_m\nu_m\tau_c(m)^2- \sum_{x:D_{r,x,c}(\nu)>0} \frac{N_{r,x,c}(\nu)^2}{D_{r,x,c}(\nu)} \right].$ Then the computable finite-sample certificates satisfy $B_{n,c}(\nu_{n,M,c})\leq\rho_n(c)\leq U_{n,M}(c) \leq\Lambda_{n,M}(c) \leq B_{n,c}(\nu_{n,M,c})+\frac{C_0(c)}{4M^2}.$ The upper procedure draws $r$ from $\pi^{n,M,c}$ , assigns uniformly conditional on $r$ , and reports $\delta_{n,M,c}(r,x)$ . The lower certificate completes the rational prior by the unrestricted posterior mean $N_{r,x,c}(\nu)/D_{r,x,c}(\nu)$ , omitting zero-denominator observations. There are $\binom{n+2^K-1}{2^K-1}$ response-count risk constraints, $\binom{n+K-1}{K-1}$ allocation orbits, and $\binom{n+2K-1}{2K-1}$ allocation--observation orbits; hence one exact rational linear program with $1+\binom{n+K-1}{K-1} +(2M+1)\binom{n+2K-1}{2K-1}$ primal variables, followed by finite rational sums and maxima, evaluates the certificates. For every positive normalizer $a_n$ and every integer sequence $(M_n)$ satisfying $a_n/M_n^2\to0$ , $a_n\{U_{n,M_n}(c)-B_{n,c}(\nu_{n,M_n,c})\}\longrightarrow0.$ If $a_nd_n(c)$ has a finite limit, the upper- and lower-certificate improvements converge to that same limit. At the universally established scale $a_n=n^{4/3}$ , the choice $M_n=n$ is sufficient. These conclusions neither assert convergence of primal optimizers or dual priors nor identify a sharp second-order constant or an analytic limiting feedback rule.
Theorem T-9 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrast_risk_continuity For every integer $n\geq1$ , every $K\geq2$ , and every pair of nonzero zero-sum contrasts $c,c'\in\mathbb R^K$ , put $\eta(c,c')=\frac{L_{c-c'}}2 =\frac12\sum_{a\in\mathcal A_K}|c_a-c'_a|.$ Then $\left|\sqrt{\rho_n(c)}-\sqrt{\rho_n(c')}\right| \leq \eta(c,c').$ The bound is finite-sample, permits arbitrary dependent assignment laws and arbitrary possibly biased estimators, and requires no positivity condition.
Theorem T-10 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.real_contrast_grid_certificate_transfer Fix $K\geq2$ and a nonzero real zero-sum contrast $c\in\mathbb R^K$ . Let $c^{(n)}\in\mathbb Q^K\setminus\{0\}$ be zero-sum rational contrasts such that $c^{(n)}\to c$ , let $M_n\geq1$ be integers, and put $\eta_n=\frac{L_{c-c^{(n)}}}{2}.$ For each $n$ , let $B_n=B_{n,c^{(n)}}(\nu_{n,M_n,c^{(n)}}), \qquad U_n=U_{n,M_n}(c^{(n)})$ be the exact rational prior and procedure certificates from the rational-contrast grid theorem, and define $R_n^-=\left\{\max(0,\sqrt{B_n}-\eta_n)\right\}^2, \qquad R_n^+=(\sqrt{U_n}+\eta_n)^2.$ Then $R_n^-\leq\rho_n(c)\leq R_n^+.$ The upper endpoint is attained as a valid worst-case-risk certificate by using the rational finite-program design and projecting its barycenter rule onto $[-L_c/2,L_c/2]$ . The lower endpoint is a valid Bayes-risk certificate against every design and estimator, obtained from the same rational response-count prior. Moreover, $\begin{aligned} 0\leq R_n^+-R_n^- &\leq \frac{5C_0(c^{(n)})}{4M_n^2} +4\sqrt{C_0(c^{(n)})}\,\eta_n n^{-1/2} +3\eta_n^2\\ &=O\!\left(\frac{C_0(c^{(n)})}{M_n^2} +\eta_n n^{-1/2}+\eta_n^2\right). \end{aligned}$ For every fixed real $c$ , the rational zero-sum approximants may be chosen with $\eta_n=o(n^{-5/6})$ . With that choice and $M_n=n$ , $R_n^+-R_n^-=o(n^{-4/3}).$ Equivalently, if $\underline d_n=\frac{C_0(c)}n-R_n^+, \qquad \overline d_n=\frac{C_0(c)}n-R_n^-,$ then $\underline d_n\leq d_n(c)\leq\overline d_n, \qquad \overline d_n-\underline d_n=o(n^{-4/3}).$ For rational $c$ , one may take $c^{(n)}=c$ ; the certificates are then direct exact-rational finite programs and their gap for $M_n=n$ is at most $C_0(c)/(4n^2)$ . Thus scale-matched finite-program procedure and prior certificates exist directly for every rational contrast and, through the displayed approximation brackets, for every real contrast. This conclusion does not assert convergence of $n^{4/3}d_n(c)$ , convergence of the finite-program optimizers or priors, sharp second-order optimality of $q^\star$ , a sharp constant, an analytic limiting rule, or a limiting operator.
Theorem T-11 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3_grid_certificate_sandwich For $K=3$ , $c=c^\dagger$ , and integers $n,M\geq1$ , let $(\pi^{n,M},w^{n,M},\Lambda_{n,M}^\dagger)$ be an exact rational optimal primal certificate for the grid-action program, and let $\nu_{n,M}\in\Delta(\mathcal M_{n,3})$ be the response-count multiplier in a complete exact rational optimal dual certificate. For $\pi_r^{n,M}>0$ , define $\kappa_{r,x}^{n,M}(g)=\frac{w_{r,x,g}^{n,M}}{\pi_r^{n,M}}, \qquad \delta_{n,M}(r,x)=\sum_{g\in\Gamma_M}g\kappa_{r,x}^{n,M}(g),$ and put $\delta_{n,M}(r,x)=0$ when $\pi_r^{n,M}=0$ . Define the computable upper certificate $U_{n,M}=\max_{m\in\mathcal M_{n,3}} \sum_{r\in\mathcal R_{n,3}}\pi_r^{n,M}\sum_xP_m(x\mid r) \{\delta_{n,M}(r,x)-\tau_{c^\dagger}(m)\}^2.$ For any rational prior $\nu\in\Delta(\mathcal M_{n,3})$ , put $D_{r,x}(\nu)=\sum_m\nu_mP_m(x\mid r), \qquad N_{r,x}(\nu)=\sum_m\nu_mP_m(x\mid r)\tau_{c^\dagger}(m),$ and define the computable lower certificate $B_n(\nu)=\min_{r\in\mathcal R_{n,3}} \left[ \sum_m\nu_m\tau_{c^\dagger}(m)^2- \sum_{x:D_{r,x}(\nu)>0}\frac{N_{r,x}(\nu)^2}{D_{r,x}(\nu)} \right].$ Then $B_n(\nu_{n,M})\leq\rho_n^\dagger\leq U_{n,M} \leq\Lambda_{n,M}^\dagger \leq B_n(\nu_{n,M})+\frac{1}{4M^2}.$ The upper procedure first draws $r$ from $\pi^{n,M}$ , assigns uniformly conditional on $r$ , and reports $\delta_{n,M}(r,x)$ ; the lower certificate uses the unrestricted posterior mean $N_{r,x}(\nu)/D_{r,x}(\nu)$ , with zero-denominator observations omitted. Consequently, for any positive normalizer $a_n$ and integers $M_n$ satisfying $a_n/M_n^2\to0$ , $a_n\{U_{n,M_n}-B_n(\nu_{n,M_n})\}\longrightarrow0.$ If $a_nd_n(c^\dagger)$ has a finite limit, both certificate improvements converge to that same limit. This theorem supplies a finite- $n$ multivariate rule and an unrestricted rational-prior converse at every such selected normalization; it does not select the normalization, decide whether the independent $q^\star$ design is second-order optimal, or produce an analytic Hamilton--Jacobi--Bellman feedback formula.
Theorem T-12 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.second_order_rate_and_certificate_frontier Fix $K\geq2$ and a nonzero zero-sum contrast $c$ with $|S_c|\geq3$ , and define $x_n(c)=n^{4/3}d_n(c)$ . Then $0<\liminf_{n\to\infty}x_n(c) \leq\limsup_{n\to\infty}x_n(c) \leq43C_0(c)<\infty.$ Consequently, $d_n(c)=\Theta(n^{-4/3})$ , and the subsequential-limit set of $x_n(c)$ is a nonempty compact subset of a compact interval contained in $(0,\infty)$ . If a positive regularly varying normalizer $(a_n)$ makes $a_nd_n(c)$ converge to a finite positive limit, then its index is $4/3$ . Such a normalizer exists if and only if $x_n(c)$ is slowly varying. In the branch where $x_n(c)$ does not converge, this is equivalent to the existence of a nonconvergent slowly varying correction multiplying $n^{4/3}$ . If $c$ is rational, take $M=n$ in the exact-rational grid certificate. The normalized grid-value improvement, barycenter-procedure improvement, prior-lower-bound improvement, and $x_n(c)$ have pairwise distance $o(1)$ . They therefore have the same subsequential-limit set and converge simultaneously, with a common limit whenever any one converges. If $c$ is real, there are rational zero-sum contrasts $c^{(n)}$ with $\eta_n=\frac{L_{c-c^{(n)}}}{2}=o(n^{-5/6})$ such that, again with $M=n$ , the transferred risk bracket has width $o(n^{-4/3})$ . The two corresponding normalized improvement endpoints and $x_n(c)$ have the same subsequential-limit set and convergence behavior. These certificates retain every response-type count vector and require no exposure-positivity, allocation-support, rank, or overlap restriction on the unrestricted minimax design. For $K=3$ and $c=c^\dagger$ , the general rational certificates specialize to the exact three-arm certificates, while the displayed scalar sign-score compression is not minimax-preserving even at $n=3$ . Thus the established results determine the second-order exponent through an internal coarse two-arm converse and give scale-matched computable procedures and prior converses, but they do not determine whether $x_n(c)$ converges, its sharp limit or exact subsequential-limit set, whether a nonconvergent $x_n(c)$ is slowly varying, sharp second-order optimality of $q^\star$ , convergence of optimizers or priors, or a limiting face--corner Hamilton--Jacobi--Bellman, cone-spectral, separable, or other operator.
Theorem T-13 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.attainment_and_k3_certified_converse Fix $K\geq2$ and a nonzero zero-sum contrast $c$ . First, if $|S_c|=2$ and $\rho_{n,2}=\rho_n((1,-1))$ , then $\rho_n(c)=C_0(c)\rho_{n,2}.$ An exact finite- $n$ saddle exists: take any minimax procedure and least-favorable complete-schedule prior supplied internally by the finite two-arm orbit game, map its two arms to the active sign pair, scale the estimator by $\sqrt{C_0(c)}$ , and push the prior forward by the sign-group schedule embedding. This existence statement does not specify the published scalar posterior mean or least-favorable prior. Moreover, $C_0(c)\kappa_c \leq\liminf_{n\to\infty}n^{4/3}d_n(c) \leq\limsup_{n\to\infty}n^{4/3}d_n(c) \leq43C_0(c),$ where $\kappa_c>0$ is the explicit constant in the universal clipped-shrinkage theorem. Second, for arbitrary support, the independent $q^\star$ design and the explicit clipped-shrinkage estimator satisfy, for every sufficiently large $n$ , $\max_zR_n(\mathcal D^\star,\widehat\tau_n^{\mathrm{sh}};z) \leq C_0(c)\{n^{-1}-\kappa_cn^{-4/3}\}.$ Together with the internal coarse two-arm converse and sign-group embedding, this proves $0<C_0(c)\kappa_c \leq\liminf_{n\to\infty}n^{4/3}d_n(c) \leq\limsup_{n\to\infty}n^{4/3}d_n(c) \leq43C_0(c).$ Third, choose rational nonzero zero-sum contrasts $c^{(n)}\in\mathbb Q^K$ with $c^{(n)}\to c$ , put $\eta_n=\frac{L_{c-c^{(n)}}}{2},$ and choose integers $M_n\geq1$ . Let $B_n=B_{n,c^{(n)}}(\nu_{n,M_n,c^{(n)}}), \qquad U_n=U_{n,M_n}(c^{(n)}),$ and define $R_n^-=\{\max(0,\sqrt{B_n}-\eta_n)\}^2, \qquad R_n^+=(\sqrt{U_n}+\eta_n)^2.$ Then $R_n^-\leq\rho_n(c)\leq R_n^+.$ The projected rational-certificate procedure has worst-case risk at most $R_n^+$ , while the transferred rational prior certifies $R_n^-$ against every design and estimator. If $\underline d_n=\frac{C_0(c)}n-R_n^+, \qquad \overline d_n=\frac{C_0(c)}n-R_n^-,$ then $\underline d_n\leq d_n(c)\leq\overline d_n$ and $\begin{aligned} 0\leq\overline d_n-\underline d_n &=R_n^+-R_n^-\\ &\leq\frac{5C_0(c^{(n)})}{4M_n^2} +4\sqrt{C_0(c^{(n)})}\,\eta_nn^{-1/2} +3\eta_n^2. \end{aligned}$ One may choose $\eta_n=o(n^{-5/6})$ ; with $M_n=n$ , the bracket has width $o(n^{-4/3})$ . If $c$ is rational, one may take $c^{(n)}=c$ , and the direct exact-rational gap is at most $C_0(c)/(4n^2)$ . For $c=c^\dagger$ , $C_0(c^\dagger)=1$ , and for every $n,M\geq1$ , $B_n(\nu_{n,M})\leq\rho_n(c^\dagger)\leq U_{n,M} \leq B_n(\nu_{n,M})+\frac1{4M^2}.$ The available exact-rational calculations through $n=5$ are a finite three-arm showcase for this formula and do not infer a cross- $n$ limit. These conclusions establish exact value and saddle transfer for two active arms, explicit order attainment for every fixed support, and scale-matched computable procedure and converse certificates for every real zero-sum contrast, directly exact-rational for rational contrasts. They do not determine, when $|S_c|\geq3$ , convergence or the sharp constant of $n^{4/3}d_n(c)$ , sharp second-order optimality of $q^\star$ , convergence of finite-program optimizers or priors, an analytic limiting feedback rule, or a limiting Hamilton--Jacobi--Bellman, spectral, separable, or other operator.
Theorem T-14 verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.coarse_two_arm_minimax_lower For every integer $n\geq8$ , the unrestricted two-arm binary finite-population minimax risk satisfies the fully internal lower bound $\rho_n((1,-1)) \geq \frac{(1-n^{-1/3})^2} {n/(1-n^{-2/3}/4)+40n^{2/3}} \geq \frac1n-43n^{-4/3}.$ The bound holds against every assignment law, including dependent assignments, and every possibly biased estimator. It is obtained from a prior on complete fixed response schedules and does not use a published two-arm minimax value, Airy expansion, optimizer, or least-favorable prior.

Citeds (5)

Cited result lem:published-two-arm-minimax-airy verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.PublishedTwoArmMinimaxAiry For $K=2$ and $c=(1,-1)$ , Sudijono, Dobriban, and Tchetgen Tchetgen's unrestricted binary finite-population minimax risk equals the scalar minimax value based on $X_2=p_++\operatorname{Binomial}(r_0,1/2)$ for estimating $(p_+-p_-)/n$ , where $p_++p_-+r_0=n$ . They further prove $\rho_n(c)=n^{-1}-C_A n^{-4/3}+o(n^{-4/3})$ , with $C_A=1.617\ldots$ .
Cited result lem:published-hull-two-arm-bounded-scope verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.PublishedHullTwoArmBoundedScope Hull's unrestricted minimax theorem concerns a finite population with two treatment arms and potential outcomes bounded in a known interval. Within that two-arm model it optimizes over arbitrary assignment mechanisms and measurable estimators. It does not state a fixed- $K\geq3$ bounded-outcome orbit-game theorem.
Cited result lem:published-kallus-msod-inference-scope verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.PublishedKallusMsodInferenceScope Kallus's framework fixes $K=2$ , even $n$ , balanced assignments, treatment-label blinding, and a fixed unbiased sample-average-treatment-effect estimator. The ordinary mixed-strategy optimal design minimizes the worst-case variance contribution over a specified conditional-mean class. It may fail to give enough assignment-probability uniformity for Fisher randomization inference. Separately, the inference-constrained mixed-strategy optimal design is minimax for that variance criterion among designs satisfying the stated per-assignment probability cap.
Cited result lem:published-aronow-lopatto-bounded-unbiased verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.PublishedAronowLopattoBoundedUnbiased For a finite population of $N$ fixed values $y_j\in[\ell_j,u_j]$ , put $r_j=(u_j-\ell_j)/2$ . For every sampling design having inclusion probabilities $\pi_j>0$ and every design-unbiased estimator of the finite-population total, the worst-case squared error is at least $\sum_{j=1}^N r_j^2\frac{1-\pi_j}{\pi_j}.$ The midpoint-differenced Horvitz--Thompson estimator attains equality exactly when the inclusion indicators are pairwise independent for coordinates with $r_j>0$ . When every $r_j>0$ , jointly over sampling designs with a stated expected-sample-size budget $b$ and design-unbiased estimators, the value is the minimum of the displayed expression; it is attained by independent sampling with $\pi_j^\star=\min\{1,\gamma r_j\}$ , where $\gamma$ is chosen so that $\sum_j\pi_j^\star=b$ . This result concerns unit-inclusion sampling and design-unbiased estimation of a bounded total, not mutually exclusive multi-arm assignment or a decision class containing possibly biased estimators.
Cited result lem:published-two-arm-scalar-attainment-scope verified CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.PublishedTwoArmScalarAttainmentScope For $K=2$ and $c=(1,-1)$ , the published reduced scalar minimax problem has a unique minimizer $f_n^\star$ , which is the posterior mean of $(p_+-p_-)/n$ given $X_2$ under a least-favorable prior $\pi_n^\star$ on $(p_+,p_-,r_0)$ , and the Bayes risk of $\pi_n^\star$ equals the scalar minimax value. A separate published reduction theorem equates the unrestricted full binary two-arm minimax value to that scalar value and proves that balanced Bernoulli assignment with $f_n^\star(\sum_i S_i)$ attains the full-game value and is admissible. The published Airy theorems give an explicit scalar nonlinear-shrinkage upper sequence and an asymptotically least-favorable scalar prior sequence at scale $n^{-4/3}$ . The cited source does not lift $\pi_n^\star$ to a prior on complete response schedules.

Auxiliary Lean lemmas (machine-checked) (523)

Definition AllocVec —
Definition C0 —
Definition cDaggerQ —
Definition ClusterValue —
Definition Contrast —
Definition contrastDistance —
Definition CountVec —
Definition dN —
Definition GridLPFeasible —
Definition GridPi —
Definition GridWeight —
Definition HasOrbitSaddle —
Definition IsRationalPrior —
Definition kappaC —
Definition labeledRisk —
Definition lambdaC —
Definition LosslessSymmetrization —
Definition lowerCertificate —
Definition ObsVec —
Definition Procedure —
Definition RatContrast —
Definition ratContrastToReal —
Definition RegularlyVarying —
Definition rho2 —
Definition rhoNDagger —
Definition Sc —
Definition Schedule —
Definition scheduleCounts —
Definition secondOrderScale —
Definition shrinkageProcedure —
Definition tauC —
Definition tauCount —
Definition twoArmContrast —
Definition TwoArmLikelihoodNoncollapse —
Definition upperCertificate —
Definition AsymptoticallyCoincident —
Definition cDagger —
Definition ExactGridPrimalDualCertificate —
Definition ExactInvariantProcedureCorrespondence —
Definition fullDataRuleRiskBound —
Definition GridWeightInRange —
Definition hullKappa —
Definition IsGridBarycenter —
Definition IsInvariantProcedure —
Definition MinimaxEnvelopeBound —
Definition OrbitAverageDomination —
Definition OrbitProcedure —
Definition orbitRisk —
Definition PositiveSequence —
Definition RationalCertificateCluster —
Definition scalarBayesCertificate —
Definition SlowlyVarying —
Definition Estimator —
Definition GridDualFeasible —
Definition GridDualMultipliers —
Definition IsMeasurableHullEstimator —
Definition IsPermutationAverage —
Definition IsProjectedUpperProcedureCertificate —
Definition IsTransferredPriorCertificate —
Definition k3ScalarMinimaxValue —
Definition qStarDesign —
Definition RationalGridWeightNonnegative —
Definition RealContrastTransferCertificate —
Definition RealizesOrbitProcedure —
Definition SlowlyVaryingEventuallyPositive —
Definition Unit —
Definition HullPublishedTheoremAt —
Definition BinaryScheduleDomain —
Definition HullBoundedScheduleDomain —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finCountEquivNatCount —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.natCountEquivAntidiag —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.boundedCountVectorCardinality —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3TauCountRat_sq_le_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPValueK3_le_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrastTypeScore_sub_abs_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.tauC_sub_abs_le_contrastDistance —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.tauC_mem_naturalInterval —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteDesign_E_abs_le_sqrt_E_sq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteDesign_root_mse_triangle_const —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clip_sq_dist_le —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.transferProcedure —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.transferred_root_risk_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.sqrt_worstCaseRisk_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.sqrt_rhoN_eq_root_minimax —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.root_minimax_one_sided_contrast_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finite_orbit_game_has_saddle —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmContrast_C0 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmLikelihoodNoncollapse_exists —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArm_secondOrder_lower_witness —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.hull_not_subset_binary —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.GV —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gvEquiv —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gvDecode —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gvDot_eq —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.objCoeff —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.sum_objCoeff —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.GR —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rowCoeff —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rowBound —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.encodePoint —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decodeU —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decodePi —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decodeW —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dot_norm —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dot_piNonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dot_wNonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dot_occ —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dot_risk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decode_encode_u —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_objective_encode —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_objective —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_feasible_iff —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decode_encode_pi —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decode_encode_w —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_encode_feasible —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPFeasible_u_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPFeasible_exists —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_bounded_below —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLP_rational_value_exists —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPValueRat —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLP_u_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPValue_le_rat —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLP_dual_bound_real —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridLPValue_eq_rat —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dualRiskRow_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dualRiskRow_sum_eq_one —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.firstOrderUnitScore —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.firstOrderUnitScore_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.firstOrderUnitScore_secondMoment —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrastWeightedProcedure_risk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrast_has_admissible_arm_count —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.firstOrder_minimax_limit —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarBayesRisk_le_rho2_of_scheduleKernel —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clusterValue_iff_mapClusterPt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.subsequentialLimitSet_nonempty_compact —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.allocationObservationEquivNatCount —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.allocation_observation_cardinality —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3MeanLevel_sq_ge_half_abs —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3MeanLevel_sq_ge_three_abs_sub_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarMomentMinimum_wedge_lower —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.feasible_total_half_integer —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.low_level_attainer —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.high_level_attainer —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarMomentMinimum_wedge_upper —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarMomentMinimum_wedge —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.coarseTwoArmNumericalBound —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmEffectClass_indicator_partition —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.hasEffectTriple_counts_sum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmS_of_positive_effect —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmS_of_negative_effect —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmS_of_zero_effect —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScheduleOfScoreChoices —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScheduleOfScoreChoices_score —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScheduleOfScoreChoices_effects —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreScheduleFiberEquiv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreScheduleFiber_card —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalDenominator_eq_choose —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmChooseFiber_identity —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScoreFiber_mass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmX_eq_score_card —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScoreFiber_mass_eq_binomial —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreSupportEquiv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreSupport_card —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmXMass_eq_sum_scoreMass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScore_mass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScore_mass_eq_binomial —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmXMass_eq_binomial_of_support —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScore_mass_eq_zero_of_lt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScore_mass_eq_zero_of_gt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmXMass_eq_zero_of_outside —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmXMass_eq_binomial_sum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalScheduleKernel_sum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalConditionalScore_mass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.hasTwoArmScalarKernel_canonical —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmTau_eq_effectTriple —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalScheduleKernel_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalScoreMass_sum —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalNuLift —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scoreCount —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment_jointMass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCanonicalScore_mass_eq_statisticFactor —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreFactorization —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scoreToObserved —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scoreToObserved_twoArmS —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scoreFullEstimator —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.effectTarget —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment_fullRisk_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreFactorization_fiberCarrierMass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment_statisticMass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment_statisticRisk_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalScheduleKernel_mul_sq_target —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment_fullRisk_eq_labeled —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalNuLift_expectedRisk —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.extendFinEstimator —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.extendFinEstimator_of_lt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreExperiment_statisticRisk_eq_scalar —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRaoBlackwellThroughX —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmLc_eq_two —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.effectTarget_mem_twoArmRange —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.observedScore —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarClippedEstimator —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scoreFullEstimator_scalarClipped —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarPriorRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarPriorRisk_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scalarBayesRisk_eq_sInf_scalarPriorRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clippedFullRisk_eq_statisticRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clippedScalarPriorRisk_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.canonicalNuLift_clippedRisk_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.fullScheduleBayesRisk_canonicalNuLift_eq —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBandwidth —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBandwidth_pos —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBandwidth_le_half —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBandwidth_sq —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorWeight —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorWeight_bounds —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreAverage —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTarget —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorWeight_hasDerivAt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTarget_hasDerivAt —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliUnitDesign —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliVectorDesign —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliUnit_signed_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreAverage_mean —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTargetDeriv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTargetDeriv_mean —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihood —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihoodDeriv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihood_eq_design —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihood_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihood_sum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihood_hasDerivAt —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliRawScore —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihoodDeriv_eq_mul_score —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliUnit_rawScore_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliUnit_rawScore_var —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliRawScore_sq_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihood_pos —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulli_guardedScore_eq_rawScore —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulli_fisherInformation —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTargetDeriv_mean_lower —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulli_fisherInformation_upper —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCountKernel —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCountKernel_mass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCount_statewiseSquaredLoss_eq_statisticRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCount_expectedLoss_eq_scalarPriorRisk_extend —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmCount_expectedLoss_eq_scalarPriorRisk_restrict —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteDesignBayesRisk_twoArmCount_eq_scalarBayesRisk —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothResponseTypeDesign —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmObservedScore —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothResponseType_score_mass —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.responseVectorEffectTriple —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmClampedParameter —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmClampedParameter_abs_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmClampedParameter_eq —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothEffectDesign —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.effectTriple_nonempty —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothEffectDesign_p_measurable —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothEffectKernel —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothEffectKernel_apply —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothEffectKernel_singletonReal —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.inducedFiniteDesign_twoArmSmoothEffect_p —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmResponseEffect —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothResponseType_effect_score_mass —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.effectTarget_responseVectorEffectTriple —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothMixedCountLoss_eq_inducedSquaredRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothRealBayesRisk_eq_scalarBayesRisk —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_sum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_integral_eq_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_integrable_count —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmBernoulliLikelihoodDeriv_integrable_count —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_hasDerivAt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_pos —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihoodDeriv_sum_eq_zero —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihoodDeriv_integral_eq_zero —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliIntegral_hasDerivAt —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_absolutelyContinuous —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTarget_absolutelyContinuous —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteCountProduct_integrable_of_sections —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteCountProduct_integrable_of_continuousSections —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_hasDerivAt_ae —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTarget_hasDerivAt_ae —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularBernoulliLikelihood_continuousOn —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTargetDeriv_continuousOn —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSensitivityField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmErrorSqField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmRegularLikelihoodScore_continuousOn —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmFisherScoreField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPriorScoreField_integrable_section —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPriorScoreField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreCrossField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteCountProduct_aestronglyMeasurable_of_sections —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreSqField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmErrorScoreField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmDerivativeBalanceField_integrable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmFiniteVanTreesRegularity —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmSmoothVanTreesErrorLowerBound —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmScoreAverage_eq_count —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorTarget_eq_of_scoreCount_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_sum_prod_mul_apply —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_marginal —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_effectTarget —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_mean_unit_posterior —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_first_moment —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_square_completion —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_product_formula —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_responseSchedule —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_scheduleResponse —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_responseScheduleEquiv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_hasEffectTriple_iff —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_product_eq_of_effectTriple_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_smoothEffect_canonical_mixture —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_zeroAssignDesign —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_zeroAssign_labeledRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_effectRisk_eq_responseRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_error_le_effectRisk —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_smoothPriorMeasure —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_smoothPriorMeasure_isProbability —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_fraction_le_mixedLoss —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_mixedLoss_clip_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArmPosteriorCompat_fraction_le_rho2 —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.signGroupScale —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.signGroupScale_pos —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.signGroupScale_sq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.positiveCoefficientSum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.negativeCoefficientSum —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.signGroupAssignment —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.embeddedSignSchedule —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.inactiveZeroObservation —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.embeddedSignSchedule_observation —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.embeddedSignSchedule_target —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.fairBoolDesign —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.signLatentDesign —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.signLatentDesign_E_fst —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scaledLatentEstimator —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scaledLatentEstimator_mem_Icc —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.inducedTwoArmProcedure —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.inducedTwoArmProcedure_statewiseRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.inducedTwoArmProcedure_worstCaseRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.embeddedTwoArmLowerBound —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.exists_positive_contrast_arm —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.exists_negative_contrast_arm —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.positiveContrastArm —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.negativeContrastArm —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.positiveContrastArm_pos —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.negativeContrastArm_neg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.support_eq_positive_negative_of_card_two —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.coefficient_eq_zero_of_card_two —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.positiveContrastArm_coeff —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.negativeContrastArm_coeff —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoToActiveAssignment —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.activeToTwoAssignment —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.positiveContrastArm_ne_negativeContrastArm —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.activeToTwo_twoToActive —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.activeSchedule —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.activeSchedule_observation —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.activeSchedule_target —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.liftTwoArmProcedure —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.liftTwoArmProcedure_statewiseRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.supportTwoUpperBound —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.contrastWeightedProcedure_upperRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.minimaxEnvelopeBound_of_contrastWeightedProcedure —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3SignedScore_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3SignedScore_var —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.tauC_cDagger_eq_k3Mu_average —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3SignedScore_average_mse —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.decodeGridDualMultipliers —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_optimal_primal_dual_exists —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarRule —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarRule_mem_Icc —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridProgram_dualObjective_eq_normalization —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.GridRowIndex —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridRowEquiv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.sum_grid_rows —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.exists_exact_grid_primal_dual_certificate —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rationalContrast_admissibleArmCount —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.tauCountRat_cast —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rationalOrbitModel —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rationalPriorOf —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rationalGridDesignOf —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.posteriorResidual_eq_secondMoment_sub —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.posteriorResidual_eq_allocationBayesRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.Lc_ratContrastToReal —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.hRat_cast —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.hRat_sq_eq_C0 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.tauCount_mem_gridInterval —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gammaMC_mem_gridInterval —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.posteriorMean_mem_gridInterval —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.lowerCertificate_eq_posteriorResidual_sInf —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.orbitModel_minimax_eq_orbitGameValue —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.lowerCertificate_le_rhoN —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridBarycenter_eq_conditionalBarycenter —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.upperCertificate_le_gridObjective —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rhoN_le_upperCertificate —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridDualObjective_le_fixedGridBayesRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.gridObjective_le_lowerCertificate_add_mesh —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarBoolDesign —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarProductDesign —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarBoolDesign_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarBoolDesign_var —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarRule_risk_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarMinimaxValue_le_exact —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarLeastFavorableWeight —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarPriorRisk —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.fin3FunEquiv —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarLeastFavorableWeight_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarLeastFavorableWeight_sum —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarPriorRisk_decomposition —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarRule_homogeneous_risk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarPriorRisk_rule —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarPriorRisk_lower —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3ScalarMinimaxValue_ge_exact —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_000 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_001 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_010 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_011 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_100 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_101 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_110 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le_111 —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_risk_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_boundary_risk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_worstCaseRisk —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataFin3FunEquiv —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataR1 —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataS1 —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataSMinus —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataTable —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.cDaggerQ_zero —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.cDaggerQ_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.cDaggerQ_two —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.cDagger_apply —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.qStarDesign_cDagger_p —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataTable_scaled_mem —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clip_cDagger_eq_self —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRule_coe —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3BoundarySchedule —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataTargetQ —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataAssignmentProbQ —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataEstimateQ —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRiskQ —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataTargetQ_cast —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataAssignmentProbQ_cast —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataEstimateQ_cast —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.k3FullDataRiskQ_cast —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedUpperUnitScore —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedTypeScore —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedUpperUnitScore_mem_Icc —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedUpperUnitScore_mean —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedUpperUnitScore_secondMoment —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.lambdaC_le_abs_normalizedTypeScore —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedTypeScore_sq_lower —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedScore_tail_bound —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteDesign_cov_monotone_comp_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteDesign_cov_perturbation_lower —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedTypeScore_mem_Icc —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clippedScore_monotone —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clippedScore_abs_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clippedScore_eq_self —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.clippedScore_sub_abs_le —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedScore_mean_variance —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.normalizedScore_bounds —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.finiteDesign_mse_shrinkage_upper —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.shrunkenClippedScore_abs_le_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.shrinkagePowerIdentities —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.shrinkageProcedure_risk_bound_of_tail —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.dyadic_ratio_limit_eq_one —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.shrinkage_tail_prefactor_tendsto_zero —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.secondOrderScale_mul_inverse —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.responseCountOrbitCard_pos —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.responseCountFiber_card —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.scheduleCounts_permute_eq —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfRationalPrior —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfRationalPrior_permute —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfRationalPrior_E_permute —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfRationalPrior_E_count —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.lowerCertificate_le_schedulePriorRisk —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.lowerCertificate_nonneg —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.transferredSchedulePrior_lowerBound —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.projectedUpperProcedureCertificate —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.fixedRealContrastTransferCertificate —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.exists_ratContrast_close —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.coordinate_le_two_contrastDistance —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.realContrastTransferCertificate_bounds —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.secondOrderScalePositive —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.eventuallyPositivePatch —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.eventuallyPositivePatch_eventually_eq —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.secondOrderScalePositive_regularlyVarying —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.regularlyVarying_div —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.asymptoticallyCoincident_of_scaled_sandwich —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.secondOrderScale_mesh_tendsto_zero —
Definition embeddedSignSchedule —
Definition IsRationalGridCertificateSequence —
Definition liftTwoArmProcedure —
Definition orbitToInvariantProcedure —
Definition RationalGridCertificateAsymptotics —
Definition responseCountOrbitCard —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.selectRationalGridCertificateSequence —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.rationalGridCertificateAsymptotics_proof —
Definition CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfCountPrior —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfCountPrior_E_count —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.schedulePriorOfCountPrior_E_permute —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.orbitSaddle_labeledBayesLower —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.bayesianInformation_ae_section_nonnegative —
Definition secondOrderRateAndCertificateOpenFrontier —
Definition Assign —
Definition contingencyFiber —
Definition HullEstimator —
Definition hullRisk —
Definition HullSchedule —
Definition pMinus —
Definition pPlus —
Definition twoArmNegativeEffectType —
Definition twoArmPositiveEffectType —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArm_responseType_cases —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArm_responseTypes_pairwise_distinct —
Lemma CausalSmith.Experimentation.MultiarmSecondorderMinimaxFrontier.twoArm_tauCount_eq_effect_difference —
Definition attainmentAndK3CertifiedConverseScope —

Symbol realizations (Lean @realizes) (53)

$\rho_n(c)$ verified realized by rhoN
d_n(c) verified realized by dN, MinimaxEnvelopeBound
G_n(c) verified realized by orbitGameValue
$\widehat\tau_{\mathrm{cHT}}^\star$ verified realized by contrastWeightedProcedure
u verified realized by GridLPFeasible
$\Lambda_{n,M}(c)$ verified realized by gridLPValue
$c^\dagger$ verified realized by cDaggerQ
$\Lambda_{n,M}^\dagger$ verified realized by gridLPValueK3
$\rho_n^\dagger$ verified realized by rhoNDagger
$C_{\mathrm{so}}(c)$ verified realized by PositiveLimitBranch
C_A verified realized by AiryConstant
$P_m(x\mid r)$ verified realized by orbitLik
X_2 verified realized by twoArmX
$\mathcal A_K$ verified realized by Arm
i verified realized by Unit
t verified realized by RespType
$\mathcal T_K$ verified realized by RespType
z verified realized by Schedule
A verified realized by Assign
$Y_i^{\mathrm{obs}}$ verified realized by ObservedOutcome, obsOutcome
K verified realized by AdmissibleArmCount
c verified realized by RealContrast
n verified realized by variable@57
a verified realized by variable@57
S_c verified realized by Sc
L_c verified realized by Lc
$q^\star$ verified realized by qStar
Y_i(a) verified realized by potentialOutcome
$\tau_c(z)$ verified realized by tauC
$\widehat\tau$ verified realized by Estimator
$\mathcal D$ verified realized by Procedure
$R_n(\mathcal D,\widehat\tau;z)$ verified realized by labeledRisk
C_0(c) verified realized by C0
m verified realized by CountVec
$\mathcal M_{n,K}$ verified realized by CountVec
r verified realized by AllocVec
$\mathcal R_{n,K}$ verified realized by AllocVec
x verified realized by ObsVec
h verified realized by Contingency
$\mathcal H(m,r,x)$ verified realized by contingencyFiber
$\tau_c(m)$ verified realized by tauCount
$\delta$ verified realized by OrbitEstimator
$\pi$ verified realized by OrbitProcedure
M verified realized by variable@234
$\Gamma_M$ verified realized by gammaMC
j verified realized by gammaMC
$\Gamma_{M,c}$ verified realized by gammaMC
g verified realized by gammaMC
w_{r,x,g} verified realized by GridWeight, GridWeightInRange
p_+ verified realized by pPlus
p_- verified realized by pMinus
r_0 verified realized by rZero
a_n verified realized by PositiveSequence

Proof map 16 statements · 24 edges

Second-order Minimax Risk in Multi-arm Binary Randomization

Abstract

Introduction

Related work

Setup and the finite orbit game

First-order risk and the second-order rate

Finite-program certificates for rational and real contrasts

Three-arm diagnostics and support-specific conclusions

Discussion, limitations, and future work

Design implications for multi-arm binary experiments

Limitations and future work

Appendices

Proofs and auxiliary lemmas

Verification note

Environment.

Matched environments.

Presentation-only environments.

What is not checked.

Proofs of the main results

References