Formalization: A Minimax Lower Bound for Interior Dose-Response Estimation

The published-rate frontier sequence is nonnegative, enforcing the declared range ρ_n ∈ [0,∞) of the setup symbol rho_n: each branch (n:ℝ)^(·) is an rpow of the nonnegative base (n:ℝ) ≥ 0, hence ≥ 0, and the max preserves it. @realizes rho_n

The signed two-point mean channel has quadratic KL divergence on the central half of its mean range. This local wrapper links the paper lemma to the reusable Causalean implementation.

Le Cam's two-point reduction: a finite KL budget gives a positive, budget-dependent constant multiplying the squared parameter separation.

Certified β-frontier bracket. Assume the strict-slack baseline. For every β > 0 there is c > 0 with R_n ≥ c n^{-2α/(2α+1)} eventually; and — for EVERY n ≥ 1, not merely eventually, matching the note's unconditional ρ_n algebra — in the smooth-covariate regime s ≥ d/4 (d ≤ 4s) the benchmark ρ_n collapses to that oracle exponent, while in the deficient regime 0 < s < d/4 (4s < d) ρ_n has a strictly smaller exponent. The regime identities are stated outside the ∀ᶠ n (eventually) clause so they are NOT weakened to large n only. No same-class upper endpoint is packaged.

Smooth-covariate collapse: if 0 < α, 0 < s, and d ≤ 4 s (i.e. s ≥ d/4), then ρ_n = n^{-2α/(2α+1)} for every n ≥ 1.

Deficient-covariate collapse: if 0 < α, 0 < s, and 4 s < d (i.e. 0 < s < d/4), then ρ_n = n^{-2/(1 + d/(4s) + 1/α)} for every n ≥ 1, and the deficient exponent is strictly smaller than the oracle exponent.

Uniform bound on the identifying functional over the model class: |θ_P(t_0)| ≤ M · M · vol([0,1]^d). Used for BddAbove of the worst-case risk.

The two-point Le Cam reduction in minimax form: two class members with bounded KL between their n-fold product laws force the pointwise minimax risk to be at least c_K times their squared target separation.

The all-β oracle lower floor. Assume the strict-slack baseline. For every β > 0 there is c_or > 0, depending only on the fixed model radii and the slack baseline (not on n), such that for all sufficiently large n, R_n(P_{α,β,s}(M,c_0,ε_0,t_0), t_0) ≥ c_or n^{-2α/(2α+1)}. The construction perturbs only the treatment regression and leaves the treatment density fixed, so the bound is independent of whether β ≥ α.

The unit covariate cube is a measurable set.

A nonnegative density on a measurable restriction that integrates to one defines a probability measure after weighting the restricted measure.

On a finite measure space, a measurable real function that is almost surely bounded in absolute value is integrable.

The map sending an observation to its outcome, treatment, and covariate tuple is measurable.

The observed outcome coordinate is a measurable function of the observation.

The observed treatment coordinate is a measurable function of the observation.

The observed covariate coordinate is a measurable function of the observation.

For fixed treatment and covariates, forming an observation from an outcome is measurable.

Forming an observation from covariates, treatment, and outcome is measurable.

Base-coordinate marginal of a bind whose fibre is reattached by a map.

SUBSTRATE GATE (the only one of the genuine construction). For a positive Hölder order α and radius M, there is a fixed amplitude λ ∈ (0, M/4] such that for every sign ζ ∈ {−1, +1} and every bandwidth h ∈ (0,1], the scaled smooth bump slice a ↦ ζ·λ·h^α·doseBump((a−t_0)/h) lies in the univariate order-α radius-M Hölder ball on the evaluation window [t_0−ε_0, t_0+ε_0]. This is the classical h^α-normalized Hölder estimate for ContDiffBump; everything else in the construction is proved. The amplitude bound λ ≤ M/4 is the constraint consumed by the two-point KL band (|μ_ζ| ≤ λ ≤ B/2 with B = M/2).

Applying the outcome channel at a treatment-covariate pair gives the two-point outcome law pushed forward to the observed-data space at that same pair.

If the two-point outcome laws are valid probability laws, then the witness outcome channel is a Markov kernel.

If the witness covariate and treatment measures are probability measures, then their joint treatment-covariate measure is also a probability measure.

The genuine nested X → A → Y joint law equals the single bind of the shared (A,X)-marginal against the outcome channel. This is the bridge that lets the regression, ignorability, and KL leaves work over one Measure.bind.

The treatment bump equals one at the center of the bump.

The treatment bump is everywhere nonnegative.

The treatment bump is everywhere bounded above by one.

The absolute value of the treatment bump is everywhere bounded by one.

The treatment bump vanishes at every point whose distance from the center is at least one in the normalized coordinate.

The treatment bump is a measurable real-valued function.

The witness conditional mean is measurable as a function of treatment and covariates.

For fixed covariates, the two-point outcome law pushed to observed triples varies measurably with the treatment value.

The probability assigned by the pushed-forward two-point outcome law to any measurable event is a measurable function of covariates and treatment.

Integrating the treatment law against the two-point outcome law gives an outcome kernel that is measurable as a function of covariates.

If the covariate density is nonnegative on the cube and integrates to one there, then the witness covariate measure is a probability measure.

If the treatment density is nonnegative and integrates to one on the unit interval, then the witness treatment measure is a probability measure.

The all-zero covariate vector lies in the unit covariate cube in every dimension.

If the treatment-window radius is nonnegative, the window contains its center point.

The unit covariate cube is compact in every finite dimension.

The unit covariate cube has finite Lebesgue volume in every finite dimension.

A univariate function that belongs to a Hölder ball of a given radius also belongs to any larger-radius Hölder ball of the same smoothness order on the same set.

A multivariate function that belongs to a Hölder ball of a given radius also belongs to any larger-radius Hölder ball of the same smoothness order on the same set.

A constant multivariate function belongs to a Hölder ball whenever the absolute size of the constant is at most the ball radius and the radius is nonnegative.

Bump-support concentration: the doseAMeasure-integral of ψ((a−t_0)/h)² is at most 2(M−η_0)·h, because ψ is supported in (t_0−h, t_0+h) ⊆ window (for 0 < h ≤ ε_0) where the treatment density q0 is bounded by M−η_0.

Single-observation KL budget: the joint KL between the two witness data laws is at most 16 λ² (M−η_0) / B² · h^{2α+1}. GENUINE and bump-driven (NONZERO): it is the (A,X)-integral of the per-fibre two-point-channel KL, not a Dirac collapse.

The two single-observation witness laws are mutually absolutely continuous and have integrable log-likelihood ratio (each fibre channel has positive mass on both atoms {B,−B}, so the rn-derivative is bounded). These are the hypotheses of productKL_tensorization.

n-fold KL budget via product tensorization: the KL between the n-fold product witness laws is at most n · (single-obs budget).

If the component laws are probability measures and the two-point means are bounded inside the outcome range, then the witness data law is a probability measure.

Under the valid two-point outcome construction, marginalizing the witness data law to covariates recovers the witness covariate measure.

Under the stated outcome and treatment-law assumptions, the witness covariate density is tied to the covariate marginal of the witness data law.

The witness potential-outcome process is consistent with the observed outcome at the realized treatment for every observation.

If the two-point outcome support is contained in the target bound, then the witness data law almost surely has outcomes inside that bounded interval.

If the two-point outcome support is contained in the target bound, then the witness law satisfies the bounded-outcome condition at that bound.

Generic existence of an i.i.d. sample with a prescribed probability law. Discharges the i.i.d.-sampling existential of IidSampling; this is exactly Causalean.Stat.HasIIDSample, which every probability measure satisfies.

The genuine witness's data law puts the covariate X in the cube [0,1]^d a.s.

The genuine witness's data law puts the treatment A in [0,1] a.s.

The genuine two-point witness lies in the anisotropic Hölder dose-response class. Only the treatment regression is perturbed (by the bump), the slack baseline (p_0, q_0) is frozen, and the two semantic ties hold for the genuine joint law.

Under a valid two-point outcome construction, marginalizing the witness data law to treatment and covariates gives exactly the witness treatment-covariate measure.

The witness treatment-covariate measure is the product of the treatment measure and the covariate measure.

If the treatment density is nonnegative and both component densities integrate to one, then the joint treatment-covariate measure has product density.

Under the stated positivity, normalization, and bounded-mean assumptions, the witness treatment density is a valid conditional treatment density for the witness law.

The A-marginal of the shared (A,X)-law is doseAMeasure q0.

The realized-treatment level is a.s. not equal to any fixed level a: the treatment has a Lebesgue density q0, so {O | O.A = a} is doseDataMeasure-null.

Semantic tie 1 (genuine regression). Under the genuine joint law, the conditional mean of Y given (A,X) equals μ_ζ(A,X), because the outcome channel twoPointMean B (μ_ζ(a,x)) has mean exactly μ_ζ(a,x). This forbids the Dirac/ decoupled laundering: μ is the real regression of the data law.

Semantic tie 2 (ignorability). The potential outcome Y(a) equals the regression mean μ_ζ(a,X) almost surely (the realized A hits the fixed level a only on a q0-null set), so Y(a) is X-measurable and Y(a) ⟂ A | X.

For a normalized covariate density, the witness target value at the center treatment equals the signed bump amplitude.

For a normalized covariate density, the separation between the positive and negative witness target values is twice the bump amplitude without the sign.

Under baseline-submodel slack, every fixed beta > 0 has the oracle lower floor eventually. In the smooth-covariate regime the floor matches the published rho_n exponent; in the deficient regime the published comparator has a strictly smaller exponent and no same-class upper endpoint is asserted.

Certified deficient-covariate regime theorem. Assume the strict-slack baseline. For every β > 0 and every 0 < s < d/4 (4s < d) there is c > 0 such that, for all sufficiently large n, R_n ≥ c n^{-2α/(2α+1)}, while ρ_n = n^{-2/(1+d/(4s)+1/α)} has a strictly smaller exponent. The same-class upper endpoint C ρ_n is NOT discharged.

Oracle regime reduction. Under the assumptions of sharp_pointwise_lower_bound, if s ≥ d/4 (d ≤ 4s) then ρ_n = n^{-2α/(2α+1)} and hence R_n ≥ c n^{-2α/(2α+1)} = c ρ_n for some c > 0. The certified lower floor reduces to the classical interior pointwise nonparametric regression barrier.

Certified smooth-covariate regime theorem. Assume the strict-slack baseline. For every β > 0 and every s ≥ d/4 (d ≤ 4s) there is c > 0 such that, for all sufficiently large n, R_n ≥ c n^{-2α/(2α+1)} = c ρ_n. The certified lower floor lands on the same exponent as ρ_n; NO same-class upper bound is claimed.

Certified kernel converse. Assume the strict-slack baseline. For every β > 0 there is c > 0, depending only on the fixed model radii and the slack baseline, such that for all sufficiently large n, R_n(P_{α,β,s}(M,c_0,ε_0,t_0), t_0) ≥ c n^{-2α/(2α+1)} — the classical interior treatment-regression pointwise rate, uniformly over every β > 0.