Introduction

Modified treatment policies define interventions by transforming the treatment value that a unit would naturally receive. This idea is central in causal analyses where feasible interventions depend on the observed treatment process, including stochastic and modified policies for continuous or longitudinal treatments (Muñoz et al., 2011; Haneuse et al., 2013; Young et al., 2014; Díaz et al., 2021). Applied implementations and tutorials make such policies operational, including threshold rules motivated by support and positivity considerations (Williams et al., 2023; Hoffman et al., 2024). This paper studies one exact rule: the lower-threshold clamp $d_{\delta }(a)=\max \{a,\delta \}$, which assigns all natural values below $\delta$, the intervention threshold, to the boundary point $\delta$.

The statistical feature of this policy is the atom created by the clamp. In a continuous-treatment experiment, values above $\delta$ remain on their natural continuous scale, while the lower tail collapses to a point mass at $\delta$. The corresponding observed-data target in Definition 7 is $${\theta }_{\delta }(P)={\mathbb{E}}_P\{Y1(A>\delta )\}+\sum _{x\in \mathcal{X}}{p}_x{q}_x(\delta ){\mu }_x^P(\delta ),$$ where $q_x(\delta )$, the lower-tail conditional treatment mass, is the probability collapsed by the clamp in stratum $x$, and ${\mu }_x^P(\delta )$, the threshold regression value, is the outcome regression at the boundary. This decomposition separates an ordinary retained-course mean from an atom-regression product. The latter term is the source of the boundary inference problem.

The observed model in Definition 5 fixes a finite stratum space, bounded outcomes, a Hölder regression class with known smoothness exponent $\beta$ and radius $L$, and a polynomial treatment-density envelope $$c_{-}{a}^{\kappa }\le {\pi }_x(a)\le c_{+}{a}^{\kappa }$$ near the lower endpoint, with known thinning exponent $\kappa$. The deterministic threshold sequence ${\delta }_n$ may move with sample size. The information-balance bandwidth $h_n$ in Definition 8 is defined by the local balance $$n{h}_n^{2\beta +1}({\delta }_n+h_n{)}^{\kappa }\asymp 1.$$ This balance combines the usual local-polynomial bias scale $h_n^{\beta }$ with the effective local design mass under polynomial thinning. Since the clamp weights the boundary regression by an atom of order ${\delta }_n^{\kappa +1}$, the resulting frontier rate in Definition 9 is $$r_n=n^{-1/2}+{\delta }_n^{\kappa +1}{h}_n^{\beta }.$$

The first main result is the minimax clamp frontier. Theorem 1 establishes matching upper and lower bounds for absolute-error risk over the observed Hölder model $\mathcal{M}$. The upper bound is attained by the total-Gram stabilized estimator of Algorithm 1, which estimates the retained mean, the lower-tail atom masses, and the boundary regression values on separate deterministic split blocks. The local-polynomial component is used on realized designs with sufficient total Gram curvature and is paired with a bounded fallback on singular designs. The lower bound combines two same-class experiments: a global Bernoulli mean shift for the $n^{-1/2}$ term and a localized Bernoulli perturbation at the moving threshold for the atom-weighted term.

The second main result gives honest inference at the same scale. Theorem 2 proves that the bias-aware interval in Algorithm 2 has uniform coverage over $\mathcal{M}$ and worst-case expected length of order $r_n$, and that every uniformly honest observed-sample interval has worst-case expected length at least a constant multiple of $r_n$. The interval uses bounded-outcome concentration for the retained mean and atom-mass estimates, and a Hölder bias plus empirical weight-energy radius for the threshold regression. This connects the clamp problem to honest nonparametric confidence-interval theory (Low, 1997; Armstrong et al., 2018; Armstrong et al., 2016) while preserving the atom-weighted structure of the target.

The phase diagram in Theorem 3 expands the compact rate $r_n$. Two deterministic scales organize the regimes: the risk-transition scale $${\delta }_{\mathrm{crit},n}=n^{-1/\{2(\beta \kappa +2\beta +\kappa +1)\}},$$ and the bandwidth-transition scale $${\delta }_{\mathrm{edge},n}=n^{-1/(2\beta +\kappa +1)}.$$ Thresholds with ${\delta }_n/{\delta }_{\mathrm{crit},n}\to 0$, or with ${\delta }_n/{\delta }_{\mathrm{crit},n}\to c_0\in (0,\infty )$, have root-$n$ order. Vanishing thresholds above the critical scale enter an atom-dominated regime with rate $${\delta }_n^{\kappa +1}(n{\delta }_n^{\kappa }{)}^{-\beta /(2\beta +1)}.$$ Thresholds converging to a fixed positive value have the ordinary positive-density pointwise Hölder exponent $\beta /(2\beta +1)$, weighted by an atom mass bounded away from zero. At threshold zero, the atom vanishes and the target reduces to the observed mean.

The causal interpretation is obtained through a full-data class tailored to the same lower-threshold policy. Members of ${\mathcal{M}}^{\mathrm{F}}$ in Definition 20 have observed margins in $\mathcal{M}$ and satisfy simultaneous structural consistency, conditional exchangeability, and continuity of the structural response mean on the threshold range. Proposition 2 identifies the causal clamp mean ${\psi }_{\delta }(P^{\mathrm{F}})$ with the observed target ${\theta }_{\delta }(P)$. Proposition 3 shows that the observed-margin map covers $\mathcal{M}$ and aligns the decision criteria. Consequently, Theorem 4 transports the same minimax risk and honest-length frontier to the causal clamp mean on ${\mathcal{M}}^{\mathrm{F}}$.

The paper also studies a continuity-only counterpart. In Definition 24, the density, stratum-mass, and polynomial-thinning restrictions are retained, while the regression condition supplies a memberwise continuous extension on the threshold interval. Theorem 5 characterizes the resulting frontier as $$s_n=n^{-1/2}+{\delta }_n^{\kappa +1}.$$ The fallback estimator and Hoeffding interval attain this scale, and the minimax risk and honest expected length are bounded below at the same order. The elbow occurs at ${\delta }_n\asymp n^{-1/[2(\kappa +1)]}$, where the collapsed atom mass reaches the sampling scale. For fixed positive thresholds, the theorem records persistent minimax risk and honest length under qualitative continuity.

The results sit at the intersection of continuous-treatment causal inference, degenerate-design regression, and honest nonparametric inference. Existing work on continuous-treatment dose-response and modified-policy estimands develops identification, efficient estimation, targeted learning, threshold-response functionals, stochastic shifts, and support-aware policy analysis under their corresponding regularity structures (Imbens, 2000; Hirano et al., 2005; Imai et al., 2004; Kennedy et al., 2017; van der Laan et al., 2022; Díaz et al., 2026; Koo et al., 2026; McCoy et al., 2026). Degenerate-design regression theory gives pointwise rates when the design density vanishes near the target (Gaïffas, 2005; Gaïffas, 2007), and classical minimax and local-polynomial theory provide the positive-density benchmark (Stone, 1982; Fan et al., 1996; Tsybakov, 2009). The contribution here is the threshold-atom problem generated by the exact deterministic clamp: the local regression value is learned under thinning and then multiplied by the policy-induced atom mass.

Standing conditions.

One set of conditions is in force throughout and is not repeated in the discussion surrounding each statement. The class constants $J,\beta ,L,\kappa ,c_{-},c_{+},p_{\min },\bar{\delta },\alpha$, the deterministic threshold path $({\delta }_n)$, and the deterministic sample-split blocks are fixed before the sample is drawn and are available to every procedure; observations are independent and identically distributed from the observed law; the local-polynomial degree is the one induced by the smoothness exponent; and estimators and interval endpoints take values in the outcome range $[0,1]$. All uniformity statements range over laws sharing these constants, and all risks, coverage probabilities, and expected lengths are computed under the $n$-fold product law of the observed data. The formal statements repeat these conditions in their hypotheses because each is separately checked, but a reader may treat them as one standing regime-and-sampling condition.

Each model, target, estimator, interval, and criterion has exactly one canonical definition: the model class is Definition 5, the target is Definition 7, and the estimator and interval are Algorithms 1 and 2. A few later environments introduce split-indexed aliases of those objects, written with an explicit block subscript, so that the finite-sample criteria can name a particular split; each such environment says which canonical definition it specializes and adds no new content. Section A is the verification supplement: it holds the auxiliary design, coercivity, and local-polynomial algebra on which the main proofs rest, and nothing in it is needed to read the main results.

Selected formal statements are proved in the verification supplement, with the precise verification scope and reproduction instructions given in Section A.

The rest of the paper proceeds as follows. Section 2 positions the paper within modified-policy causal inference, continuous-treatment estimation, degenerate-design regression, and honest confidence intervals. Section 3 defines the observed model, clamp policy, target, bandwidth, and frontier rate. Section 4 constructs the total-Gram estimator and bias-aware interval. Section 5 states the minimax risk, honest-length, phase-diagram, and calibration results. Section 6 gives the full-data causal lift. Section 7 treats the continuity-only model. Section 8 discusses scope, limitations, and open questions, including exact-modulus calibration.

Related work

Structural and longitudinal causal frameworks provide the counterfactual language for interventions and treatment regimes (Pearl, 2009; Robins et al., 2000; van der Laan et al., 2003; van der Laan et al., 2011). Within that language, theoretical work develops identification and estimation for stochastic and modified policies in continuous-treatment settings, including rules whose assigned dose depends on the observed natural value (Muñoz et al., 2011; Haneuse et al., 2013; Young et al., 2014; Díaz et al., 2021). Software and applied guidance make these policies operational and illustrate threshold interventions motivated by positivity and support constraints (Williams et al., 2023; Hoffman et al., 2024). The present paper studies the observed-data experiment generated by a lower-threshold modified treatment policy in which the deterministic rule clamps natural treatment values below the threshold to the threshold itself. That exact clamp creates a point mass at the boundary of the post-policy distribution, so the estimand combines an ordinary retained natural-course component with a boundary regression value multiplied by the mass collapsed by the policy.

A second line of work studies continuous-treatment dose-response functionals and the role of overlap in semiparametric and nonparametric inference (Imbens, 2000; Hirano et al., 2005; Imai et al., 2004; Kennedy et al., 2017; Bonvini et al., 2026). Recent natural-value-policy work develops root-$n$ targeted-learning inference under its regularity and nuisance-rate conditions (Díaz et al., 2026), while positivity-violation work separates an in-support point-identified component from a Lipschitz-bounded extrapolation component (Koo et al., 2026). Threshold-response methods target support-restricted stochastic functionals with efficient estimation and simultaneous bands (van der Laan et al., 2022), and stochastic-shift work targets effect modification with valid confidence intervals (McCoy et al., 2026). The present estimand has a different post-policy law: its deterministic clamp maps the entire lower tail to one moving boundary point, producing a Dirac component whose weight and regression value must be learned together. Under a global two-sided polynomial envelope, the delivered results characterize minimax risk and uniformly honest expected length for this unsmoothed clamp functional.

The nonparametric component is closely related to regression with degenerate design. For smoothness $s$ and a design density regularly varying with exponent $\gamma$, Gaïffas (2005) obtains the pointwise rate $n^{-s/(1+2s+\gamma )}$, up to slowly varying factors; the classical positive-density Hölder rate $n^{-s/(2s+1)}$ from Stone (1982) and the local-polynomial constructions surveyed by Fan et al. (1996) provide the interior reference. In the present boundary regime, the information balance $n{h}_n^{2\beta +1}({\delta }_n+h_n{)}^{\kappa }\asymp 1$ recovers the degenerate-design bandwidth when ${\delta }_n$ is at the edge scale. Farther into the interior it gives $h_n\asymp (n{\delta }_n^{\kappa }{)}^{-1/(2\beta +1)}$. The clamp target then multiplies the resulting boundary-regression error by the collapsed mass of order ${\delta }_n^{\kappa +1}$, yielding the atom term ${\delta }_n^{\kappa +1}{h}_n^{\beta }$ alongside the retained-mean $n^{-1/2}$ term. This mass weighting, absent from ordinary pointwise regression, generates the critical and atom-dominated threshold regimes. Two antecedents deserve a closer comparison, because they bear directly on the oracle conditions used here and on the sharp-constant question left open below.

The first is adaptation under degenerate design. Gaïffas (2007) studies pointwise estimation from inhomogeneous data when the smoothness is unknown, and obtains the adaptive rate with the usual logarithmic payment: the exponent matches the known-smoothness degenerate-design rate with $n$ replaced by $n/\log n$, so adaptation costs a logarithmic factor rather than a change of exponent. The procedures in the present paper are class-indexed in the oracle sense: the smoothness exponent and radius, the thinning exponent and envelope constants, and the threshold path are fixed before sampling. Atom weighting changes the adaptation question rather than inheriting it. The clamp target multiplies the boundary-regression error by the collapsed mass ${\delta }_n^{\kappa +1}$, so a logarithmic loss in the regression component is invisible whenever the retained-mean term $n^{-1/2}$ dominates, and is felt only above the critical threshold scale; moreover the mass itself depends on the thinning exponent, so an adaptive treatment would have to adapt to $\kappa$ as well as to $\beta$. The frontier reported here is the known-class benchmark for that question: it holds with $\beta$, $L$, and $\kappa$ fixed and available to the procedure, which is the reference an adaptive treatment would be measured against. Section 8 takes up adaptation to unknown $\beta$, $L$, and $\kappa$ as a question for future work.

The second is the modulus of continuity. The classical account of minimax rates through a modulus is due to Donoho et al. (1991), who show that the difficulty of estimating a linear functional over a convex class is governed by the between-class modulus of the functional with respect to a statistical distance, and Cai et al. (2004) develop the corresponding theory for confidence intervals, where expected length over a subclass is governed by a within-class or between-class modulus rather than by the minimax rate alone. The honest expected-length lower bound established here is proved by direct two-point and localized-perturbation arguments, and it matches the risk frontier in order. Relating that frontier to a modulus is a well-posed next step, and the structure of the clamp target says what such a modulus would have to accommodate: the functional is linear in the outcome regression, while its weight $q_x(\delta )$ is determined by the design, so the appropriate family is a design-indexed one rather than the convex class of a single fixed linear functional. The sharp-constant question stated in the discussion is exactly this calibration question.

The inference results also build on the theory of honest confidence intervals. Low (1997) shows how expected length lower bounds constrain honest adaptation over nonparametric classes, and Armstrong et al. (2018); Armstrong et al. (2016) develop bias-aware interval methods that balance stochastic error and worst-case approximation bias. Those ideas are especially natural here because the clamp target contains a boundary regression term whose bias is controlled by smoothness and whose variance is controlled by the local design mass. The paper establishes matched minimax absolute-risk and uniformly honest expected-length rates for this target, so the estimation and interval scales agree under the same polynomial-thinning conditions.

Finally, the deterministic lower-threshold clamp studied here has a mixed-measure post-intervention law: values already above the threshold remain on their continuous natural scale, while values below the threshold collapse to the boundary point. That perspective explains why the lower-threshold clamp differs from smooth dose-response estimation: the intervention produces an atom whose mass is estimable from treatment frequencies, while its contribution to the mean depends on a boundary response value. The analysis below isolates this atom-regression product and derives the rates for both the Hölder model and the continuity-only model, thereby connecting modified-policy causal estimands to degenerate-design regression and honest nonparametric inference.

Setup and assumptions

We work with deterministic sequences and fixed constants throughout. Generic finite constants may change from line to line and depend only on the displayed model parameters. All asymptotic notation is along the sample size $n$; indices $x$ range over the finite stratum space $\mathcal{X}$, and all probabilities and expectations are taken under the observed law $P$ unless a different law is displayed. The observed unit is $O=(X,A,Y)$, with baseline stratum $X$, treatment $A$, and outcome $Y$. The deterministic regime constants include the smoothness exponent $\beta$, Hölder radius $L$, thinning exponent $\kappa$, density constants $c_{-}$ and $c_{+}$, stratum-mass floor $p_{\min }$, threshold envelope $\bar{\delta }$, noncoverage level $\alpha$, and deterministic threshold sequence ${\delta }_n$.

The procedures below use fixed sample splitting and product sampling notation. The split is deterministic, so the probability statements condition on no random partitioning device.

Definition 1 fixes the block structure used by the estimator and interval. The lower bounds on the block sizes keep each empirical component on the same sample-size scale while leaving the blocks independent under product sampling.

The notation in Definition 2 is used for risk and coverage statements in the observed and causal formulations. It records that full-data sampling still feeds estimators through the observed coordinates.

The sampling statement fixes the i.i.d. observed-data experiment used by the estimators and risk criteria.

The independent random-design sampling condition in Assumption 1 is the standard sampling structure for the boundary-regression experiment (Gaïffas, 2005). It lets the mass above the threshold, the empirical lower-threshold atom, and the local regression fit be analyzed with product-law concentration under the same observed law.

The next conditions describe the treatment distribution within strata and the regression function at the boundary created by the lower-threshold clamp.

The conditional absolute-continuity requirement in Assumption 2 is standard for continuous-treatment modified policies (Díaz et al., 2021). It introduces the stratum density ${\pi }_x$ and the stratum mass $p_x$, so the probability collapsed by a threshold can be expressed as an ordinary integral.

The finite-stratum overlap condition in Assumption 3 is standard in stratified continuous-treatment analyses (Díaz et al., 2021). It ensures that every stratum contributes a nonvanishing share of the sample, which keeps the local design problem from being driven by empty baseline cells.

Assumption 4 is specific to this analysis. It states that treatment support near the lower boundary thins at a polynomial order, so the amount of information available to learn the boundary regression value is governed by the same exponent in every stratum. The two sides of the envelope do different work, and it helps to keep them apart. The lower bound $c_{-}{a}^{\kappa }\le {\pi }_x(a)$ guarantees local information: it lower bounds the expected count and the local Gram eigenvalue in the threshold window, and it lower bounds the collapsed atom mass $q_x(\delta )$. It is therefore the side used for Gram coercivity and for atom-mass lower bounds. The upper bound ${\pi }_x(a)\le c_{+}{a}^{\kappa }$ controls upper-risk quantities: the same window mass, the local Gram entries, and the collapsed atom mass entering the bias and length calculations. The lower-bound experiments use the canonical density ${\pi }_x(a)=(\kappa +1)a^{\kappa }$, which the regime constraint $c_{-}\le \kappa +1\le c_{+}$ admits, and the likelihood divergences are computed for that canonical density and its localized perturbations. Densities proportional to $a^{\kappa }$, and bounded multiplicative perturbations $a^{\kappa }{r}_x(a)$ with $0<c_{-}\le r_x(a)\le c_{+}$, are representative members.

The degree $\ell$ in Definition 3 is the order used by the local polynomial at the threshold. It matches the Taylor-remainder smoothness imposed next.

The Taylor-remainder formulation in Assumption 5 is a standard Hölder outcome-regression smoothness condition for bias-aware nonparametric inference (Armstrong et al., 2016). In this model, the boundary value of the regression ${\mu }_x^P$ is multiplied by the threshold mass, so the condition supplies the deterministic approximation control used by the local-polynomial estimator.

These ingredients define the observed model class and the lower-threshold clamp target.

Definition 4 records the continuity-only class ${\mathcal{M}}_{\mathrm{cont}}$ alongside the Hölder class. The later continuity-only section uses the selected continuous regression extension ${\mu }_{x,\mathrm{cont}}^P$ when smoothness is replaced by qualitative continuity on the threshold envelope.

The observed class $\mathcal{M}$ in Definition 5 collects the design and regression restrictions under one membership statement. This compact notation is the domain for the estimator, honest interval, and minimax criteria in the next two sections.

The map $d_{\delta }$ in Definition 6 leaves treatments above the threshold unchanged and moves lower treatments to the threshold. The resulting target therefore combines a natural-course component above $\delta$ with a boundary-regression component at $\delta$.

In Definition 7, $q_x(\delta )$ is the conditional probability mass collapsed by the clamp in stratum $x$, ${\nu }_{\delta }(P)$ is the retained mean above the threshold, and ${\theta }_{\delta }(P)$ is the observed-data clamp target. The expression isolates the atom-regression product that drives the boundary inference problem.

At sample size $n$ the target is written ${\theta }_{{\delta }_n}(P)$: it is the functional of Definition 7 evaluated at the deterministic threshold $\delta ={\delta }_n$.

The remaining definitions set the bandwidth, rate scale, and range projection used by the estimation and interval statements.

The bandwidth $h_n$ in Definition 8 is chosen by the local information available in a window to the right of the threshold. The factor $({\delta }_n+h{)}^{\kappa }$ carries the polynomial thinning from Assumption 4 into the window-mass calculation.

The rate $r_n$ in Definition 9 combines the regular root-$n$ contribution from the retained mean and threshold mass with the atom-weighted Hölder approximation scale. This is the benchmark used for the minimax risk and honest expected-length results.

Definition 10 records the range projection used later for estimators and confidence endpoints. Because the target is built from an outcome regression taking values in $[0,1]$, the projection keeps the reported numerical object on the same outcome scale.

Estimation and honest intervals

The estimator separates the three empirical tasks in ${\theta }_{{\delta }_n}(P)$: estimating the retained mean above the threshold, estimating the mass collapsed by the clamp, and estimating the boundary regression value that multiplies that mass. The deterministic split from Definition 1 assigns one block to the retained-course mean, a second to the lower-threshold atom masses, and a third to the local-polynomial fit at the boundary. This separation lets the interval treat the three components with direct concentration bounds for bounded variables. The local regression step follows the boundary-polynomial logic of Fan et al. (1996), with a stabilization rule tailored to the total Gram matrix in each stratum.

The realized-design notation is as follows. It records the rescaled local coordinate, the $v_{\ell }(u)$ monomial basis, the stratum-specific local count, and the event on which the total Gram matrix has enough curvature to support the intercept fit.

In Definition 12, the local count $N_x$ and the total localized Gram matrix $G_x$ are computed on the third split block $I_2$. The normalized population moment matrix $M_{\ell ,\kappa ,\rho }$ supplies the reference curvature induced by the polynomial thinning law in Assumption 4. The constant ${\lambda }_{\star }$ converts that reference curvature into a uniform empirical design threshold, and the good-design event ${\Omega }_x$ is the realized condition under which the intercept weights $w_{ix}$ are used.

With these quantities fixed, the point estimator is a plug-in estimator for the decomposition in Definition 7. It estimates each summand on its own split block and applies the range projection introduced in Definition 10.

The estimator ${\hat{\theta }}_n$ in Algorithm 1 uses the local-polynomial intercept only on ${\Omega }_x$. On the complementary realized designs, the fallback value $1/2$ keeps the atom-regression factor inside the bounded outcome range. The projection in the last line keeps the estimator on the same scale as ${\theta }_{{\delta }_n}(P)$.

For statements indexed by a full deterministic split $B_n$, it is useful to record the same construction with the split made explicit.

Definition 13 is the version used by the finite-sample risk and length criteria below. It makes explicit that the sampling split is fixed before the data are observed, while the good-design events and intercept weights are realized from the third block.

The confidence interval adds deterministic bias allowances to concentration radii. The stochastic terms use Hoeffding-type calibration for bounded outcomes and indicators (Hoeffding, 1963), while the local regression radius combines a Hölder bias bound with the empirical ${\ell }_2$ size of the intercept weights. This bias-aware construction is in the spirit of honest nonparametric inference with smoothness restrictions (Armstrong et al., 2018; Armstrong et al., 2016).

Here $J$, the number of strata, enters only through the Hoeffding multiplier $t_{\alpha }$. The interval ${\mathrm{CI}}_n$ in Algorithm 2 treats the retained mean, atom masses, and local-regression intercepts as separate empirical components. On ${\Omega }_x$, $D_x$ carries both the uncertainty in ${\hat{b}}_x$ and the bias-plus-noise radius for the threshold regression. On ${\Omega }_x^c$, the contribution covers the full feasible atom-regression range, which is the branch used to preserve the same bounded-outcome scale on singular realized designs.

The split-indexed interval records the same accounting with $B_n$ attached to each empirical quantity.

Definition 14 fixes the interval whose coverage and expected length are evaluated in the main statistical results. Its two branches mirror Algorithm 2: the regular branch combines atom-mass uncertainty with local-polynomial bias and noise, and the fallback branch uses the bounded outcome range when the realized design lacks sufficient Gram curvature.

The length criterion is pointwise in the observed sample and then averaged under the observed product law.

Finally, the finite-sample criteria place the split-indexed estimator and interval on the common worst-case scale used in the minimax comparisons.

Definition 16 evaluates the point estimator and interval over the observed-law class $\mathcal{M}$ from Definition 5. The risk uses absolute error for the clamp target ${\theta }_{{\delta }_n}(P)$, and the length functional applies Definition 15 to the bias-aware interval from Definition 14.

Main statistical results

Section 4 constructed the split-sample total-Gram estimator and its bias-aware interval for the observed clamp target in Definition 7. We now state the corresponding performance guarantees. The product sampling law in Definition 2 and the interval-length convention in Definition 15 put point estimators, confidence intervals, and minimax benchmarks on the same finite-sample scale.

The scalar calibration at the end of the section uses a concrete Bernoulli pair. We record it before the main bounds so that the same notation can be used in the rate check.

The triangular alternative in Definition 17 is centered at the moving threshold and has height proportional to the bandwidth. It gives a transparent version of the same local perturbation that drives the lower-bound component of the general theory.

The minimax benchmarks optimize over all observed-sample procedures. The first criterion measures worst-case absolute accuracy for estimating the unsmoothed clamp target, and the second measures worst-case expected length among intervals with uniform coverage.

Thus $R_n^{\star }$ is the best possible worst-case absolute accuracy for the observed clamp target, while $L_n^{\star }$ is the best possible worst-case length under uniform honesty. These criteria match the honesty framework used in nonparametric confidence intervals (Low, 1997; Armstrong et al., 2018).

The main risk result gives matching upper and lower bounds at the rate in Definition 9. The upper bound is attained by the stabilized total-Gram estimator from Algorithm 1; the lower bound is witnessed by both a global mean shift and a localized threshold perturbation.

The two lower-bound constructions in Theorem 1 isolate the two terms in $r_n$. A global Bernoulli shift produces the parametric contribution, while a Hölder-sized bump at the threshold produces the atom-weighted local contribution. The estimator in Algorithm 1 attains this combined scale, so the observed clamp problem has the same order from the attainable and unavoidable sides.

The corresponding interval result uses the same rate scale. Its statement combines finite-sample coverage for the constructed interval with a minimax lower bound for every honest observed-sample interval.

The interval in Theorem 2 is calibrated to the same two sources of difficulty as the estimator. The retained-course and atom-mass terms are controlled by bounded-outcome concentration, while the threshold regression term carries the local Hölder bias and the realized Gram behavior encoded in Algorithm 2. The lower bound shows that uniform honesty over $\mathcal{M}$ requires expected length of order $r_n$, in line with modulus-based accounts of optimal confidence sets (Low, 1997; Armstrong et al., 2016).

The two theorems above leave the rate in the compact form $r_n$. The next result expands that expression into the threshold regimes induced by the information-balance bandwidth. Two deterministic scales organize those regimes, the risk-transition scale ${\delta }_{\mathrm{crit},n}$ and the bandwidth-transition scale ${\delta }_{\mathrm{edge},n}$, $${\delta }_{\mathrm{crit},n}=n^{-1/\{2(\beta \kappa +2\beta +\kappa +1)\}},{\delta }_{\mathrm{edge},n}=n^{-1/(2\beta +\kappa +1)},$$ and they are restated inside Theorem 3. Table 1 collects the regimes, separating the geometry of the treatment support at the threshold from the resulting risk order.

Theorem 3 identifies ${\delta }_{\mathrm{crit},n}$ as the boundary at which the local atom contribution reaches the parametric scale, and ${\delta }_{\mathrm{edge},n}$ as the smaller edge scale for the bandwidth algebra. Regular and critical threshold paths retain root-$n$ order. Vanishing thresholds above the critical scale inherit a degenerate-design nonparametric rate of the kind studied by Gaïffas (2005): the estimation point rides toward the thinned endpoint $a=0$, so the local design mass degenerates at the polynomial rate implied by Assumption 4. A fixed positive threshold is an interior-design geometry. There the target point ${\delta }_0$ is interior to the treatment support and the density at that point is bounded away from zero, so the exponent $\beta /(2\beta +1)$ is the ordinary positive-density pointwise Hölder rate (Stone, 1982; Donoho et al., 1991) multiplied by an atom weight ${\delta }_0^{\kappa +1}$ bounded away from zero. The estimator still uses a one-sided window, because the clamp evaluates the regression at ${\delta }_0$ from above, but that is a property of the procedure rather than of the design geometry. At the exact zero threshold, the clamp target reduces to the observed mean and the total-Gram estimator reduces to the clipped split-sample outcome mean.

The section closes with a scalar calibration. It specializes the localized perturbation to one stratum, linear thinning, and Lipschitz smoothness, giving an explicit exponent check for the critical scale.

In Proposition 1, $K_n$ is the product Kullback–Leibler scale for the localized Bernoulli alternative, while ${\Delta }_n$ is the induced clamp-target separation. The balance $h_n\asymp (n{\delta }_n{)}^{-1/3}$ keeps the localized alternative statistically contiguous at constant information scale, and the identity ${\delta }_n\asymp n^{-1/10}$ reproduces the critical exponent from Theorem 3 when $\beta =\kappa =1$.

Causal interpretation through full-data lifts

The observed-data results in Theorems 1 and 2 concern the clamp functional ${\theta }_{\delta }(P)$ under the law of $(X,A,Y)$. This section supplies a full-data causal interpretation for the same target. One notational convention applies throughout this section and the continuity-only section. Every estimator and interval used in this paper is a measurable function of the observed sample, so its risk, coverage probability, and expected length are the same whether computed under the $n$-fold product law of the observed margin or under the $n$-fold product law of the full-data package. We therefore state all such quantities under the observed-margin product law $P^{\otimes n}$, and use the full-data product law only where the statement genuinely concerns full-data objects. The structural full-data overlay augments the observed experiment with a full-data law $P^{\mathrm{F}}$, a member-specific latent-response space $\mathcal{U}$, a latent response variable $U$, structural response maps $g_x$, and a potential-outcome process $\{Y(a):a\in [0,1]\}$. Within that overlay, the lower-threshold policy changes the realized treatment through the same clamp map introduced in Definition 6; the statistical analysis remains based on the observed sample.

The analysis rests on three full-data restrictions and one response-mean definition. The first two restrictions are standard causal conditions for a latent-response representation, and the third records the continuity required on the threshold interval used by the clamp policy.

Assumption 6 is the structural consistency condition: a common latent response generates all potential outcomes and the observed outcome at the realized treatment, in the simultaneous NPSEM-style formulation used for modified treatment policies (Díaz et al., 2021). The common null set makes the equality simultaneous over doses, which is the form needed when the clamp map evaluates the response at $d_{\delta }(A)$.

Assumption 7 is the latent-response randomization condition (Díaz et al., 2021). Within each stratum, it lets the conditional treatment density weight the same latent-response distribution that determines the structural mean.

The full-data response mean $m_x^{\mathrm{F}}(a)$ is the fiberwise structural analogue of the observed regression extension. It isolates the average structural response at dose $a$ among units in stratum $x$, which is the quantity the bridge result compares with ${\mu }_x^P$ on the threshold interval.

Assumption 8 is specific to this analysis. It places the continuity requirement directly on the structural response mean over the range where clamping can create a threshold evaluation, matching the interval on which the observed regression extension is used.

These conditions define the full-data causal class and the decision criteria used below.

The full-data class ${\mathcal{M}}^{\mathrm{F}}$ keeps the observed law inside $\mathcal{M}$ while allowing each member to carry its own standard Borel latent-response space. This memberwise formulation is the natural setting for comparing causal and observed decision problems without imposing a common latent carrier across all laws.

The causal clamp mean ${\psi }_{\delta }(P^{\mathrm{F}})$ is the mean outcome under the same lower-threshold modified treatment policy used to define the observed clamp functional in Definition 7. Thus the causal estimand changes the interpretation of the response while preserving the policy rule and threshold range.

The causal minimax risk $R_{n,\mathrm{F}}^{\star }$ and causal honest-length criterion $L_{n,\mathrm{F}}^{\star }$ compare observed-sample estimators and intervals under the full-data class through their induced observed-sample behavior. The procedures therefore use the same data as in the observed experiment while the target is the causal clamp mean.

The next proposition gives the identification step. It connects the structural mean under the full-data law to the observed regression extension and then identifies the causal clamp mean with the observed clamp target.

Proposition 2 is the fiberwise bridge. Conditional exchangeability converts observed outcomes within a stratum and treatment set into an integral of the structural response mean against the treatment density; continuity and the observed regression extension then identify that mean on the threshold interval. The final identity shows that the causal lower-threshold policy mean is exactly the clamp functional analyzed in the observed experiment.

A second structural fact records that the full-data class covers the observed model class exactly at the margin. The construction uses regular conditional distributions on standard Borel spaces, a standard measurable-disintegration tool (Kallenberg, 2002).

Proposition 3 establishes the decision-theoretic equivalence needed for causal transport. Every observed law in $\mathcal{M}$ has a full-data representative in ${\mathcal{M}}^{\mathrm{F}}$, and the risk and honest-length criteria agree at each fixed threshold. Thus lower bounds and attainable procedures for the observed clamp target can be read as statements about the causal clamp mean on the declared full-data class.

One technical convention remains for comparing expected lengths. The next definition fixes the order-preserving way real bounds are read inside the extended nonnegative reals.

The embedding in Definition 23 lets the real constants in the rate bounds be compared with honest expected lengths when those lengths are represented in the extended nonnegative-real order. This is the scale used in the final causal length statement.

The main causal conclusion now follows by combining the bridge identity, observed-margin surjectivity, and the observed-data minimax and honest-interval results.

Theorem 4 transports the observed minimax and honest-length frontier to the causal clamp mean on ${\mathcal{M}}^{\mathrm{F}}$. The same total-Gram estimator and the same bias-aware interval remain observed-sample procedures, and their guarantees hold uniformly over the full-data class because Proposition 2 aligns the target while Proposition 3 aligns the induced decision criteria. The rate is therefore the same $r_n$ characterized in Definition 9 and Theorem 3: root-$n$ and boundary-driven regimes carry over with the causal interpretation supplied by the full-data lift.

Continuity-only clamp inference

Sections 3 and 5 impose a Hölder modulus for the regression at the lower threshold. We now record the enlarged continuity-only experiment, which keeps the same density, stratum-mass, and lower-tail thinning restrictions while replacing the Hölder regression member with qualitative continuity.

The class in Definition 24 preserves the design restrictions that govern the amount of mass near the lower endpoint. Its regression condition supplies a selected point value at the clamp threshold through continuity, rather than through a quantitative Hölder modulus.

The corresponding target has the same retained-course plus threshold-atom decomposition as ${\theta }_{\delta }(P)$ in Definition 7; the atom value is read from the continuous extension in the continuity-only class.

The continuity-only scale is determined by the ordinary sampling fluctuation and the amount of mass moved by the lower-threshold clamp.

The estimator and interval make the enlarged model explicit. They use the retained-course mean above the threshold and a fixed midpoint value for the lower-tail atom, with a bounded-sum allowance for the empirical lower-tail mass.

The concentration step for Algorithm 4 uses the standard bounded-sum inequality of Hoeffding (1963), applied separately to the retained-course and atom-mass split statistics.

The causal version of the continuity-only analysis uses the same structural ingredients as Definition 20, with the observed margin placed in ${\mathcal{M}}_{\mathrm{cont}}$.

For later comparison, the next definition fixes the observed margin of a full-data law as the pushforward onto the observed coordinates.

The decision-theoretic criteria are recorded for both the observed continuity-only class and its full-data counterpart.

The first causal statement aligns the full-data response mean with the observed continuous extension and therefore identifies the causal clamp mean with the continuity-only observed target.

Proposition 4 shows that qualitative continuity turns almost-everywhere regression information into the pointwise threshold value needed after the clamp collapses the lower tail. Under the stated full-data continuity conditions, the structural response mean agrees with the observed continuous extension throughout the threshold envelope, and the causal clamp mean equals the observed continuity clamp target.

The next statement records that the continuity-only full-data class covers the observed continuity-only class exactly at the margin. Its construction uses regular conditional distributions on standard Borel spaces, as supplied by Kallenberg (2002).

Proposition 5 puts the observed and causal continuity-only decision problems on the same footing. Every observed continuity-only law has a full-data representative satisfying the causal restrictions, and the minimax risk and honest expected-length criteria agree exactly for each sample size and threshold in the stated range.

The main continuity-only conclusion now combines the bridge identity, observed-margin surjectivity, and the fallback estimator and interval.

Theorem 5 characterizes the continuity-only rate for both observed and causal formulations. The fallback estimator and Hoeffding interval attain the rate $s_n$, while the minimax risk and honest expected length are bounded below at the same order. The honest-length comparison follows the nonparametric confidence-interval lower-bound phenomenon of Low (1997), specialized here to the threshold-atom decision problem.

The rate has a simple interpretation. When ${\delta }_n$ is at the elbow $n^{-1/[2(\kappa +1)]}$, the lower-tail mass term ${\delta }_n^{\kappa +1}$ matches $n^{-1/2}$, so the continuity-only problem has a root-$n$ scale. At ${\delta }_n=0$, the atom term vanishes and the bounded retained-course mean again gives the root-$n$ scale. For thresholds converging to a fixed positive value, the theorem records persistent minimax risk and honest length, reflecting the positive amount of treatment mass assigned a threshold regression value governed by qualitative continuity.

Discussion, limitations, and open questions

The results characterize how lower-tail overlap and smoothness jointly determine inference for threshold modified treatment policies with continuous treatments. In the Hölder model $\mathcal{M}$, Theorems 1 and 2 give the common rate $r_n$, combining the ordinary sampling term $n^{-1/2}$ with the local extrapolation term ${\delta }_n^{\kappa +1}{h}_n^{\beta }$. The phase diagram in Theorem 3 explains how this expression changes with the threshold sequence: moving the clamp closer to the lower endpoint changes both the mass affected by the policy and the amount of information available near the threshold.

The continuity-only comparison in Theorem 5 separates quantitative smoothness from qualitative continuity. With a Hölder radius and exponent fixed in the model, the local polynomial construction uses observations in a shrinking neighborhood of ${\delta }_n$ and pays the bias term $h_n^{\beta }$. With continuity alone, the delivered rate is $s_n=n^{-1/2}+{\delta }_n^{\kappa +1}$, because the threshold regression contribution is controlled through boundedness and continuity rather than through a quantitative modulus. Taken together, the two frontiers show that atom mass and smoothness enter different parts of the problem: the mass below the threshold weights the regression value to be learned, while the smoothness condition determines how sharply that value can be inferred from nearby continuous-treatment observations.

The causal lifting results give the same statistical rates a full-data interpretation. Under the consistency, exchangeability, and response-continuity conditions in Assumptions 6, 7, and 8, Proposition 2 and Theorem 4 transfer the observed-data Hölder frontier to the causal clamp mean. The continuity-only bridge in Proposition 4 and Theorem 5 gives the analogous interpretation for the qualitative-continuity class. These statements make the role of the full-data class explicit: once the observed margin belongs to the relevant observed model and the stated causal restrictions hold, the observed clamp functional and the causal clamp mean have matching minimax criteria.

The exact-modulus perspective below records a more calibrated object behind the order results. The intervals in Algorithm 2 use analytic upper bounds on local bias and stochastic error. A sharper implementation can instead optimize directly over affine weights in the realized local design, in the spirit of modulus-based honest inference (Armstrong et al., 2018; Armstrong et al., 2016). The next three definitions name the realized affine-modulus handle, the integrated split-specific radius, and the resulting interval used to formulate the sharp calibration question.

Definitions 30, 31, and 32 record one concrete, fully specified object against which the sharp-constant question can be posed: an interval centered at the total-Gram estimator whose half-width is the realized affine-modulus radius, intersected with $[0,1]$. It is important to be precise about what that object does and does not calibrate. Its radius solves the affine-modulus problem for the threshold-regression component alone: it is the supremum over strata of unweighted realized regression radii, it does not include the retained-mean deviation radius, the atom-mass deviation radii, or the atom weights $p_x{q}_x({\delta }_n)$ by which those components enter the target, and it does not aggregate across strata in the way the target does. On the singular realized designs where the affine constraints are infeasible the realized radius is zero, so the object is not by itself a coverage-guaranteed procedure; the honest interval with a proved coverage guarantee is Algorithm 2. The zero-threshold regime makes the gap explicit: there the target is an ordinary bounded mean with frontier $n^{-1/2}$, while this radius continues to solve a local-regression problem whose natural scale is $h_n^{\beta }$. The question below is therefore posed for a component-level modulus; extending it to a modulus for the complete clamp functional, with the regular components, atom weights, their uncertainty, stratum aggregation, and a singular-design fallback included, is the substantive part of the problem and is open. The integrated radius averages the design-dependent affine optimization under the observed-design product law for the third split block and is defined stratum by stratum, replacing the analytic threshold-regression radius by a realized optimization problem.

Limitations.

The scope of the results is the fixed finite-stratum design described in Definition 5, with outcomes bounded in $[0,1]$, known smoothness radius and exponent in the Hölder model, and a known polynomial-thinning exponent governing lower-tail overlap. The causal interpretation is attached to the declared full-data classes in Definitions 20 and 27, and the continuity-only conclusions are attached to the continuity-only observed and full-data classes in Definitions 24 and 27. Within that scope, the theorems characterize the minimax risk and honest expected-length orders for the lower-threshold clamp target and its causal lift. A further limitation concerns evidence rather than scope. The case made here for the stabilized estimator and the bias-aware interval is an asymptotic-order case, supported at the level of a single analytic calibration cell in Proposition 1. Practical questions about total-Gram stabilization on singular realized designs and interval behavior at moderate sample sizes concern quantities whose finite-sample behavior need not resemble the eventual regime: the frequency of the Gram fallback, the realized coverage, and the realized expected length all depend on constants this analysis does not track. A systematic finite-sample evaluation across the regular, critical, atom-dominated, fixed-threshold, and near-zero regimes, reporting those quantities and comparing against unstabilized local polynomial and regular modified-policy benchmarks where the targets and conditions permit a fair comparison, is left to future work and is not attempted here.

Open questions.

Two questions remain open. The first is adaptation: the frontier established here is the known-class benchmark, with $\beta$, $L$, and $\kappa$ fixed and available to the procedure, and whether the same frontier is attainable when those constants are unknown, and at what logarithmic cost, is not settled by the present analysis. The second concerns exact constants rather than rates. Modulus-based confidence-interval theory describes the difficulty of such problems through the local modulus of a functional over a smoothness class; Cai et al. (2004) gives a general adaptation framework for confidence intervals, while Armstrong et al. (2018); Armstrong et al. (2016) develop honest inference procedures that use modulus calculations for bias-aware intervals. The honest expected-length lower bound proved here is obtained by direct two-point and localized-perturbation arguments, and it matches the risk frontier in order. The clamp target is linear in the outcome regression but includes the design-dependent weight $q_x({\delta }_n)$, so sharp modulus calibration is treated as a separate question. The exact-modulus constructions in Definitions 30, 31, and 32 identify one concrete candidate for studying sharp honest length: use the total-Gram center, compute the realized affine-modulus radius on the third split block, and compare its worst integrated radius to the minimax expected-length benchmark.

A positive answer to Remark 1 would identify the sharp asymptotic full-length constant for the exact-modulus interval across the phase regimes covered by the displayed threshold condition. It would also connect the mixed regular-and-local perturbation comparison directly to the integrated realized radius, sharpening the order-optimal honest interval of Theorem 2 into an exact asymptotic calibration.

Appendices

Proofs and verification note

This appendix collects the auxiliary statements that support the main risk, honest-length, causal-bridge, and continuity-only conclusions. The statements are organized by mathematical role: the clamp pushforward and support facts, the bandwidth and local-design algebra, the stabilized estimator and interval bounds, the phase calculations, the population moment identities underlying total-Gram stabilization, and the uniqueness statement for the continuity-only regression extension.

The first two propositions record the measure-theoretic behavior of the exact lower-threshold clamp. They connect the deterministic policy in Definition 6 to the mixed law that appears in the observed clamp functional in Definition 7.

Proposition 6 supplies the exact mixed-measure decomposition induced by the clamp. The continuous part retains the natural treatment values above the threshold, and the lower tail becomes a point mass at the threshold whose order is ${\delta }^{\kappa +1}$ under Assumption 4. This atom-size calculation is the source of the weighting in Definition 9 and Definition 26.

Proposition 7 records the support property used by the causal bridge in Proposition 2. The polynomial lower envelope gives positive conditional treatment measure in every neighborhood of the clamp value, so the structural response at the clamped treatment is evaluated on the support generated by the observed design.

Lemmas 10, 11, 12, and 13 establish the bandwidth normalization and the realized local-polynomial algebra used by the total-Gram estimator. These statements translate the thinning envelope into local sample information and deterministic weight identities.

Lemma 10 turns the infimum definition of $h_n$ into an eventual equality at the information boundary. This equality is the normalization used in the risk upper bound and in the expected weight-energy calculation.

Lemma 11 gives the population conditioning bound for the local monomial design. The lower bound is scaled by the local window mass, so it remains compatible with the global polynomial-thinning envelope in the lower-endpoint window.

Lemma 12 is the deterministic local-polynomial fact used on the good-design event. Moment reproduction gives the intercept interpretation at the threshold, while the absolute-weight and squared-weight bounds control the Hölder bias and stochastic radius in Algorithm 2.

Lemma 13 integrates the squared-weight scale over the random design. Under the balance equation, the average stochastic radius has the same $h^{\beta }$ order as the deterministic local-polynomial bias term.

The following two lemmas are the finite-sample ingredients for the stabilized procedures. They combine bounded-outcome concentration, deterministic fallback branches, and the local-design controls above.

Lemma 14 gives the finite-sample coverage statement for the interval in Algorithm 2. The interval combines split-block concentration with a bias-aware local radius; on singular local designs, the bounded fallback radius covers the whole possible threshold-mean contribution in the affected stratum.

Lemma 15 supplies the estimator side of the upper bound in Theorem 1. The retained-course and atom-mass empirical terms contribute the $n^{-1/2}$ scale, while Lemma 13 aligns the threshold-regression stochastic scale with the Hölder bias scale after multiplication by the collapsed mass.

The phase analysis rests on a separate bandwidth comparison. The result below identifies the edge scale where the boundary form of the local design transitions to the interior form.

Lemma 16 is the algebraic input behind the phase diagram in Theorem 3. It shows that the bandwidth is comparable to the boundary scale near the edge and to the usual thinned interior bandwidth when the threshold is asymptotically beyond that edge.

The total-Gram probability bound depends on population coercivity and elementary local-window identities. Lemmas 1, 2, 3, 4, 5, and 6 isolate these deterministic and integral facts before they are assembled in Lemma 17.

Lemma 1 gives a uniform lower eigenvalue bound for the shifted reference moment matrices. The bound is uniform in the shift parameter, which is what permits thresholds ranging from the boundary region to fixed positive interior points.

Lemma 2 verifies the positivity of the Gram constant used in Definition 12. This makes the good-design event meaningful uniformly over the polynomial-thinning envelope.

Lemma 3 gives a simple lower bound on local design mass under the power envelope. It is used to compare the realized local count with $h(\delta +h{)}^{\kappa }$.

Lemma 4 expresses the local-window probability through the stratum mass and conditional density. It is the population counterpart of the count $N_x$ in Definition 12.

Lemma 5 identifies each population Gram entry with a one-dimensional integral over the local treatment window. Combined with the density envelope, it connects the empirical Gram matrix in Definition 12 to the shifted reference matrices.

Lemma 6 converts the normalized shifted-moment lower bound into the physical treatment scale. This is the population inequality used before applying concentration to the realized Gram matrix.

Lemmas 7, 8, and 9 record pointwise properties of the exact intercept weights on the good-design event. They are deterministic consequences of the total Gram matrix and the affine reproduction equations.

Lemma 7 gives the explicit intercept-weight representation for observations in the local stratum-window. It is the algebraic formula behind the threshold regression term in Algorithm 1.

Lemma 8 states the monomial reproduction property in the same notation used by the estimator. This property turns local-polynomial weighting into an estimate of the regression value at the threshold.

Lemma 9 gives the pointwise weight control implied by Gram coercivity. Together with Lemma 12, it bounds both individual leverage and aggregate variance.

Lemma 17 assembles the local-window identities, shifted coercivity, and concentration into the good-design probability bound used by the stabilized procedures. The exponential bound makes the singular-design fallback negligible on the scales used in Theorems 1 and 2, while the on-event identities preserve the usual local-polynomial intercept behavior.

The last auxiliary lemma for the appendix supports the continuity-only causal and observed targets. It establishes pointwise uniqueness of the continuous conditional-regression extension on the threshold envelope.

Lemma 18 justifies the pointwise notation for the continuity-only regression extension used in Definition 25. The positive treatment-density envelope gives enough support throughout $[0,\bar{\delta }]$ for almost-everywhere equality of continuous versions to determine the extension at every threshold value.

Verification note.

The Lean kernel checks were run at repository commit 6f1bf59b1255336e29c22bdea39edd530cddc960 using the CausalSmith Lean toolchain leanprover/lean4:v4.33.0; the executable reports Lean 4.33.0, target x86_64-unknown-linux-gnu, Lean commit d8b18978322de05a8f3dba51ef03cf5461676c17. The kernel-checked assumption, class, definition, and algorithm anchors in this manuscript are the objects introduced earlier in the paper. The kernel-checked result and open-question anchors are the objects introduced earlier in the paper. Presentation-synthesized restatements such as Definitions 1, 2, 3, 4, 10, 11, 13, 14, 15, 16, 17, 18, 19, 23, 29, 31, and 32 are reader-facing anchors whose status is recorded separately in the verification contract. Remark 1 contributes a kernel-checked, well-formed encoding of the open sharp-constant question in the formal layer. Statement- and proof-faithfulness audits compare the displayed manuscript statements and proof narratives with those checked declarations; those audits are separate from Lean kernel checking. The theorem-local verification footnotes identify the published conditional-distribution input from Kallenberg (2002) and the bounded-difference concentration input from Hoeffding (1963) at the exact statements where those cited conclusions are invoked.

Reproducing the verification.

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Proofs of the main results

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