CausalSmith Working Papers

For fixed interaction order $\beta\ge1$ and fixed assignment probability $p\in(0,1)$, this paper establishes a finite-population design-based minimax mean squared error frontier, up to constants depending on $(\beta,p)$, for estimating the all-treated-versus-all-control effect under known bounded-degree low-order polynomial interference and common-probability Bernoulli assignment. The constants depend only on $(\beta,p)$ and are uniform in $n$, $d$, and $B$; they may deteriorate as $p$ approaches $0$, $1$, or a zero of the Bernoulli contrast. For radius $B$, population size $n$, and degree bound $d$, the paper's coefficient-mass and uniformly bounded-outcome minimax risks share the scale $$B^2\min\!\left\{1,\frac{dA_d}{n}\right\},$$ where $A_d$, the complete-block score energy, is the design second moment of the complete-block score used by the score-weighted neighborhood inverse-probability estimator (SNIPE, the shifted-neighbourhood inverse-probability estimator introduced by \citet{CortezRodriguezEichhornYu2023}). Equivalently, $$dA_d\asymp_{\beta,p} d\binom d{k_\star(d,\beta,p)},$$ where $k_\star(d,\beta,p)$, the largest interaction order with nonzero all-one-versus-all-zero Bernoulli contrast, is the exposed order in that contrast. A clipped SNIPE estimator attains the minimax rate on both bounded classes, while unprojected SNIPE attains the linear-regime variance scale. Complete directed $d$-blocks, with $\lfloor n/d\rfloor$ active blocks in the block construction of Definition 15, supply the matching lower-bound construction. When $d\mid n$, the same blocks give the exact worst-case risk $B^2A_d/m=B^2dA_d/n$, where $m=n/d$ is the number of complete blocks, for unprojected SNIPE. The complete-block local linear benchmark is solved exactly for block-local design-unbiased weights on fixed complete directed blocks under the coefficient-mass schedule class, with minimax risk $B^2A_d/m$ for a population of $m$ complete $d$-blocks. For the fair-coin design, even-order Bernoulli contrasts cancel, $A_d=4\sum_{1\le r\le \min\{\beta,d\},\,r\text{ odd}}\binom dr$, and first-order interference yields the frontier $B^2\min\{1,d^2/n\}$.

This paper characterizes the finite-sample minimax risk for estimating the scalar optimal-treatment value with a binary treatment, binary outcome, and a discrete covariate alphabet of size $d$. Under consistency, conditional exchangeability, i.i.d. observed sampling, and a fixed overlap parameter $\epsilon\in(0,1/2)$, the causal value equals the observed-law functional that sums, over covariate cells, the larger of the two arm-specific outcome regressions. For sample size $n$, the fixed-sample minimax squared risk satisfies $$\mathfrak R_{n,d,\epsilon} \asymp_\epsilon \min\left\{1,\frac{d}{n\log(ed)}\right\}.$$ The upper bound is attained by an explicit Jackson factorial estimator: pilot counts form local rectangles for each four-coordinate treatment--outcome cell table, a tensor Jackson polynomial approximates the cellwise maximum contribution, centered factorial moments lift the polynomial into unbiased count statistics under Poisson splitting, and Rao--Blackwellization returns an all-data fixed-sample estimator. The lower bound embeds normalized two-sample $L_1$ distance into an equal-propensity slice of the observed model and uses moment matching for the absolute-value cusp. The matched rate gives uniform consistency exactly when $d=o\{n\log(en)\}$ and gives the parametric $n^{-1}$ minimax scale exactly along bounded-alphabet sequences.

This paper establishes finite-sample minimax bounds for a scalar average treatment effect in a real-outcome observed-data model with a finite discrete adjustment variable. The model imposes consistency, conditional exchangeability, and fixed overlap parameter $\epsilon$; allows arbitrary cell masses; fixes the known outcome scale $M$, with conditional means in an interval of radius $M/2$; bounds conditional second central moments by $M^2$; and restricts the maximal cell-effect deviation to at most $\sigma M$, where $\sigma$ is the heterogeneity radius. For sample size $n$ and alphabet size $d$, the number of adjustment cells, the paper constructs two clipped estimators and a clipped three-branch selector: a heavy--light signed Chebyshev factorial estimator for rare cells, an occupancy-weighted treated-control estimator for empirically crossed cells, and a constant-zero branch. The known-radius selector satisfies $$\sup_{P\in\mathcal P_{d,\epsilon,M,\sigma}} E\{(\widehat\tau_n^\star-\tau(P))^2\} \le C_\epsilon M^2 \left[ \frac1n+ \min\left\{ 1,\frac{d^2}{n^2\log^2(en)},\sigma^2+\frac d{n^2} \right\} \right].$$ The same model class admits the lower benchmark $$c_\epsilon M^2 \left[ \frac1n+\min\left(1,\frac d{n^2}\right) +\sigma^2\min\left\{1,\frac{d^2}{n^2\log^2(en)}\right\} \right].$$ These two bounds match in order at exact homogeneity, at the unrestricted-radius endpoint, for every fixed positive radius, in the saturated alphabet regime, and at the parametric-dominance elbows. The results give a radius-indexed minimax bracket for point-estimation mean-squared error under fixed overlap, known outcome scale $M$, and known heterogeneity radius $\sigma$. The selector attaining the upper bound is a known-radius procedure: it uses $\sigma$ as an input to choose deterministically between its polynomial, collision, and zero branches, and the regime theorem isolates the residual shrinking-radius region generated by the displayed upper and lower benchmarks.

This paper studies honest confidence sets for a transported complier effect in a two-sample encouragement design. Outcomes, receipt, encouragement, and covariates are observed in a source population, while covariates are observed in a target population. Under the transported complier-effect model class with binary receipt, bounded outcomes, instrument overlap, monotonicity, and target-to-source covariate domination, the target complier effect $\theta_T$, the target complier-conditional outcome contrast, is identified as a transported reduced-form ratio and lies in the compact causal range $\Theta=[-1,1]$. The expected-length frontier is governed by the effective strength $t_n=n\mu_n^2/\kappa_n$, where $\mu_n$ is the transported first-stage mean and $\kappa_n$ is the Kish second moment of the transport weights. For every fixed threshold $t_0>0$, oracle honest confidence sets have minimax expected length of order $\min\{1,t_0^{-1/2}\}$, both globally and conditional on each admissible deterministic source-covariate, transport-weight, and propensity geometry. The lower bound is matched by an oracle Anderson--Rubin/Fieller score inversion. In uniform finite-cell designs, empirical target cell frequencies yield a score inversion that is sample-only in the uniform class with balanced assignment, while the regular nonuniform extension additionally uses known source cell probabilities and a known cell-varying propensity, the covariate being observed as a finite-cell label and the cell count obeying $k_n=o(\sqrt n)$, attaining the same order; the same conclusion extends to regular nonuniform source cells with known cell probabilities and known cell-varying propensity.

This paper establishes two new unconditional logarithmic minimax lower bounds for distance-compressed boundary regression. In the unsigned experiment, a rule observes outcomes and scalar Euclidean distances from each boundary query point. Over the compact nonparametric law class $\mathcal P_{\mathrm{NP}}(L,q)$ in Definition 9, with $q\geq1$ and $L\geq4$, both the CTY common-map risk in Definition 12 and the larger point-indexed outer risk in Definition 14 have minimax expected boundary sup-loss bounded below on the scale $$a_n=\left({\log n\over n}\right)^{1/4}.$$ The support-boundary hypercube constructs separated boundary perturbations with matching scalar-distance information at the queried point, yielding the same lower-bound scale even when the rule may use point-indexed Borel sections. Economically, this experiment isolates the risk cost of summarizing proximity by scalar distance after angular location and treatment-side information have been removed: a distance-only rule cannot tell which boundary arc or treatment side generated nearby observations, so recovering the entire boundary curve incurs a logarithmic testing penalty. The second unconditional lower bound holds in a signed-distance experiment with known treatment geometry, already on a fixed rectangular subexperiment. Conditional on the three CTY-style analytic inputs collected in Definition 33, the stabilized local-polynomial estimator and that lower bound characterize the signed-distance expected outer-risk rate on the same scale for every fixed polynomial order $p$, moment exponent $\nu\ge2$, and envelope $L\ge L_0(p)$. Thus the unsigned contribution is lower-bound sharpness, while the signed two-sided frontier is explicitly conditional on $\mathsf{AI}_{p,\nu,L}$.

This paper studies estimation of the partially linear coefficient $\theta_0$, the treatment coefficient, when the treatment innovation is independent of covariates, sub-Gaussian, and separated from Gaussianity by a fixed nonzero cumulant. The identifying resource is a zero of the treatment-innovation moment-generating function: the associated polynomial-exponential weight annihilates real shifts induced by treatment-code error, and a contour average of observable residual transforms identifies $\theta_0$ under the stated boundary zero-freeness, nuisance zero-freeness, and positive-count conditions. The statistical theorem is a fixed-code result. For fixed primitive constants, i.i.d. product sampling, the partially linear conditional-mean restrictions, bounded coefficient and regression ranges, sub-Gaussian treatment and outcome-noise envelopes, a nonempty fixed-code class, and the displayed current $L^1(P_X)$ treatment-code radius gate, where $P_X$ is the covariate marginal, a finite translated-dyadic contour bank and a total Borel statistic attain matched $c/n$ and $C/n$ mean-squared-error bounds on the non-Gaussian spectral class. For $\gamma\in(1/2,1)$, where $\gamma$ is the probability level, the same statistic controls the generalized lower $(1-\gamma)$-quantile of absolute error by order $(\gamma n)^{-1/2}$. The same fixed-separation rate holds on the aligned Jin--Mackey--Syrgkanis ACE comparison class. The ACE comparison records the published finite-order ACE upper guarantee and compares it with the contour upper guarantee on the common clipped-code class. The paper also gives a bounded-outcome Gaussian diagnostic and explicit sine-ratio reductions for mixture benchmarks. The represented-data construction is a conditional reproducibility layer for the same ordinary Borel statistic. When a compiled bounded spectral adapter satisfies the full canonical build-and-compilation specification for the parameter record, transported fixed records, and base treatment-code sequence, its represented-data execution realizes the statistic used in the statistical risk statements.

This paper studies design-based variance estimation for random group-formation experiments in finite populations. Units are assigned to disjoint groups of a fixed common size $M$, the group-size parameter, and treatment is randomized across realized groups. The target is the finite-population partition-average marginal effect, defined by averaging composition-indexed potential outcomes over the uniform $M$-slice of possible groups. The exact variance of the group-level difference-in-means estimator admits a Kneser covariance representation: disjoint group formation diagonalizes in Johnson harmonic degrees, and the scalar equal-group CR2 statistic differs in expectation from the exact variance by the Kneser covariance of the arm-difference table. In triangular arrays with growing group count $G_n$, the number of realized groups, stable treatment fraction, bounded potential outcomes, and grouped-unit sampling fraction $N_n/n$ converging to $\rho$, the limiting grouped-unit fraction, the exact design variance $\sigma_n^2$ of the difference-in-means estimator satisfies, after scaling by $G_n$, $$G_n\sigma_n^2=R_n-\rho E_{\tau,1,n}+o(1),$$ where $R_n$ is the independent-group leading variance scale and $E_{\tau,1,n}$ is the degree-one Johnson energy of the arm contrast. Consequently, the equal-group CR2 statistic estimates $R_n/G_n$, is asymptotically conservative for the exact variance in the stated one-sided probability sense, and is ratio-consistent exactly when the dense correction is negligible relative to the corrected variance scale. At zero sampling density, ratio consistency holds under $N_n/n\to0$, including regimes with $N_n^2/n\to\infty$. The paper also identifies the scalar statistic returned by a versioned clubSandwich CR2 calculation, gives an eight-unit witness with exact variance $1/7$ and expected CR2 value $2/7$, and proves a positive-density lower bound: for a fixed group size $M\ge2$, a strictly positive limiting grouped-unit fraction $\rho>0$, an outcome envelope $B\ge1$, and a variance floor $c_\sigma$ below the separated independent-arm limit $1/[Mp(1-p)]-2\rho/M$, every measurable one-realization variance statistic has nonvanishing worst-case relative error over the bounded schedule class with those parameters.

This paper studies nonlinear causal representation learning from an observational environment and one unknown-target perfect intervention for each scalar latent variable. In a positive $C^3$ compact-cube latent model with shared $C^2$ diffeomorphic mixing, causal minimality, and fixed own-coordinate derivative signs, likelihood ratios between interventional and observational laws define observable one-dimensional ratio coordinates. Comparing their laws across environments with Gaussian maximum mean discrepancy yields a directed ratio-discrepancy graph. The first result establishes that, in every nonempty fixed-sign stratum, the mechanisms for which these ratio laws separate all transported ancestral covers form an open dense set. On this cover-separated population class, a decoder recovers the transitive closure of the target-permuted latent DAG, constructs conditional-rank coordinates that agree with monotone transforms of the intervened latent variables, aligns intervention labels with coordinates, and prunes parents exactly by conditional independence under the observational law. Within the same structural class, equality of the observed environment-law family identifies the observed world up to componentwise $C^2$ coordinate changes and simultaneous relabeling of graph and targets. A sample-split procedure adds familywise-valid confidence edges for ancestral discoveries, conditional on a simultaneous first-stage $L^1$ likelihood-ratio error event. On the overlapping smooth compact-support faithful regime, the result gives an arbitrary-dimensional one-perfect-intervention-per-node counterpart to the bivariate law-separation route of \citet{vonKugelgenEtAl2023UnknownInterventions}.

This paper studies proxy-based causal inference for latent-class mean treatment effects in a fixed finite-class model. Under the Virk--Mazaheri--Wu proxy moment structure with fixed latent cardinality $k$ (the number of latent classes), fixed proxy dimensions, bounded observable contributions, supplied latent-arm positivity and proxy-rank margins, the target is the quotient latent-effect law $\nu_P$, the population-weighted distribution of latent-class average treatment effects with masses aggregated at coincident class means. The main population result establishes a gap-free Lipschitz modulus from the observable five-block summary $S(P)$, the vector of proxy moment summaries, to $\nu_P$ in $W_1$, the one-Wasserstein distance. The modulus covers homogeneous, partially colliding, and separated effect configurations within the same uniformly conditioned class. The statistical results turn this modulus into root-$n$ law estimation and confidence reporting. A nearest-summary repair estimator and a structured-lattice estimator, calibrated by the supplied constants $k,d_x,d_z,L,\pi_0,\sigma_0$, are Borel sample maps and achieve uniform root-$n$ $W_1$ tail bounds over the model class. In the fixed-dimensional unit-cost exact-real model, the lattice estimator has polynomial candidate growth in $n$ and centers an honest finitely represented exact-real Wasserstein outer confidence set. The associated cluster report merges empirically unresolved support points and reports compatible aggregate mass intervals. On separated gap-local strata, effect-ordered latent masses have clipped inverse-gap risk $\min\{1,(\sqrt n\,g)^{-1}\}$, where $g$ is the local effect-gap scale. Local and same-class lower bounds from explicit two-class proxy experiments establish root-$n$ sharpness for quotient-law estimation and the inverse-gap rate for ordered weights on the displayed two-class specialization.

This paper characterizes the observable-data minimax risk for stationary off-policy evaluation in finite partially observed Markov decision processes. The data are one behavior-policy trajectory of length $T$, the trajectory horizon, with bounded rewards and binary actions; policies are known functions of the observed state; the joint observed--latent Markov state satisfies stationary start, sequential ignorability, one-step policy overlap $L=\exp(\zeta)$, where $\zeta$ is the log-overlap scale, geometric total-variation contraction $\alpha=\exp(-1/t_0)$, where $t_0$ is the mixing scale, and latent stationary overlap $d_e(s)\le C d_b(s)$, where $C$ is the stationary-overlap constant and $d_e$ and $d_b$ are the target and behavior stationary joint-state laws. For sufficiently large $T$, and cardinality-uniformly over finite observed and latent alphabets, the minimax squared-error risk is comparable, uniformly over $C\ge1$, to $$T^{-1}+T^{-\beta}q_C^{2(1-\beta)}, \qquad q_C=\frac{C-1}{C}, \qquad \beta=\frac{2}{2+t_0\zeta},$$ where $q_C$ is the overlap-distance radius, the normalized distance of the stationary density-ratio bound from unit overlap, and $\beta$ is the rate exponent set by the mixing and log-overlap scales, up to constants depending only on $t_0$ and $\zeta$. For every fixed $C>1$, the surface has the Hu--Wager partial-history exponent $T^{-\beta}$; the parametric term changes the rate on shrinking overlap-distance scales $q_C\lesssim T^{-1/2}$. A clipped partial-history importance weighting (PHIW) estimator with radius-calibrated history depth attains this surface when the depth is calibrated from $(t_0,\zeta,C)$. A signed-depth finite POMDP pair supplies matching hard alternatives while satisfying the same overlap and contraction conditions. A finite controlled insulin-policy demonstration shows that bounded latent stationary occupancy and bounded action overlap can hold together in a stylized treatment model.

This paper characterizes sharp bounds on the survivor-complier strict-benefit probability in a finite ordered-outcome instrumental-variables model with treatment-induced selection. The target is the probability $\theta$, the survivor-complier benefit probability, that treatment raises the ordered outcome among units whose treatment receipt is shifted by the instrument and whose outcome would be observed under either treatment state. Under conditional instrument independence, consistency and exclusion through received treatment, instrument overlap, treatment monotonicity, covariate-specific weak selection monotonicity, and positive aggregate survivor-complier mass, observed instrument contrasts identify two cellwise selected-complier outcome capacities. Their smaller total is the survivor-complier mass, including cells with zero selected-complier capacity gap. The sharp identified set is the interval obtained by solving a branch-free exact-mass partial-transport problem in each covariate cell and aggregating the resulting threshold-cut values. The lower and upper endpoints have closed finite formulas, and endpoint-attaining full latent laws establish sharpness. The same construction gives sparse endpoint witnesses computable in $O(|\mathcal X|K)$ operations for finite covariate support $\mathcal X$ and $K$ ordered outcome levels, with $O(|\mathcal X|K^2)$ work when full coupling matrices are materialized. A three-level synthetic witness yields the sharp interval $[0,0.7]$. For i.i.d. samples from the fixed finite-slate structural class with fixed finite support, prescribed screening thresholds $\eta_n$ satisfying $\eta_n\to0$ and $\sqrt n\,\eta_n\to\infty$, a common instrument-overlap bound, and a uniform positive lower bound on aggregate survivor-complier mass, a projected screened plug-in estimator has a fixed-law Hadamard directional limit and a deterministic guarded confidence interval contains the full sharp identified interval uniformly.

We study independent, heterogeneous Bernoulli assignment in bipartite experiments, where intervention units are assigned treatment and outcome units may depend on assignments in known intervention neighborhoods. For the finite-population contrast between all-treated and all-control neighborhood outcomes, we analyze an exposure-weighted Hájek estimator under bipartite neighborhood interference. With bounded potential outcomes, we derive a graph-and-design-dependent envelope that upper-bounds the design-based variance scale of its linearization and can be minimized subject to a positivity-constrained expected-treatment budget. Under the full assumptions of Theorem 4 and Theorem 5, the envelope-optimal design supports asymptotic normality and a graph-only Wald scale yields asymptotically conservative coverage. We also give conditions under which heterogeneous probabilities strictly improve the envelope relative to homogeneous assignment and study a separable overlap-based surrogate under an admissible budget and bounded outcome degree. A complementary unbounded-degree construction shows that degree dispersion and comparable surrogate weights alone do not control this approximation. Together, the results provide an outcome-model-free design criterion with convex, inferential, and separation guarantees, and identify approximation questions for surrogate criteria.

Continuous-treatment modified treatment policies can map a range of natural treatment values to a single assigned dose. This paper studies the exact lower-threshold clamp, which sends treatments below a deterministic threshold $\delta_n$, the policy threshold, to $\delta_n$ and leaves larger treatments unchanged. Under a fixed finite-stratum observed-data model with bounded outcomes, Hölder outcome regression with known exponent $\beta$ and radius $L$, and polynomial lower-tail treatment-density thinning $c_-a^\kappa\leq\pi_x(a)\leq c_+a^\kappa$ with known exponent $\kappa$, the post-policy law contains a moving atom. The target is the observed clamp mean $\theta_{\delta_n}(P)$, the sum of the retained mean above the threshold and an atom-regression product at the boundary. The paper establishes matching minimax absolute-risk and uniformly honest expected-length rates $$r_n=n^{-1/2}+\delta_n^{\kappa+1}h_n^\beta, \qquad n h_n^{2\beta+1}(\delta_n+h_n)^\kappa\asymp1,$$ attained by a split-sample total-Gram stabilized local-polynomial estimator and a bias-aware interval. The phase diagram identifies the regular, critical, atom-dominated, fixed-threshold, and zero-threshold regimes induced by $\delta_n$. A full-data lift gives the same frontier a causal interpretation under simultaneous consistency, conditional exchangeability, and structural-mean continuity on the threshold range. A continuity-only model yields the separate frontier $n^{-1/2}+\delta_n^{\kappa+1}$, with an explicit elbow and fixed-positive-threshold behavior.

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 $\ell_1$ norm of the contrast, and the contrast-weighted allocation $q_a^\star\propto |c_a|$ 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.

This paper studies offline policy learning for deterministic treatment rules when overlap can deteriorate near either propensity boundary in the same region where the treatment contrast is small. The target is observed-law welfare regret, defined directly from the conditional treatment contrast. The law class imposes bounded outcomes, positivity, a margin condition for small treatment contrasts, a zero-effect convention, and a joint overlap-decay restriction that ties weak treatment-arm information to the small-contrast region. Under the margin-window normalization and the auxiliary calibrations used by the two-point construction, the paper proves an observed-law minimax lower bound $$M_n(\alpha,\gamma)\ge c\,n^{-r_\star(\alpha,\gamma)}, \qquad r_\star(\alpha,\gamma) = \frac{1+\alpha}{2+\alpha+\beta_{\alpha,\gamma}}, \qquad \beta_{\alpha,\gamma} = \begin{cases} 0,&\gamma=0,\\ \alpha\gamma/(\alpha+1),&\gamma>0. \end{cases}$$ The exponent is obtained by balancing margin mass, local contrast size, and the probability of observing the informative treatment arm. The paper also gives a conditional analysis of a specified clipped cross-fitted AIPW empirical welfare rule with supplied nuisance estimates satisfying explicit rate, boundedness, cross-fitting, and localized empirical-process conditions. In the nonbinding nuisance-and-clipping regime, this conditional upper exponent matches the lower-bound exponent up to logarithmic factors; in the binding regime, the analysis gives the rule's nuisance-limited exponent.

This paper studies how to place measurement rounds in a finite-population rollout experiment when the target is the full-adoption contrast but the rollout budget stops at a treated fraction $q<1$. Under static rollout consistency, a degree-$\beta$ polynomial restriction on the rollout mean curve, and a common round-mean variance envelope---restrictions that low-order interference motivates but that we impose rather than derive from a microfounded interference model---unbiased linear estimation reduces to a polynomial extrapolation problem from $[0,q]$ to $1$. For a fixed schedule, the worst-case variance over the positive-semidefinite covariance class meeting this diagonal envelope equals the variance scale times the squared $\ell^1$ (total-variation) norm of the polynomial-exact weights. The resulting diagonal-envelope amplification criterion, defined over linear unbiased estimators, attains its minimax value up to multiplicative constants at shifted Chebyshev-Lobatto measurement fractions when $q\le q_{\max}<1$ and the number of rollout intervals satisfies $k\ge c\beta$ for some $c>1$. The minimax amplification is of order $$\left(\frac{(1+\sqrt{1-q})^2}{q}\right)^{2\beta},$$ up to constants depending only on $q_{\max}$ and the oversampling ratio. For the exact nested rollout covariance problem, the same Chebyshev schedule is rate-feasible through the envelope upper bound, and exact optimality under the true rollout covariance structure is posed as a separate covariance-specific design question.

We study a population identification question for a bivariate latent linear non-Gaussian model in which the number $m$ of middle source slots and an axis normalization are maintained inputs. The representation-level target asks whether a truncated joint-cumulant vector through order $L$ can also arise from the opposite axis-normalized arrow convention with the same $m$. At $L=2m+2$, the unordered finite loading-slope support is generically recovered and the opposite cumulant-map fiber is empty, while the same-arrow parametrization retains ambiguity beyond middle-slot relabelling. The closure of the opposite-arrow compatibility locus has codimension exactly one in each arrow image variety. For $m\ge3$, generic real opposite-fiber exclusion already holds at order $2m+1$; loading-support recovery uses the apolar order $2m+2$. There exist Euclidean-open parameter neighborhoods whose intersections with the feasible regions are nonempty, sharpening the population geometry of the maintained representation class.

This paper studies the population treatment coefficient from pooling a staggered-adoption panel in a unit-and-time fixed-effect Poisson pseudo-likelihood. We show that this projection coefficient can be negative even when every cohort-time proportional treatment effect is positive. The failure reflects heterogeneous effects interacting with the fixed-effect PPML score: a four-cohort design with equal shares and flat untreated means provides an explicit sign reversal, while a positive-weight proportional treatment-on-the-treated target remains positive. The result characterizes the interpretation of the deterministic population projection targeted by the pooled criterion.

This paper studies when a covariance relaxation for interference-aware randomized design is exactly implementable by a finite $\{\pm1\}$ assignment law. The setting is a design-based finite-population model with two equal homophilous communities, sign-symmetric assignment, and a block-weighted graph. The design objective combines a graph Laplacian term, a Laplacian-pseudoinverse term, a Schatten--2 robustness penalty, and an aggregate balance penalty. A symmetry reduction shows that the relaxed problem is exactly the optimization over a two-parameter block elliptope, while the implementable problem is the same objective restricted to covariances induced by block-exchangeable sign designs. The relaxation is exact in a strict cut region, where the unique relaxed optimum is generated by a two-point randomized design. Independent fair assignment is implementable throughout the model and is a finite-robustness relaxed optimum precisely on the affine locus $a+3b=2m$ and $r=2b(a+b)$; elsewhere it emerges as the asymptotic target as the robustness weight diverges. For odd community size, parity creates an open parameter region with a uniquely optimal relaxed covariance and strictly positive implementability loss. Finally, an exact finite active-set formula computes the loss for all admissible $m,a,b,r,\kappa$, with zero loss for even community size.

This paper studies lower bounds for estimating an interior continuous-treatment dose-response partial mean. The target is $$\theta_P(t_0)=\int \mu_P(t_0,x)p_{X,P}(x)\,dx,$$ interpretable as a causal dose-response mean under consistency, no unmeasured confounding, and local positivity. Over the same Hölder dose class $\mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0)$, under iid sampling, bounded outcomes, an interior evaluation dose, the stated anisotropic Hölder restrictions, and a strict-slack baseline condition, we prove the same-class minimax lower bound $$R_n\{\mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0),t_0\} \ge c\,n^{-2\alpha/(2\alpha+1)}$$ for all sufficiently large $n$. The lower-bound exponent is the one-dimensional treatment-regression exponent and holds for every fixed $\beta>0$, although the constant and slack-baseline feasibility may depend on $\beta$. We compare this lower floor with the published higher-order influence-function benchmark $$\rho_n= n^{-2\alpha/(2\alpha+1)} \vee n^{-2/(1+d/(4s)+1/\alpha)}.$$ When $d\le 4s$, the lower-bound exponent equals the exponent of this published comparator, whose upper theorem concerns a distinct localized-regularity class. When $4s<d$, the benchmark is governed by the covariate-smoothness term and has a strictly smaller exponent than the same-class lower-floor exponent. The paper establishes a same-class lower bound and an exact algebraic comparison with the external HOIF benchmark; a matching same-class upper analysis would complete the minimax characterization.