Faithful Federated Tabular Data Synthesis

# Faithful Federated Tabular Data Synthesis
Source: [https://arxiv.org/html/2610.33403](https://arxiv.org/html/2610.33403)
Zilong Cao††thanks:Email:nwu\_czl@stumail\.nwu\.edu\.cnHai Zhang††thanks:Email:zhanghai@nwu\.edu\.cnAffiliation:School of Mathematics, Northwest UniversityXuan Bi††thanks:Corresponding author\. Email:xbi@umn\.eduAffiliation:Carlson School of Management, University of Minnesota

###### Abstract

Synthetic data is an important data source in the AI era\. This paper studies the generation of mixed\-type tabular synthetic data in federated settings, where two key challenges arise: ensuring distributional fidelity and effectively handling mixed\-type data\. We propose Federated Integrative Two\-Step Synthesis \(FITS\), a framework for mixed\-type tabular data synthesis\. At its core is a two\-step federated algorithm for estimating conditional cumulative distribution functions \(CDFs\) using local linear regression\. Building on this algorithm, we develop a federated synthesis procedure based on the probability chain rule to ensure distributional fidelity\. To handle mixed\-type data, FITS introduces a continualization method that lifts discrete variables into a continuous space and recovers their discrete distributions without loss at both the density\-function and CDF levels\. This transformation provides a unified continuous representation of mixed\-type variables, facilitating the application of local linear regression and synthetic data generation\. Our theoretical analysis characterizes the conditional density and cumulative distribution functions of the continualized mixed\-type variables and establishes the uniform convergence of the resulting federated conditional CDF estimators\. Experiments demonstrate that FITS\-generated synthetic data improves downstream task performance for all clients, with the benefits depending on how the synthetic data is used\.

## 1Introduction

Synthetic data generation has received increasing attention as the AI era progresses, with synthetic data becoming an important data source in many fields, such as large language models\([Wang et al\., 2023](https://arxiv.org/html/2610.33403#bib.bib16)\), imaging\([Kim et al\., 2022](https://arxiv.org/html/2610.33403#bib.bib17)\), and medical research\([Kokosi and Harron, 2022](https://arxiv.org/html/2610.33403#bib.bib18)\)\. However, synthetic data generation \(or, broadly speaking, data synthesis\) is itself a nontrivial methodological problem, rather than a simple consequence of assuming a generative distribution\([Alaa et al\., 2022](https://arxiv.org/html/2610.33403#bib.bib19)\)\. In addition to the intrinsic challenges of generating synthetic data, a fundamental issue is the limited availability of real data, which hinders the production of high\-quality synthetic data\. Such scarcity frequently arises from data silos, where data holders prioritize privacy and legal considerations\. This severely limits or even prevents access to their data\. Federated learning \(FL,[McMahan et al\. \(2016\)](https://arxiv.org/html/2610.33403#bib.bib25);[Kairouz and McMahan \(2021\)](https://arxiv.org/html/2610.33403#bib.bib6)\) provides an efficient and privacy\-preserving framework for synthetic data generation, addressing this dilemma through a distributed paradigm that eliminates centralized storage and avoids centralizing raw data while transmitting only processed data from clients to ensure security\. Numerous federated synthetic data generation efforts focus on images and other data formats\([Little et al\., 2023](https://arxiv.org/html/2610.33403#bib.bib20)\); nonetheless, the development of synthetic tabular data generation in federated learning remains limited, despite its substantial importance across various domains\.

Existing studies have explored generative adversarial networks\([Zhao et al\., 2021](https://arxiv.org/html/2610.33403#bib.bib21)\), diffusion models\([Sattarov and Schreyer, 2025](https://arxiv.org/html/2610.33403#bib.bib26)\), and marginal\-based synthesis\([Li et al\., 2026](https://arxiv.org/html/2610.33403#bib.bib22)\)\. However, several major challenges remain unresolved\. \(i\) Communication costs remain substantial\. Federated synthetic data generation methods, especially deep learning\-based models, require extensive communication to learn the generative model\. Several reports\([Zhu et al\., 2019](https://arxiv.org/html/2610.33403#bib.bib23);[Kairouz and McMahan, 2021](https://arxiv.org/html/2610.33403#bib.bib6);[National Institute of Standards and Technology, 2025](https://arxiv.org/html/2610.33403#bib.bib24)\)emphasize that reducing communication is essential for improving both the efficiency and safety of federated learning\. \(ii\) Lack of faithfulness remains a concern\([Alaa et al\., 2022](https://arxiv.org/html/2610.33403#bib.bib19)\)\. Generative adversarial networks and diffusion models lack theoretical interpretability, which may lead to synthetic data that is unfaithful\. For example, many deep\-learning\-based models cannot guarantee that the synthetic data is faithful to the original distribution\. Marginal\-based synthesis, on the other hand, relies only on low\-order marginal information and may have limitations in approximating high\-dimensional distributions\. \(iii\) Existing methods remain inefficient in handling mixed\-type data\. For example, black\-box approaches, such as generative adversarial networks and diffusion models, typically use one\-hot encodings or continuous embeddings for discrete variables; however, one\-hot encodings are not genuinely continuous and substantially increase dimensionality, whereas non\-invertible continuous embeddings destroy the discrete structure and compromise interpretability\. Marginal\-based synthesis, by contrast, discretizes continuous variables to facilitate the computation of marginal statistics, thereby losing information carried by continuous variables\. Nonparametric methods provide established approaches to handling mixed\-type data\.[Li and Racine \(2008\)](https://arxiv.org/html/2610.33403#bib.bib3)estimate conditional CDFs and quantiles using generalized product kernels that combine continuous and discrete kernels\. Alternatively,[Nagler \(2018\)](https://arxiv.org/html/2610.33403#bib.bib8)shows that suitably chosen independent noisy jitter allows nonparametric density estimators designed for continuous data to be applied to mixed data\. These approaches provide important foundations for mixed\-data estimation but do not develop a federated synthesis procedure\.

To address these challenges, we propose a federated framework, termed federated integrative two\-step synthesis for mixed\-type tabular data \(FITS\), which supports the generation of multivariate mixed\-type tabular data based on the chain rule\. FITS is characterized by a two\-step communication procedure\. In the first step, clients transmit weighted Hessian\-type matrices to the central server, which computes the inverse of the aggregated matrix and sends only its first row back to the clients\. In the second step, clients send their local response\-smoothed CDF contribution vectors to the central server\. The server then constructs the global conditional cumulative distribution functions \(CDFs\), which are subsequently used for inverse sampling and data synthesis\. For mixed\-type data, we employ an efficient technique that lifts discrete variables into a continuous space while allowing inversion back to the original discrete space without incurring additional dimensional cost\. FITS includes a core federated algorithm for estimating conditional cumulative distribution functions via local linear regression\. In theory, local linear regression \(LLR,[Fan et al\. \(1996\)](https://arxiv.org/html/2610.33403#bib.bib1)\) can effectively reduce estimator bias and has been studied for conditional CDF estimation\([Hall et al\., 1999](https://arxiv.org/html/2610.33403#bib.bib7);[Xie, 2023](https://arxiv.org/html/2610.33403#bib.bib4)\)and widely used in various applications\([Cai et al\., 2011](https://arxiv.org/html/2610.33403#bib.bib13);[Cheng and Wu, 2013](https://arxiv.org/html/2610.33403#bib.bib15);[Chakrabortty and Cai, 2017](https://arxiv.org/html/2610.33403#bib.bib12);[Cheng et al\., 2020](https://arxiv.org/html/2610.33403#bib.bib14)\)\. Furthermore, we propose client\-designed and penalty\-adaptive kernels to enable effective LLR fitting in federated settings with mixed\-type data\. At the client level, the client\-designed kernel offers both fixed\-bandwidth and K\-nearest\-neighbor \(KNN\) variants, with the choice determined by each client\. Under this design, each client can specify kernel components, including the kernel function and bandwidth parameters\. The penalty\-adaptive kernel, by contrast, operates at the kernel\-function level and penalizes continualized samples whose categories do not match that of the query\.

FITS offers several distinctive advantages for federated synthesis of mixed\-type tabular data\. First, unlike iterative federated optimization, FITS requires only the upload of processed data and a*two\-step*procedure to achieve distribution estimation\. Second, by estimating conditional CDFs and combining them through the chain rule, FITS directly targets the underlying joint distribution, providing a principled basis for distributionally faithful data synthesis\. Third, we utilize an invertible continualization technique that maps continuous and discrete variables into a unified continuous domain without increasing the data dimension\. This transformation preserves the recoverability of the original discrete variables while providing a foundation for applying established methods developed for continuous data and facilitates synthetic data generation through a unified procedure\. Finally, FITS supports client\-designed kernels that accommodate client\-specific data characteristics and a penalty\-adaptive mechanism that reduces kernel\-fitting error caused by category mismatches after continualization\.

Our contributions can be summarized as follows:

From a methodological perspective, we propose a novel FITS framework for federated mixed\-type tabular data synthesis\. It supports federated conditional CDF estimation and synthetic data generation\. In addition, FITS introduces an invertible discrete\-to\-continuous lifting technique, the client\-designed kernel for the FL scheme, and the penalty\-adaptive kernel for the mixed\-type data\.

In theory, we characterize the proposed continualization technique at both the density and distribution\-function levels, showing, in particular, its blockwise smoothness\. Moreover, under this blockwise structure, we establish uniform convergence rates for the federated local linear conditional CDF estimator using Gaussian kernels with unbounded support\. In particular, direct kernel smoothing near blockwise boundaries can yield inconsistent estimates; we therefore develop a modified estimation procedure that restores consistency at these points\. Finally, we establish the uniform consistency of the per\-step conditional CDF estimates used in the chain\-rule generation process\.

We evaluate FITS on simulated and real datasets under imbalanced client sample sizes and compare it with two federated data synthesis baselines, HeteroFedSyn\([Li et al\., 2026](https://arxiv.org/html/2610.33403#bib.bib22)\)and FedTabDiff\([Sattarov and Schreyer, 2025](https://arxiv.org/html/2610.33403#bib.bib26)\), using three downstream models: linear regression, logistic regression, and FT\-Transformer \(a deep learning model,\([Gorishniy et al\., 2021](https://arxiv.org/html/2610.33403#bib.bib28)\)\), covering both regression and classification tasks\. The results show that FITS\-generated synthetic data can substantially improve downstream performance and provide the best distributional fidelity compared to synthetic data generated by the other methods\.

The remainder of the paper is organized as follows\. Section[2](https://arxiv.org/html/2610.33403#S2)introduces the federated setup, notation, and local linear formulation of conditional CDF estimation\. Section[3](https://arxiv.org/html/2610.33403#S3)presents FITS for continuous data, including two\-step federated aggregation, client\-designed kernel schemes, and synthetic data generation through the chain rule and inverse sampling\. Section[4](https://arxiv.org/html/2610.33403#S4)extends the framework to mixed\-type data through continualization of discrete variables and penalty\-adaptive kernels\. Section[5](https://arxiv.org/html/2610.33403#S5)establishes uniform convergence results for both continuous and continualized mixed\-type data\. Section[6](https://arxiv.org/html/2610.33403#S6)evaluates the utility of synthetic data through simulation and real\-data studies\. The final section summarizes the findings and outlines future research directions\. The appendices provide auxiliary theoretical tools, detailed proofs, and supplementary experiments\.

## 2Background

Subsection[2\.1](https://arxiv.org/html/2610.33403#S2.SS1)establishes the federated learning setup and the general notation used throughout the paper\. Subsection[2\.2](https://arxiv.org/html/2610.33403#S2.SS2)introduces local linear regression, explains how it estimates conditional CDFs, and describes its adaptation to the federated learning scheme\.

### 2\.1Federated setup and notation

We consider a federated learning system consisting of a central server \(CS\) andKKclients\. Clientk∈\{1,…,K\}k\\in\\\{1,\\ldots,K\\\}holds a client\-specific hold\-out dataset

ℋk=\{𝐡i\(k\)\}i=1mk,𝐡i\(k\)=\(𝐡i,:d\(k\),hi,d\+1\(k\)\)∈ℝd\+1,\\mathcal\{H\}\_\{k\}=\\bigl\\\{\{\\bf h\}\_\{i\}^\{\(k\)\}\\bigr\\\}\_\{i=1\}^\{m\_\{k\}\},\\qquad\{\\bf h\}\_\{i\}^\{\(k\)\}=\\bigl\(\{\\bf h\}\_\{i,:d\}^\{\(k\)\},\\,h\_\{i,d\+1\}^\{\(k\)\}\\bigr\)\\in\\mathbb\{R\}^\{d\+1\},where𝐡i,:d\(k\)=\(hi,1\(k\),…,hi,d\(k\)\)∈𝒳⊂ℝd\{\\bf h\}\_\{i,:d\}^\{\(k\)\}=\(h\_\{i,1\}^\{\(k\)\},\.\.\.,h\_\{i,d\}^\{\(k\)\}\)\\in\\mathcal\{X\}\\subset\\mathbb\{R\}^\{d\}is the covariate vector andhi,d\+1\(k\)∈𝒵⊂ℝh\_\{i,d\+1\}^\{\(k\)\}\\in\\mathcal\{Z\}\\subset\\mathbb\{R\}is the response\. We writeM=∑k=1KmkM=\\sum\_\{k=1\}^\{K\}m\_\{k\}for the total hold\-out sample size\. Throughout the paper, we assume that the samples\{𝐡i\(k\):i∈\[mk\],k∈\[K\]\}\\\{\{\\bf h\}\_\{i\}^\{\(k\)\}:i\\in\[m\_\{k\}\],\\ k\\in\[K\]\\\}are i\.i\.d\. draws from a common joint distribution\.

Let𝐗∈𝒳\{\\bf X\}\\in\\mathcal\{X\}andZ∈𝒵Z\\in\\mathcal\{Z\}denote the covariate vector and response, respectively\. The estimand of interest is the conditional distribution function

F⁡\(z∣𝒙\)=ℙ⁡\(Z≤z∣𝐗=𝒙\),z∈𝒵,𝒙∈𝒳\.F\(z\\mid\{\\bm\{x\}\}\)=\\mathbb\{P\}\(Z\\leq z\\mid\{\\bf X\}=\{\\bm\{x\}\}\),\\qquad z\\in\\mathcal\{Z\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\.\(1\)It is the canonical object from which arbitrary conditional functionals, including means, quantiles, and probability masses, can be derived, and samples can be generated by inverse\-CDF sampling\.

The federated system is subject to the following communication and data\-access constraints\. The clients communicate with the CS in synchronous rounds; in each round, a client may upload one or more deterministic functions of its local dataℋk\\mathcal\{H\}\_\{k\}\(and any quantities previously received from the CS\), and the CS may broadcast a deterministic function of the received uploads to all clients\. The following raw\-data objects are*never*transmitted: the individual samples\{𝐡i\(k\)\}i\\\{\{\\bf h\}\_\{i\}^\{\(k\)\}\\\}\_\{i\}, the local response vectors\{hi,d\+1\(k\)\}i\\\{h\_\{i,d\+1\}^\{\(k\)\}\\\}\_\{i\}, and the local design matrices used in weighted regression\. The federated estimator developed in Section[3](https://arxiv.org/html/2610.33403#S3)respects these constraints by transmitting only*aggregated sufficient statistics*—a\(d\+1\)×\(d\+1\)\(d\{\+\}1\)\\times\(d\{\+\}1\)design summary in the first round and anLL\-dimensional CDF contribution vector in the second round—each of which is a many\-to\-one function of the underlying raw data\.

##### Coordinate grids\.

For each coordinatej∈\[d\]j\\in\[d\]we fix an evaluation grid𝐠j=\(g1​j,…,gG​j\)⊤∈ℝG\{\\bf g\}\_\{j\}=\(g\_\{1j\},\\ldots,g\_\{Gj\}\)^\{\\top\}\\in\\mathbb\{R\}^\{G\}, and write𝒢=\{𝐠j\}j=1d\+1\\mathcal\{G\}=\\\{\{\\bf g\}\_\{j\}\\\}\_\{j=1\}^\{d\+1\}\. The grids serve two purposes: in Section[3](https://arxiv.org/html/2610.33403#S3), they parameterize the evaluation points ofF^\(⋅∣𝒙\)\\widehat\{F\}\(\\cdot\\mid\{\\bm\{x\}\}\)\. And the grids are used to generate synthetic data via linear interpolation\.

### 2\.2Local\-linear formulation of the conditional CDF

We estimateF⁡\(z∣𝒙\)F\(z\\mid\{\\bm\{x\}\}\)by applying local linear regression to a smoothed CDF score of the response with respect to the continuous covariates\. Let𝒦:ℝ→\[0,1\]\\mathcal\{K\}:\\mathbb\{R\}\\to\[0,1\]denote a continuous CDF kernel\. The local linear estimatorF^​\(z∣𝒙\)=β^0\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)=\\hat\{\\beta\}\_\{0\}ofF⁡\(z∣𝒙\)F\(z\\mid\{\\bm\{x\}\}\)at fixed𝒙\{\\bm\{x\}\}andzzis defined as the intercept of the weighted least\-squares fit

\(β^0,𝜷^:d\)=argmin\(β0,𝜷:d\)∑i=1M\(𝒦i\(z\)−β0−𝜷:d⊤\(𝒙−𝐡i,:d\)\)2ϖi,\(\\hat\{\\beta\}\_\{0\},\\hat\{\\bm\{\\beta\}\}\_\{:d\}\)=\\arg\\min\_\{\(\\beta\_\{0\},\\bm\{\\beta\}\_\{:d\}\)\}\\sum\_\{i=1\}^\{M\}\\Bigl\(\\mathcal\{K\}\_\{i\}\(z\)\-\\beta\_\{0\}\-\\bm\{\\beta\}\_\{:d\}^\{\\top\}\(\{\\bm\{x\}\}\-\{\\bf h\}\_\{i,:d\}\)\\Bigr\)^\{2\}\\varpi\_\{i\},\(2\)where𝜷:d=\(β1,…,βd\)∈ℝd\\bm\{\\beta\}\_\{:d\}=\(\\beta\_\{1\},\\ldots,\\beta\_\{d\}\)\\in\\mathbb\{R\}^\{d\}denotes thedd\-dimensional slope vector,ϖi\\varpi\_\{i\}is a nonnegative kernel weight on the covariate coordinates, and𝐡i,:d\{\\bf h\}\_\{i,:d\}is the covariate vector of theiith observation\. The choice ofϖi\\varpi\_\{i\}is left abstract here and is generally specified as the kernel evaluation of the query point𝒙\{\\bm\{x\}\}at the sample point𝐡i,:d\{\\bf h\}\_\{i,:d\}\. For each query point𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and each evaluation thresholdz∈𝒵z\\in\\mathcal\{Z\}, define the smoothed response indicator with bandwidthwz\(i\)\>0w^\{\(i\)\}\_\{z\}\>0, generally settingwz\(i\)=wz\>0w^\{\(i\)\}\_\{z\}=w\_\{z\}\>0, as

𝒦i​\(z\):=𝒦⁡\(z−hi,d\+1wz\(i\)\),i∈\[M\],\\mathcal\{K\}\_\{i\}\(z\):=\\mathcal\{K\}\\left\(\\frac\{z\-h\_\{i,d\+1\}\}\{w^\{\(i\)\}\_\{z\}\}\\right\),\\qquad i\\in\[M\],\(3\)where\{𝐡i\}i=1M\\\{\{\\bf h\}\_\{i\}\\\}\_\{i=1\}^\{M\}denotes the pooled hold\-out sample under a temporary common index, with𝐡i=\(𝐡i,:d,hi,d\+1\)\{\\bf h\}\_\{i\}=\(\{\\bf h\}\_\{i,:d\},h\_\{i,d\+1\}\)\. Under standard kernel assumptions,𝔼⁡\[𝒦i​\(z\)∣𝐗i=𝒙\]→F⁡\(z∣𝒙\)\\mathbb\{E\}\[\\mathcal\{K\}\_\{i\}\(z\)\\mid\{\\bf X\}\_\{i\}=\{\\bm\{x\}\}\]\\to F\(z\\mid\{\\bm\{x\}\}\)aswz\(i\)→0w^\{\(i\)\}\_\{z\}\\to 0\.

The local\-linear estimator\([Fan, 1992](https://arxiv.org/html/2610.33403#bib.bib2)\)has two well\-known statistical advantages over the Nadaraya–Watson type local\-constant estimator: the leading bias term contributed by the marginal density of the covariates vanishes, and the estimator is automatically bias\-corrected at the boundary of the covariate support\. These properties carry over to the federated setting without modification, because, as shown below, the federated estimator coincides exactly with the centralized local linear estimator on the pooled sample\.

For the federated decomposition, we record the closed\-form solution of \([2](https://arxiv.org/html/2610.33403#S2.E2)\)\. Write

𝑼i:=𝜾\(𝐡i,:d−𝒙\)=\(1,\(𝐡i,:d−𝒙\)⊤\)⊤∈ℝd\+1,𝐖:=diag\(ϖ1,…,ϖM\),\\bm\{U\}\_\{i\}:=\\bm\{\\iota\}\(\{\\bf h\}\_\{i,:d\}\-\{\\bm\{x\}\}\)=\\bigl\(1,\(\{\\bf h\}\_\{i,:d\}\-\{\\bm\{x\}\}\)^\{\\top\}\\bigr\)^\{\\top\}\\in\\mathbb\{R\}^\{d\+1\},\\qquad\{\\bf W\}:=\\mathrm\{diag\}\(\\varpi\_\{1\},\\ldots,\\varpi\_\{M\}\),where𝜾⁡\(𝐲\)=\(1,𝐲\)⊤\\bm\{\\iota\}\(\{\\bf y\}\)=\(1,\{\\bf y\}\)^\{\\top\}, and let𝑼∈ℝM×\(d\+1\)\\bm\{U\}\\in\\mathbb\{R\}^\{M\\times\(d\+1\)\}stack the𝑼i⊤\\bm\{U\}\_\{i\}^\{\\top\}as rows\. Set𝓚⁡\(z\):=\(𝒦1​\(z\),…,𝒦M​\(z\)\)⊤\\bm\{\\mathcal\{K\}\}\(z\):=\(\\mathcal\{K\}\_\{1\}\(z\),\\ldots,\\mathcal\{K\}\_\{M\}\(z\)\)^\{\\top\}and𝐞1=\(1,0,…,0\)⊤∈ℝd\+1\{\\bf e\}\_\{1\}=\(1,0,\\ldots,0\)^\{\\top\}\\in\\mathbb\{R\}^\{d\+1\}\. The intercept of \([2](https://arxiv.org/html/2610.33403#S2.E2)\) admits the closed form

β^0=𝐞1⊤\(𝑼⊤𝐖𝑼\)−1𝑼⊤𝐖𝓚\(z\)=∑i=1Mϖi​𝐞1⊤​\(𝑼⊤​𝐖​𝑼\)−1​𝑼i⏟=:αi𝒦i\(z\)\.\\hat\{\\beta\}\_\{0\}=\{\\bf e\}\_\{1\}^\{\\top\}\\bigl\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\\bigr\)^\{\-1\}\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{\\mathcal\{K\}\}\(z\)=\\sum\_\{i=1\}^\{M\}\\underbrace\{\\varpi\_\{i\}\\,\{\\bf e\}\_\{1\}^\{\\top\}\\bigl\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\\bigr\)^\{\-1\}\\bm\{U\}\_\{i\}\}\_\{\\displaystyle=:\\,\\alpha\_\{i\}\}\\,\\mathcal\{K\}\_\{i\}\(z\)\.\(4\)The local\-linear estimator is thus a linear combination of the smoothed response indicators with weights\{αi\}i=1M\\\{\\alpha\_\{i\}\\\}\_\{i=1\}^\{M\}that depend on𝒙\{\\bm\{x\}\}and the design𝑼\\bm\{U\}but not onzz\. This separability is the structural property that enables the two\-step federated learning procedure developed in Section[3](https://arxiv.org/html/2610.33403#S3)\.

## 3The Proposed Method

This section develops our framework FITS for continuous data, focusing on federated conditional CDF estimation and synthetic data generation\. Subsection[3\.1](https://arxiv.org/html/2610.33403#S3.SS1)formulates the two\-step FL procedure for estimating LLR\-based conditional CDFs, develops the core algorithm, and analyzes its communication cost and equivalence to the pooled setting\. Subsection[3\.2](https://arxiv.org/html/2610.33403#S3.SS2)presents two interchangeable bandwidth schemes \(fixed\-bandwidth and KNN\-kernel\)\. Subsection[3\.3](https://arxiv.org/html/2610.33403#S3.SS3)proposes a method for federated synthetic data generation that combines the chain rule with inverse sampling from conditional CDFs\.

### 3\.1Two\-step federated aggregation

![Refer to caption](https://arxiv.org/html/2610.33403v1/Figure/FITS_twostep.png)Figure 1:FITS: two\-step aggregation\.We describe our two\-step federated aggregation procedure first, whose purpose is to perform federated estimation of the closed\-form expression in \([4](https://arxiv.org/html/2610.33403#S2.E4)\)\. The first step calculates the clients’ kernel\-weighted Hessian matrices, and the CS aggregates them and broadcasts the first\-row vector of the inverse of the aggregated Hessian matrix\. In the second step, clients upload their vectors of conditional CDF estimates, after which the CS aggregates and post\-processes them to obtain the final conditional CDF estimation vector\. Figure[1](https://arxiv.org/html/2610.33403#S3.F1)illustrates the two\-step procedure\.

##### Step 1: aggregation of the kernel\-weighted Hessian\.

For each clientkk, define the local block matrix

𝐒\(k\)​\(𝒙\):=∑iϖi\(k\)​𝑼i\(k\)​𝑼i\(k\)⊤∈ℝ\(d\+1\)×\(d\+1\),\{\\bf S\}^\{\(k\)\}\(\{\\bm\{x\}\}\):=\\sum\_\{i\}\\varpi\_\{i\}^\{\(k\)\}\\,\\bm\{U\}\_\{i\}^\{\(k\)\}\\bm\{U\}\_\{i\}^\{\(k\)\\top\}\\in\\mathbb\{R\}^\{\(d\+1\)\\times\(d\+1\)\},\(5\)where𝑼i\(k\)=𝜾\(𝐡i,:d\(k\)−𝒙\)\\bm\{U\}\_\{i\}^\{\(k\)\}=\\bm\{\\iota\}\(\{\\bf h\}\_\{i,:d\}^\{\(k\)\}\-\{\\bm\{x\}\}\)andϖi\(k\)\\varpi\_\{i\}^\{\(k\)\}is the \(still abstract\) covariate\-kernel weight on theiith local sample\. We write𝐒\(k\)\{\\bf S\}^\{\(k\)\}in block form as

𝐒\(k\)​\(𝒙\)=\(S0\(k\)𝐒1\(k\)⊤𝐒1\(k\)𝐒2\(k\)\),S0\(k\)=∑iϖi\(k\),𝐒1\(k\)=∑iϖi\(k\)​𝐮i\(k\),𝐒2\(k\)=∑iϖi\(k\)​𝐮i\(k\)​𝐮i\(k\)⊤,\{\\bf S\}^\{\(k\)\}\(\{\\bm\{x\}\}\)=\\begin\{pmatrix\}S\_\{0\}^\{\(k\)\}&\{\\bf S\}\_\{1\}^\{\(k\)\\top\}\\\\ \{\\bf S\}\_\{1\}^\{\(k\)\}&\{\\bf S\}\_\{2\}^\{\(k\)\}\\end\{pmatrix\},\\qquad S\_\{0\}^\{\(k\)\}=\\sum\_\{i\}\\varpi\_\{i\}^\{\(k\)\},\\ \{\\bf S\}\_\{1\}^\{\(k\)\}=\\sum\_\{i\}\\varpi\_\{i\}^\{\(k\)\}\{\\bf u\}\_\{i\}^\{\(k\)\},\\ \{\\bf S\}\_\{2\}^\{\(k\)\}=\\sum\_\{i\}\\varpi\_\{i\}^\{\(k\)\}\{\\bf u\}\_\{i\}^\{\(k\)\}\{\\bf u\}\_\{i\}^\{\(k\)\\top\},where𝐮i\(k\):=𝐡i,:d\(k\)−𝒙\{\\bf u\}\_\{i\}^\{\(k\)\}:=\{\\bf h\}\_\{i,:d\}^\{\(k\)\}\-\{\\bm\{x\}\}\. The global weighted Hessian satisfies

𝑼⊤​𝐖​𝑼=∑k=1K𝐒\(k\)​\(𝒙\)\.\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}=\\sum\_\{k=1\}^\{K\}\{\\bf S\}^\{\(k\)\}\(\{\\bm\{x\}\}\)\.\(6\)In the first communication round, each client uploads𝐒\(k\)​\(𝒙\)\{\\bf S\}^\{\(k\)\}\(\{\\bm\{x\}\}\)—an object of size\(d\+1\)​\(d\+2\)/2\(d\{\+\}1\)\(d\{\+\}2\)/2when symmetry is exploited—to the CS\. The CS aggregates the uploads to form𝐒⁡\(𝒙\)=∑k𝐒\(k\)​\(𝒙\)\{\\bf S\}\(\{\\bm\{x\}\}\)=\\sum\_\{k\}\{\\bf S\}^\{\(k\)\}\(\{\\bm\{x\}\}\)and computes the vector

𝒒⁡\(𝒙\):=\(𝐒⁡\(𝒙\)\+λridge​𝐈d\+1\)−1​𝐞1∈ℝd\+1,\\bm\{q\}\(\{\\bm\{x\}\}\):=\\bigl\(\{\\bf S\}\(\{\\bm\{x\}\}\)\+\\lambda\_\{\\rm ridge\}\\mathbf\{I\}\_\{d\+1\}\\bigr\)^\{\-1\}\{\\bf e\}\_\{1\}\\ \\in\\mathbb\{R\}^\{d\+1\},\(7\)whereλridge≥0\\lambda\_\{\\rm ridge\}\\geq 0is a small ridge stabilizer, withλridge=0\\lambda\_\{\\rm ridge\}=0corresponding to the unregularized solution𝒒​\(𝒙\)⊤=𝐞1⊤​\(𝑼⊤​𝐖​𝑼\)−1\\bm\{q\}\(\{\\bm\{x\}\}\)^\{\\top\}=\{\\bf e\}\_\{1\}^\{\\top\}\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\)^\{\-1\}\. The CS then broadcasts𝒒⁡\(𝒙\)\\bm\{q\}\(\{\\bm\{x\}\}\)to all clients\.

##### Step 2: aggregation of conditional CDF estimates\.

Given𝒒⁡\(𝒙\)\\bm\{q\}\(\{\\bm\{x\}\}\), the federated weight in \([4](https://arxiv.org/html/2610.33403#S2.E4)\) factorizes as

αi\(k\)=ϖi\(k\)​𝒒​\(𝒙\)⊤​𝑼i\(k\),\\alpha\_\{i\}^\{\(k\)\}=\\varpi\_\{i\}^\{\(k\)\}\\,\\bm\{q\}\(\{\\bm\{x\}\}\)^\{\\top\}\\bm\{U\}\_\{i\}^\{\(k\)\},\(8\)which is computable locally fromϖi\(k\)\\varpi\_\{i\}^\{\(k\)\},𝑼i\(k\)\\bm\{U\}\_\{i\}^\{\(k\)\}, and the broadcast𝒒⁡\(𝒙\)\\bm\{q\}\(\{\\bm\{x\}\}\)\. Each client then computes the local response\-smoothed CDF vector at the target grid𝒛=\(z1,…,zL\)∈ℝL\\bm\{z\}=\(z\_\{1\},\\ldots,z\_\{L\}\)\\in\\mathbb\{R\}^\{L\},

𝑭^\(k\)​\(𝒛∣𝒙\):=∑iαi\(k\)​𝒦​\(𝒛−hi,d\+1\(k\)​𝟏Lwz\(k\)\)∈ℝL,\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\):=\\sum\_\{i\}\\alpha\_\{i\}^\{\(k\)\}\\,\\mathcal\{K\}\\\!\\left\(\\frac\{\\bm\{z\}\-h\_\{i,d\+1\}^\{\(k\)\}\\mathbf\{1\}\_\{L\}\}\{w^\{\(k\)\}\_\{z\}\}\\right\)\\in\\mathbb\{R\}^\{L\},\(9\)where𝟏L∈ℝL\\mathbf\{1\}\_\{L\}\\in\\mathbb\{R\}^\{L\}is the all\-ones vector,wz\(k\)w^\{\(k\)\}\_\{z\}is thekkth client’s response bandwidth, and𝒦\\mathcal\{K\}is applied componentwise\. The client then uploads𝑭^\(k\)​\(𝒛∣𝒙\)\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)to the CS\. The CS aggregates,

𝑭^raw​\(𝒛∣𝒙\)=∑k=1K𝑭^\(k\)​\(𝒛∣𝒙\)∈ℝL,\\widehat\{\\bm\{F\}\}^\{\\rm raw\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)=\\sum\_\{k=1\}^\{K\}\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)\\in\\mathbb\{R\}^\{L\},\(10\)and applies a post\-processing operatorℳcdf\\mathcal\{M\}\_\{\\rm cdf\}that clips to\[0,1\]\[0,1\], isotonizes inzz, and normalizes the right endpoint to11to obtain the final estimate𝑭^​\(𝒛∣𝒙\)\\widehat\{\\bm\{F\}\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)\.

Algorithm 1FITS: Two\-Step Federated Estimation of the Conditional CDF0:Query covariate

𝒙∈𝒳⊂ℝd\{\{\\bm\{x\}\}\}\\in\{\\mathcal\{X\}\}\\subset\\mathbb\{R\}^\{d\}; target grid

𝒛=\(z1,…,zL\)∈ℝL\\bm\{z\}=\(z\_\{1\},\\dots,z\_\{L\}\)\\in\\mathbb\{R\}^\{L\}; response bandwidth

wzw\_\{z\}; clients

k=1,…,Kk=1,\\dots,Kwith continuous hold\-out data

ℋk=\{𝐡i\(k\)\}i=1mk\{\\mathcal\{H\}\}\_\{k\}=\\\{\{\{\\bf h\}\}\_\{i\}^\{\(k\)\}\\\}\_\{i=1\}^\{m\_\{k\}\}, where

𝐡i\(k\)=\(𝐡i,:d\(k\),hi,d\+1\(k\)\)\{\{\\bf h\}\}\_\{i\}^\{\(k\)\}=\(\{\{\\bf h\}\}\_\{i,:d\}^\{\(k\)\},\{h\}\_\{i,d\+1\}^\{\(k\)\}\); kernel\-bandwidth scheme parameter

r0∈ℕr\_\{0\}\\in\\mathbb\{N\}\.

1:—————– Step 1: aggregation of the kernel\-weighted Hessian —————–

2:for

k=1,…,Kk=1,\\dots,Kin paralleldo

3:According to Eq\. \([14](https://arxiv.org/html/2610.33403#S3.E14)\), run

𝒲k​\(𝒙,ℋk,r0\)\\mathcal\{W\}\_\{k\}\(\\bm\{x\};\\mathcal\{H\}\_\{k\},r\_\{0\}\)to get the index set of selected samples

𝒩ck\(k\)⊆\[mk\]\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}\\subseteq\[m\_\{k\}\]and the samples’ kernel weights

ϖi\(k\),i∈𝒩ck\(k\)\\varpi\_\{i\}^\{\(k\)\},\\ i\\in\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}\.

4:According to the summation Eq\. \([6](https://arxiv.org/html/2610.33403#S3.E6)\), client

kkobtains

𝐒\(k\)​\(𝒙\)\{\\bf S\}^\{\(k\)\}\(\{\\bm\{x\}\}\)and uploads it to CS\.

5:endfor

6:// CS aggregates and broadcasts

7:CS computes and broadcasts

q⁡\(𝒙\)q\(\{\\bm\{x\}\}\)according to Eq\. \([7](https://arxiv.org/html/2610.33403#S3.E7)\)\.

8:—————– Step 2: aggregation of conditional CDF estimates —————–

9:for

k=1,…,Kk=1,\\dots,Kin paralleldo

10:For each

i∈𝒩ck\(k\)i\\in\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}, client

kkcomputes the federated local linear weight

αi\(k\)\\alpha^\{\(k\)\}\_\{i\}according to Eq\. \([8](https://arxiv.org/html/2610.33403#S3.E8)\)\.

11:Compute and upload the local response\-smoothed CDF vector

𝑭^\(k\)​\(𝒛∣𝒙\)\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)according to Eq\. \([9](https://arxiv.org/html/2610.33403#S3.E9)\)\.

12:endfor

13:// CS aggregates and post\-processes

14:According to Eq\. \([10](https://arxiv.org/html/2610.33403#S3.E10)\), CS gets

𝑭^raw​\(𝒛∣𝒙\)\\widehat\{\\bm\{F\}\}^\{\\rm raw\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)and applies CDF post\-processing

ℳcdf\\mathcal\{M\}\_\{\\rm cdf\}to it\.

15:return

𝑭^​\(𝒛∣𝒙\)=ℳcdf​\[𝑭^raw​\(𝒛∣𝒙\)\]\\widehat\{\\bm\{F\}\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)=\\mathcal\{M\}\_\{\\rm cdf\}\[\\widehat\{\\bm\{F\}\}^\{\\rm raw\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)\]\.

###### Remark 3\.1\(Communication cost\)\.

The two\-step procedure requires, per query𝐱\{\\bm\{x\}\}: \(i\) Round 1\. Each client uploads a symmetric\(d\+1\)×\(d\+1\)\(d\{\+\}1\)\\times\(d\{\+\}1\)matrix, costing\(d\+1\)​\(d\+2\)/2\(d\{\+\}1\)\(d\{\+\}2\)/2scalars per client; the CS broadcasts a\(d\+1\)\(d\{\+\}1\)\-dimensional vector𝐪⁡\(𝐱\)\\bm\{q\}\(\{\\bm\{x\}\}\)\. \(ii\) Round 2\. Each client uploads anLL\-dimensional CDF vector with elements𝐅^\(k\)​\(zi∣𝐱\),i∈\[L\]\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(z\_\{i\}\\mid\{\\bm\{x\}\}\),\\ i\\in\[L\]\. The per\-query communication is thereforeO⁡\(d2\+L\)O\(d^\{2\}\+L\)per client, independent of the local sample sizemkm\_\{k\}\.

The full procedure is given in Algorithm[1](https://arxiv.org/html/2610.33403#alg1)and the communication cost isO⁡\(d2\+L\)O\(d^\{2\}\+L\)as stated in Remark[3\.1](https://arxiv.org/html/2610.33403#S3.Thmrem1)\. Here, in Algorithm[1](https://arxiv.org/html/2610.33403#alg1),𝒲k​\(𝒙,ℋk,r0\)\\mathcal\{W\}\_\{k\}\(\\bm\{x\};\\mathcal\{H\}\_\{k\},r\_\{0\}\)denotes the client\-designed procedure described in Subsection[3\.2](https://arxiv.org/html/2610.33403#S3.SS2), where clients can choose their own kernel\-fitting schemes\. It returns the selected local samples together with their weightsϖ\\varpi, and allows each client to adaptively choose a kernel\-fitting scheme, either the fixed\-bandwidth scheme or the KNN\-kernel scheme\. The fixed\-bandwidth scheme uses all local hold\-out samples, so that𝒩ck\(k\)=\[mk\]\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}=\[m\_\{k\}\]\. By contrast, the KNN\-kernel scheme searches for the neighbors of𝒙\{\\bm\{x\}\}and uses a subset𝒩ck\(k\)⊂\[mk\]\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}\\subset\[m\_\{k\}\]\.

Combining \([6](https://arxiv.org/html/2610.33403#S3.E6)\)–\([8](https://arxiv.org/html/2610.33403#S3.E8)\) with \([4](https://arxiv.org/html/2610.33403#S2.E4)\) yields the following algebraic equivalence between the federated and pooled estimators\.

###### Proposition 3\.1\(Federated–pooled equivalence\)\.

Fix any query𝐱∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and thresholdz∈𝒵z\\in\\mathcal\{Z\}\. Suppose the kernel weights\{ϖi\(k\):i∈\[mk\],k∈\[K\]\}\\\{\\varpi\_\{i\}^\{\(k\)\}:i\\in\[m\_\{k\}\],\\ k\\in\[K\]\\\}used by the federated algorithm coincide with those that would be assigned to the same observations under the centralized rule on the pooled sample, and thatλridge=0\\lambda\_\{\\rm ridge\}=0and𝐔⊤​𝐖​𝐔\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}is invertible\. Then

F^raw​\(z∣𝒙\)=∑k=1KF^\(k\)​\(z∣𝒙\)=β^0,\\widehat\{F\}^\{\\rm raw\}\(z\\mid\{\\bm\{x\}\}\)\\ =\\ \\sum\_\{k=1\}^\{K\}\\widehat\{F\}^\{\(k\)\}\(z\\mid\{\\bm\{x\}\}\)\\ =\\ \\hat\{\\beta\}\_\{0\},whereβ^0\\hat\{\\beta\}\_\{0\}is the centralized local linear estimator \([4](https://arxiv.org/html/2610.33403#S2.E4)\) computed on the pooled sample\.

###### Proof\.

Substituting \([6](https://arxiv.org/html/2610.33403#S3.E6)\) into \([7](https://arxiv.org/html/2610.33403#S3.E7)\) gives𝒒​\(𝒙\)⊤=𝐞1⊤​\(𝑼⊤​𝐖​𝑼\)−1\\bm\{q\}\(\{\\bm\{x\}\}\)^\{\\top\}=\{\\bf e\}\_\{1\}^\{\\top\}\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\)^\{\-1\}whenλridge=0\\lambda\_\{\\rm ridge\}=0\. Hence, \([8](https://arxiv.org/html/2610.33403#S3.E8)\) equals the centralized weightαi\\alpha\_\{i\}in \([4](https://arxiv.org/html/2610.33403#S2.E4)\)\. Summing \([9](https://arxiv.org/html/2610.33403#S3.E9)\) overkkreproduces the right\-hand side of \([4](https://arxiv.org/html/2610.33403#S2.E4)\)\. ∎

Proposition[3\.1](https://arxiv.org/html/2610.33403#S3.Thmprop1)establishes that Algorithm[1](https://arxiv.org/html/2610.33403#alg1)is not an approximation of the centralized estimator but an exact reformulation of it under federated data access\. The practical implication is that all statistical properties derived for the centralized estimator—in particular the uniform convergence rates of Section[5](https://arxiv.org/html/2610.33403#S5)—transfer to the federated estimator without further argument\.

### 3\.2Client\-designed kernel

In this subsection, we propose two client\-designed kernel schemes: the fixed\-bandwidth scheme and the KNN\-kernel scheme, supporting the𝒲k​\(𝒙,ℋk,r0\)\\mathcal\{W\}\_\{k\}\(\\bm\{x\};\\mathcal\{H\}\_\{k\},r\_\{0\}\)procedure in line 3 of Algorithm[1](https://arxiv.org/html/2610.33403#alg1)\. Clients may choose either scheme to select local samples and compute their kernel weights\. Moreover, both schemes leverage the client’s local data to standardize the selected samples and the query point before kernel fitting, thereby improving the stability and effectiveness of kernel fitting\. The standardization procedure is performed entirely on the client side: the marginal means, standard deviations, and raw covariate scales are not transmitted, in accordance with the data\-access constraints in Subsection[2\.1](https://arxiv.org/html/2610.33403#S2.SS1)\.

*Fixed\-bandwidth scheme\.*Under the fixed\-bandwidth scheme, each client uses all of its local hold\-out samples, so that𝒩ck\(k\)=\[mk\]\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}=\[m\_\{k\}\], together with a client\-specific covariate bandwidthwx\(k\)\>0w\_\{x\}^\{\(k\)\}\>0\. After the covariate coordinates have been locally standardized using the client’s marginal means and standard deviations, the weight assigned to observationiiis

ϖi\(k\)=𝜿⁡\(𝒙¯\(k\)−𝐡¯i\(k\)wx\(k\)\)​ρi\(k\),i∈\[mk\]\\varpi\_\{i\}^\{\(k\)\}=\\bm\{\\kappa\}\\\!\\left\(\\frac\{\\bar\{\\bm\{x\}\}^\{\(k\)\}\-\\bar\{\\bf h\}\_\{i\}^\{\(k\)\}\}\{w\_\{x\}^\{\(k\)\}\}\\right\)\\rho\_\{i\}^\{\(k\)\},\\qquad i\\in\[m\_\{k\}\]\(11\)where𝜿⁡\(⋅\)=∏j=1dκj​\(⋅\)\\bm\{\\kappa\}\(\\cdot\)=\\prod\_\{j=1\}^\{d\}\\kappa\_\{j\}\(\\cdot\)is the product kernel and barred quantities denote locally Z\-score standardized coordinates\.ρi\(k\)\\rho\_\{i\}^\{\(k\)\}is the category penalty introduced in Subsection[4\.4](https://arxiv.org/html/2610.33403#S4.SS4)and is used to reduce the kernel\-fitting error caused by mismatched categories; if there is no discrete variable,ρi\(k\)=1\\rho\_\{i\}^\{\(k\)\}=1\. This scheme is conceptually simple and directly inherits the usual fixed\-bandwidth local linear interpretation, but it may lead to highly uneven effective sample sizes due to client imbalance\. Similarly, the effective sample size problem also affects the pooled method\. A possible approach is based on KNN\-type methods\([Fan and Gijbels, 1992](https://arxiv.org/html/2610.33403#bib.bib11);[Benezet et al\., 2025](https://arxiv.org/html/2610.33403#bib.bib10)\)\.

*KNN\-kernel scheme\.*Under the KNN\-kernel scheme, the effective bandwidth is chosen adaptively at each client based on the distance to theckc\_\{k\}\-th nearest neighbor of the query point\. The local KNN budgets are allocated in proportion to the hold\-out sample sizes:

ck=max⁡\{c0,min⁡\(⌊mkM​r0⌉,mk\)\},c\_\{k\}=\\max\\\!\\left\\\{c\_\{0\},\\ \\min\\\!\\left\(\\big\\lfloor\\frac\{m\_\{k\}\}\{M\}r\_\{0\}\\big\\rceil,\\ m\_\{k\}\\right\)\\right\\\},\(12\)wherer0∈ℕr\_\{0\}\\in\\mathbb\{N\}is a global KNN budget andc0\>1c\_\{0\}\>1is the minimum number of nearest neighbors\. Hence, the total number of observations contributing to the federated local fit is approximatelyr0r\_\{0\}, matching the budget of a centralized KNN rule with total sample sizeMM\. After locally standardizing the covariate coordinates, clientkkselects theckc\_\{k\}nearest neighbors of the query point and sets

rck\(k\)​\(𝒙¯\)=maxi∈𝒩ck\(k\)⁡‖𝐡¯i\(k\)−𝒙¯\(k\)‖\.r\_\{c\_\{k\}\}^\{\(k\)\}\(\\bar\{\\bm\{x\}\}\)=\\max\_\{i\\in\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\}\}\\\|\\bar\{\\bf h\}\_\{i\}^\{\(k\)\}\-\\bar\{\\bm\{x\}\}^\{\(k\)\}\\\|\.The corresponding kernel weight is

ϖi\(k\)=𝜿⁡\(𝐡¯i\(k\)−𝒙¯\(k\)η​rck\(k\)​\(𝒙¯\)\)​ρi\(k\),i∈𝒩ck\(k\),\\varpi\_\{i\}^\{\(k\)\}=\\bm\{\\kappa\}\\\!\\left\(\\frac\{\\bar\{\\bf h\}\_\{i\}^\{\(k\)\}\-\\bar\{\\bm\{x\}\}^\{\(k\)\}\}\{\\eta\\,r\_\{c\_\{k\}\}^\{\(k\)\}\(\\bar\{\\bm\{x\}\}\)\}\\right\)\\rho\_\{i\}^\{\(k\)\},\\qquad i\\in\\mathcal\{N\}\_\{c\_\{k\}\}^\{\(k\)\},\(13\)whereη\>0\\eta\>0is a fixed kernel\-scale parameter\. Typically, a bounded\-support kernel𝜿⁡\(⋅\)\\bm\{\\kappa\}\(\\cdot\)is used for the KNN\-kernel scheme\.

Based on these schemes, we define a kernel\-fitting function for each client that outputs the index set of selected samples and their kernel weights:

𝒲k​\(𝒙,ℋk,r0\)→Nck\(k\),\{ϖi\(k\)\}iNck\(k\),k∈\[K\],\\mathcal\{W\}\_\{k\}\(\\bm\{x\};\\mathcal\{H\}\_\{k\},r\_\{0\}\)\\to N\_\{c\_\{k\}\}^\{\(k\)\},\\\{\\varpi\_\{i\}^\{\(k\)\}\\\}\_\{i\}^\{N\_\{c\_\{k\}\}^\{\(k\)\}\},\\ k\\in\[K\],\(14\)wherer0∈ℕr\_\{0\}\\in\\mathbb\{N\},r0\>0r\_\{0\}\>0selects the KNN\-kernel scheme, whereasr0=0r\_\{0\}=0selects the fixed\-bandwidth scheme\.

###### Remark 3\.2\.

A separate design choice concerns the scale on which the local linear expansion is formed\. The kernel weightsϖi\(k\)\\varpi\_\{i\}^\{\(k\)\}are evaluated in the locally standardized covariate space, but the local linear design vector𝐔i\(k\)=𝛊\(𝐡i,:d\(k\)−𝐱\)\\bm\{U\}\_\{i\}^\{\(k\)\}=\\bm\{\\iota\}\(\{\\bf h\}\_\{i,:d\}^\{\(k\)\}\-\{\\bm\{x\}\}\)is constructed in the original coordinate space\. This mismatch is intentional\. Standardization stabilizes the kernel\-fitting step under heterogeneous client scales; keeping the design in the native scale ensures that the local linear expansion targetsF⁡\(z∣𝐱\)F\(z\\mid\{\\bm\{x\}\}\)in the original covariate system, so that the federated estimator remains an estimator of the population conditional distribution on𝒳\{\\mathcal\{X\}\}rather than on a client\-dependent rescaling thereof\.

### 3\.3Federated Synthetic Data Generation

![Refer to caption](https://arxiv.org/html/2610.33403v1/Figure/FedSyn.png)Figure 2:FITS: generation of a single synthetic sample by the chain rule\.We combine the chain rule with inverse sampling from the federated conditional CDF estimator𝑭^​\(𝒛∣𝒙\)\\widehat\{\\bm\{F\}\}\(\\bm\{z\}\\mid\{\\bm\{x\}\}\)to generate synthetic data\. Without loss of generality, we assume that the chain order follows the variable sequence from𝐗1\{\\bf X\}\_\{1\}to𝐗d\+1\{\\bf X\}\_\{d\+1\}and denote𝐗:j=\(𝐗1,…,𝐗j\)\{\\bf X\}\_\{:j\}=\(\{\\bf X\}\_\{1\},\\ldots,\{\\bf X\}\_\{j\}\)\. Thejj\-th sampling step is based on the one\-dimensional conditional distribution function given the previously sampledj−1j\-1dimensions:

x^j=𝑭^−1\(U∣𝒙^:j−1=\(x^1,…,x^j−1\)\),\\hat\{x\}\_\{j\}=\\widehat\{\\bm\{F\}\}^\{\-1\}\(U\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}=\(\\hat\{x\}\_\{1\},\\ldots,\\hat\{x\}\_\{j\-1\}\)\),\(15\)whereU∼Unif⁡\[0,1\]U\\sim\\mathrm\{Unif\}\[0,1\],x^i\\hat\{x\}\_\{i\}denotes the sampled value of theii\-th coordinate, and𝑭^−1\(⋅∣𝒙^:j−1\)\\widehat\{\\bm\{F\}\}^\{\-1\}\(\\cdot\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}\)denotes the estimated inverse conditional CDF given the event𝒙^:j−1=\(x^1,…,x^j−1\)\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}=\(\\hat\{x\}\_\{1\},\\ldots,\\hat\{x\}\_\{j\-1\}\)\. For notational convenience, we denote𝑭^\(z∣𝒙^:0\)\\widehat\{\\bm\{F\}\}\(z\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:0\}\)as the unconditional CDF, which is equivalent to the CDF estimate for the first variable𝑭^​\(z\)\\widehat\{\\bm\{F\}\}\(z\)\. For conditional CDF estimation, we estimate the conditional CDF using Algorithm[1](https://arxiv.org/html/2610.33403#alg1)and invert the estimate using linear interpolation\. For the first dimension, we use the following smooth KDE\-based CDF estimator:

𝑭^​\(z\)=1M​∑k=1K𝑭^\(k\)​\(z\)=1M​∑k=1K\[∑i=1mk𝒦⁡\(z−hi,1\(k\)wz\(k\)\)\],\\widehat\{\\bm\{F\}\}\(z\)=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{K\}\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(z\)=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{K\}\\left\[\\sum\_\{i=1\}^\{m\_\{k\}\}\\mathcal\{K\}\\left\(\\frac\{z\-h\_\{i,1\}^\{\(k\)\}\}\{w\_\{z\}^\{\(k\)\}\}\\right\)\\right\],\(16\)where𝒦⁡\(⋅\)\\mathcal\{K\}\(\\cdot\)is the kernel CDF\. The federated CDF estimation in this procedure is straightforward: each client uploads the estimated values𝑭^\(k\)​\(zℓ\)\\widehat\{\\bm\{F\}\}^\{\(k\)\}\(z\_\{\\ell\}\)on the grid points𝒛=\(z1,…,zL\)\\bm\{z\}=\(z\_\{1\},\\ldots,z\_\{L\}\)\. Values between adjacent grid points are obtained by linear interpolation\. Specifically, ifU∈\(𝑭^\(zℓ−1∣𝒙^:j−1\),𝑭^\(zℓ∣𝒙^:j−1\)\),U\\in\(\\widehat\{\\bm\{F\}\}\(z\_\{\\ell\-1\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}\),\\widehat\{\\bm\{F\}\}\(z\_\{\\ell\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}\)\),then

x^j=zℓ−1\+U−𝑭^\(zℓ−1∣𝒙^:j−1\)𝑭^\(zℓ∣𝒙^:j−1\)−𝑭^\(zℓ−1∣𝒙^:j−1\)\(zℓ−zℓ−1\),ℓ∈\[L\]\.\\hat\{x\}\_\{j\}=z\_\{\\ell\-1\}\+\\frac\{U\-\\widehat\{\\bm\{F\}\}\(z\_\{\\ell\-1\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}\)\}\{\\widehat\{\\bm\{F\}\}\(z\_\{\\ell\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}\)\-\\widehat\{\\bm\{F\}\}\(z\_\{\\ell\-1\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}\)\}\(z\_\{\\ell\}\-z\_\{\\ell\-1\}\),\\qquad\\ell\\in\[L\]\.\(17\)
Algorithm 2FITS: Federated Synthetic Data Generation0:Number of synthetic samples

BB; coordinate grids

𝒢=\{𝒈j\}j=1d\+1\\mathcal\{G\}=\\\{\\bm\{g\}\_\{j\}\\\}\_\{j=1\}^\{d\+1\}in the chain order; clients

k=1,…,Kk=1,\\ldots,Kwith hold\-out data

ℋk=\{𝐡i\(k\)\}i=1mk\\mathcal\{H\}\_\{k\}=\\\{\{\\bf h\}\_\{i\}^\{\(k\)\}\\\}\_\{i=1\}^\{m\_\{k\}\}, and the inputs required by Algorithm[1](https://arxiv.org/html/2610.33403#alg1)\.

1:for

j=1,2,…,d\+1j=1,2,\\ldots,d\+1do

2:for

b=1,…,Bb=1,\\ldots,Bdo

3:Estimate the CDFs in a federated manner

\{𝑭^​\(𝒈1\)​by Eq\. \([16](https://arxiv.org/html/2610.33403#S3.E16)\),j=1,𝑭^\(𝒈j∣𝒙^b,:j−1=\(x^b,1,…,x^b,j−1\)\)by Algorithm[1](https://arxiv.org/html/2610.33403#alg1),j\>1\.\\begin\{cases\}\\widehat\{\\bm\{F\}\}\(\\bm\{g\}\_\{1\}\)\\text\{ by Eq\.~\\eqref\{eq:first\_dim\_kde\}\},&j=1,\\\\\[2\.84526pt\] \\widehat\{\\bm\{F\}\}\(\\bm\{g\}\_\{j\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{b,:j\-1\}=\(\\hat\{x\}\_\{b,1\},\.\.\.,\\hat\{x\}\_\{b,j\-1\}\)\)\\text\{ by Algorithm~\\ref\{alg:two\_step\_fllr\}\},&j\>1\.\\end\{cases\}
4:CS draws

ub​j∼Unif⁡\(0,1\)u\_\{bj\}\\sim\\mathrm\{Unif\}\(0,1\)and samples

x^b,j←𝑭^−1\(ub​j∣𝒙^:j−1=\(x^1,…,x^j−1\)\)by Eq\. \([15](https://arxiv.org/html/2610.33403#S3.E15)\) and Eq\. \([17](https://arxiv.org/html/2610.33403#S3.E17)\)\.\\hat\{x\}\_\{b,j\}\\leftarrow\\widehat\{\\bm\{F\}\}^\{\-1\}\(u\_\{bj\}\\mid\\hat\{\{\\bm\{x\}\}\}\_\{:j\-1\}=\(\\hat\{x\}\_\{1\},\.\.\.,\\hat\{x\}\_\{j\-1\}\)\)\\text\{ by Eq\.~\\eqref\{eq:inv\_sampling\} and Eq\.~\\eqref\{eq:interpolation\}\}\.
5:endfor

6:endfor

7:returnSynthetic data

ℬ=\{𝒙^b=\(x^b,1,…,x^b,d\+1\)\}b=1B\\mathcal\{B\}=\\\{\\hat\{\{\\bm\{x\}\}\}\_\{b\}=\(\\hat\{x\}\_\{b,1\},\.\.\.,\\hat\{x\}\_\{b,d\+1\}\)\\\}\_\{b=1\}^\{B\}\.

###### Remark 3\.3\(Cost of synthetic data generation\)\.

ForDDvariables,BBsynthetic records,KKclients, andLLresponse grid points, each conditional CDF query withrrconditioning variables requires each client to upload\(r\+1\)​\(r\+2\)/2\(r\+1\)\(r\+2\)/2Hessian entries andLLCDF values\. Thus, direct execution of the conditional generation steps in Algorithm[2](https://arxiv.org/html/2610.33403#alg2)requiresO⁡\(K​B​\(D3\+D​L\)\)O\\\!\\left\(KB\(D^\{3\}\+DL\)\\right\)uploaded scalars in total\.

## 4Generalization to Mixed\-Type Data

In this section, we propose a continualization method for discrete variables\. In Subsection[4\.1](https://arxiv.org/html/2610.33403#S4.SS1), the proposed continualization procedure lifts discrete variables into a continuous space and allows exact inversion back to the original discrete space without information loss\. In particular, we characterize continualization at the density\-function level, thereby unifying mixed\-type data within a continuous framework in Subsection[4\.2](https://arxiv.org/html/2610.33403#S4.SS2)\. For illustration, we provide a bivariate example in Subsection[4\.3](https://arxiv.org/html/2610.33403#S4.SS3)\. For practical implementation, we further develop a penalty\-adaptive kernel method for mixed\-type covariates in Subsection[4\.4](https://arxiv.org/html/2610.33403#S4.SS4)\. This method effectively mitigates the impact of mismatches between continualized discrete samples used for kernel fitting and the query point\.

### 4\.1Continualization of discrete variables

The lifting is constructed independently for each discrete coordinate; hence, this subsection focuses on the univariate case\. Discrete variables include ordered variables and unordered \(nominal/categorical\) variables\. For ordered variables, we lift the discrete support into continuous adjacent intervals\. For unordered variables, we fix an arbitrary one\-to\-one numerical coding to define the continualization intervals\.

##### Ordered discrete variables\.

LetΥ\\Upsilonbe a discrete random variable with finite ordered support

Υ=\{υ1,…,υL\},υ1<⋯<υL,pℓ=ℙ⁡\(Υ=υℓ\),∑ℓ=1Lpℓ=1\.\\varUpsilon=\\\{\\upsilon\_\{1\},\\ldots,\\upsilon\_\{L\}\\\},\\qquad\\upsilon\_\{1\}<\\cdots<\\upsilon\_\{L\},\\qquad p\_\{\\ell\}=\\mathbb\{P\}\(\\Upsilon=\\upsilon\_\{\\ell\}\),\\quad\\sum\_\{\\ell=1\}^\{L\}p\_\{\\ell\}=1\.Setυ0:=υ1−1\\upsilon\_\{0\}:=\\upsilon\_\{1\}\-1and define the disjoint intervalsIℓI\_\{\\ell\}and an interval\-valued random variableIΥI\_\{\\Upsilon\}corresponding toΥ\\Upsilon:

Iℓ=\(υℓ−1,υℓ\],IΥ=\(⌊Υ⌋,Υ\],ℓ∈\[L\],I\_\{\\ell\}=\(\\upsilon\_\{\\ell\-1\},\\upsilon\_\{\\ell\}\],\\qquad I\_\{\\Upsilon\}=\(\\lfloor\\Upsilon\\rfloor,\\Upsilon\],\\qquad\\ell\\in\[L\],where⌊Υ⌋\\lfloor\\Upsilon\\rfloorrepresents the predecessorυℓ−1\\upsilon\_\{\\ell\-1\}ofΥ=υℓ\\Upsilon=\\upsilon\_\{\\ell\}in the ordered support, and\|Iℓ\|=υℓ−υℓ−1\|I\_\{\\ell\}\|=\\upsilon\_\{\\ell\}\-\\upsilon\_\{\\ell\-1\}\. LetU∼Unif⁡\(0,\|IΥ\|\)U\\sim\\mathrm\{Unif\}\(0,\|I\_\{\\Upsilon\}\|\)be independent ofΥ\\Upsilon, and define the continualized variable

ThenYYtakes values in⋃ℓ=1LIℓ\\bigcup\_\{\\ell=1\}^\{L\}I\_\{\\ell\}, and on each intervalIℓI\_\{\\ell\}, the masspℓp\_\{\\ell\}is spread uniformly\. The density and distribution function ofYYare

fY\(y\)=∑ℓ=1L\(pℓ/\|Iℓ\|\)𝕀\{y∈Iℓ\},FY\(y\)=∑r<ℓpr\+pℓ\(y−υℓ−1\)\|Iℓ\|,y∈Iℓ,f\_\{Y\}\(y\)=\\sum\_\{\\ell=1\}^\{L\}\(p\_\{\\ell\}/\|I\_\{\\ell\}\|\)\\,\\mathbb\{I\}\\\{y\\in I\_\{\\ell\}\\\},\\qquad F\_\{Y\}\(y\)=\\sum\_\{r<\\ell\}p\_\{r\}\+p\_\{\\ell\}\\frac\{\(y\-\\upsilon\_\{\\ell\-1\}\)\}\{\|I\_\{\\ell\}\|\},\\quad y\\in I\_\{\\ell\},sofYf\_\{Y\}is piecewise constant andFYF\_\{Y\}is piecewise affine\.

##### Inverse map via the ceiling operator\.

The original discrete variable is recovered from its continualization by

Υ=ceil⁡\(Y\):=υℓ​on the event​Y∈Iℓ\.\\Upsilon=\\mathrm\{ceil\}\(Y\):=\\upsilon\_\{\\ell\}\\text\{ on the event \}Y\\in I\_\{\\ell\}\.\(19\)Equivalently,ceil\\mathrm\{ceil\}is the canonical inverse map between the partition\{Iℓ\}ℓ=1L\\\{I\_\{\\ell\}\\\}\_\{\\ell=1\}^\{L\}of⋃ℓIℓ\\bigcup\_\{\\ell\}I\_\{\\ell\}and the supportΥ\\varUpsilon\.

##### Unordered \(nominal\) discrete variables\.

For an unordered categorical variable with support\{c1,…,cL\}\\\{c\_\{1\},\\ldots,c\_\{L\}\\\}, the same construction applies after fixing an arbitrary one\-to\-one codingcℓ↔ℓc\_\{\\ell\}\\leftrightarrow\\ell\. We set

Iℓ=\(ℓ−1,ℓ\],Y=ℓ−Uon the eventΥ=cℓ,U∼Unif\(0,1\),I\_\{\\ell\}=\(\\ell\-1,\\ell\],\\qquad Y=\\ell\-U\\text\{ on the event \}\\Upsilon=c\_\{\\ell\},\\qquad U\\sim\\mathrm\{Unif\}\(0,1\),and recover the original category through

Υ=cat⁡\(Y\):=c⌈Y⌉\.\\Upsilon=\\mathrm\{cat\}\(Y\):=c\_\{\\lceil Y\\rceil\}\.\(20\)Because theceil\\mathrm\{ceil\}operator is also included in thecat\\mathrm\{cat\}operation, we usecat\\mathrm\{cat\}as the unified inverse operator\. The coding is used to define the partition and inverse map and introduces a Euclidean metric\. We summarize the properties of the univariate construction here\. The more general multidimensional construction is presented in Subsection[4\.2](https://arxiv.org/html/2610.33403#S4.SS2), together with the corresponding theoretical results, Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)and Corollary[5\.2](https://arxiv.org/html/2610.33403#S5.Thmcor2)\.

###### Proposition 4\.1\(Univariate continualization\)\.

LetΥ\\Upsilonbe a discrete variable onΥ=\{υ1,…,υL\}\\varUpsilon=\\\{\\upsilon\_\{1\},\\ldots,\\upsilon\_\{L\}\\\}\(ordered or nominal\) and letYYbe its continualization as in \([18](https://arxiv.org/html/2610.33403#S4.E18)\)\. Then:

1. 1\.Mass preservation\.ℙ⁡\(Y∈Iℓ\)=pℓ=ℙ⁡\(Υ=υℓ\)\\mathbb\{P\}\(Y\\in I\_\{\\ell\}\)=p\_\{\\ell\}=\\mathbb\{P\}\(\\Upsilon=\\upsilon\_\{\\ell\}\)for everyℓ∈\[L\]\\ell\\in\[L\]\.
2. 2\.Exact inverse\.For ordered variables,ceil⁡\(Y\)=Υ\\mathrm\{ceil\}\(Y\)=\\Upsilonalmost surely; for nominal variables,cat⁡\(Y\)=Υ\\mathrm\{cat\}\(Y\)=\\Upsilonalmost surely\.
3. 3\.Piecewise regularity\.The densityfYf\_\{Y\}is piecewise constant on the partition\{Iℓ\}ℓ\\\{I\_\{\\ell\}\\\}\_\{\\ell\}, and the distribution functionFYF\_\{Y\}is continuous on⋃ℓIℓ\\bigcup\_\{\\ell\}I\_\{\\ell\}and piecewise affine within eachIℓI\_\{\\ell\}\.

Properties \(i\)–\(iii\) follow directly from the definition and therefore require no further proof\.

### 4\.2Multidimensional continualization and block structure

We now extend the univariate construction to the mixed\-type covariate vector𝐗\{\\bf X\}\. Write

𝐗=\(𝐗\(c\),𝐗\(d\)\),\{\\bf X\}=\\bigl\(\{\\bf X\}^\{\(c\)\},\{\\bf X\}^\{\(d\)\}\\bigr\),where𝐗\(c\)∈𝒳c⊂ℝd−d~\{\\bf X\}^\{\(c\)\}\\in\\mathcal\{X\}\_\{c\}\\subset\\mathbb\{R\}^\{d\-\\tilde\{d\}\}collects thed−d~d\-\\tilde\{d\}continuous coordinates and𝐗\(d\)\{\\bf X\}^\{\(d\)\}collects thed~\\tilde\{d\}discrete coordinates\. For each discrete coordinatej∈\[d~\]j\\in\[\\tilde\{d\}\], letΥj=\{υj,1,…,υj,τj\}\\varUpsilon\_\{j\}=\\\{\\upsilon\_\{j,1\},\\ldots,\\upsilon\_\{j,\\tau\_\{j\}\}\\\}denote its support, and let𝒴j=⋃ℓ=1τjIj,ℓ\\mathcal\{Y\}\_\{j\}=\\bigcup\_\{\\ell=1\}^\{\\tau\_\{j\}\}I\_\{j,\\ell\}denote the union of its continualization intervals as defined in Section[4\.1](https://arxiv.org/html/2610.33403#S4.SS1)\. The product support of the discrete covariates is

Υ=∏j=1d~Υj\.\\varUpsilon=\\prod\_\{j=1\}^\{\\tilde\{d\}\}\\varUpsilon\_\{j\}\.For each𝝊=\(υ1,ℓ1,…,υd~,ℓd~\)∈Υ\\bm\{\\upsilon\}=\(\\upsilon\_\{1,\\ell\_\{1\}\},\\ldots,\\upsilon\_\{\\tilde\{d\},\\ell\_\{\\tilde\{d\}\}\}\)\\in\\varUpsilon, define the associated block

I\(𝝊\)=I1,ℓ1×⋯×Id~,ℓd~⊂𝒴:=∏j=1d~𝒴j\.I\(\\bm\{\\upsilon\}\)=I\_\{1,\\ell\_\{1\}\}\\times\\cdots\\times I\_\{\\tilde\{d\},\\ell\_\{\\tilde\{d\}\}\}\\ \\subset\\ \\mathcal\{Y\}:=\\prod\_\{j=1\}^\{\\tilde\{d\}\}\\mathcal\{Y\}\_\{j\}\.\(21\)Under the left\-open right\-closed convention, the blocks\{I⁡\(𝝊\)\}𝝊∈Υ\\\{I\(\\bm\{\\upsilon\}\)\\\}\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}form a partition of𝒴\\mathcal\{Y\}, and the indicator𝕀\{𝐘∈I\(𝝊\)\}\\mathbb\{I\}\\\{\{\\bf Y\}\\in I\(\\bm\{\\upsilon\}\)\\\}is equivalent to𝕀\{𝐗\(d\)=𝝊\}\\mathbb\{I\}\\\{\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}\\\}\. The blockI⁡\(𝝊\)I\(\\bm\{\\upsilon\}\)also defines a corresponding volume\|I⁡\(𝝊\)\|\|I\(\\bm\{\\upsilon\}\)\|, which is used to normalize the lifted density function\.

Let𝐘\{\\bf Y\}denote the joint continualization of𝐗\(d\)\{\\bf X\}^\{\(d\)\}, obtained by applying \([18](https://arxiv.org/html/2610.33403#S4.E18)\) coordinatewise with mutually independent uniform random variables\. The resulting continualized vector is

𝐗~=\(𝐗\(c\),𝐘\)∈𝒳c×𝒴=:𝒳~\.\\widetilde\{\\bf X\}=\\bigl\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\\bigr\)\\in\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}=:\\widetilde\{\\mathcal\{X\}\}\.\(22\)Throughout the paper,𝒙\{\\bm\{x\}\}denotes a generic point in the original mixed\-type covariate space𝒳\\mathcal\{X\}, while𝒙~=\(𝒙\(c\),𝐲\)\\tilde\{\\bm\{x\}\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)denotes the corresponding point in𝒳~\\widetilde\{\\mathcal\{X\}\}\. The inverse discrete map

𝒙~↦\(𝒙\(c\),cat1​\(y1\),…,catd~​\(yd~\)\)\\tilde\{\\bm\{x\}\}\\mapsto\\bigl\(\{\\bm\{x\}\}^\{\(c\)\},\\mathrm\{cat\}\_\{1\}\(y\_\{1\}\),\\ldots,\\mathrm\{cat\}\_\{\\tilde\{d\}\}\(y\_\{\\tilde\{d\}\}\)\\bigr\)recovers the original mixed\-type variable exactly\.

The block structure naturally separates𝒳~\\widetilde\{\\mathcal\{X\}\}into an interior region and a boundary region\. Define

𝒳~∘:=𝒳c×𝒴∘=𝒳c×⋃𝝊∈ΥI​\(𝝊\)∘,\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}:=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}^\{\\circ\}=\\mathcal\{X\}\_\{c\}\\times\\bigcup\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}I\(\\bm\{\\upsilon\}\)^\{\\circ\},\(23\)whereI​\(𝝊\)∘I\(\\bm\{\\upsilon\}\)^\{\\circ\}denotes the topological interior ofI⁡\(𝝊\)I\(\\bm\{\\upsilon\}\)and𝒴∘\\mathcal\{Y\}^\{\\circ\}denotes the union of the interiors of all blocks\. Its complement is

𝒳~∙:=𝒳c×𝒴∙,𝒴∙:=𝒴∖𝒴∘\.\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}:=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}^\{\\bullet\},\\qquad\\mathcal\{Y\}^\{\\bullet\}:=\\mathcal\{Y\}\\setminus\\mathcal\{Y\}^\{\\circ\}\.This set is the union of all block boundaries in the𝐲\{\\bf y\}\-coordinates and will be referred to as the covariate jumping set \(block\-boundary set\), since the lifted joint density of𝐗~\\widetilde\{\\bf X\}may exhibit jump discontinuities across𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}\.

This construction yields a piecewise smooth lifted law\. On each block𝒳c×I​\(𝝊\)∘\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, the joint density of𝐗~\\widetilde\{\\bf X\}inherits the smoothness of the conditional density of𝐗\(c\)\{\\bf X\}^\{\(c\)\}given𝐗\(d\)=𝝊\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}, because the continualization is uniform within the corresponding discrete cell\. Across𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}, the density may jump, since the conditional densities of𝐗\(c\)\{\\bf X\}^\{\(c\)\}on adjacent blocks need not match\. Thus the lifted law is generally piecewise smooth rather than globally smooth\. A formal piecewise\-CrC^\{r\}regularity condition is given in Section[5](https://arxiv.org/html/2610.33403#S5), where it supports the blockwise uniform convergence analysis of conditional CDF estimation\.

The same continualization device is applied to the response whenZZis discrete\. Specifically, ifZ∈\{z1,…,zL\}Z\\in\\\{z\_\{1\},\\ldots,z\_\{L\}\\\}withz1<⋯<zLz\_\{1\}<\\cdots<z\_\{L\}, setz0:=z1−1z\_\{0\}:=z\_\{1\}\-1,Ii=\(zi−1,zi\]I\_\{i\}=\(z\_\{i\-1\},z\_\{i\}\], and defineZ~=Z−V\\tilde\{Z\}=Z\-V, whereV∼Unif⁡\(0,\|IZ\|\)V\\sim\\mathrm\{Unif\}\(0,\|I\_\{Z\}\|\)is independent of\(Z,𝐗~\)\(Z,\\widetilde\{\\bf X\}\)\. ThenZ~∈𝒵~:=⋃i=1LIi\\tilde\{Z\}\\in\\widetilde\{\\mathcal\{Z\}\}:=\\bigcup\_\{i=1\}^\{L\}I\_\{i\}, and the target in \([1](https://arxiv.org/html/2610.33403#S2.E1)\) is estimated asF⁡\(z~∣𝒙~\)F\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\)\. IfZZis already continuous, no response continualization is performed and we setz~=z\\tilde\{z\}=z\. Section[5](https://arxiv.org/html/2610.33403#S5)covers both cases\.

### 4\.3An illustrative bivariate example

![Refer to caption](https://arxiv.org/html/2610.33403v1/schematic_figure_heatmap.png)\(a\)Lifted density after binary continualization\.
![Refer to caption](https://arxiv.org/html/2610.33403v1/toy_example_discrete.png)\(b\)Original joint CDF\.
![Refer to caption](https://arxiv.org/html/2610.33403v1/toy_example_continualized.png)\(c\)Continualized joint CDF\.

Figure 3:Illustration of binary continualization in a bivariate model with one continuousX1X\_\{1\}and one binary variableX2∈\{0,1\}X\_\{2\}\\in\\\{0,1\\\}\. Panel \(a\) shows the lifted densityg\(x1,y\)=f\(x1,0\)𝕀\{y∈\(−1,0\]\}\+f\(x1,1\)𝕀\{y∈\(0,1\]\}g\(x\_\{1\},y\)=f\(x\_\{1\},0\)\\mathbb\{I\}\\\{\{y\\in\(\-1,0\]\}\\\}\+f\(x\_\{1\},1\)\\mathbb\{I\}\\\{y\\in\(0,1\]\\\}, wheref⁡\(x1,0\)=p​f​\(x1∣0\)f\(x\_\{1\},0\)=pf\(x\_\{1\}\\mid 0\)andf⁡\(x1,1\)=\(1−p\)​f​\(x1∣1\)f\(x\_\{1\},1\)=\(1\-p\)f\(x\_\{1\}\\mid 1\)\. The liney=0y=0marks the internal block interface across which the lifted density may jump\. Panel \(b\) shows the original joint CDFForig​\(x1,t\)=ℙ⁡\(X1≤x1,X2≤t\)F\_\{\\mathrm\{orig\}\}\(x\_\{1\},t\)=\\mathbb\{P\}\(X\_\{1\}\\leq x\_\{1\},X\_\{2\}\\leq t\), which is stepwise in the discrete coordinate\. Panel \(c\) shows the lifted CDF after settingY=X2−UY=X\_\{2\}\-U, whereU∼Unif⁡\(0,1\)U\\sim\\mathrm\{Unif\}\(0,1\)is independent of\(X1,X2\)\(X\_\{1\},X\_\{2\}\); on\(−1,0\]\(\-1,0\]and\(0,1\]\(0,1\], it is affine inyy, with formsp​ϕ0​\(x1\)​\(1\+y\)p\\bm\{\\phi\}\_\{0\}\(x\_\{1\}\)\(1\+y\)andp​ϕ0​\(x1\)\+\(1−p\)​ϕ1​\(x1\)​yp\\bm\{\\phi\}\_\{0\}\(x\_\{1\}\)\+\(1\-p\)\\bm\{\\phi\}\_\{1\}\(x\_\{1\}\)y, respectively\.To fix ideas, consider a bivariate covariate𝒙=\(x1,x2\)\{\\bm\{x\}\}=\(x\_\{1\},x\_\{2\}\), wherex1x\_\{1\}is continuous on a compact set𝒳1⊂ℝ\\mathcal\{X\}\_\{1\}\\subset\\mathbb\{R\}andx2∈\{0,1\}x\_\{2\}\\in\\\{0,1\\\}is binary withℙ⁡\(x2=0\)=p\\mathbb\{P\}\(x\_\{2\}=0\)=p\. Suppose that the joint mass\-densityf⁡\(x1,x2\)f\(x\_\{1\},x\_\{2\}\)is continuous inx1x\_\{1\}for each value ofx2x\_\{2\}and the marginal densityf⁡\(x1\)\>0f\(x\_\{1\}\)\>0is continuous on𝒳1\\mathcal\{X\}\_\{1\}\.

Then

f\(x1,x2\)=f\(x1,0\)𝕀\{x2=0\}\+f\(x1,1\)𝕀\{x2=1\}\.f\(x\_\{1\},x\_\{2\}\)=f\(x\_\{1\},0\)\\mathbb\{I\}\\\{x\_\{2\}=0\\\}\+f\(x\_\{1\},1\)\\mathbb\{I\}\\\{x\_\{2\}=1\\\}\.
Under binary continualization, letY=X2−UY=X\_\{2\}\-U, whereU∼Unif⁡\(0,1\)U\\sim\\mathrm\{Unif\}\(0,1\)is independent ofX2X\_\{2\}\. The continualized variable takes values in\(−1,0\]∪\(0,1\]\(\-1,0\]\\cup\(0,1\], with blocksI\(0\)=\(−1,0\]I\(0\)=\(\-1,0\]andI⁡\(1\)=\(0,1\]I\(1\)=\(0,1\], and the lifted covariate is𝒙~=\(x1,y\)∈𝒳1×\(−1,1\]\\tilde\{\\bm\{x\}\}=\(x\_\{1\},y\)\\in\\mathcal\{X\}\_\{1\}\\times\(\-1,1\]\. The lifted joint density has the blockwise form

g\(x1,y\)=f\(x1,0\)𝕀\{y∈I\(0\)\}\+f\(x1,1\)𝕀\{y∈I\(1\)\}\.g\(x\_\{1\},y\)=f\(x\_\{1\},0\)\\mathbb\{I\}\\\{y\\in I\(0\)\\\}\+f\(x\_\{1\},1\)\\mathbb\{I\}\\\{y\\in I\(1\)\\\}\.
Thusggis continuous, and as smooth asf⁡\(⋅,j\)f\(\\cdot,j\), within each of the two open blocks𝒳1×\(−1,0\)\\mathcal\{X\}\_\{1\}\\times\(\-1,0\)and𝒳1×\(0,1\)\\mathcal\{X\}\_\{1\}\\times\(0,1\), but may jump across the internal interface𝒳1×\{0\}\\mathcal\{X\}\_\{1\}\\times\\\{0\\\}wheneverf⁡\(x1,0\)≠f⁡\(x1,1\)f\(x\_\{1\},0\)\\neq f\(x\_\{1\},1\)\. Figure[3](https://arxiv.org/html/2610.33403#S4.F3)illustrates this lifted density and the corresponding change from a stepwise discrete\-coordinate CDF to a piecewise affine continualized CDF\. The same blockwise interpretation applies to conditional quantities\. For example, on the block corresponding tox2=0x\_\{2\}=0,

g\(x1∣y\)=f\(x1∣0\)=f⁡\(x1,0\)p,y∈\(−1,0\],g\(x\_\{1\}\\mid y\)=f\(x\_\{1\}\\mid 0\)=\\frac\{f\(x\_\{1\},0\)\}\{p\},\\qquad y\\in\(\-1,0\],and

g\(y∣x1\)=ℙ\(X2=0∣x1\)=f⁡\(x1,0\)f⁡\(x1\),y∈\(−1,0\]\.g\(y\\mid x\_\{1\}\)=\\mathbb\{P\}\(X\_\{2\}=0\\mid x\_\{1\}\)=\\frac\{f\(x\_\{1\},0\)\}\{f\(x\_\{1\}\)\},\\qquad y\\in\(\-1,0\]\.
These lifted conditional functions are continuous inx1x\_\{1\}within the block and constant in the continualized coordinateyy\. The block𝒳1×\(0,1\]\\mathcal\{X\}\_\{1\}\\times\(0,1\]is treated analogously, withppreplaced by1−p1\-pandf⁡\(x1,0\)f\(x\_\{1\},0\)replaced byf⁡\(x1,1\)f\(x\_\{1\},1\)\. Finally, applying the inverse categorization map, here the ceiling map in \([19](https://arxiv.org/html/2610.33403#S4.E19)\), collapses𝒳1×\(−1,0\]\\mathcal\{X\}\_\{1\}\\times\(\-1,0\]and𝒳1×\(0,1\]\\mathcal\{X\}\_\{1\}\\times\(0,1\]back to𝒳1×\{0\}\\mathcal\{X\}\_\{1\}\\times\\\{0\\\}and𝒳1×\{1\}\\mathcal\{X\}\_\{1\}\\times\\\{1\\\}, respectively\. Hence continualization creates a blockwise continuous representation without losing the original mixed\-type information\.

### 4\.4Penalty\-adaptive kernel

The continualization in Section[4\.1](https://arxiv.org/html/2610.33403#S4.SS1)maps each discrete valueυℓ\\upsilon\_\{\\ell\}to the intervalIℓ=\(υℓ−1,υℓ\]I\_\{\\ell\}=\(\\upsilon\_\{\\ell\-1\},\\upsilon\_\{\\ell\}\], and the continualized query valueyqy\_\{q\}lies somewhere inside the interval associated with its category\. Ifyqy\_\{q\}, a continualized coordinate of the query point, is close to an interval boundary, a Euclidean KNN search in the continualized space may select observations that are geometrically close but belong to different discrete categories\. For the fixed\-bandwidth scheme, kernel evaluation atyqy\_\{q\}may also assign excessive weight to mismatched neighboring observations, thereby affecting the effective fitting weights of observations with matched categories\. Such neighbors are undesirable because locality should respect both the continualized geometry and the recovered discrete labels\.

![Refer to caption](https://arxiv.org/html/2610.33403v1/Figure/continualization_boundary_effect.png)Figure 4:Boundary mismatches\.We address this boundary effect by replacing each discrete covariate coordinate of the query point with the center of its continualization interval before standardization, neighbor selection, and kernel evaluation:

yq⋄←aq,catq​\(yq\)⋄:=midpoint of​Iq,catq​\(yq\),q∈𝒟,y\_\{q\}^\{\\diamond\}\\leftarrow a\_\{q,\\mathrm\{cat\}\_\{q\}\(y\_\{q\}\)\}^\{\\diamond\}:=\\text\{midpoint of \}I\_\{q,\\mathrm\{cat\}\_\{q\}\(y\_\{q\}\)\},\\qquad q\\in\\mathcal\{D\},\(24\)where𝒟\\mathcal\{D\}is the index set of discrete variables\. Because the lifted law is uniform within each discrete block \(Proposition[4\.1](https://arxiv.org/html/2610.33403#S4.Thmprop1)and Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3)\), this query\-side substitution does not alter the target conditional law within the corresponding block\. The corresponding theoretical guarantee is provided in Corollary[5\.3](https://arxiv.org/html/2610.33403#S5.Thmcor3)\. Its role is purely geometric: it moves the query point away from block interfaces so that the selected neighborhood is consistent with the query’s discrete label\.

We further multiply the continuous kernel weight by a category\-aware penalty\. Define

ρi\(k\)=∏q∈𝒟ρi,q\(k\),\\rho\_\{i\}^\{\(k\)\}=\\prod\_\{q\\in\\mathcal\{D\}\}\\rho\_\{i,q\}^\{\(k\)\},\(25\)with per\-coordinate factor

ρi,q\(k\)=\{exp⁡\(−λq​\|catq​\(hi,q\(k\)\)−catq​\(yq⋄\)\|\),kindq=ordinal,𝕀\{catq\(hi,q\(k\)\)=catq\(y⋄q\)\}\+γq\(1−𝕀\{⋯\}\),kindq=nominal\.\\rho\_\{i,q\}^\{\(k\)\}=\\begin\{cases\}\\exp\\\!\\left\(\-\\lambda\_\{q\}\\bigl\|\\mathrm\{cat\}\_\{q\}\(h\_\{i,q\}^\{\(k\)\}\)\-\\mathrm\{cat\}\_\{q\}\(y^\{\\diamond\}\_\{q\}\)\\bigr\|\\right\),&\\mathrm\{kind\}\_\{q\}=\\mathrm\{ordinal\},\\\\\[2\.84526pt\] \\mathbb\{I\}\\bigl\\\{\\mathrm\{cat\}\_\{q\}\(h\_\{i,q\}^\{\(k\)\}\)=\\mathrm\{cat\}\_\{q\}\(y^\{\\diamond\}\_\{q\}\)\\bigr\\\}\+\\gamma\_\{q\}\\bigl\(1\-\\mathbb\{I\}\\\{\\cdots\\\}\\bigr\),&\\mathrm\{kind\}\_\{q\}=\\mathrm\{nominal\}\.\\end\{cases\}\(26\)For ordinal variables, the penalty decays with the category distance\|catq​\(hi,q\(k\)\)−catq​\(yq⋄\)\|\|\\mathrm\{cat\}\_\{q\}\(h\_\{i,q\}^\{\(k\)\}\)\-\\mathrm\{cat\}\_\{q\}\(y^\{\\diamond\}\_\{q\}\)\|at rateλq\>0\\lambda\_\{q\}\>0, preserving the ordering information of the original support\. For nominal variables, a category mismatch receives a multiplicative discountγq∈\(0,1\]\\gamma\_\{q\}\\in\(0,1\], independent of the numerical coding of the labels\.

Thus, the mixed\-type adaptation consists of two local deterministic modifications of the weighting procedure: a query\-side interval\-center substitution and a multiplicative category penalty\. Together, they make the final kernel weight reflect both geometric proximity in the continualized space and categorical proximity in the recovered discrete label space\. No changes are made to the two\-step aggregation logic or to the standardization\-versus\-design scale separation described above\. In particular, the algebraic identity in Proposition[3\.1](https://arxiv.org/html/2610.33403#S3.Thmprop1)continues to hold for the mixed\-type estimator, provided that the centralized local linear estimator is defined using the same weighting modifications\.

## 5Theoretical Analysis

In this section, we develop uniform convergence results for the federated local linear conditional CDF estimator introduced in Section[3](https://arxiv.org/html/2610.33403#S3)\. The convergence is uniform not only over the covariates but also over the response variable\. Specifically, the uniformity over the response applies to the function estimatorF^​\(z∣𝒙\)\\hat\{F\}\(z\\mid\{\\bm\{x\}\}\)for allz∈ℝz\\in\\mathbb\{R\}and provides a theoretical guarantee for the quality of data generation based onF^​\(z∣𝒙\)\\hat\{F\}\(z\\mid\{\\bm\{x\}\}\)\. By Proposition[3\.1](https://arxiv.org/html/2610.33403#S3.Thmprop1), the federated estimator coincides exactly with the centralized local linear estimator computed on the pooled sample whenever the kernel weights are matched\. The convergence analysis can therefore be carried out for the centralized estimator with total sample sizeM=∑k=1KmkM=\\sum\_\{k=1\}^\{K\}m\_\{k\}, and the resulting rates transfer directly to the federated estimator\. Throughout this section, we consider the asymptotic regimeM→∞M\\to\\inftyand let the bandwidthswx\(k\),wz\(k\)→0w^\{\(k\)\}\_\{x\},w^\{\(k\)\}\_\{z\}\\to 0at rates to be specified\.

Subsection[5\.1](https://arxiv.org/html/2610.33403#S5.SS1)states the regularity assumptions on the covariate marginal density, the conditional distribution function, and the kernels\. Subsection[5\.2](https://arxiv.org/html/2610.33403#S5.SS2)establishes uniform convergence in the basic setting where all covariates and the response are continuous, and serves as the analytical backbone for the mixed\-variable case\. Subsection[5\.3](https://arxiv.org/html/2610.33403#S5.SS3)characterizes the piecewise smoothness of the lifted joint and conditional distributions induced by continualization, providing the regularity needed for the mixed\-variable analysis\. Subsection[5\.4](https://arxiv.org/html/2610.33403#S5.SS4)then extends the uniform convergence results to the continualized mixed\-type setting, both on interior blocks and on the covariate jumping set, and establishes a strictly sharper rate for the conditional probability mass estimator when the response is discrete\.

### 5\.1Assumptions

We work with three groups of assumptions: a geometric and smoothness condition on the covariate marginal density, a smoothness condition on the conditional distribution function, and a regularity condition on the kernels\. The first two assumptions follow the standard local linear regression setup\([Xie, 2023](https://arxiv.org/html/2610.33403#bib.bib4)\)and are stated on a generic covariate domain𝒳⊂ℝd\\mathcal\{X\}\\subset\\mathbb\{R\}^\{d\}; the same conditions will be invoked blockwise in Subsection[5\.4](https://arxiv.org/html/2610.33403#S5.SS4)\. The kernel assumption is generalized to the Gaussian kernel with unbounded support\.

###### Assumption 1\(Marginal density\)\.

The covariate domain𝒳⊂ℝd\\mathcal\{X\}\\subset\\mathbb\{R\}^\{d\}is compact and convex, and satisfies an internal cone condition: there exist constantsλ0,λ1∈\(0,1\]\\lambda\_\{0\},\\lambda\_\{1\}\\in\(0,1\]such that, for every𝐱∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and everyw∈\(0,λ0\]w\\in\(0,\\lambda\_\{0\}\], there exists𝐱′∈𝒳\{\\bm\{x\}\}^\{\\prime\}\\in\\mathcal\{X\}withB⁡\(𝐱′,λ1​w\)⊂B⁡\(𝐱,w\)∩𝒳B\(\{\\bm\{x\}\}^\{\\prime\},\\lambda\_\{1\}w\)\\subset B\(\{\\bm\{x\}\},w\)\\cap\\mathcal\{X\}, whereB⁡\(𝐱,δ\)B\(\{\\bm\{x\}\},\\delta\)denotes the Euclidean ball centered at𝐱\{\\bm\{x\}\}with radiusδ\\delta\. The covariate marginal densityffsatisfies:

1. 1\.inf𝒙∈𝒳f⁡\(𝒙\)\>0\\inf\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}f\(\{\\bm\{x\}\}\)\>0;
2. 2\.f∈C2​\(𝒳∘\)f\\in C^\{2\}\(\\mathcal\{X\}^\{\\circ\}\), with all second\-order partial derivatives uniformly continuous and bounded on the interior domain𝒳∘\\mathcal\{X\}^\{\\circ\}\.

###### Assumption 2\(Conditional distribution\)\.

The conditional distribution functionF⁡\(z∣𝐱\)=ℙ⁡\(Z≤z∣𝐗=𝐱\)F\(z\\mid\{\\bm\{x\}\}\)=\\mathbb\{P\}\(Z\\leq z\\mid\{\\bf X\}=\{\\bm\{x\}\}\)is twice continuously differentiable on𝒮=𝒵×𝒳\\mathcal\{S\}=\\mathcal\{Z\}\\times\\mathcal\{X\}, where𝒵⊆ℝ\\mathcal\{Z\}\\subseteq\\mathbb\{R\}, and all partial derivatives of order up to two are bounded and uniformly continuous on𝒮\\mathcal\{S\}\.

###### Assumption 3\(Kernel functions\)\.

The kernel functions used to define the weightsϖi=𝛋⁡\(⋅\)\\varpi\_\{i\}=\\bm\{\\kappa\}\(\\cdot\)with the fixed bandwidthwxw\_\{x\}used in Section[3](https://arxiv.org/html/2610.33403#S3)are taken to be Gaussian:κ\(u\)=ϕ\(u\):=\(2π\)−1/2e−u2/2\\kappa\(u\)=\\phi\(u\):=\(2\\pi\)^\{\-1/2\}e^\{\-u^\{2\}/2\},𝒦⁡\(u\)=Φ⁡\(u\):=∫−∞uϕ⁡\(v\)​𝑑v\\mathcal\{K\}\(u\)=\\varPhi\(u\):=\\int\_\{\-\\infty\}^\{u\}\\phi\(v\)\\,dv, and𝛋⁡\(𝐭\)=∏j=1dϕ⁡\(tj\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)=\\prod\_\{j=1\}^\{d\}\\phi\(t\_\{j\}\)\.

Assumption[1](https://arxiv.org/html/2610.33403#Thmass1)is the standard interior cone condition under which uniform kernel approximation arguments apply\. Assumption[2](https://arxiv.org/html/2610.33403#Thmass2)is the second\-order smoothness condition required for local linear bias control\. Assumption[3](https://arxiv.org/html/2610.33403#Thmass3)generalizes the kernel family from kernels with bounded support to kernels with unbounded support, specifically the Gaussian kernel\.

We adopt a common\-bandwidth regime across clients in federated learning\. All clients use the same covariate bandwidthwx\(k\)=wx\>0w^\{\(k\)\}\_\{x\}=w\_\{x\}\>0and the same response bandwidthwz\(k\)=wz\>0w\_\{z\}^\{\(k\)\}=w\_\{z\}\>0\. They satisfywx,wz→0w\_\{x\},w\_\{z\}\\to 0andM​wxd/\|log⁡wx\|→∞Mw\_\{x\}^\{d\}/\|\\log w\_\{x\}\|\\to\\inftyasM→∞M\\to\\infty\. This is the standard fixed\-bandwidth regime fordd\-dimensional local linear smoothing\.

### 5\.2Uniform convergence under continuous covariates

To isolate the analytic core, we first consider the case where all covariates are continuous\. The federated estimatorF^​\(z∣𝒙\)\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)at the query point\(𝒙,z\)∈𝒳×ℝ\(\{\\bm\{x\}\},z\)\\in\\mathcal\{X\}\\times\\mathbb\{R\}equals the centralized local linear intercept \([4](https://arxiv.org/html/2610.33403#S2.E4)\) on the pooled sample, and so its analysis reduces to the standard local linear conditional CDF analysis\. We state the result here in a form that will be reused blockwise in Subsection[5\.4](https://arxiv.org/html/2610.33403#S5.SS4)\.

###### Theorem 5\.1\(Uniform convergence under continuous covariates\)\.

Let Assumptions[1](https://arxiv.org/html/2610.33403#Thmass1)–[3](https://arxiv.org/html/2610.33403#Thmass3)hold\. Then

supz∈ℝ,𝒙∈𝒳\|F^​\(z∣𝒙\)−F⁡\(z∣𝒙\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\bigl\|\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)\-F\(z\\mid\{\\bm\{x\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(27\)The deterministic termwx2\+wz2w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}is the oracle bias and the stochastic term\|log⁡wx\|/\(M​wxd\)\\sqrt\{\|\\log w\_\{x\}\|/\(Mw\_\{x\}^\{d\}\)\}controls the empirical fluctuation\.

A sketch of the proof is as follows; the full argument is given in Appendix[B](https://arxiv.org/html/2610.33403#A2)\. Decompose the estimation error as

F^​\(z∣𝒙\)−F⁡\(z∣𝒙\)=F^​\(z∣𝒙\)−β0⋄​\(z,𝒙,wz,wx\)⏟stochastic\+β0⋄​\(z,𝒙,wz,wx\)−F⁡\(z∣𝒙\)⏟bias,\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)\-F\(z\\mid\{\\bm\{x\}\}\)=\\underbrace\{\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)\-\\beta\_\{0\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\}\_\{\\text\{stochastic\}\}\+\\underbrace\{\\beta\_\{0\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-F\(z\\mid\{\\bm\{x\}\}\)\}\_\{\\text\{bias\}\},\(28\)whereβ0⋄​\(z,𝒙,wz,wx\)\\beta\_\{0\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)is the intercept of the minimizer of the expected objective corresponding to the empirical objective in \([2](https://arxiv.org/html/2610.33403#S2.E2)\)\. Under Assumption[2](https://arxiv.org/html/2610.33403#Thmass2), a second\-order Taylor expansion ofF⁡\(z∣𝒙\)F\(z\\mid\{\\bm\{x\}\}\)in both𝒙\{\\bm\{x\}\}andzzshows that the conditional bias is uniformlyO⁡\(wx2\+wz2\)O\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\), with the first\-order term in𝒙\{\\bm\{x\}\}eliminated by the local linear construction\. The stochastic term is handled by an empirical process argument over the Euclidean class\{\(𝒙′,z′\)↦𝜿\(\(𝒙′−𝒙\)/wx\)𝒦\(\(z′−z\)/wz\):𝒙∈𝒳,z∈ℝ\}\\\{\(\{\\bm\{x\}\}^\{\\prime\},z^\{\\prime\}\)\\mapsto\\bm\{\\kappa\}\(\(\{\\bm\{x\}\}^\{\\prime\}\-\{\\bm\{x\}\}\)/w\_\{x\}\)\\mathcal\{K\}\(\(z^\{\\prime\}\-z\)/w\_\{z\}\):\{\\bm\{x\}\}\\in\\mathcal\{X\},\\,z\\in\\mathbb\{R\}\\\}, whose entropy is controlled through Lemmas[A\.1](https://arxiv.org/html/2610.33403#A1.Thmlem1)–[A\.4](https://arxiv.org/html/2610.33403#A1.Thmlem4)in Appendix[A](https://arxiv.org/html/2610.33403#A1)\.

The following corollary records the implication for pointwise estimation, which is the form most directly useful in downstream applications\.

###### Corollary 5\.1\(Conditional CDF estimation bound\)\.

Under the conditions of Theorem[5\.1](https://arxiv.org/html/2610.33403#S5.Thmthm1), for any fixed\(z,𝐱\)\(z,\{\\bm\{x\}\}\)in the interior ofℝ×𝒳\\mathbb\{R\}\\times\\mathcal\{X\},

\|F^​\(z∣𝒙\)−F⁡\(z∣𝒙\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\),\|\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)\-F\(z\\mid\{\\bm\{x\}\}\)\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\),with the same constants as in \([27](https://arxiv.org/html/2610.33403#S5.E27)\)\.

When at least one covariate coordinate is discrete and is handled through the continualization of Section[4\.1](https://arxiv.org/html/2610.33403#S4.SS1), neither Assumption[1](https://arxiv.org/html/2610.33403#Thmass1)nor Assumption[2](https://arxiv.org/html/2610.33403#Thmass2)holds globally on𝒳~\\widetilde\{\\mathcal\{X\}\}: the lifted density generally jumps across𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}, and the lifted conditional distribution function inherits this jump structure\. The remainder of this section adapts Theorem[5\.1](https://arxiv.org/html/2610.33403#S5.Thmthm1)to the lifted setting by exploiting the blockwise regularity of the lifted laws\.

### 5\.3Regularity of the lifted joint and conditional laws

The lifted covariate vector𝐗~=\(𝐗\(c\),𝐘\)\\widetilde\{\\bf X\}=\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)defined in Section[4\.2](https://arxiv.org/html/2610.33403#S4.SS2)has a joint density on𝒳~=𝒳c×𝒴\\widetilde\{\\mathcal\{X\}\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}that is piecewise smooth rather than globally smooth: on each open block𝒳c×I​\(𝝊\)∘\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}it inherits the regularity of the conditional law of𝐗\(c\)\{\\bf X\}^\{\(c\)\}given𝐗\(d\)=𝝊\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}, while across the covariate jumping set𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}it may exhibit finite jumps\. This subsection makes that clear and then derives the corresponding piecewise structure of the lifted conditional distribution function, which is the regularity input required for the uniform convergence analysis in Subsection[5\.4](https://arxiv.org/html/2610.33403#S5.SS4)\.

#### 5\.3\.1Lifted joint density: blockwise continuity and one\-sided limits

We first describe the lifted joint densityggat the level ofC0C^\{0\}\-regularity, before turning to higher\-order smoothness\.

###### Theorem 5\.2\(Continuity of the lifted joint density\)\.

Suppose that for every𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilonthe mixed density\-mass section𝐱\(c\)↦f⁡\(𝐱\(c\),𝛖\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)is continuous on𝒳c\\mathcal\{X\}\_\{c\}\. Letggdenote the joint density of the continualized variable vector𝐗~\\widetilde\{\\bf X\}\. Then the following hold\.

1. 1\.Blockwise representation\.For every𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilonand every\(𝒙\(c\),𝐲\)∈𝒳c×I​\(𝝊\)∘\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},soggis continuous on each open block𝒳c×I​\(𝝊\)∘\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}and independent of𝐲\{\\bf y\}on that block\.
2. 2\.Jumps across𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}\.Let\(𝒙0\(c\),𝐲0\)∈𝒳~∙\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}, and let𝝊,𝝊′∈Υ\\bm\{\\upsilon\},\\bm\{\\upsilon\}^\{\\prime\}\\in\\varUpsilonbe two indices of two blocks sharing the boundary on which𝐲0\{\\bf y\}\_\{0\}lies\. Then the one\-sided limits ofggat\(𝒙0\(c\),𝐲0\)\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)from inside𝒳c×I⁡\(𝝊\)\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)and from inside𝒳c×I⁡\(𝝊′\)\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}^\{\\prime\}\)exist and equalf⁡\(𝒙0\(c\),𝝊\)/\|I⁡\(𝝊\)\|f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\)/\|I\(\\bm\{\\upsilon\}\)\|andf⁡\(𝒙0\(c\),𝝊′\)/\|I⁡\(𝝊′\)\|f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)/\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|, respectively; these limits coincide if and only iff⁡\(𝒙0\(c\),𝝊\)/\|I⁡\(𝝊\)\|=f⁡\(𝒙0\(c\),𝝊′\)/\|I⁡\(𝝊′\)\|f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\)/\|I\(\\bm\{\\upsilon\}\)\|=f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)/\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\.
3. 3\.Piecewise continuity\.The functionggis piecewise continuous on𝒳~∘\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}, and at every interface point\(𝒙1\(c\),𝐲1\)∈𝒳~∙\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\},g\|𝒳c×I⁡\(𝝊\)g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}admits a continuous extension to the closed block𝒳c×I⁡\(𝝊\)¯\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}, given by the same formula in \(i\)\.

The proof is given in Appendix[C\.2](https://arxiv.org/html/2610.33403#A3.SS2)\. Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)captures the basic geometric picture: continualization unfolds the discrete mass onΥ\\varUpsiloninto a continuous density on𝒳~\\widetilde\{\\mathcal\{X\}\}that is constant in𝐲\{\\bf y\}within each block, has well\-defined one\-sided limits at every block boundary, and exhibits a finite jump exactly when the adjacent blockwise density valuesf⁡\(𝒙\(c\),𝝊\)/\|I⁡\(𝝊\)\|f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)/\|I\(\\bm\{\\upsilon\}\)\|differ at the corresponding𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\. Raising the smoothness assumption on the sectionsf⁡\(⋅,𝝊\)f\(\\cdot,\\bm\{\\upsilon\}\)toCrC^\{r\}extends thisC0C^\{0\}picture toCrC^\{r\}in a directly analogous manner\. The proof, given in Appendix[C\.3](https://arxiv.org/html/2610.33403#A3.SS3), applies Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)to each partial derivative of order at mostrr\.

###### Corollary 5\.2\(PiecewiseCrC^\{r\}regularity of the lifted density\)\.

Letr≥0r\\geq 0be an integer and assume that for every𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilonthe section𝐱\(c\)↦f⁡\(𝐱\(c\),𝛖\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)belongs toCr​\(𝒳c\)C^\{r\}\(\\mathcal\{X\}\_\{c\}\), with all partial derivatives of order up torrcontinuous on𝒳c\\mathcal\{X\}\_\{c\}uniformly in𝛖\\bm\{\\upsilon\}\. Then:

1. 1\.For every𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon,g∈Cr​\(𝒳c×I​\(𝝊\)∘\)g\\in C^\{r\}\\bigl\(\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\\bigr\), with all𝐲\{\\bf y\}\-derivatives identically zero and all𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivatives inherited fromf⁡\(⋅,𝝊\)/\|I⁡\(𝝊\)\|f\(\\cdot,\\bm\{\\upsilon\}\)/\|I\(\\bm\{\\upsilon\}\)\|\.
2. 2\.Across𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}, the one\-sided limits ofggand of its𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivatives from adjacent blocks need not coincide; the magnitude of any such discontinuity follows the formula of Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)\(ii\) applied to the corresponding derivative\.
3. 3\.The collection\{g\|𝒳c×I⁡\(𝝊\)\}𝝊∈Υ\\\{g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}\\\}\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}constitutes a piecewiseCrC^\{r\}structure on𝒳~\\widetilde\{\\mathcal\{X\}\}\. The restrictiong\|𝒳c×I⁡\(𝝊\)g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}admits aCrC^\{r\}extension to𝒳c×I⁡\(𝝊\)¯\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}\.

#### 5\.3\.2Lifted conditional distribution: blockwiseC2C^\{2\}structure

With the regularity of the joint density in hand, we now turn to the lifted conditional distribution functionF⁡\(z∣𝒙~\)=ℙ⁡\(Z≤z∣𝐗~=𝒙~\)F\(z\\mid\\tilde\{\\bm\{x\}\}\)=\\mathbb\{P\}\(Z\\leq z\\mid\\widetilde\{\\bf X\}=\\tilde\{\\bm\{x\}\}\)\. Two features of this function are critical for the analysis of Subsection[5\.4](https://arxiv.org/html/2610.33403#S5.SS4): it is independent of𝐲\{\\bf y\}within each block \(so𝐲\{\\bf y\}\-derivatives contribute no bias to local linear smoothing\), and, in the case of an ordered discrete response, it is exactly affine in the continualized response coordinatez~\\tilde\{z\}on each response block \(so thez~\\tilde\{z\}\-direction Taylor remainder vanishes whenever the response smoothing window stays within a single block\)\.

###### Theorem 5\.3\(Lifted conditional distribution\)\.

Let𝐗~=\(𝐗\(c\),𝐘\)\\widetilde\{\\bf X\}=\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)be the continualized covariate random vector\.

1. 1\.Continuous response\.SupposeZZis continuous and, for every𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the section\(z,𝒙\(c\)\)↦F⁡\(z∣𝒙\(c\),𝝊\)\(z,\{\\bm\{x\}\}^\{\(c\)\}\)\\mapsto F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)belongs toC2​\(𝒵×𝒳c\)C^\{2\}\(\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}\)with second\-order partial derivatives uniformly continuous and bounded on𝒵×𝒳c\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}, uniformly in𝝊\\bm\{\\upsilon\}\. Then F\(z∣𝒙\(c\),𝐲\)=∑𝝊∈ΥF\(z∣𝒙\(c\),𝝊\)𝕀\{𝐲∈I\(𝝊\)\}F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\,\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\(29\)belongs toC2​\(𝒵×𝒳c×I​\(𝝊\)∘\)C^\{2\}\\bigl\(\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\\bigr\)for every𝝊\\bm\{\\upsilon\}, is independent of𝐲\{\\bf y\}on each block \(so all𝐲\{\\bf y\}\-derivatives vanish identically\), and may jump across𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}\.
2. 2\.Discrete response after continualization\.SupposeZZis ordered discrete on\{z1,…,zL\}\\\{z\_\{1\},\\ldots,z\_\{L\}\\\}and, for everyi∈\[L\]i\\in\[L\]and𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the conditional mass𝒙\(c\)↦P⁡\(zi∣𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)belongs toC2​\(𝒳c\)C^\{2\}\(\\mathcal\{X\}\_\{c\}\), uniformly iniiand𝝊\\bm\{\\upsilon\}\. Then the lifted conditional density is f\(z~∣𝒙\(c\),𝐲\)=∑𝝊∈Υ∑i=1LP⁡\(zi∣𝒙\(c\),𝝊\)\|Ii\|𝕀\{z~∈Ii\}𝕀\{𝐲∈I\(𝝊\)\},f\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\sum\_\{i=1\}^\{L\}\\frac\{P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\}\|\}\\,\\mathbb\{I\}\\\{\\tilde\{z\}\\in I\_\{i\}\\\}\\,\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\},\(30\)a valid conditional density, and the lifted conditional CDF satisfies, forz~∈Ii\\tilde\{z\}\\in I\_\{i\}and𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\), F⁡\(z~∣𝒙\(c\),𝐲\)=∑k=1i−1P⁡\(zk∣𝒙\(c\),𝝊\)\+z~−zi−1\|Ii\|​P​\(zi∣𝒙\(c\),𝝊\)\.F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{k=1\}^\{i\-1\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\+\\frac\{\\tilde\{z\}\-z\_\{i\-1\}\}\{\|I\_\{i\}\|\}\\,P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\.\(31\)ConsequentlyF⁡\(z~∣𝒙\(c\),𝐲\)F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)isC2C^\{2\}on eachIi∘×𝒳c×I​\(𝝊\)∘I\_\{i\}^\{\\circ\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, is independent of𝐲\{\\bf y\}on each block, and is*exactly affine*inz~\\tilde\{z\}on eachIi∘I\_\{i\}^\{\\circ\}\.
3. 3\.Covariate jumping set\.In both cases \(i\) and \(ii\), at every covariate jumping point\(𝒙\(c\),𝐲0\)∈𝒳~∙\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\},FFadmits one\-sidedC2C^\{2\}limits from every block whose closure contains the point in the sense of Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)\(ii\); the values ofFFand its𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivatives on adjacent blocks need not agree\.

The two structural features mentioned at the start of this discussion—block\-internal𝐲\{\\bf y\}\-invariance in both cases and exact affineness inz~\\tilde\{z\}in the discrete response case—are not byproducts of the construction; they are direct consequences of the uniform spread of probability mass within each continualized cell, and they are precisely what allows the uniform convergence theory of Subsection[5\.4](https://arxiv.org/html/2610.33403#S5.SS4)to proceed blockwise from the analytic backbone of Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3)\.

### 5\.4Uniform convergence under continualized mixed\-type variates

We now establish the uniform convergence rate of the federated local linear conditional CDF estimator under continualized mixed\-type variates\. The strategy is to apply Theorem[5\.1](https://arxiv.org/html/2610.33403#S5.Thmthm1)blockwise on each piecewiseC2C^\{2\}region identified in Subsection[5\.3](https://arxiv.org/html/2610.33403#S5.SS3), and then to combine the blockwise bounds using the finiteness ofΥ\\varUpsilon\. However, points near block boundaries require special care\.

###### Theorem 5\.4\(Uniform convergence on shrinking interior blocks\)\.

Let𝐱~=\(𝐱\(c\),𝐲\)∈𝒳~∘\\tilde\{\\bm\{x\}\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}and suppose the following assumptions hold: \(1\) the continuous covariates satisfy Assumption[1](https://arxiv.org/html/2610.33403#Thmass1)on𝒳c\\mathcal\{X\}\_\{c\}; \(2\) Assumption[3](https://arxiv.org/html/2610.33403#Thmass3)holds for both the covariate and response kernels; \(3\) for every𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon,𝐱\(c\)↦f⁡\(𝐱\(c\),𝛖\)∈C2​\(𝒳c\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\in C^\{2\}\(\\mathcal\{X\}\_\{c\}\)withinf𝐱\(c\)f⁡\(𝐱\(c\),𝛖\)\>0\\inf\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\>0, uniformly in𝛖\\bm\{\\upsilon\}; \(4\) for every𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the conditional CDFF⁡\(z∣𝐱\(c\),𝛖\)F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)satisfies Assumption[2](https://arxiv.org/html/2610.33403#Thmass2)on𝒵×𝒳c\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}, uniformly in𝛖\\bm\{\\upsilon\}\. Then:

1. 1\.Continuous response\.IfZZis continuous, define 𝒴ϵ∘:=\{𝐲∈𝒴∘:minj∈\[d~\]1≤ℓ≤τj−1⁡\|yj−υj,ℓ\|≥ϵ\},𝒳~ϵ∘:=𝒳c×𝒴ϵ∘,\\mathcal\{Y\}\_\{\\epsilon\}^\{\\circ\}:=\\left\\\{\{\\bf y\}\\in\\mathcal\{Y\}^\{\\circ\}:\\min\_\{\\begin\{subarray\}\{c\}j\\in\[\\tilde\{d\}\]\\\\ 1\\leq\\ell\\leq\\tau\_\{j\}\-1\\end\{subarray\}\}\|y\_\{j\}\-\\upsilon\_\{j,\\ell\}\|\\geq\\epsilon\\right\\\},\\qquad\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}:=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\_\{\\epsilon\}^\{\\circ\},whereϵ≥Cϵ​wx​\|log⁡wx\|\\epsilon\\geq C\_\{\\epsilon\}w\_\{x\}\\sqrt\{\|\\log w\_\{x\}\|\}andCϵ\>2C\_\{\\epsilon\}\>2\. Then supz∈ℝ,𝒙~∈𝒳~ϵ∘\|F^​\(z∣𝒙~\)−F⁡\(z∣𝒙~\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\\,\\tilde\{\\bm\{x\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\}\\bigl\|\\widehat\{F\}\(z\\mid\\tilde\{\\bm\{x\}\}\)\-F\(z\\mid\\tilde\{\\bm\{x\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(32\)
2. 2\.Discrete response after continualization\.IfZZis ordered discrete on\{z1,…,zL\}\\\{z\_\{1\},\\ldots,z\_\{L\}\\\}, define 𝒵~ϵ′∘:=\{z~∈𝒵~∘:min0≤i≤L⁡\|z~−zi\|≥ϵ′\},\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}:=\\left\\\{\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}^\{\\circ\}:\\min\_\{0\\leq i\\leq L\}\|\\tilde\{z\}\-z\_\{i\}\|\\geq\\epsilon^\{\\prime\}\\right\\\},whereϵ′≥Cϵ′​wz​\|log⁡wz\|\\epsilon^\{\\prime\}\\geq C\_\{\\epsilon^\{\\prime\}\}w\_\{z\}\\sqrt\{\|\\log w\_\{z\}\|\}andCϵ′\>2C\_\{\\epsilon^\{\\prime\}\}\>2\. Then supz~∈𝒵~ϵ′∘𝒙~∈𝒳~ϵ∘\|F^​\(z~∣𝒙~\)−F⁡\(z~∣𝒙~\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{\\begin\{subarray\}\{c\}\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}\\\\ \\tilde\{\\bm\{x\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\\end\{subarray\}\}\\bigl\|\\widehat\{F\}\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(33\)

###### Remark 5\.1\.

The shrinking radiiϵ\\epsilonandϵ′\\epsilon^\{\\prime\}are of the same order asO⁡\(wx​\|log⁡wx\|\)O\(w\_\{x\}\|\\log w\_\{x\}\|\)andO⁡\(wz​\|log⁡wz\|\)O\(w\_\{z\}\|\\log w\_\{z\}\|\)\. Whenwx=wz=n−α,α\>0w\_\{x\}=w\_\{z\}=n^\{\-\\alpha\},\\quad\\alpha\>0, the shrinking radii are of orderO⁡\(α​log⁡nnα\)→0O\(\\frac\{\\alpha\\log n\}\{n^\{\\alpha\}\}\)\\to 0asn→∞n\\to\\infty\. In addition, the theorem shows the interval\-center substitution moves the query away from the excluded interface neighborhoods\(𝒴ϵ∘\)c\(\\mathcal\{Y\}\_\{\\epsilon\}^\{\\circ\}\)^\{c\}effectively in practice\.

The proof is given in Appendix[C\.5](https://arxiv.org/html/2610.33403#A3.SS5)\. Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4)directly applies only to the shrinking interior𝒳ϵ∘\\mathcal\{X\}\_\{\\epsilon\}^\{\\circ\}\. To extend the result to the excluded interface neighborhoods and the boundary points, for each𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, fix a representative

𝐲ϵ∘∈I⁡\(𝝊\)∩𝒴ϵ∘,\{\\bf y\}\_\{\\epsilon\}^\{\\circ\}\\in I\(\\bm\{\\upsilon\}\)\\cap\\mathcal\{Y\}\_\{\\epsilon\}^\{\\circ\},which exists for all sufficiently smallϵ\\epsilon\. Under the left\-open right\-closed convention, every𝐲∈𝒴\{\\bf y\}\\in\\mathcal\{Y\}, including a boundary point𝐲∙\{\\bf y\}^\{\\bullet\}, belongs to a unique blockI⁡\(𝝊\)I\(\\bm\{\\upsilon\}\)\. By Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3),

F⁡\(z∣𝒙\(c\),𝐲\)=F⁡\(z∣𝒙\(c\),𝐲ϵ∘\),𝐲∈I⁡\(𝝊\)\.F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\_\{\\epsilon\}^\{\\circ\}\),\\qquad\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\.Thus, although Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4)does not directly control the estimator at points within the excluded interface neighborhoods, their conditional distributions, including those at the boundary points, can be recovered by evaluating the estimator at the corresponding interior representatives\. In what follows, with the continuous covariates𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}fixed, we therefore use

𝒙~ϵ∘:=\(𝒙\(c\),𝐲ϵ∘\)∈𝒳~ϵ∘\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}:=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\_\{\\epsilon\}^\{\\circ\}\)\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}as the representative of any𝒙~=\(𝒙\(c\),𝐲\)\\tilde\{\\bm\{x\}\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)belonging to the same block\.

###### Corollary 5\.3\(Uniform convergence near covariate jumping points\)\.

Under the conditions of Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4), define

𝒳~ϵ∙:=𝒳~∖𝒳~ϵ∘,\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\bullet\}:=\\widetilde\{\\mathcal\{X\}\}\\setminus\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\},which includes the excluded interface neighborhoods and the covariate jumping points\. For each𝐱~ϵ∙=\(𝐱\(c\),𝐲\)∈𝒳~ϵ∙\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\bullet\}, let𝐱~ϵ∘∈𝒳~ϵ∘\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}be its interior representative defined above\. Then the following hold\.

1. 1\.IfZZis continuous, supz∈ℝ𝒙~ϵ∙∈𝒳~ϵ∙\|F^​\(z∣𝒙~ϵ∘\)−F⁡\(z∣𝒙~ϵ∙\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{\\begin\{subarray\}\{c\}z\\in\\mathbb\{R\}\\\\ \\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\bullet\}\\end\{subarray\}\}\\bigl\|\\widehat\{F\}\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\)\-F\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(34\)
2. 2\.IfZZis ordered discrete and is continualized, supz~∈𝒵~ϵ′∘𝒙~ϵ∙∈𝒳~ϵ∙\|F^​\(z~∣𝒙~ϵ∘\)−F⁡\(z~∣𝒙~ϵ∙\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{\\begin\{subarray\}\{c\}\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}\\\\ \\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\bullet\}\\end\{subarray\}\}\\bigl\|\\widehat\{F\}\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(35\)

In particular, evaluating the federated estimator at the corresponding interior representative yields a uniformly consistent estimate of the conditional CDF throughout the excluded interface neighborhoods, including the jumping points, at the shrinking\-interior rate\.

The proof is short: by Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3),

F⁡\(z∣𝒙~ϵ∙\)=F⁡\(z∣𝒙~ϵ∘\)F\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}\)=F\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\)for every𝒙~∈𝒳~ϵ∙\\tilde\{\\bm\{x\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\bullet\}and the corresponding interior representative𝒙~ϵ∘∈𝒳~ϵ∘\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\. Consequently,

F^​\(z∣𝒙~ϵ∘\)−F⁡\(z∣𝒙~ϵ∙\)=F^​\(z∣𝒙~ϵ∘\)−F⁡\(z∣𝒙~ϵ∘\),\\widehat\{F\}\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\)\-F\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\bullet\}\)=\\widehat\{F\}\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\)\-F\(z\\mid\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}\),and the right\-hand side is bounded uniformly by Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4)\.

Corollary[5\.3](https://arxiv.org/html/2610.33403#S5.Thmcor3)extends the shrinking\-interior result to covariate interface neighborhoods, including the jumping points, through a deterministic blockwise reparametrization\. This relies on the𝐲\{\\bf y\}\-invariance of the conditional CDF within each continualization block\. Similarly, the response points near the boundariesz~∈𝒵~ϵ′∙:=𝒵~∖𝒵~ϵ′∘\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\bullet\}:=\\widetilde\{\\mathcal\{Z\}\}\\setminus\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}also need to be analyzed carefully\. We cannot obtain a convergence result analogous to \([35](https://arxiv.org/html/2610.33403#S5.E35)\), but the original conditional probability massespi​\(𝒙~\):=ℙ⁡\(Z=zi∣𝒙~\)p\_\{i\}\(\\tilde\{\\bm\{x\}\}\):=\\mathbb\{P\}\(Z=z\_\{i\}\\mid\\tilde\{\\bm\{x\}\}\)can be estimated by a finite difference ofF^\\widehat\{F\}at the two interior representative points, with the resulting estimators attaining a strictly sharper rate\.

###### Corollary 5\.4\(Uniform convergence for conditional probability masses\)\.

Under the conditions of Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4)for the discrete response case, fix for eachi∈\[L\]i\\in\[L\]two interior evaluation pointsz~i,−∘,z~i,\+∘∈Ii∩𝒵~ϵ′∘\\tilde\{z\}\_\{i,\-\}^\{\\circ\},\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\\in I\_\{i\}\\cap\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}withz~i,−∘<z~i,\+∘\\tilde\{z\}\_\{i,\-\}^\{\\circ\}<\\tilde\{z\}\_\{i,\+\}^\{\\circ\}, satisfying the separation condition

inf1≤i≤Lz~i,\+∘−z~i,−∘\|Ii\|≥cfor some​c∈\(0,1\]\.\\inf\_\{1\\leq i\\leq L\}\\frac\{\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\-\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\}\{\|I\_\{i\}\|\}\\geq c\\quad\\text\{for some \}c\\in\(0,1\]\.\(36\)Define the conditional mass estimator

p^i​\(𝒙~\):=\|Ii\|z~i,\+∘−z~i,−∘​\(F^​\(z~i,\+∘∣𝒙~\)−F^​\(z~i,−∘∣𝒙~\)\)\.\\widehat\{p\}\_\{i\}\(\\tilde\{\\bm\{x\}\}\):=\\frac\{\|I\_\{i\}\|\}\{\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\-\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\}\\Bigl\(\\widehat\{F\}\(\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\-\\widehat\{F\}\(\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\\Bigr\)\.Then

sup1≤i≤L𝒙~∈𝒳ϵ∘\|p^i​\(𝒙~\)−pi​\(𝒙~\)\|=Op​\(wx2\+\|log⁡wx\|M​wxd\)\+o⁡\(wz2\)\.\\sup\_\{\\begin\{subarray\}\{c\}1\\leq i\\leq L\\\\ \\tilde\{\\bm\{x\}\}\\in\\mathcal\{X\}\_\{\\epsilon\}^\{\\circ\}\\end\{subarray\}\}\\bigl\|\\widehat\{p\}\_\{i\}\(\\tilde\{\\bm\{x\}\}\)\-p\_\{i\}\(\\tilde\{\\bm\{x\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\+o\(w\_\{z\}^\{2\}\)\.\(37\)

The three results of this section—Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4), Corollary[5\.3](https://arxiv.org/html/2610.33403#S5.Thmcor3), and Corollary[5\.4](https://arxiv.org/html/2610.33403#S5.Thmcor4)—together establish that the federated estimator developed in Section[3](https://arxiv.org/html/2610.33403#S3)attains the uniform rateOp​\(wx2\+wz2\+\|log⁡wx\|/\(M​wxd\)\)O\_\{p\}\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\|\\log w\_\{x\}\|/\(Mw\_\{x\}^\{d\}\)\}\)on the full continualized covariate domain in both the continuous\- and discrete\-response cases\.

## 6Experiments

We compare FITS with two baseline methods, the marginal\-based HeteroFedSyn\([Li et al\., 2026](https://arxiv.org/html/2610.33403#bib.bib22)\)and the diffusion model FedTabDiff\([Sattarov et al\., 2024](https://arxiv.org/html/2610.33403#bib.bib9)\)\. We evaluate the synthetic data generators with three models, including linear regression and logistic regression for parameter estimation, and a deep learning model \(FT\-Transformer,\([Gorishniy et al\., 2021](https://arxiv.org/html/2610.33403#bib.bib28)\)\) for prediction\. The simulation study uses a controlled mixed\-type data\-generating mechanism with known downstream parameters, allowing direct assessment of parameter recovery under imbalanced federated sample sizes\. The real\-data study uses two mixed\-type tabular data sets, one for regression and one for binary classification, to examine these generators’ utility in practice\.

Throughout this section, we ask not only whether synthetic data should replace local data, but also whether synthetic samples generated by FITS provide a statistically useful complement to client\-level local data, especially for clients with limited sample sizes\. This distinction is important: in federated settings, the value of synthetic data depends not only on the quality of the generator but also on how synthetic and local data is combined in the downstream tasks\. Table[1](https://arxiv.org/html/2610.33403#S6.T1)summarizes the datasets and experimental designs\.

Table 1:Experimental designs for the simulation and real\-data studies\.### 6\.1Experimental setup

#### 6\.1\.1Federated split with sample\-size imbalance

All experiments useK=10K=10clients\. Unless otherwise stated, the training data is randomly permuted and split according to the fixed proportion vector\(p1,…,p10\)=\(30%,20%,10%,9%,8%,7%CLOSE\(p\_\{1\},\\ldots,p\_\{10\}\)=\(30\\%,20\\%,10\\%,9\\%,8\\%,7\\%,6%,5%,4%,1%\)\.,6\\%,5\\%,4\\%,1\\%\)\.Thus clientkkreceives approximatelypk​ntrainp\_\{k\}n\_\{\\rm train\}observations\. This creates a deliberately imbalanced federated system: the largest client contains about30%30\\%of the training data, whereas the smallest client contains about1%1\\%\.

This design is intended to stress\-test the downstream data usage modes under client\-size imbalance\. If all clients had comparable sample sizes, the relative advantage of synthetic data would be harder to separate from ordinary sampling variability\. In contrast, under the present split, estimators from the local datasets only are expected to be high\-variance on the smallest clients and comparatively stable on the largest clients\. This makes it possible to assess whether synthetic data primarily helps small clients and whether it degrades performance on large clients\. The main experiments focus on sample\-size imbalance while keeping the federated split otherwise random\.

#### 6\.1\.2Federated generator configurations

For FITS, we use the Gaussian\-kernel scheme and the KNN\-kernel scheme, abbreviated as FITS\-G and FITS\-K respectively\. The general settings include that the target bandwidth parameterwzw\_\{z\}is set to0\.050\.05and the conditional CDF grid size isG=300G=300\. For covariates’ kernel\-fitting, FITS\-G takes the Gaussian kernel with Scott’s rule kernel bandwidthw\(k\)x=nt​r​a​i​n−1/\(d\+4\)w^\{\(k\)\}\_\{x\}=n\_\{train\}^\{\-1/\(d\+4\)\}, wherent​r​a​i​nn\_\{train\}is the size of the full training dataset and the dimensionddincreases as the covariates are added along the chain process\. FITS\-K uses the Epanechnikov kernel withη=1\.05\\eta=1\.05and the global KNN budget is dimension\-adaptive: whendcond=1d\_\{\\rm cond\}=1,r0=350r\_\{0\}=350; whendcond=2d\_\{\\rm cond\}=2,r0=250r\_\{0\}=250and otherwise,r0=200r\_\{0\}=200\. For HeteroFedSyn and FedTabDiff, Appendix[E](https://arxiv.org/html/2610.33403#A5)gives the detailed configurations\.

#### 6\.1\.3Downstream data usage modes

For each downstream task, we compare four data usage modes introduced in Section[D\.1](https://arxiv.org/html/2610.33403#A4.SS1), which are used to explore how synthetic and local data is combined in the downstream tasks\. Mode 1 \(M1\) uses only synthetic data\. Mode 2 \(M2\) uses only the client’s local data \(and therefore does not use federated learning\)\. Mode 3 \(M3\) pools local observations with a subset of the synthetic observations directly in a mixed\-data objective\. Mode 4 \(M4\) combines local and synthetic observations through a loss\-reweighted objective, where the synthetic component receives a tunable weight\.

### 6\.2Simulation experiments

#### 6\.2\.1Simulation design

The simulation study uses a mixed\-type tabular data\-generating mechanism with six covariates,\(XB,XG1,XG2,XO,XL,XG3\),\(X\_\{B\},X\_\{G\_\{1\}\},X\_\{G\_\{2\}\},X\_\{O\},X\_\{L\},X\_\{G\_\{3\}\}\),and two task\-specific responses,YlinY\_\{\\rm lin\}for linear regression andYlogY\_\{\\rm log\}for logistic regression\. The variableXBX\_\{B\}is binary,XOX\_\{O\}is ordinal, and the remaining covariates are continuous\. The dependence structure among the covariates and the model settings are shown in Section[D\.2](https://arxiv.org/html/2610.33403#A4.SS2)\.

Operationally, the generator first produces a shared synthetic covariate table according to the chain order\(XB,XG1,XG2,XO,XL,XG3\)\(X\_\{B\},X\_\{G\_\{1\}\},X\_\{G\_\{2\}\},X\_\{O\},X\_\{L\},X\_\{G\_\{3\}\}\)\. The two task\-specific responses,YlinY\_\{\\rm lin\}andYlogY\_\{\\rm log\}, are then generated on top of this same covariate table\. This design ensures that the linear and logistic downstream comparisons differ only in the response\-generating mechanism, while sharing the same generated covariate distribution\. This shared\-covariate design removes sampling variation in the covariate distribution as a confounder when comparing the regression and classification tasks\. More details are described in Table[S1](https://arxiv.org/html/2610.33403#A4.T1)\.

#### 6\.2\.2Baseline comparisons

We conduct three sets of experiments in this section\. First, we compare FITS\-G with two baselines, HeteroFedSyn and FedTabDiff, across all modes and clients\. We report the parameter estimation errors of linear and logistic regression, as well as the classification accuracy of FT\-Transformer\. Second, we compare the Gaussian\-kernel and KNN\-kernel schemes, denoted by FITS\-G and FITS\-K, respectively\. Third, we evaluate FITS\-G under the four modes and their respective parameter settings to identify effective ways to use FITS\-generated synthetic data\. All results are reported as the mean \(standard deviation\) over 50 repeats, with lower errors and higher accuracy indicating better performance\.

Table 2:Simulation: linear\-regression parameter estimation error\.Table 3:Simulation: logistic\-regression parameter estimation error\.Table 4:Simulation: FT\-Transformer classification accuracy on the test data\.Tables[2](https://arxiv.org/html/2610.33403#S6.T2),[3](https://arxiv.org/html/2610.33403#S6.T3), and[4](https://arxiv.org/html/2610.33403#S6.T4)present the first set of experiments comparing the synthetic data generation methods\. Unless stated otherwise, we always use all available synthetic data and each client’s local data for model training in M3\. M4 also uses all local and synthetic data but employs a weighted loss function, in whichλ=0\.5\\lambda=0\.5specifies the weight assigned to each synthetic observation\. For each mode and client, the best result among the compared methods is shown in bold\. Overall, FITS\-generated synthetic data yields better performance than the baselines\. Clients with smaller local datasets benefit more from synthetic data, which can even serve as an effective substitute for their local data in downstream model training\. For example, in Table[2](https://arxiv.org/html/2610.33403#S6.T2), training solely on FITS\-G synthetic data \(M1\) begins to outperform training solely on local data \(M2\) at Client 6 \(6%\), with parameter estimation errors of0\.15840\.1584and0\.17510\.1751, respectively\. Similarly, in Table[3](https://arxiv.org/html/2610.33403#S6.T3), this advantage begins at Client 3 \(9%\)\. Table[4](https://arxiv.org/html/2610.33403#S6.T4)also shows that FITS\-G improves FT\-Transformer classification accuracy across all clients\. Regarding data usage, combining synthetic and local data through M3 or M4 is more effective than using synthetic data alone through M1\. Across these tables, M3 and M4 generally outperform M1; for FT\-Transformer classification in Table[4](https://arxiv.org/html/2610.33403#S6.T4), they also outperform M2\.

Next, we compare FITS\-G and FITS\-K in Tables[S3](https://arxiv.org/html/2610.33403#A6.T3),[S4](https://arxiv.org/html/2610.33403#A6.T4), and[S5](https://arxiv.org/html/2610.33403#A6.T5)\. The performance differences between the Gaussian\-kernel and KNN\-kernel schemes are negligible across the downstream tasks\. These findings support the effectiveness of the proposed KNN\-kernel scheme for FITS\-based synthetic data generation\.

The preceding experiments show that data usage modes affect downstream performance differently\. In the third set of experiments, we therefore vary the parameters of M3 and M4 for FITS\-G to investigate how synthetic data can be used more effectively\. In Tables[S6](https://arxiv.org/html/2610.33403#A6.T6),[S7](https://arxiv.org/html/2610.33403#A6.T7), and[S8](https://arxiv.org/html/2610.33403#A6.T8), the best result across all modes and parameter settings is shown in bold for each client\. These results suggest that M4 is an effective way to exploit synthetic data, provided that the synthetic\-observation loss weightλ\\lambdais chosen appropriately\.

#### 6\.2\.3Distributional fidelity

Synthetic data needs to exhibit distributional fidelity\. In this section, we use a distributional metric called Energy Distance \(ED,[Székely and Rizzo \(2013\)](https://arxiv.org/html/2610.33403#bib.bib29)\) to assess the fidelity of this synthetic data\. Compared with the commonly used KL divergence, ED can be estimated directly from samples without requiring density estimation and avoids the infinite divergence that may occur due to mismatched supports between original and synthetic distributions\.

DE​\(P^,Q^\)=2n​m​∑i=1n∑j=1m‖𝒙i−𝒚j‖2−1n2​∑i=1n∑k=1n‖𝒙i−𝒙k‖2−1m2​∑j=1m∑ℓ=1m‖𝒚j−𝒚ℓ‖2,D\_\{E\}\(\\hat\{P\},\\hat\{Q\}\)=\\sqrt\{\\frac\{2\}\{nm\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{m\}\\\|\\bm\{x\}\_\{i\}\-\\bm\{y\}\_\{j\}\\\|\_\{2\}\-\\frac\{1\}\{n^\{2\}\}\\sum\_\{i=1\}^\{n\}\\sum\_\{k=1\}^\{n\}\\\|\\bm\{x\}\_\{i\}\-\\bm\{x\}\_\{k\}\\\|\_\{2\}\-\\frac\{1\}\{m^\{2\}\}\\sum\_\{j=1\}^\{m\}\\sum\_\{\\ell=1\}^\{m\}\\\|\\bm\{y\}\_\{j\}\-\\bm\{y\}\_\{\\ell\}\\\|\_\{2\}\},whereP^\\hat\{P\}is the empirical distribution of the sample\{𝒙i\}i=1n\\\{\\bm\{x\}\_\{i\}\\\}\_\{i=1\}^\{n\}drawn fromPP; similarly,Q^\\hat\{Q\}is the empirical distribution of the sample\{𝒚i\}i=1m\\\{\\bm\{y\}\_\{i\}\\\}\_\{i=1\}^\{m\}drawn fromQQ\. We evaluate the distributional distance between the synthetic data and the full data including the training samples and test samples\. We use the full data as the reference because it includes the training data used for synthesis and, owing to its larger sample size, is expected to provide a more reliable approximation to the underlying data distribution\. This choice allows us to assess synthetic\-data fidelity against a reference that incorporates the training observations while more broadly representing the underlying distribution\. In the experiment, continuous and ordered variables are standardized using the means and standard deviations computed from the real reference data\. Unordered categorical variables are one\-hot encoded and each one\-hot block is divided by2\\sqrt\{2\}, giving unit Euclidean distance between distinct categories\.

In Table[5](https://arxiv.org/html/2610.33403#S6.T5), we compute three Energy Distances to assess the fidelity of the covariate distribution, the joint distribution of the covariates and the regression response, and the joint distribution of the covariates and the classification response\. FITS\-G gives the lowest ED values among all synthetic data generators, providing the best distributional fidelity for the full\-dimensional tabular data\.

Table 5:Simulation: Energy Distance between synthetic and full data\.
#### 6\.2\.4Additional experiments

Appendix[F](https://arxiv.org/html/2610.33403#A6)provides supplementary results for Section[6\.2\.2](https://arxiv.org/html/2610.33403#S6.SS2.SSS2)\. For the first set of experiments, which compares FITS\-G with HeteroFedSyn and FedTabDiff across all modes and clients, Tables[S21](https://arxiv.org/html/2610.33403#A6.T21),[S23](https://arxiv.org/html/2610.33403#A6.T23), and[S22](https://arxiv.org/html/2610.33403#A6.T22)report the test MSE of linear regression, the test MSE of FT\-Transformer regression, and the test accuracy of logistic regression, respectively\. For the second set of experiments, which compares the Gaussian\-kernel and KNN\-kernel schemes, FITS\-G and FITS\-K, the corresponding results are presented in Tables[S24](https://arxiv.org/html/2610.33403#A6.T24),[S26](https://arxiv.org/html/2610.33403#A6.T26), and[S25](https://arxiv.org/html/2610.33403#A6.T25)\. For the third set of experiments, which examines FITS\-G across the four modes and their respective parameter settings to identify effective ways to use synthetic data, the corresponding results are provided in Tables[S27](https://arxiv.org/html/2610.33403#A6.T27),[S29](https://arxiv.org/html/2610.33403#A6.T29), and[S28](https://arxiv.org/html/2610.33403#A6.T28)\.

### 6\.3Real\-data experiments

#### 6\.3\.1Datasets and preprocessing

We next evaluate the method on two real mixed\-type tabular data sets\. The first is the UCI Bike Sharing data set, used for a linear regression task with response variablecnt\. After preprocessing, the dataset contains17,37917\{,\}379observations, of which13,90313\{,\}903are used for training and3,4763\{,\}476for testing\. As in the simulation study, the training set is randomly split across ten clients according to the fixed proportion vector\(p1,…,p10\)\(p\_\{1\},\.\.\.,p\_\{10\}\), yielding client sizes of\{4171,2781,1391,1251,1112,973,834,695,556,139\}\\\{4171,2781,1391,1251,1112,973,834,695,556,139\\\}\. The preprocessed predictors consist of four continuous variables and eight discrete variables\. The response is standardized before fitting the generator and downstream linear models\. The second real\-data experiment uses the Adult Income dataset for binary classification\. After preprocessing, the dataset contains45,22245\{,\}222observations, with36,17736\{,\}177used for training and9,0459\{,\}045for testing\. The training dataset is randomly split according to the fixed proportion vector\(p1,…,p10\)\(p\_\{1\},\.\.\.,p\_\{10\}\), yielding client sizes of\{10853,7235,3618,3256,2894,2532,2171,1809,1447,362\}\\\{10853,7235,3618,3256,2894,2532,2171,1809,1447,362\\\}\. The predictors include five continuous variables and nine discrete variables\. The binary income response is encoded as0/10/1\. For both data sets, missing or invalid entries are removed during preprocessing\. Continuous predictors are standardized using complete\-data means and standard deviations\. Appendix[D\.3](https://arxiv.org/html/2610.33403#A4.SS3)provides more detailed information about the datasets and the chain order in FITS\. Unlike the simulation study, the true downstream parameters are unavailable in the real\-data experiments\. Therefore, parameter error is computed relative to the full\-data reference estimator fitted on the complete preprocessed training and test data\. Prediction metrics are computed on the test set\.

#### 6\.3\.2Bike dataset

We evaluate regression performance on the Bike dataset\. As in the simulation study, this evaluation comprises three sets of experiments and an assessment of distributional fidelity\.

Tables[6](https://arxiv.org/html/2610.33403#S6.T6),[7](https://arxiv.org/html/2610.33403#S6.T7), and[8](https://arxiv.org/html/2610.33403#S6.T8)present the first set of experiments comparing the synthetic data generation methods, withλ\\lambdafixed at0\.10\.1for M4\. The results show that FITS achieves the best performance across the regression tasks and models considered on this dataset\. The second set of experiments compares FITS\-G and FITS\-K, as reported in Tables[S9](https://arxiv.org/html/2610.33403#A6.T9),[S10](https://arxiv.org/html/2610.33403#A6.T10), and[S11](https://arxiv.org/html/2610.33403#A6.T11)\. The two schemes yield broadly comparable performance on the regression tasks\. The third set of experiments, presented in Tables[S12](https://arxiv.org/html/2610.33403#A6.T12),[S13](https://arxiv.org/html/2610.33403#A6.T13), and[S14](https://arxiv.org/html/2610.33403#A6.T14), further suggests that M4, with an appropriately chosenλ\\lambda, provides an effective way to combine synthetic data with real local data\. In particular, M4 withλ=1\\lambda=1is equivalent to M3 withr=∞r=\\infty\. Regarding distributional fidelity, the Energy Distance results in Table[9](https://arxiv.org/html/2610.33403#S6.T9)indicate that FITS more faithfully preserves the joint distribution of the Bike data\.

Table 6:Bike: linear\-regression parameter estimation error\.Table 7:Bike: linear\-regression MSE on the test data\.Table 8:Bike: FT\-Transformer regression MSE on the test data\.Table 9:Bike: Energy Distance between synthetic and full real data\.
#### 6\.3\.3Adult dataset

We evaluate classification performance on the Adult dataset using the same experimental structure\. Tables[10](https://arxiv.org/html/2610.33403#S6.T10),[11](https://arxiv.org/html/2610.33403#S6.T11), and[12](https://arxiv.org/html/2610.33403#S6.T12)compare the synthetic data generation methods, withλ\\lambdafixed at0\.050\.05for M4\. The comparisons between FITS\-G and FITS\-K are presented in Tables[S15](https://arxiv.org/html/2610.33403#A6.T15),[S16](https://arxiv.org/html/2610.33403#A6.T16), and[S17](https://arxiv.org/html/2610.33403#A6.T17)\. Finally, Tables[S18](https://arxiv.org/html/2610.33403#A6.T18),[S19](https://arxiv.org/html/2610.33403#A6.T19), and[S20](https://arxiv.org/html/2610.33403#A6.T20)report the performance of FITS\-G across the four modes and their respective parameter settings\.

In contrast to the findings on the other datasets, the results here show that HeteroFedSyn achieves lower parameter estimation errors than FITS for logistic regression \(Table[10](https://arxiv.org/html/2610.33403#S6.T10)\)\. However, the two methods yield comparable test classification accuracy \(Table[11](https://arxiv.org/html/2610.33403#S6.T11)\), while FITS achieves the best overall performance with the more expressive FT\-Transformer model \(Table[12](https://arxiv.org/html/2610.33403#S6.T12)\)\.

One possible explanation is that two\-way marginals capture much of the distributional structure relevant to logistic regression on this dataset\. Preserving these pairwise relationships may therefore allow HeteroFedSyn to achieve better logistic\-regression parameter estimates than FITS\. In contrast, FT\-Transformer can exploit richer, higher\-order dependencies, making the preservation of more distributional information beyond two\-way marginals more important\. HeteroFedSyn may preserve these dependencies less effectively, which could explain its weaker performance relative to FITS with FT\-Transformer\. The Energy Distance results in Table[13](https://arxiv.org/html/2610.33403#S6.T13)are consistent with this interpretation: although HeteroFedSyn also achieves relatively low distances, FITS exhibits the highest overall distributional fidelity\.

Table 10:Adult: logistic\-regression parameter estimation error\.Table 11:Adult: logistic\-regression classification accuracy on the test data\.Table 12:Adult: FT\-Transformer classification accuracy on the test data\.Table 13:Adult: Energy Distance between synthetic and full real data\.

## 7Conclusion

In this paper, we propose FITS, a federated nonparametric framework for generating synthetic mixed\-type tabular data\. The framework is built on three main components\. First, we develop a two\-step federated local linear procedure for conditional CDF estimation\. The first communication step aggregates local weighted Hessian\-type matrices and broadcasts the corresponding intercept coefficient vector, while the second step aggregates response\-dependent local CDF contributions\. Under fixed query points and weights, this construction recovers the pooled local linear estimator without centralizing raw observations\. Second, we introduce a continualization scheme for discrete variables, together with an explicit inverse mapping to the original discrete domain\. This construction lifts mixed mass\-density structures to a piecewise continuous representation, allowing continuous and discrete variables to be handled within a unified smoothing framework\. Third, we develop client\-designed kernels with fixed\-bandwidth and KNN\-based schemes, together with a penalty\-adaptive kernel that adjusts penalties for mismatches in continualized discrete covariates to accommodate mixed\-type data\. These designs improve the practical stability of the estimator in federated settings with mixed covariate types\.

On the theoretical side, we characterize the proposed continualization at the level of distribution laws\. The results show that the continualized representation preserves probability mass while inducing a piecewise continuous joint density and corresponding conditional CDFs on the lifted domain\. For the federated conditional CDF estimator, we establish uniform convergence over both response and covariate directions under the continualized support structure and Gaussian\-kernel smoothing\. These results provide the theoretical guarantees underlying the FITS framework\.

As the final goal for FITS, we construct the synthetic\-data generator based on the chain rule and inverse conditional\-CDF sampling, and evaluate its utility through downstream learning tasks under several modes of combining local and synthetic data\. The experiments highlight three factors governing the utility of synthetic data: client sample size, model capacity, and training mode\. For clients with limited data, training on synthetic samples alone can outperform training solely on local observations\. The benefits also depend on the downstream model: expressive models such as FT\-Transformer can exploit synthetic data effectively\. Across the experiments, weighted augmentation through M4 provides an effective way to combine synthetic and local observations, with the weightλ\\lambdaplaying a crucial role\.

Several questions remain for future work\. One direction is to develop more principled weighting schemes for combining local and synthetic data in downstream learning\. Another is to characterize the utility boundary of federated synthetic data, including when synthetic samples are beneficial, redundant, or potentially harmful for a given client\. It is also of interest to study stronger privacy guarantees, communication\-efficient grid construction, and ordering strategies for the chain\-rule generator\. These extensions would further clarify the statistical and practical scope of federated synthetic data\.

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## Supplementary Materials for "Faithful Federated Tabular Data Synthesis"

## Appendix AUseful tools

This appendix collects the empirical\-process tools used in the proofs of Appendix[B](https://arxiv.org/html/2610.33403#A2)\. We state them without proof and refer to[Xie \[2023\]](https://arxiv.org/html/2610.33403#bib.bib4)for detailed arguments\.

###### Lemma A\.1\(Lemma B\.1 in[Xie \[2023\]](https://arxiv.org/html/2610.33403#bib.bib4)\)\.

Letκ:ℝ→ℝ\\kappa:\\mathbb\{R\}\\to\\mathbb\{R\}be a function of bounded variation\. Then the following class is Euclidean:

\{κ\(\(⋅−x\)w\):x∈ℝ,w\>0\}\.\\big\\\{\\kappa\(\\frac\{\(\\cdot\-x\)\}\{w\}\):x\\in\\mathbb\{R\},w\>0\\big\\\}\.

###### Lemma A\.2\(Lemma B\.2 in[Xie \[2023\]](https://arxiv.org/html/2610.33403#bib.bib4)\)\.

Any uniformly bounded and finite\-dimensional vector space of functions is Euclidean\.

###### Lemma A\.3\(Lemma B\.5 in[Xie \[2023\]](https://arxiv.org/html/2610.33403#bib.bib4)\)\.

Let𝒢1\\mathcal\{G\}\_\{1\}be a class of functions that is uniformly bounded byM1M\_\{1\}and Euclidean with coefficients\(A1,v1\)\(A\_\{1\},v\_\{1\}\)and𝒢2\\mathcal\{G\}\_\{2\}a class of functions that is uniformly bounded byM2M\_\{2\}and Euclidean with coefficients\(A2,v2\)\(A\_\{2\},v\_\{2\}\)\. Then the class𝒢1𝒢2=\{g1⋅g2:g1∈𝒢1,g2∈𝒢2\}\\mathcal\{G\}\_\{1\}\\mathcal\{G\}\_\{2\}=\\\{g\_\{1\}\\cdot g\_\{2\}:g\_\{1\}\\in\\mathcal\{G\}\_\{1\},\\,g\_\{2\}\\in\\mathcal\{G\}\_\{2\}\\\}is uniformly bounded byM1​M2M\_\{1\}M\_\{2\}and Euclidean with coefficients\(A1​A2​\(M1\+M2\)v1\+v2,v1\+v2\)\.\\left\(A\_\{1\}A\_\{2\}\(M\_\{1\}\+M\_\{2\}\)^\{v\_\{1\}\+v\_\{2\}\},\\,v\_\{1\}\+v\_\{2\}\\right\)\.

###### Lemma A\.4\(Lemma 2 in[Xie \[2023\]](https://arxiv.org/html/2610.33403#bib.bib4)\)\.

Letξ1,…,ξn\\xi\_\{1\},\\ldots,\\xi\_\{n\}be an iid sample of a random vectorξ\\xiinℝd\\mathbb\{R\}^\{d\}\. Let𝒢n\\mathcal\{G\}\_\{n\}be a sequence of classes of measurable real\-valued functions defined onℝd\\mathbb\{R\}^\{d\}\. Assume that there is a uniformly bounded Euclidean class𝒢\\mathcal\{G\}with coefficientsAAandvvsuch that𝒢n⊂𝒢\\mathcal\{G\}\_\{n\}\\subset\\mathcal\{G\}for allnn\. Letσn2\\sigma\_\{n\}^\{2\}be a positive sequence such that

σn2≥supg∈𝒢nE⁡\[g​\(ξ\)2\]\.\\sigma\_\{n\}^\{2\}\\geq\\sup\_\{g\\in\\mathcal\{G\}\_\{n\}\}E\[g\(\\xi\)^\{2\}\]\.Then

ϵ=supg∈𝒢n\|∑i=1n\(g⁡\(ξi\)−E​g​\(ξi\)\)\|=Op​\(n​σn2​\|log⁡σn\|\+\|log⁡σn\|\)\.\\epsilon=\\sup\_\{g\\in\\mathcal\{G\}\_\{n\}\}\\left\|\\sum\_\{i=1\}^\{n\}\\bigl\(g\(\\xi\_\{i\}\)\-Eg\(\\xi\_\{i\}\)\\bigr\)\\right\|=O\_\{p\}\\\!\\left\(\\sqrt\{n\\sigma\_\{n\}^\{2\}\|\\log\\sigma\_\{n\}\|\}\+\|\\log\\sigma\_\{n\}\|\\right\)\.In particular, if

n​σn2\|log⁡σn\|→∞,\\frac\{n\\sigma\_\{n\}^\{2\}\}\{\|\\log\\sigma\_\{n\}\|\}\\to\\infty,then

ϵ=Op​\(n​σn2​\|log⁡σn\|\)\.\\epsilon=O\_\{p\}\\\!\\left\(\\sqrt\{n\\sigma\_\{n\}^\{2\}\|\\log\\sigma\_\{n\}\|\}\\right\)\.

###### Lemma A\.5\.

DenoteΩ\(𝐱,wx\)=∫𝛊\(𝐭\)𝛊\(𝐭\)⊤𝛋\(𝐭\)𝕀\{𝐱\+wx𝐭∈𝒳\}d𝐭\.\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)=\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\.Under Assumptions[1](https://arxiv.org/html/2610.33403#Thmass1)and[3](https://arxiv.org/html/2610.33403#Thmass3), there exists a constantC\>0C\>0such that, for all𝐱∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and all sufficiently smallwx\>0w\_\{x\}\>0, the eigenvalues ofΩ⁡\(𝐱,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)and𝚵⁡\(𝐱,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)lie in\[1/C,C\]\[1/C,C\]\.

## Appendix BProof of Theorem[5\.1](https://arxiv.org/html/2610.33403#S5.Thmthm1)

Throughout this and the subsequent appendices, we work under Assumptions[1](https://arxiv.org/html/2610.33403#Thmass1)–[3](https://arxiv.org/html/2610.33403#Thmass3)with the pooled sample of sizeM=∑k=1KmkM=\\sum\_\{k=1\}^\{K\}m\_\{k\}, treatingF^​\(z∣𝒙\)\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)as the centralized local linear estimator on\{\(𝐡i,:d,hi,d\+1\)\}i=1M\\\{\(\{\\bf h\}\_\{i,:d\},h\_\{i,d\+1\}\)\\\}\_\{i=1\}^\{M\}\. By Proposition[3\.1](https://arxiv.org/html/2610.33403#S3.Thmprop1), this is the same object as the federated estimatorF^raw​\(z∣𝒙\)\\widehat\{F\}^\{\\rm raw\}\(z\\mid\{\\bm\{x\}\}\)in Algorithm[1](https://arxiv.org/html/2610.33403#alg1)under matched kernel weights, so all rates established in this appendix transfer to the federated estimator without further argument\.

### B\.1Setup and notation used throughout the proofs

We collect here the notation that recurs in all subsequent proofs\. Recall the smoothed response indicator from \([3](https://arxiv.org/html/2610.33403#S2.E3)\):

𝒦i​\(z\):=𝒦⁡\(z−hi,d\+1wz\),i∈\[M\],\\mathcal\{K\}\_\{i\}\(z\):=\\mathcal\{K\}\\\!\\left\(\\frac\{z\-h\_\{i,d\+1\}\}\{w\_\{z\}\}\\right\),\\qquad i\\in\[M\],where𝒦=Φ\\mathcal\{K\}=\\varPhiis the standard normal CDF\. The local linear regression \([2](https://arxiv.org/html/2610.33403#S2.E2)\) targets the population parameter

𝜷⋆​\(z,𝒙\)=\(β0⋆​\(z,𝒙\),β1⋆​\(z,𝒙\),…,βd⋆​\(z,𝒙\)\)⊤=\(F⁡\(z∣𝒙\),∇𝒙F​\(z∣𝒙\)⊤\)⊤,\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)=\\bigl\(\\beta\_\{0\}^\{\\star\}\(z,\{\\bm\{x\}\}\),\\beta\_\{1\}^\{\\star\}\(z,\{\\bm\{x\}\}\),\\ldots,\\beta\_\{d\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\\bigr\)^\{\\top\}=\\bigl\(F\(z\\mid\{\\bm\{x\}\}\),\\,\\nabla\_\{\{\\bm\{x\}\}\}F\(z\\mid\{\\bm\{x\}\}\)^\{\\top\}\\bigr\)^\{\\top\},\(38\)where∇𝒙F​\(z∣𝒙\)=\(∂F⁡\(z∣𝒙\)/∂x1,…,∂F⁡\(z∣𝒙\)/∂xd\)⊤\\nabla\_\{\{\\bm\{x\}\}\}F\(z\\mid\{\\bm\{x\}\}\)=\\bigl\(\\partial F\(z\\mid\{\\bm\{x\}\}\)/\\partial x\_\{1\},\\ldots,\\partial F\(z\\mid\{\\bm\{x\}\}\)/\\partial x\_\{d\}\\bigr\)^\{\\top\}is the gradient with respect to𝒙\{\\bm\{x\}\}\. The interceptβ0⋆​\(z,𝒙\)=F⁡\(z∣𝒙\)\\beta\_\{0\}^\{\\star\}\(z,\{\\bm\{x\}\}\)=F\(z\\mid\{\\bm\{x\}\}\)is the estimand of interest\.

For eachi∈\[M\]i\\in\[M\], define the centered covariate increment and covariate\-kernel weight at the query point𝒙\{\\bm\{x\}\}:

𝐮i:=𝐡i,:d−𝒙∈ℝd,ϖi:=𝜿wx\(𝐡i,:d−𝒙\)=∏j=1dκ\(hi​j−xjwx\),\{\\bf u\}\_\{i\}:=\{\\bf h\}\_\{i,:d\}\-\{\\bm\{x\}\}\\ \\in\\mathbb\{R\}^\{d\},\\qquad\\varpi\_\{i\}:=\\bm\{\\kappa\}\_\{w\_\{x\}\}\\\!\\left\(\{\\bf h\}\_\{i,:d\}\-\{\\bm\{x\}\}\\right\)=\\prod\_\{j=1\}^\{d\}\\kappa\\\!\\left\(\\frac\{h\_\{ij\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\),\(39\)and define the design matrix and the diagonal weight matrix as

𝑼:=\(1𝐮1⊤1𝐮M⊤\)∈ℝM×\(d\+1\),𝐖:=diag⁡\(ϖ1,…,ϖM\)\.\\bm\{U\}:=\\begin\{pmatrix\}1&\{\\bf u\}\_\{1\}^\{\\top\}\\\\ \\vdots&\\vdots\\\\ 1&\{\\bf u\}\_\{M\}^\{\\top\}\\end\{pmatrix\}\\in\\mathbb\{R\}^\{M\\times\(d\+1\)\},\\qquad\{\\bf W\}:=\\mathrm\{diag\}\(\\varpi\_\{1\},\\ldots,\\varpi\_\{M\}\)\.With𝑼i:=𝜾⁡\(𝐮i\)=\(1,𝐮i⊤\)⊤∈ℝd\+1\\bm\{U\}\_\{i\}:=\\bm\{\\iota\}\(\{\\bf u\}\_\{i\}\)=\(1,\{\\bf u\}\_\{i\}^\{\\top\}\)^\{\\top\}\\in\\mathbb\{R\}^\{d\+1\}, the weighted\-least\-squares normal equations yield the closed form

𝜷^​\(z,𝒙,wz,wx\)=\(𝑼⊤​𝐖​𝑼\)−1​𝑼⊤​𝐖​𝓚​\(z\),𝓚⁡\(z\):=\(𝒦1​\(z\),…,𝒦M​\(z\)\)⊤,\\widehat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\bigl\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\\bigr\)^\{\-1\}\\bm\{U\}^\{\\top\}\{\\bf W\}\\,\\bm\{\\mathcal\{K\}\}\(z\),\\qquad\\bm\{\\mathcal\{K\}\}\(z\):=\\bigl\(\\mathcal\{K\}\_\{1\}\(z\),\\ldots,\\mathcal\{K\}\_\{M\}\(z\)\\bigr\)^\{\\top\},\(40\)and the intercept used asF^​\(z∣𝒙\)=β^0​\(z,𝒙,wz,wx\)\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)=\\hat\{\\beta\}\_\{0\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)admits the linear\-functional representation

F^​\(z∣𝒙\)=β^0​\(z,𝒙,wz,wx\)=𝐞1⊤​\(𝑼⊤​𝐖​𝑼\)−1​𝑼⊤​𝐖​𝓚​\(z\)=∑i=1Mαi​\(𝒙\)​𝒦i​\(z\),\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)=\\hat\{\\beta\}\_\{0\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\{\\bf e\}\_\{1\}^\{\\top\}\\bigl\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\\bigr\)^\{\-1\}\\bm\{U\}^\{\\top\}\{\\bf W\}\\,\\bm\{\\mathcal\{K\}\}\(z\)=\\sum\_\{i=1\}^\{M\}\\alpha\_\{i\}\(\{\\bm\{x\}\}\)\\,\\mathcal\{K\}\_\{i\}\(z\),\(41\)where𝐞1=\(1,0,…,0\)⊤∈ℝd\+1\{\\bf e\}\_\{1\}=\(1,0,\\ldots,0\)^\{\\top\}\\in\\mathbb\{R\}^\{d\+1\}andαi​\(𝒙\):=ϖi​𝐞1⊤​\(𝑼⊤​𝐖​𝑼\)−1​𝑼i\\alpha\_\{i\}\(\{\\bm\{x\}\}\):=\\varpi\_\{i\}\\,\{\\bf e\}\_\{1\}^\{\\top\}\(\\bm\{U\}^\{\\top\}\{\\bf W\}\\bm\{U\}\)^\{\-1\}\\bm\{U\}\_\{i\}\.

##### Pseudo\-true parameter and the error decomposition\.

The empirical objective in \([2](https://arxiv.org/html/2610.33403#S2.E2)\) is the finite\-sample analogue of the population weighted least\-squares problem whose minimizer is the deterministic pseudo\-true parameter

𝜷⋄​\(z,𝒙,wz,wx\):=arg⁡min𝜷∈ℝd\+1​𝔼Z,𝐗​\[\(𝒦⁡\(z−Zwz\)−𝜾​\(𝐗−𝒙\)⊤​𝜷\)2​𝜿wx​\(𝐗−𝒙\)\],\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\):=\\arg\\min\_\{\\bm\{\\beta\}\\in\\mathbb\{R\}^\{d\+1\}\}\\mathbb\{E\}\_\{Z,\{\\bf X\}\}\\\!\\left\[\\Bigl\(\\mathcal\{K\}\\\!\\Bigl\(\\frac\{z\-Z\}\{w\_\{z\}\}\\Bigr\)\-\\bm\{\\iota\}\(\{\\bf X\}\-\{\\bm\{x\}\}\)^\{\\top\}\\bm\{\\beta\}\\Bigr\)^\{2\}\\bm\{\\kappa\}\_\{w\_\{x\}\}\(\{\\bf X\}\-\{\\bm\{x\}\}\)\\right\],\(42\)where𝜾⁡\(𝒙\)=\(1,x1,…,xd\)⊤∈ℝd\+1\\bm\{\\iota\}\(\{\\bm\{x\}\}\)=\(1,x\_\{1\},\\ldots,x\_\{d\}\)^\{\\top\}\\in\\mathbb\{R\}^\{d\+1\}and𝜿wx​\(𝐗−𝒙\)=∏j=1dκ⁡\(\(Xj−xj\)/wx\)\\bm\{\\kappa\}\_\{w\_\{x\}\}\(\{\\bf X\}\-\{\\bm\{x\}\}\)=\\prod\_\{j=1\}^\{d\}\\kappa\(\(X\_\{j\}\-x\_\{j\}\)/w\_\{x\}\)\. The minimizer of \([42](https://arxiv.org/html/2610.33403#A2.E42)\) admits the closed form

𝐃wx​𝜷⋄​\(z,𝒙,wz,wx\)=𝚵​\(𝒙,wx\)−1​𝝁​\(z,𝒙,wz,wx\),\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\,\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\),\(43\)where𝐃wx=diag⁡\(1,wx,…,wx\)∈ℝ\(d\+1\)×\(d\+1\)\{\\bf D\}\_\{w\_\{x\}\}=\\mathrm\{diag\}\(1,w\_\{x\},\\ldots,w\_\{x\}\)\\in\\mathbb\{R\}^\{\(d\+1\)\\times\(d\+1\)\}and

𝚵⁡\(𝒙,wx\)\\displaystyle\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\):=1wxd​𝔼𝐗​\[𝜾⁡\(𝐗−𝒙wx\)​𝜾​\(𝐗−𝒙wx\)⊤​𝜿wx​\(𝐗−𝒙\)\],\\displaystyle:=\\frac\{1\}\{w\_\{x\}^\{d\}\}\\,\\mathbb\{E\}\_\{\{\\bf X\}\}\\\!\\left\[\\bm\{\\iota\}\\\!\\Bigl\(\\frac\{\{\\bf X\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\Bigr\)\\bm\{\\iota\}\\\!\\Bigl\(\\frac\{\{\\bf X\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\Bigr\)^\{\\top\}\\bm\{\\kappa\}\_\{w\_\{x\}\}\(\{\\bf X\}\-\{\\bm\{x\}\}\)\\right\],𝝁⁡\(z,𝒙,wz,wx\)\\displaystyle\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\):=1wxd​𝔼Z,𝐗​\[𝜾⁡\(𝐗−𝒙wx\)​𝒦​\(z−Zwz\)​𝜿wx​\(𝐗−𝒙\)\]\.\\displaystyle:=\\frac\{1\}\{w\_\{x\}^\{d\}\}\\,\\mathbb\{E\}\_\{Z,\{\\bf X\}\}\\\!\\left\[\\bm\{\\iota\}\\\!\\Bigl\(\\frac\{\{\\bf X\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\Bigr\)\\mathcal\{K\}\\\!\\Bigl\(\\frac\{z\-Z\}\{w\_\{z\}\}\\Bigr\)\\bm\{\\kappa\}\_\{w\_\{x\}\}\(\{\\bf X\}\-\{\\bm\{x\}\}\)\\right\]\.The pseudo\-true parameter𝜷⋄\\bm\{\\beta\}^\{\\diamond\}is deterministic and depends on the bandwidths\(wz,wx\)\(w\_\{z\},w\_\{x\}\); it converges to𝜷⋆​\(z,𝒙\)\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)aswz,wx→0w\_\{z\},w\_\{x\}\\to 0\. This deterministic intermediate quantity gives the natural decomposition of the overall estimation error into a stochastic component and a smoothing\-induced \(oracle\) bias:

𝜷^​\(z,𝒙,wz,wx\)−𝜷⋆​\(z,𝒙\)=𝜷^​\(z,𝒙,wz,wx\)−𝜷⋄​\(z,𝒙,wz,wx\)⏟stochastic error\+𝜷⋄​\(z,𝒙,wz,wx\)−𝜷⋆​\(z,𝒙\)⏟oracle bias\.\\widehat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)=\\underbrace\{\\widehat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\}\_\{\\text\{stochastic error\}\}\+\\underbrace\{\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\}\_\{\\text\{oracle bias\}\}\.\(44\)The stochastic error captures the empirical fluctuation of the finite\-sample estimator around its deterministic population counterpart, and is controlled by an empirical\-process argument over the kernel class

𝒢M:=\{\(𝒙′,z′\)↦𝜿\(𝒙′−𝒙wx\)𝒦\(z′−zwz\):𝒙∈𝒳,z∈ℝ\},\\mathcal\{G\}\_\{M\}:=\\Bigl\\\{\(\{\\bm\{x\}\}^\{\\prime\},z^\{\\prime\}\)\\mapsto\\bm\{\\kappa\}\\\!\\Bigl\(\\frac\{\{\\bm\{x\}\}^\{\\prime\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\Bigr\)\\mathcal\{K\}\\\!\\Bigl\(\\frac\{z^\{\\prime\}\-z\}\{w\_\{z\}\}\\Bigr\):\\ \{\\bm\{x\}\}\\in\\mathcal\{X\},\\ z\\in\\mathbb\{R\}\\Bigr\\\},\(45\)which is uniformly bounded and Euclidean by Lemmas[A\.1](https://arxiv.org/html/2610.33403#A1.Thmlem1)–[A\.4](https://arxiv.org/html/2610.33403#A1.Thmlem4)\. The oracle bias captures the deterministic smoothing distortion introduced by the bandwidths\(wx,wz\)\(w\_\{x\},w\_\{z\}\)and is controlled by a Taylor expansion ofF\(⋅∣⋅\)F\(\\cdot\\mid\\cdot\)under Assumption[2](https://arxiv.org/html/2610.33403#Thmass2)\. The two components are bounded separately in Sections[B\.2](https://arxiv.org/html/2610.33403#A2.SS2)and[B\.3](https://arxiv.org/html/2610.33403#A2.SS3)below; combining them yields the conclusion of Theorem[5\.1](https://arxiv.org/html/2610.33403#S5.Thmthm1)\.

### B\.2Uniform bias bound

###### Lemma B\.1\(Uniform bias bound\)\.

Under Assumptions[1](https://arxiv.org/html/2610.33403#Thmass1)–[3](https://arxiv.org/html/2610.33403#Thmass3),

𝐃wx​\(𝜷⋄​\(z,𝒙,wz,wx\)−𝜷⋆​\(z,𝒙\)\)\\displaystyle\\bf\{\\displaystyle D\}\_\{w\_\{x\}\}\\bigl\(\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\\bigr\)=wx22Ω\(𝒙,wx\)−1∑i,j=1d∂2F⁡\(z∣𝒙\)∂xi​∂xj∫𝜾\(𝐭\)titj𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle=\\frac\{w\_\{x\}^\{2\}\}\{2\}\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\sum\_\{i,j=1\}^\{d\}\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\,t\_\{i\}t\_\{j\}\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+wz22Ω\(𝒙,wx\)−1∂2F⁡\(z∣𝒙\)∂z2\(∫v2κ\(v\)dv\)∫𝜾\(𝐭\)𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\,\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial z^\{2\}\}\\,\\Big\(\\int v^\{2\}\\kappa\(v\)\\,dv\\Big\)\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+o⁡\(wx2\+wz2\)\\displaystyle\+o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)holds uniformly overz∈ℝz\\in\\mathbb\{R\}and𝐱∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\. In particular, for the intercept term,

supz∈ℝ,𝒙∈𝒳\|β0⋄​\(z,𝒙,wz,wx\)−F⁡\(z∣𝒙\)\|=O⁡\(wx2\+wz2\)\.\\sup\_\{z\\in\\mathbb\{R\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\Bigl\|\\,\\beta\_\{0\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-F\(z\\mid\{\\bm\{x\}\}\)\\,\\Bigr\|=O\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\\right\)\.

###### Proof\.

The proof follows the same decomposition as in the proof of Theorem 1 of[Xie \[2023\]](https://arxiv.org/html/2610.33403#bib.bib4), with the only difference that the kernel𝜿\\bm\{\\kappa\}is allowed to have unbounded support\. Hence we only need to verify that the remainder terms remain uniformly negligible under the Gaussian kernel\. The original proof writes the population matrices and moments as integral expressions and then applies second\-order Taylor expansions inzzand𝒙\{\\bm\{x\}\}\. Write

𝚵\(𝒙,wx\)=∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝜿\(𝐭\)f\(𝒙\+wx𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭∈ℝ\(d\+1\)×\(d\+1\),\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)=\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\\in\\mathbb\{R\}^\{\(d\+1\)\\times\(d\+1\)\},and

𝝁\(z,𝒙,wz,wx\)=∫𝜾\(𝐭\)𝜿\(𝐭\)F~\(z∣𝒙\+wx𝐭\)f\(𝒙\+wx𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭∈ℝd\+1,\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\\in\\mathbb\{R\}^\{d\+1\},where

F~​\(z∣𝒙\)=𝔼⁡\[𝒦⁡\(z−Zwz\)\|𝒙\]\.\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\)=\\mathbb\{E\}\\\!\\left\[\\mathcal\{K\}\\\!\\left\(\\frac\{z\-Z\}\{w\_\{z\}\}\\right\)\\Bigm\|\{\\bm\{x\}\}\\right\]\.By definition of𝜷⋄\\bm\{\\beta\}^\{\\diamond\},

𝐃wx​𝜷⋄​\(z,𝒙,wz,wx\)=𝚵​\(𝒙,wx\)−1​𝝁​\(z,𝒙,wz,wx\)\.\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\.
We first expandF~​\(z∣𝒙\+wx​𝐭\)\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)in the response direction,

F~​\(z∣𝒙\+wx​𝐭\)=∫κ⁡\(v\)​F​\(z−wz​v∣𝒙\+wx​𝐭\)​𝑑v\.\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=\\int\\kappa\(v\)\\,F\(z\-w\_\{z\}v\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,dv\.\(46\)SinceFFis twice continuously differentiable inzzand𝒙\{\\bm\{x\}\}, and its second\-order derivatives are uniformly continuous onℝ×𝒳\\mathbb\{R\}\\times\\mathcal\{X\}by Assumption[2](https://arxiv.org/html/2610.33403#Thmass2), we have the second\-order expansion

F⁡\(z−wz​v∣𝒙\+wx​𝐭\)=F⁡\(z∣𝒙\+wx​𝐭\)−∂zF⁡\(z∣𝒙\+wx​𝐭\)​wz​v\+wz22​∂z2F⁡\(z∣𝒙\+wx​𝐭\)​v2\+Rz​\(z,𝒙,𝐭,v\),F\(z\-w\_\{z\}v\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\partial\_\{z\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,w\_\{z\}v\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,v^\{2\}\+R\_\{z\}\(z,\{\\bm\{x\}\},\\mathbf\{t\},v\),\(47\)where the remainder

Rz​\(z,𝒙,𝐭,v\)=wz22​v2​\(∂z2F⁡\(z~∣𝒙\+wx​𝐭\)−∂z2F⁡\(z∣𝒙\+wx​𝐭\)\)R\_\{z\}\(z,\{\\bm\{x\}\},\\mathbf\{t\},v\)=\\frac\{w\_\{z\}^\{2\}\}\{2\}v^\{2\}\\Big\(\\partial\_\{z\}^\{2\}F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\Big\)andz~∈\[z−wz​v,z\]\\tilde\{z\}\\in\[z\-w\_\{z\}v,z\]\. Becauseκ\\kappais symmetric,∫v​κ​\(v\)​𝑑v=0\\int v\\,\\kappa\(v\)\\,dv=0implies that the integral of the first\-order term∂zF⁡\(z∣𝒙\+wx​𝐭\)​wz​v\\partial\_\{z\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,w\_\{z\}vvanishes\. Becauseκ\\kappais Gaussian,

∫v2​κ​\(v\)​𝑑v<∞\.\\int v^\{2\}\\kappa\(v\)\\,dv<\\infty\.
To control the integrated remainder, fix anyη\>0\\eta\>0\. Since∂z2F\\partial\_\{z\}^\{2\}Fis uniformly bounded and the Gaussian kernel has a finite second moment, we can chooseM\>0M\>0sufficiently large such that

2​‖∂z2F‖∞​∫\|v\|\>Mv2​κ​\(v\)​𝑑v<η\.2\\\|\\partial\_\{z\}^\{2\}F\\\|\_\{\\infty\}\\int\_\{\|v\|\>M\}v^\{2\}\\kappa\(v\)\\,dv<\\eta\.For this fixedMM, the uniform continuity of∂z2F\\partial\_\{z\}^\{2\}Fimplies that there existsδ\>0\\delta\>0such that

\|z′−z\|<δ⟹\|∂z2F⁡\(z′∣𝒙′\)−∂z2F⁡\(z∣𝒙′\)\|<η\|z^\{\\prime\}\-z\|<\\delta\\quad\\Longrightarrow\\quad\\left\|\\partial\_\{z\}^\{2\}F\(z^\{\\prime\}\\mid\{\\bm\{x\}\}^\{\\prime\}\)\-\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}^\{\\prime\}\)\\right\|<\\etauniformly over𝒙′∈𝒳\{\\bm\{x\}\}^\{\\prime\}\\in\\mathcal\{X\}\. For sufficiently smallwzw\_\{z\},wz​M<δw\_\{z\}M<\\delta, and hence, whenever\|v\|≤M\|v\|\\leq M,

\|z~−z\|≤wz​\|v\|≤wz​M<δ,\|\\tilde\{z\}\-z\|\\leq w\_\{z\}\|v\|\\leq w\_\{z\}M<\\delta,so that

\|∂z2F⁡\(z~∣𝒙\+wx​𝐭\)−∂z2F⁡\(z∣𝒙\+wx​𝐭\)\|<η\.\\left\|\\partial\_\{z\}^\{2\}F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\right\|<\\eta\.Therefore,

1wz2​\|∫ℝRz​\(z,𝒙,𝐭,v\)​κ​\(v\)​𝑑v\|≤η2​∫\|v\|≤Mv2​κ​\(v\)​𝑑v\+‖∂z2F‖∞​∫\|v\|\>Mv2​κ​\(v\)​𝑑v≤C​η,\\frac\{1\}\{w\_\{z\}^\{2\}\}\\left\|\\int\_\{\\mathbb\{R\}\}R\_\{z\}\(z,\{\\bm\{x\}\},\\mathbf\{t\},v\)\\kappa\(v\)\\,dv\\right\|\\leq\\frac\{\\eta\}\{2\}\\int\_\{\|v\|\\leq M\}v^\{2\}\\kappa\(v\)\\,dv\+\\\|\\partial\_\{z\}^\{2\}F\\\|\_\{\\infty\}\\int\_\{\|v\|\>M\}v^\{2\}\\kappa\(v\)\\,dv\\leq C\\eta,uniformly over the admissible\(z,𝒙,𝐭\)\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\. Sinceη\>0\\eta\>0is arbitrary,

∫ℝRz​\(z,𝒙,𝐭,v\)​κ​\(v\)​𝑑v=o⁡\(wz2\)\\int\_\{\\mathbb\{R\}\}R\_\{z\}\(z,\{\\bm\{x\}\},\\mathbf\{t\},v\)\\kappa\(v\)\\,dv=o\(w\_\{z\}^\{2\}\)uniformly\. Therefore, by dominated convergence,

F~​\(z∣𝒙\+wx​𝐭\)=F⁡\(z∣𝒙\+wx​𝐭\)\+wz22​∂z2F⁡\(z∣𝒙\+wx​𝐭\)​∫v2​κ​\(v\)​𝑑v\+o⁡\(wz2\),\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\int v^\{2\}\\kappa\(v\)\\,dv\+o\(w\_\{z\}^\{2\}\),uniformly overz∈ℝz\\in\\mathbb\{R\},𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}, and𝒙\+wx​𝐭∈𝒳\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\.

Next, expand the map

𝐮↦F⁡\(z∣𝐮\)\\mathbf\{u\}\\mapsto F\(z\\mid\\mathbf\{u\}\)at𝐮=𝒙\\mathbf\{u\}=\{\\bm\{x\}\}in the covariate direction\. By the second\-order Taylor expansion,

F⁡\(z∣𝒙\+wx​𝐭\)=F⁡\(z∣𝒙\)\+wx​∇𝒙⊤\{F⁡\(z∣𝒙\)\}​𝐭\+wx22​𝐭⊤​∇𝒙⊤​∇𝒙\{F⁡\(z∣𝒙\)\}​𝐭\+Rx​\(z,𝒙,𝐭\),F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=F\(z\\mid\{\\bm\{x\}\}\)\+w\_\{x\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\\!\\\{F\(z\\mid\{\\bm\{x\}\}\)\\\}\\mathbf\{t\}\+\\frac\{w\_\{x\}^\{2\}\}\{2\}\\mathbf\{t\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\\!\\\{F\(z\\mid\{\\bm\{x\}\}\)\\\}\\mathbf\{t\}\+R\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\),\(48\)where

Rx​\(z,𝒙,𝐭\)=wx22​𝐭⊤​\(∇𝒙⊤​∇𝒙\{F⁡\(z∣𝒙~\)\}−∇𝒙⊤​∇𝒙\{F⁡\(z∣𝒙\)\}\)​𝐭R\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)=\\frac\{w\_\{x\}^\{2\}\}\{2\}\\mathbf\{t\}^\{\\top\}\\Big\(\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\\!\\\{F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\\}\-\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\\!\\\{F\(z\\mid\{\\bm\{x\}\}\)\\\}\\Big\)\\mathbf\{t\}and𝒙~∈\[𝒙,𝒙\+wx​𝐭\]\\tilde\{\{\\bm\{x\}\}\}\\in\[\{\\bm\{x\}\},\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\]\.

Multiplying by𝜾⁡\(𝐭\)​𝜿​\(𝐭\)\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\), using the bound‖𝜾⁡\(𝐭\)‖≲1\+‖𝐭‖\\\|\\bm\{\\iota\}\(\\mathbf\{t\}\)\\\|\\lesssim 1\+\\\|\\mathbf\{t\}\\\|, and combining thezz\- and𝒙\{\\bm\{x\}\}\-expansions, we obtain

𝝁\(z,𝒙,wz,wx\)=𝚵\(𝒙,wx\)𝐃wx𝜷⋆\(z,𝒙\)\+f\(𝒙\)wx22∑i,j=1d∂2∂xi​∂xjF\(z∣𝒙\)∫𝜾\(𝐭\)titj𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\+f\(\{\\bm\{x\}\}\)\\frac\{w\_\{x\}^\{2\}\}\{2\}\\sum\_\{i,j=1\}^\{d\}\\frac\{\\partial^\{2\}\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}F\(z\\mid\{\\bm\{x\}\}\)\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)t\_\{i\}t\_\{j\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+f\(𝒙\)wz22∂2F⁡\(z∣𝒙\)∂z2\(∫v2κ\(v\)dv\)\(∫𝜾\(𝐭\)𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\)\+o\(wx2\+wz2\),\\qquad\+f\(\{\\bm\{x\}\}\)\\frac\{w\_\{z\}^\{2\}\}\{2\}\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial z^\{2\}\}\\left\(\\int v^\{2\}\\kappa\(v\)\\,dv\\right\)\\left\(\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\\right\)\+o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\),\(49\)uniformly overz∈ℝz\\in\\mathbb\{R\}and𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\. We explain how Eq\. \([49](https://arxiv.org/html/2610.33403#A2.E49)\) is obtained as follows\.

To keep all powers ofwxw\_\{x\}explicit, we work with a remainder that does*not*carry awx2w\_\{x\}^\{2\}factor inside\. Define the Hessian\-increment remainder

R~x​\(z,𝒙,𝐭\):=𝐭⊤​\(∇𝒙⊤​∇𝒙F​\(z∣𝒙~\)−∇𝒙⊤​∇𝒙F​\(z∣𝒙\)\)​𝐭,\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\):=\\mathbf\{t\}^\{\\top\}\\Big\(\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\{\\bm\{x\}\}\)\\Big\)\\mathbf\{t\},where𝒙~\\tilde\{\{\\bm\{x\}\}\}lies on the line segment joining𝒙\{\\bm\{x\}\}and𝒙\+wx​𝐭\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\. With this convention, the second\-order Taylor expansion ofF\(z∣⋅\)F\(z\\mid\\,\\cdot\\,\)in the covariate direction reads

F⁡\(z∣𝒙\+wx​𝐭\)=F⁡\(z∣𝒙\)\+wx​∇𝒙⊤F​\(z∣𝒙\)​𝐭\+wx22​\(𝐭⊤​∇𝒙⊤​∇𝒙F​\(z∣𝒙\)​𝐭\+R~x​\(z,𝒙,𝐭\)\)\.F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=F\(z\\mid\{\\bm\{x\}\}\)\+w\_\{x\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}F\(z\\mid\{\\bm\{x\}\}\)\\,\\mathbf\{t\}\+\\frac\{w\_\{x\}^\{2\}\}\{2\}\\Big\(\\mathbf\{t\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\{\\bm\{x\}\}\)\\,\\mathbf\{t\}\+\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\\Big\)\.By the definition𝜾⁡\(𝐭\)=\(1,t1,…,td\)⊤\\bm\{\\iota\}\(\\mathbf\{t\}\)=\(1,t\_\{1\},\\dots,t\_\{d\}\)^\{\\top\}and𝐃wx=diag⁡\(1,wx,…,wx\)\{\\bf D\}\_\{w\_\{x\}\}=\\operatorname\{diag\}\(1,w\_\{x\},\\dots,w\_\{x\}\), the zeroth\- and first\-order terms combine into the identity

F⁡\(z∣𝒙\)\+wx​∇𝒙⊤F​\(z∣𝒙\)​𝐭=𝜾​\(𝐭\)⊤​𝐃wx​𝜷⋆​\(z,𝒙\)\.F\(z\\mid\{\\bm\{x\}\}\)\+w\_\{x\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}F\(z\\mid\{\\bm\{x\}\}\)\\,\\mathbf\{t\}=\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\.\(50\)Combining \([50](https://arxiv.org/html/2610.33403#A2.E50)\) with thezz\-direction expansion

F~​\(z∣𝒙\+wx​𝐭\)=F⁡\(z∣𝒙\+wx​𝐭\)\+wz22​∂z2F⁡\(z∣𝒙\+wx​𝐭\)​∫v2​κ​\(v\)​𝑑v\+o⁡\(wz2\),\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\int v^\{2\}\\kappa\(v\)\\,dv\+o\(w\_\{z\}^\{2\}\),we obtain

F~​\(z∣𝒙\+wx​𝐭\)\\displaystyle\\widetilde\{F\}\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=𝜾​\(𝐭\)⊤​𝐃wx​𝜷⋆​\(z,𝒙\)\\displaystyle=\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\(51\)\+wx22​\(𝐭⊤​∇𝒙⊤​∇𝒙F​\(z∣𝒙\)​𝐭\+R~x​\(z,𝒙,𝐭\)\)\\displaystyle\+\\frac\{w\_\{x\}^\{2\}\}\{2\}\\Big\(\\mathbf\{t\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\{\\bm\{x\}\}\)\\,\\mathbf\{t\}\+\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\\Big\)\+wz22∂z2F\(z∣𝒙\+wx𝐭\)∫v2κ\(v\)dv\+o\(wz2\)\.\\displaystyle\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\int v^\{2\}\\kappa\(v\)\\,dv\+o\(w\_\{z\}^\{2\}\)\.Substituting \([51](https://arxiv.org/html/2610.33403#A2.E51)\) into the definition of𝝁⁡\(z,𝒙,wz,wx\)\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)yields

𝝁⁡\(z,𝒙,wz,wx\)=\\displaystyle\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=𝚵⁡\(𝒙,wx\)​𝐃wx​𝜷⋆​\(z,𝒙\)\\displaystyle\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\,\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\(52\)\+wx22∫𝜾\(𝐭\)𝜿\(𝐭\)\(𝐭⊤∇𝒙⊤∇𝒙F\(z∣𝒙\)𝐭\+R~x\(z,𝒙,𝐭\)\)f\(𝒙\+wx𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\+\\frac\{w\_\{x\}^\{2\}\}\{2\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\Big\(\\mathbf\{t\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\{\\bm\{x\}\}\)\\,\\mathbf\{t\}\+\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\\Big\)f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+wz22∫v2κ\(v\)dv∫𝜾\(𝐭\)𝜿\(𝐭\)∂z2F\(z∣𝒙\+wx𝐭\)f\(𝒙\+wx𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\\!\\int v^\{2\}\\kappa\(v\)\\,dv\\\!\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+o⁡\(wz2\),\\displaystyle\+o\(w\_\{z\}^\{2\}\),uniformly overz∈ℝz\\in\\mathbb\{R\}and𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\. The first\-order Taylor expansion offfat𝒙\{\\bm\{x\}\}gives

f⁡\(𝒙\+wx​𝐭\)=f⁡\(𝒙\)\+wx​∇𝒙⊤f​\(𝒙ˇ\)​𝐭,f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=f\(\{\\bm\{x\}\}\)\+w\_\{x\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}f\(\\check\{\{\\bm\{x\}\}\}\)\\,\\mathbf\{t\},where𝒙ˇ\\check\{\{\\bm\{x\}\}\}lies between𝒙\{\\bm\{x\}\}and𝒙\+wx​𝐭\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\. Since∇𝒙f\\nabla\_\{\\bm\{x\}\}fis uniformly bounded by Assumption[1](https://arxiv.org/html/2610.33403#Thmass1),

f⁡\(𝒙\+wx​𝐭\)−f⁡\(𝒙\)=O⁡\(wx​‖𝐭‖\)\.f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-f\(\{\\bm\{x\}\}\)=O\(w\_\{x\}\\\|\\mathbf\{t\}\\\|\)\.For∂z2F\\partial\_\{z\}^\{2\}F, instead of using a first\-order Taylor expansion, we use Assumption[2](https://arxiv.org/html/2610.33403#Thmass2)to conclude that∂z2F\\partial\_\{z\}^\{2\}Fis bounded and uniformly continuous\. Hence, for any fixedM\>0M\>0,

supz,𝒙,𝐭:‖𝐭‖≤M𝒙\+wx​𝐭∈𝒳\|∂z2F\(z∣𝒙\+wx𝐭\)−∂z2F\(z∣𝒙\)\|=o\(1\),aswx→0\.\\sup\_\{\\begin\{subarray\}\{c\}z,\{\\bm\{x\}\},\\mathbf\{t\}:\\,\\\|\\mathbf\{t\}\\\|\\leq M\\\\ \{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\end\{subarray\}\}\\left\|\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\\right\|=o\(1\),\\text\{ as \}w\_\{x\}\\to 0\.For‖𝐭‖\>M\\\|\\mathbf\{t\}\\\|\>M, boundedness of∂z2F\\partial\_\{z\}^\{2\}F, together with the finite moments and exponentially decaying tails of the Gaussian kernel, makes the corresponding integral arbitrarily small by takingMMsufficiently large\. Therefore, by first takingMMlarge and then lettingwx→0w\_\{x\}\\to 0,

∫𝜾⁡\(𝐭\)​𝜿​\(𝐭\)​\{∂z2F⁡\(z∣𝒙\+wx​𝐭\)−∂z2F⁡\(z∣𝒙\)\}​f​\(𝒙\+wx​𝐭\)​𝑑𝐭=o⁡\(1\)\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\left\\\{\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\\right\\\}f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,d\\mathbf\{t\}=o\(1\)uniformly inzzand𝒙\{\\bm\{x\}\}\. Consequently, the corresponding term multiplied bywz2w\_\{z\}^\{2\}iso⁡\(wz2\)o\(w\_\{z\}^\{2\}\)\. The cross terms arising from the expansion offfare of orderswx⋅wx2=o⁡\(wx2\)w\_\{x\}\\cdot w\_\{x\}^\{2\}=o\(w\_\{x\}^\{2\}\)andwx⋅wz2=o⁡\(wx2\+wz2\)w\_\{x\}\\cdot w\_\{z\}^\{2\}=o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\), respectively, sincewx​wz2≤12​\(wx2\+wz2\)​max⁡\(wx,wz\)=o⁡\(wx2\+wz2\)\.w\_\{x\}w\_\{z\}^\{2\}\\leq\\frac\{1\}\{2\}\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\\max\(w\_\{x\},w\_\{z\}\)=o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\.Hence \([52](https://arxiv.org/html/2610.33403#A2.E52)\) reduces to

𝝁⁡\(z,𝒙,wz,wx\)=\\displaystyle\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=𝚵⁡\(𝒙,wx\)​𝐃wx​𝜷⋆​\(z,𝒙\)\\displaystyle\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\,\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\+f\(𝒙\)wx22∫𝜾\(𝐭\)𝜿\(𝐭\)𝐭⊤∇𝒙⊤∇𝒙F\(z∣𝒙\)𝐭𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\+f\(\{\\bm\{x\}\}\)\\,\\frac\{w\_\{x\}^\{2\}\}\{2\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbf\{t\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\{\\bm\{x\}\}\)\\,\\mathbf\{t\}\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+f\(𝒙\)wx22∫𝜾\(𝐭\)𝜿\(𝐭\)R~x\(z,𝒙,𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\+f\(\{\\bm\{x\}\}\)\\,\\frac\{w\_\{x\}^\{2\}\}\{2\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+f\(𝒙\)wz22∂z2F\(z∣𝒙\)∫v2κ\(v\)dv∫𝜾\(𝐭\)𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\+o\(wx2\+wz2\)\.\\displaystyle\+f\(\{\\bm\{x\}\}\)\\,\\frac\{w\_\{z\}^\{2\}\}\{2\}\\,\\partial\_\{z\}^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\\int v^\{2\}\\kappa\(v\)\\,dv\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\.
It remains to show that the term involvingR~x\\widetilde\{R\}\_\{x\}iso⁡\(wx2\)o\(w\_\{x\}^\{2\}\)uniformly\. The approach is the same: use the finite moments of the Gaussian kernel and the uniform continuity of the Hessian ofFFwith respect to𝒙\{\\bm\{x\}\}\. By Assumption[2](https://arxiv.org/html/2610.33403#Thmass2),∇𝒙⊤∇𝒙F\(z∣⋅\)\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\(z\\mid\\,\\cdot\\,\)admits a common modulus of continuityω𝐱\\omega\_\{\\mathbf\{x\}\}onℝ×𝒳\\mathbb\{R\}\\times\\mathcal\{X\}, so

‖𝜾⁡\(𝐭\)​𝜿​\(𝐭\)​R~x​\(z,𝒙,𝐭\)‖≲\(1\+‖𝐭‖\)​‖𝐭‖2​𝜿​\(𝐭\)​ω𝐱​\(wx​‖𝐭‖\)\.\\bigl\\\|\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\\bigr\\\|\\lesssim\(1\+\\\|\\mathbf\{t\}\\\|\)\\,\\\|\\mathbf\{t\}\\\|^\{2\}\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\omega\_\{\\mathbf\{x\}\}\(w\_\{x\}\\\|\\mathbf\{t\}\\\|\)\.Fixε\>0\\varepsilon\>0\. Since the Gaussian kernel has finite moments of all orders, there existsM=M⁡\(ε\)M=M\(\\varepsilon\)such that

2​‖∇𝒙⊤​∇𝒙F‖∞​∫‖𝐭‖\>M\(1\+‖𝐭‖3\)​𝜿​\(𝐭\)​𝑑𝐭<ε\.2\\,\\\|\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}F\\\|\_\{\\infty\}\\,\\int\_\{\\\|\\mathbf\{t\}\\\|\>M\}\(1\+\\\|\\mathbf\{t\}\\\|^\{3\}\)\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}<\\varepsilon\.For this fixedMM, by uniform continuity of∇𝒙⊤​∇𝒙F\\nabla\_\{\\bm\{x\}\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}Fonℝ×𝒳\\mathbb\{R\}\\times\\mathcal\{X\}, there existswx\(0\)\>0w\_\{x\}^\{\(0\)\}\>0such thatω𝐱​\(wx​M\)<ε\\omega\_\{\\mathbf\{x\}\}\(w\_\{x\}M\)<\\varepsilonfor allwx<wx\(0\)w\_\{x\}<w\_\{x\}^\{\(0\)\}\. Splitting the integral at‖𝐭‖=M\\\|\\mathbf\{t\}\\\|=M, we obtain

‖∫𝜾\(𝐭\)𝜿\(𝐭\)R~x\(z,𝒙,𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭‖≲ε∫‖𝐭‖≤M\(1\+∥𝐭∥3\)𝜿\(𝐭\)d𝐭\+ε≤Cε,\\left\\\|\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\widetilde\{R\}\_\{x\}\(z,\{\\bm\{x\}\},\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\\right\\\|\\lesssim\\varepsilon\\\!\\int\_\{\\\|\\mathbf\{t\}\\\|\\leq M\}\(1\+\\\|\\mathbf\{t\}\\\|^\{3\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}\+\\varepsilon\\leq C\\varepsilon,uniformly overz∈ℝz\\in\\mathbb\{R\}and𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\. Sinceε\\varepsilonis arbitrary, theR~x\\widetilde\{R\}\_\{x\}\-term iso⁡\(1\)o\(1\), and after multiplication by the prefactorwx22\\tfrac\{w\_\{x\}^\{2\}\}\{2\}, it iso⁡\(wx2\)o\(w\_\{x\}^\{2\}\)uniformly\. This yields \([49](https://arxiv.org/html/2610.33403#A2.E49)\)\.

We now translate the expansion of𝝁⁡\(z,𝒙,wz,wx\)\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)into an expansion of𝐃wx​𝜷⋄​\(z,𝒙,wz,wx\)=𝚵​\(𝒙,wx\)−1​𝝁​\(z,𝒙,wz,wx\)\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\. By Lemma[A\.5](https://arxiv.org/html/2610.33403#A1.Thmlem5), the eigenvalues ofΩ⁡\(𝒙,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)and𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)are uniformly bounded away from zero and infinity for all𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and all sufficiently smallwx\>0w\_\{x\}\>0, so

sup𝒙∈𝒳‖Ω​\(𝒙,wx\)−1‖\+sup𝒙∈𝒳‖𝚵​\(𝒙,wx\)−1‖<∞\.\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\\|\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\\|\+\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\\|\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\\|<\\infty\.
We next show that

𝚵⁡\(𝒙,wx\)−f⁡\(𝒙\)​Ω​\(𝒙,wx\)=O⁡\(wx\),𝚵​\(𝒙,wx\)−1​f​\(𝒙\)=Ω​\(𝒙,wx\)−1\+O⁡\(wx\),\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\-f\(\{\\bm\{x\}\}\)\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)=O\(w\_\{x\}\),\\qquad\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}f\(\{\\bm\{x\}\}\)=\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\+O\(w\_\{x\}\),\(53\)both uniformly over𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\. Indeed, by Assumption[1](https://arxiv.org/html/2610.33403#Thmass1),ffadmits a first\-order Taylor expansionf⁡\(𝒙\+wx​𝐭\)=f⁡\(𝒙\)\+wx​∇𝒙⊤f​\(𝒙ˇ\)​𝐭f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)=f\(\{\\bm\{x\}\}\)\+w\_\{x\}\\nabla\_\{\\bm\{x\}\}^\{\\top\}f\(\\check\{\{\\bm\{x\}\}\}\)\\,\\mathbf\{t\}with𝒙ˇ\\check\{\{\\bm\{x\}\}\}between𝒙\{\\bm\{x\}\}and𝒙\+wx​𝐭\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}, and∇f\\nabla fis uniformly bounded by some constantLfL\_\{f\}on the closure of𝒳\\mathcal\{X\}\. Hence

𝚵⁡\(𝒙,wx\)−f⁡\(𝒙\)​Ω​\(𝒙,wx\)\\displaystyle\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\-f\(\{\\bm\{x\}\}\)\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)=∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝜿\(𝐭\)\(f\(𝒙\+wx𝐭\)−f\(𝒙\)\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle=\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\bigl\(f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\-f\(\{\\bm\{x\}\}\)\\bigr\)\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}=wx∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝜿\(𝐭\)∇𝒙⊤f\(𝒙ˇ\)𝐭𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭,\\displaystyle=w\_\{x\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\nabla\_\{\\bm\{x\}\}^\{\\top\}f\(\\check\{\{\\bm\{x\}\}\}\)\\,\\mathbf\{t\}\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\},whose operator norm is bounded bywx​Lf​∫\(1\+‖𝐭‖\)2​‖𝐭‖​𝜿​\(𝐭\)​𝑑𝐭<∞w\_\{x\}L\_\{f\}\\int\(1\+\\\|\\mathbf\{t\}\\\|\)^\{2\}\\\|\\mathbf\{t\}\\\|\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}<\\inftysince the Gaussian kernel has finite moments of all orders\. This proves the first equality in \([53](https://arxiv.org/html/2610.33403#A2.E53)\)\. The second equality follows from the resolvent identity

𝚵−1​f​\(𝒙\)−Ω−1=𝚵−1​\(f⁡\(𝒙\)​Ω−𝚵\)​Ω−1,\\bm\{\\varXi\}^\{\-1\}f\(\{\\bm\{x\}\}\)\-\\Omega^\{\-1\}=\\bm\{\\varXi\}^\{\-1\}\\bigl\(f\(\{\\bm\{x\}\}\)\\Omega\-\\bm\{\\varXi\}\\bigr\)\\Omega^\{\-1\},combined with the uniform boundedness of𝚵−1\\bm\{\\varXi\}^\{\-1\}andΩ−1\\Omega^\{\-1\}\.

Multiplying \([49](https://arxiv.org/html/2610.33403#A2.E49)\) by𝚵​\(𝒙,wx\)−1\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}and noting that𝚵​\(𝒙,wx\)−1​\(𝚵⁡\(𝒙,wx\)​𝐃wx​𝜷⋆​\(z,𝒙\)\)=𝐃wx​𝜷⋆​\(z,𝒙\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\bigl\(\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\\bigr\)=\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\), we obtain

𝐃wx​𝜷⋄​\(z,𝒙,wz,wx\)−𝐃wx​𝜷⋆​\(z,𝒙\)\\displaystyle\\bf\{\\displaystyle D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)=wx22𝚵\(𝒙,wx\)−1f\(𝒙\)∑i,j=1d∂2F⁡\(z∣𝒙\)∂xi​∂xj∫𝜾\(𝐭\)titj𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle=\\frac\{w\_\{x\}^\{2\}\}\{2\}\\,\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}f\(\{\\bm\{x\}\}\)\\,\\sum\_\{i,j=1\}^\{d\}\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\,t\_\{i\}t\_\{j\}\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+wz22𝚵\(𝒙,wx\)−1f\(𝒙\)∂2F⁡\(z∣𝒙\)∂z2∫v2κ\(v\)dv∫𝜾\(𝐭\)𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\+o\(wx2\+wz2\)\.\\displaystyle\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\,\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}f\(\{\\bm\{x\}\}\)\\,\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial z^\{2\}\}\\int v^\{2\}\\kappa\(v\)\\,dv\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\.Applying the second identity in \([53](https://arxiv.org/html/2610.33403#A2.E53)\) to replace𝚵​\(𝒙,wx\)−1​f​\(𝒙\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}f\(\{\\bm\{x\}\}\)byΩ​\(𝒙,wx\)−1\+O⁡\(wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\+O\(w\_\{x\}\), and noting that theO⁡\(wx\)O\(w\_\{x\}\)part multiplies anO⁡\(wx2\+wz2\)O\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)quantity \(so their product iso⁡\(wx2\+wz2\)o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\), we conclude

𝐃wx​\(𝜷⋄​\(z,𝒙,wz,wx\)−𝜷⋆​\(z,𝒙\)\)=\\displaystyle\{\\bf D\}\_\{w\_\{x\}\}\\bigl\(\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\star\}\(z,\{\\bm\{x\}\}\)\\bigr\)=wx22Ω\(𝒙,wx\)−1∑i,j=1d∂2F⁡\(z∣𝒙\)∂xi​∂xj∫𝜾\(𝐭\)titj𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\\frac\{w\_\{x\}^\{2\}\}\{2\}\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\sum\_\{i,j=1\}^\{d\}\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\,t\_\{i\}t\_\{j\}\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+wz22Ω\(𝒙,wx\)−1∂2F⁡\(z∣𝒙\)∂z2∫v2κ\(v\)dv∫𝜾\(𝐭\)𝜿\(𝐭\)𝕀\{𝒙\+wx𝐭∈𝒳\}d𝐭\\displaystyle\+\\frac\{w\_\{z\}^\{2\}\}\{2\}\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\frac\{\\partial^\{2\}F\(z\\mid\{\\bm\{x\}\}\)\}\{\\partial z^\{2\}\}\\int v^\{2\}\\kappa\(v\)\\,dv\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\\\}\\,d\\mathbf\{t\}\+o⁡\(wx2\+wz2\),\\displaystyle\+o\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\),uniformly overz∈ℝz\\in\\mathbb\{R\}and𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\. This proves the first claim\. The second claim follows by taking the first coordinate of the above expansion and noting that the first diagonal entry of𝐃wx\{\\bf D\}\_\{w\_\{x\}\}is11, so that

β0⋄​\(z,𝒙,wz,wx\)−β0⋆​\(z,𝒙\)=O⁡\(wx2\+wz2\)\\beta\_\{0\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\beta\_\{0\}^\{\\star\}\(z,\{\\bm\{x\}\}\)=O\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)uniformly overz∈ℝz\\in\\mathbb\{R\}and𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}\.

∎

### B\.3Stochastic bound

###### Lemma B\.2\(Uniform stochastic bound\)\.

Under Assumptions[1](https://arxiv.org/html/2610.33403#Thmass1)–[3](https://arxiv.org/html/2610.33403#Thmass3), let

𝐃wx​𝜷^​\(z,𝒙,wz,wx\)=𝚵^​\(𝒙,wx\)−1​𝝁^​\(z,𝒙,wz,wx\)\.\{\\bf D\}\_\{w\_\{x\}\}\\hat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\.IfM​wxd\|log⁡wx\|→∞\\frac\{Mw\_\{x\}^\{d\}\}\{\|\\log w\_\{x\}\|\}\\to\\infty, then

supz∈ℝ,𝒙∈𝒳‖𝐃wx​\(𝜷^​\(z,𝒙,wz,wx\)−𝜷⋄​\(z,𝒙,wz,wx\)\)‖=Op​\(\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\{\\bf D\}\_\{w\_\{x\}\}\\bigl\(\\hat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\bigr\)\\right\\\|=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.In particular, for the intercept term,

supz∈ℝ,𝒙∈𝒳\|β^0​\(z,𝒙,wz,wx\)−β0⋄​\(z,𝒙,wz,wx\)\|=Op​\(\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\|\\hat\{\\beta\}\_\{0\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\beta\_\{0\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\right\|=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.

###### Proof\.

Let

𝚵^​\(𝒙,wx\)=1M​wxd​∑i=1M𝜾⁡\(𝐗i−𝒙wx\)​𝜾​\(𝐗i−𝒙wx\)⊤​𝜿​\(𝐗i−𝒙wx\),\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)=\\frac\{1\}\{Mw\_\{x\}^\{d\}\}\\sum\_\{i=1\}^\{M\}\\bm\{\\iota\}\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\)\\bm\{\\iota\}\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\)^\{\\top\}\\bm\{\\kappa\}\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\),and

𝝁^​\(z,𝒙,wz,wx\)=1M​wxd​∑i=1M𝜾⁡\(𝐗i−𝒙wx\)​𝒦​\(z−Ziwz\)​𝜿​\(𝐗i−𝒙wx\)\.\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\frac\{1\}\{Mw\_\{x\}^\{d\}\}\\sum\_\{i=1\}^\{M\}\\bm\{\\iota\}\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\)\\mathcal\{K\}\\\!\\left\(\\frac\{z\-Z\_\{i\}\}\{w\_\{z\}\}\\right\)\\bm\{\\kappa\}\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\)\.Recall that

𝐃wx​𝜷^​\(z,𝒙,wz,wx\)=𝚵^​\(𝒙,wx\)−1​𝝁^​\(z,𝒙,wz,wx\),𝐃wx​𝜷⋄​\(z,𝒙,wz,wx\)=𝚵​\(𝒙,wx\)−1​𝝁​\(z,𝒙,wz,wx\)\.\{\\bf D\}\_\{w\_\{x\}\}\\hat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\),\\qquad\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\.Hence

𝐃wx​\(𝜷^​\(z,𝒙,wz,wx\)−𝜷⋄​\(z,𝒙,wz,wx\)\)\\displaystyle\\bf\{\\displaystyle D\}\_\{w\_\{x\}\}\\bigl\(\\hat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\bigr\)\(54\)=\(𝚵^​\(𝒙,wx\)−1−𝚵​\(𝒙,wx\)−1\)​𝝁^​\(z,𝒙,wz,wx\)\+𝚵​\(𝒙,wx\)−1​\(𝝁^​\(z,𝒙,wz,wx\)−𝝁⁡\(z,𝒙,wz,wx\)\)\.\\displaystyle=\\bigl\(\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\-\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\bigr\)\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\+\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\bigl\(\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\bigr\)\.We therefore only need to derive uniform convergence rates for𝝁^−𝝁\\hat\{\\bm\{\\mu\}\}\-\\bm\{\\mu\}and𝚵^−𝚵\\hat\{\\bm\{\\varXi\}\}\-\\bm\{\\varXi\}, and then use the uniform invertibility of𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\.

We first study the stochastic fluctuation of𝝁^​\(z,𝒙,wz,wx\)\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\. For each coordinate indexed byπ=\(π1,…,πd\)∈\{0,1\}d\\pi=\(\\pi\_\{1\},\\dots,\\pi\_\{d\}\)\\in\\\{0,1\\\}^\{d\}with∑j=1dπj≤1\\sum^\{d\}\_\{j=1\}\\pi\_\{j\}\\leq 1, define

μ^π​\(z,𝒙,wz,wx\)=1M​wxd​∑i=1M𝒦⁡\(z−Ziwz\)​∏j=1d\(Xi​j−xjwx\)πj​κ​\(Xi​j−xjwx\),\\hat\{\\mu\}\_\{\\pi\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\frac\{1\}\{Mw\_\{x\}^\{d\}\}\\sum\_\{i=1\}^\{M\}\\mathcal\{K\}\\\!\\left\(\\frac\{z\-Z\_\{i\}\}\{w\_\{z\}\}\\right\)\\prod\_\{j=1\}^\{d\}\\left\(\\frac\{X\_\{ij\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\)^\{\\pi\_\{j\}\}\\kappa\\\!\\left\(\\frac\{X\_\{ij\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\),and let

μπ​\(z,𝒙,wz,wx\)=𝔼⁡\[μ^π​\(z,𝒙,wz,wx\)\]\.\\mu\_\{\\pi\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)=\\mathbb\{E\}\[\\hat\{\\mu\}\_\{\\pi\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\]\.
We introduce the following one\-dimensional classes:

ℱz:=\{Z↦𝒦\(z−Zwz\):z∈ℝ,wz\>0\},\\mathcal\{F\}\_\{z\}:=\\left\\\{Z\\mapsto\\mathcal\{K\}\\\!\\left\(\\frac\{z\-Z\}\{w\_\{z\}\}\\right\):\\ z\\in\\mathbb\{R\},\\ w\_\{z\}\>0\\right\\\},and for eachj=1,…,dj=1,\\dots,d,

ℱx,j\(πj\):=\{𝐗↦\(Xj−xjwx\)πjκ\(Xj−xjwx\):xj∈𝒳j,wx\>0\}\.\\mathcal\{F\}\_\{x,j\}^\{\(\\pi\_\{j\}\)\}:=\\left\\\{\{\\bf X\}\\mapsto\\left\(\\frac\{X\_\{j\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\)^\{\\pi\_\{j\}\}\\kappa\\\!\\left\(\\frac\{X\_\{j\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\):\\ x\_\{j\}\\in\\mathcal\{X\}\_\{j\},\\ w\_\{x\}\>0\\right\\\}\.Then the class indexingμ^π\\hat\{\\mu\}\_\{\\pi\}can be written as the product class

𝒢π=ℱz⋅∏j=1dℱx,j\(πj\)\.\\mathcal\{G\}\_\{\\pi\}=\\mathcal\{F\}\_\{z\}\\cdot\\prod\_\{j=1\}^\{d\}\\mathcal\{F\}\_\{x,j\}^\{\(\\pi\_\{j\}\)\}\.We now verify that each factor class is uniformly bounded and Euclidean\. It then follows from Lemmas[A\.2](https://arxiv.org/html/2610.33403#A1.Thmlem2)and[A\.3](https://arxiv.org/html/2610.33403#A1.Thmlem3)that𝒢π\\mathcal\{G\}\_\{\\pi\}is uniformly bounded and Euclidean\.

First, since𝒦=ϕ\\mathcal\{K\}=\\bm\{\\phi\}is the Gaussian distribution function,

0≤𝒦⁡\(u\)≤1for all​u∈ℝ,0\\leq\\mathcal\{K\}\(u\)\\leq 1\\qquad\\text\{for all \}u\\in\\mathbb\{R\},soℱz\\mathcal\{F\}\_\{z\}is uniformly bounded by11\. Moreover,ϕ\\bm\{\\phi\}is the integral of the Gaussian kernel; hence it is of bounded variation\. Therefore, by Lemma[A\.1](https://arxiv.org/html/2610.33403#A1.Thmlem1), the location\-scale classℱz\\mathcal\{F\}\_\{z\}is Euclidean\.

Next considerℱx,j\(πj\)\\mathcal\{F\}\_\{x,j\}^\{\(\\pi\_\{j\}\)\}\. Sinceπj∈\{0,1\}\\pi\_\{j\}\\in\\\{0,1\\\}, there are only two cases\.

Ifπj=0\\pi\_\{j\}=0, then the generating function is

u↦κ⁡\(u\)=ϕ⁡\(u\),u\\mapsto\\kappa\(u\)=\\phi\(u\),which is uniformly bounded onℝ\\mathbb\{R\}, and

ϕ′​\(u\)=−u​ϕ​\(u\)∈L1​\(ℝ\)\.\\phi^\{\\prime\}\(u\)=\-u\\phi\(u\)\\in L^\{1\}\(\\mathbb\{R\}\)\.Henceϕ\\phiis of bounded variation, and thusℱx,j\(0\)\\mathcal\{F\}\_\{x,j\}^\{\(0\)\}is Euclidean by Lemma[A\.1](https://arxiv.org/html/2610.33403#A1.Thmlem1)\.

Ifπj=1\\pi\_\{j\}=1, then the generating function is

u↦u​κ​\(u\)=u​ϕ​\(u\)\.u\\mapsto u\\kappa\(u\)=u\\phi\(u\)\.This function is also uniformly bounded onℝ\\mathbb\{R\}, since

supu∈ℝ\|u​ϕ​\(u\)\|<∞\.\\sup\_\{u\\in\\mathbb\{R\}\}\|u\\phi\(u\)\|<\\infty\.Moreover,

dd​u​\{u​ϕ​\(u\)\}=\(1−u2\)​ϕ​\(u\)∈L1​\(ℝ\),\\frac\{d\}\{du\}\\\{u\\phi\(u\)\\\}=\(1\-u^\{2\}\)\\phi\(u\)\\in L^\{1\}\(\\mathbb\{R\}\),sou​ϕ​\(u\)u\\phi\(u\)is of bounded variation\. Thereforeℱx,j\(1\)\\mathcal\{F\}\_\{x,j\}^\{\(1\)\}is also Euclidean\.

Consequently, for everyj=1,…,dj=1,\\dots,d, the classℱx,j\(πj\)\\mathcal\{F\}\_\{x,j\}^\{\(\\pi\_\{j\}\)\}is uniformly bounded and Euclidean\. Since𝒢π\\mathcal\{G\}\_\{\\pi\}is a finite product of uniformly bounded Euclidean classes, it follows from the closure property of Euclidean classes under finite products that𝒢π\\mathcal\{G\}\_\{\\pi\}is itself uniformly bounded and Euclidean; see Lemmas[A\.2](https://arxiv.org/html/2610.33403#A1.Thmlem2)and[A\.3](https://arxiv.org/html/2610.33403#A1.Thmlem3)\. And the variance bound of the class𝒢π\\mathcal\{G\}\_\{\\pi\}is of orderwxdw\_\{x\}^\{d\},

supg∈𝒢π𝔼⁡\[g​\(Z,𝐗\)2\]\\displaystyle\\sup\_\{g\\in\\mathcal\{G\}\_\{\\pi\}\}\\mathbb\{E\}\[g\(Z,\{\\bf X\}\)^\{2\}\]=supz∈ℝ,𝒙∈𝒳𝔼⁡\[𝒦2​\(z−Zwz\)​∏j=1d\(Xj−xjwx\)2​πj​κ2​\(Xj−xjwx\)\]\\displaystyle=\\sup\_\{z\\in\\mathbb\{R\},\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\mathbb\{E\}\\\!\\left\[\\mathcal\{K\}^\{2\}\\\!\\left\(\\frac\{z\-Z\}\{w\_\{z\}\}\\right\)\\prod\_\{j=1\}^\{d\}\\left\(\\frac\{X\_\{j\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\)^\{2\\pi\_\{j\}\}\\kappa^\{2\}\\\!\\left\(\\frac\{X\_\{j\}\-x\_\{j\}\}\{w\_\{x\}\}\\right\)\\right\]≤sup𝒙∈𝒳∫ℝd∏j=1dtj2​πj​𝜿​\(𝐭\)2​f​\(𝒙\+wx​𝐭\)​wxd​𝐝𝐭\\displaystyle\\leq\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\prod\_\{j=1\}^\{d\}t\_\{j\}^\{2\\pi\_\{j\}\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)^\{2\}\\,f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,w\_\{x\}^\{d\}\\,d\\mathbf\{t\}≤wxd​‖f‖∞​∫ℝd∏j=1dtj2​πj​𝜿​\(𝐭\)2​𝐝𝐭\\displaystyle\\leq w\_\{x\}^\{d\}\\\|f\\\|\_\{\\infty\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\prod\_\{j=1\}^\{d\}t\_\{j\}^\{2\\pi\_\{j\}\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)^\{2\}\\,d\\mathbf\{t\}=wxd​‖f‖∞​∏j=1d∫ℝu2​πj​κ​\(u\)2​𝑑u\\displaystyle=w\_\{x\}^\{d\}\\\|f\\\|\_\{\\infty\}\\prod\_\{j=1\}^\{d\}\\int\_\{\\mathbb\{R\}\}u^\{2\\pi\_\{j\}\}\\kappa\(u\)^\{2\}\\,du≲wxd\.\\displaystyle\\lesssim w\_\{x\}^\{d\}\.Therefore, applying Lemma[A\.4](https://arxiv.org/html/2610.33403#A1.Thmlem4)to

∑i=1M\{g⁡\(Zi,𝐗i\)−𝔼​g​\(Zi,𝐗i\)\}\\sum\_\{i=1\}^\{M\}\\\{g\(Z\_\{i\},\{\\bf X\}\_\{i\}\)\-\\mathbb\{E\}g\(Z\_\{i\},\{\\bf X\}\_\{i\}\)\\\}with variance scaleσn2≍wxd\\sigma\_\{n\}^\{2\}\\asymp w\_\{x\}^\{d\}and then dividing byM​wxdMw\_\{x\}^\{d\}yields

supz∈ℝ,𝒙∈𝒳\|μ^π​\(z,𝒙,wz,wx\)−μπ​\(z,𝒙,wz,wx\)\|=Op​\(M​wxd​\|log⁡wx\|M​wxd\)=Op​\(\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\|\\hat\{\\mu\}\_\{\\pi\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\mu\_\{\\pi\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\right\|=O\_\{p\}\\\!\\left\(\\frac\{\\sqrt\{Mw\_\{x\}^\{d\}\|\\log w\_\{x\}\|\}\}\{Mw\_\{x\}^\{d\}\}\\right\)=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.Since the number of coordinatesπ\\piis finite, we conclude that

supz∈ℝ,𝒙∈𝒳‖𝝁^​\(z,𝒙,wz,wx\)−𝝁⁡\(z,𝒙,wz,wx\)‖=Op​\(\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\\hat\{\\bm\{\\mu\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\mu\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\right\\\|=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(55\)
We now turn to𝚵^​\(𝒙,wx\)\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)\. Each entry of𝚵^​\(𝒙,wx\)\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)is of the form

1M​wxd​∑i=1Mq⁡\(𝐗i−𝒙wx\)​𝜿​\(𝐗i−𝒙wx\),\\frac\{1\}\{Mw\_\{x\}^\{d\}\}\\sum\_\{i=1\}^\{M\}q\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\)\\bm\{\\kappa\}\\\!\\left\(\\frac\{\{\\bf X\}\_\{i\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\\right\),whereq⁡\(𝐭\)q\(\\mathbf\{t\}\)is a polynomial of total degree at most22\. Thus the relevant one\-dimensional building blocks areum​ϕ​\(u\)u^\{m\}\\phi\(u\)form=0,1,2m=0,1,2\. For each suchmm,

supu∈ℝ\|um​ϕ​\(u\)\|<∞,∫ℝu2​m​ϕ​\(u\)2​𝑑u<∞,\\sup\_\{u\\in\\mathbb\{R\}\}\|u^\{m\}\\phi\(u\)\|<\\infty,\\qquad\\int\_\{\\mathbb\{R\}\}u^\{2m\}\\phi\(u\)^\{2\}\\,du<\\infty,andu↦um​ϕ​\(u\)u\\mapsto u^\{m\}\\phi\(u\)is of bounded variation because its derivative is a polynomial timesϕ⁡\(u\)\\phi\(u\)and hence belongs toL1​\(ℝ\)L^\{1\}\(\\mathbb\{R\}\)\. Therefore the same empirical\-process argument as above yields

sup𝒙∈𝒳‖𝚵^​\(𝒙,wx\)−𝚵⁡\(𝒙,wx\)‖∞=Op​\(\|log⁡wx\|M​wxd\)\.\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)\-\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\right\\\|\_\{\\infty\}=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.\(56\)Since the dimension of𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)is fixed, all matrix norms are equivalent\. In particular, there exists a constantCd<∞C\_\{d\}<\\inftysuch that‖A‖2≤Cd​‖A‖∞\\\|A\\\|\_\{2\}\\leq C\_\{d\}\\\|A\\\|\_\{\\infty\}for every matrixAAof the same dimension\. Hence \([56](https://arxiv.org/html/2610.33403#A2.E56)\) implies

sup𝒙∈𝒳‖𝚵^​\(𝒙,wx\)−𝚵⁡\(𝒙,wx\)‖2=Op​\(\|log⁡wx\|M​wxd\),\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)\-\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\right\\\|\_\{2\}=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\),under the22\-norm\. By Lemma[A\.5](https://arxiv.org/html/2610.33403#A1.Thmlem5), the eigenvalues of𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)are uniformly bounded away from zero and infinity; that is, there exists a constantC\>1C\>1such that

1/C≤λmin​\(𝚵⁡\(𝒙,wx\)\)≤λmax​\(𝚵⁡\(𝒙,wx\)\)≤C1/C\\leq\\lambda\_\{\\min\}\\bigl\(\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\bigr\)\\leq\\lambda\_\{\\max\}\\bigl\(\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\bigr\)\\leq Cfor all𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and all sufficiently smallwxw\_\{x\}\. Hence, by Weyl’s inequality, the eigenvalues of𝚵^​\(𝒙,wx\)\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)also lie in\[1/\(2​C\),2​C\]\[1/\(2C\),2C\]uniformly over𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}with probability approaching11\. In particular,

sup𝒙∈𝒳‖𝚵^​\(𝒙,wx\)−1‖2=Op​\(1\)\.\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\\|\_\{2\}=O\_\{p\}\(1\)\.
By Lemma[A\.5](https://arxiv.org/html/2610.33403#A1.Thmlem5)and Eq\. \([56](https://arxiv.org/html/2610.33403#A2.E56)\), we obtain

sup𝒙∈𝒳‖𝚵^​\(𝒙,wx\)−1−𝚵​\(𝒙,wx\)−1‖2\\displaystyle\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\-\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\right\\\|\_\{2\}=sup𝒙∈𝒳‖𝚵^​\(𝒙,wx\)−1​\(𝚵^​\(𝒙,wx\)−𝚵⁡\(𝒙,wx\)\)​𝚵​\(𝒙,wx\)−1‖2\\displaystyle=\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\Big\(\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)\-\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\Big\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\right\\\|\_\{2\}\(57\)≤sup𝒙∈𝒳‖𝚵^​\(𝒙,wx\)−1‖2​‖𝚵^​\(𝒙,wx\)−𝚵⁡\(𝒙,wx\)‖2​‖𝚵​\(𝒙,wx\)−1‖2\\displaystyle\\leq\\sup\_\{\{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\right\\\|\_\{2\}\\left\\\|\\hat\{\\bm\{\\varXi\}\}\(\{\\bm\{x\}\},w\_\{x\}\)\-\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\right\\\|\_\{2\}\\left\\\|\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)^\{\-1\}\\right\\\|\_\{2\}=Op​\(\|log⁡wx\|M​wxd\)\.\\displaystyle=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.
Substituting \([55](https://arxiv.org/html/2610.33403#A2.E55)\), \([57](https://arxiv.org/html/2610.33403#A2.E57)\), and the above bound into \([54](https://arxiv.org/html/2610.33403#A2.E54)\), we conclude that

supz∈ℝ,𝒙∈𝒳‖𝐃wx​\(𝜷^​\(z,𝒙,wz,wx\)−𝜷⋄​\(z,𝒙,wz,wx\)\)‖=Op​\(\|log⁡wx\|M​wxd\)\.\\sup\_\{z\\in\\mathbb\{R\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\left\\\|\{\\bf D\}\_\{w\_\{x\}\}\\bigl\(\\hat\{\\bm\{\\beta\}\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\diamond\}\(z,\{\\bm\{x\}\},w\_\{z\},w\_\{x\}\)\\bigr\)\\right\\\|=O\_\{p\}\\\!\\left\(\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.This proves the first claim\.

The intercept statement follows immediately by taking the first coordinate of the above vector bound, since the first diagonal entry of𝐃wx\{\\bf D\}\_\{w\_\{x\}\}is11\. ∎

### B\.4Completion of the proof of Theorem[5\.1](https://arxiv.org/html/2610.33403#S5.Thmthm1)

Combining Lemma[B\.1](https://arxiv.org/html/2610.33403#A2.Thmlem1)and Lemma[B\.2](https://arxiv.org/html/2610.33403#A2.Thmlem2)yields

supz∈ℝ,𝒙∈𝒳\|F^​\(z∣𝒙\)−F⁡\(z∣𝒙\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\),\\sup\_\{z\\in\\mathbb\{R\},\\ \{\\bm\{x\}\}\\in\\mathcal\{X\}\}\\bigl\|\\widehat\{F\}\(z\\mid\{\\bm\{x\}\}\)\-F\(z\\mid\{\\bm\{x\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\tfrac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\),which is \([27](https://arxiv.org/html/2610.33403#S5.E27)\)\. ∎

### B\.5Proof of Lemma[A\.5](https://arxiv.org/html/2610.33403#A1.Thmlem5)

###### Proof\.

The argument is analogous to the proof of Lemma 11 in[Fan and Guerre \[2016\]](https://arxiv.org/html/2610.33403#bib.bib5), except that here𝜿⁡\(𝐭\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)is allowed to have unbounded support, as is the case for the Gaussian kernel\.

We first considerΩ⁡\(𝒙,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)\. For the upper bound, note that

‖Ω⁡\(𝒙,wx\)‖≤∫‖𝜾⁡\(𝐭\)​𝜾​\(𝐭\)⊤‖​𝜿​\(𝐭\)​𝑑𝐭\.\\\|\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)\\\|\\leq\\int\\\|\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\\|\\,\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}\.Since𝜾⁡\(𝐭\)​𝜾​\(𝐭\)⊤\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}grows at most quadratically in‖𝐭‖\\\|\\mathbf\{t\}\\\|and𝜿⁡\(𝐭\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)is Gaussian, the integral on the right\-hand side is finite\. Hence the largest eigenvalue ofΩ⁡\(𝒙,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)is uniformly bounded from above over𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and allwx\>0w\_\{x\}\>0\.

For the lower bound, since𝜿⁡\(𝐭\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)is continuous and strictly positive onℝd\\mathbb\{R\}^\{d\},

cκ:=inf𝐭∈B⁡\(0,1\)𝜿⁡\(𝐭\)\>0\.c\_\{\\kappa\}:=\\inf\_\{\\mathbf\{t\}\\in B\(0,1\)\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\>0\.Therefore,

Ω\(𝒙,wx\)⪰cκ∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝕀\{𝒙\+wx𝐭∈𝒳,𝐭∈B\(0,1\)\}d𝐭\.\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)\\succeq c\_\{\\kappa\}\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\},\\ \\mathbf\{t\}\\in B\(0,1\)\\\}\\,d\\mathbf\{t\}\.Setting𝐭=𝜼−𝒙wx\\mathbf\{t\}=\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}, we have

∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝕀\{𝒙\+wx𝐭∈𝒳,𝐭∈B\(0,1\)\}d𝐭\\displaystyle\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\mathbb\{I\}\\\{\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\},\\ \\mathbf\{t\}\\in B\(0,1\)\\\}\\,d\\mathbf\{t\}=∫𝜾⁡\(𝜼−𝒙wx\)​𝜾​\(𝜼−𝒙wx\)⊤​𝕀​\{𝜼∈𝒳,𝜼∈B⁡\(𝒙,wx\)\}​d​𝜼wxd\.\\displaystyle=\\int\\bm\{\\iota\}\(\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\)\\bm\{\\iota\}\(\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\)^\{\\top\}\\mathbb\{I\}\\\{\\bm\{\\eta\}\\in\\mathcal\{X\},\\ \\bm\{\\eta\}\\in B\(\{\\bm\{x\}\},w\_\{x\}\)\\\}\\,\\frac\{d\\bm\{\\eta\}\}\{w\_\{x\}^\{d\}\}\.By the geometric condition in Assumption[1](https://arxiv.org/html/2610.33403#Thmass1), there existsλ1\>0\\lambda\_\{1\}\>0such that, for all𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and all sufficiently smallwx\>0w\_\{x\}\>0, the set\{𝜼∈B\(𝒙,wx\)\}\\\{\\bm\{\\eta\}\\in B\(\{\\bm\{x\}\},w\_\{x\}\)\\\}contains a ballB⁡\(𝒙′,λ1​wx\)B\(\{\\bm\{x\}\}^\{\\prime\},\\lambda\_\{1\}w\_\{x\}\)for some𝒙′∈B⁡\(𝒙,wx\)\{\\bm\{x\}\}^\{\\prime\}\\in B\(\{\\bm\{x\}\},w\_\{x\}\)\. Let𝐲x=𝒙′−𝒙wx\{\\bf y\}\_\{x\}=\\frac\{\{\\bm\{x\}\}^\{\\prime\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\. Then𝐲x∈B⁡\(0,1\)\{\\bf y\}\_\{x\}\\in B\(0,1\)\. Hence,

∫𝜾\(𝜼−𝒙wx\)𝜾\(𝜼−𝒙wx\)⊤𝕀\{𝜼∈B\(𝒙,wx\)∩𝒳\}d​𝜼wxd\\displaystyle\\int\\bm\{\\iota\}\(\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\)\\bm\{\\iota\}\(\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\)^\{\\top\}\\mathbb\{I\}\\\{\\bm\{\\eta\}\\in B\(\{\\bm\{x\}\},w\_\{x\}\)\\cap\\mathcal\{X\}\\\}\\,\\frac\{d\\bm\{\\eta\}\}\{w\_\{x\}^\{d\}\}⪰∫𝜾\(𝜼−𝒙wx\)𝜾\(𝜼−𝒙wx\)⊤𝕀\{𝜼∈B\(𝒙′,λ1wx\)\}d​𝜼wxd\\displaystyle\\succeq\\int\\bm\{\\iota\}\(\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\)\\bm\{\\iota\}\(\\frac\{\\bm\{\\eta\}\-\{\\bm\{x\}\}\}\{w\_\{x\}\}\)^\{\\top\}\\mathbb\{I\}\\\{\\bm\{\\eta\}\\in B\(\{\\bm\{x\}\}^\{\\prime\},\\lambda\_\{1\}w\_\{x\}\)\\\}\\,\\frac\{d\\bm\{\\eta\}\}\{w\_\{x\}^\{d\}\}⪰∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝕀\{𝐭∈B\(𝐲x,λ1\)\}d𝐭\.\\displaystyle\\succeq\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\mathbb\{I\}\\\{\\mathbf\{t\}\\in B\(\{\\bf y\}\_\{x\},\\lambda\_\{1\}\)\\\}\\,d\\mathbf\{t\}\.It follows that the smallest eigenvalue ofΩ⁡\(𝒙,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)is bounded below by

cκinf𝐲∈B⁡\(0,1\)min‖𝝃‖=1𝝃⊤\(∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝕀\{𝐭∈B\(𝐲,λ1\)\}d𝐭\)𝝃\.c\_\{\\kappa\}\\inf\_\{\{\\bf y\}\\in B\(0,1\)\}\\min\_\{\\\|\\bm\{\\xi\}\\\|=1\}\\bm\{\\xi\}^\{\\top\}\\left\(\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\mathbb\{I\}\\\{\\mathbf\{t\}\\in B\(\{\\bf y\},\\lambda\_\{1\}\)\\\}\\,d\\mathbf\{t\}\\right\)\\bm\{\\xi\}\.For each fixed\(𝐲,𝝃\)\(\{\\bf y\},\\bm\{\\xi\}\)with‖𝝃‖=1\\\|\\bm\{\\xi\}\\\|=1,

𝝃⊤\(∫𝜾\(𝐭\)𝜾\(𝐭\)⊤𝕀\{𝐭∈B\(𝐲,λ1\)\}d𝐭\)𝝃=∫B⁡\(𝐲,λ1\)\(𝝃⊤𝜾\(𝐭\)\)2d𝐭\.\\bm\{\\xi\}^\{\\top\}\\left\(\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\mathbb\{I\}\\\{\\mathbf\{t\}\\in B\(\{\\bf y\},\\lambda\_\{1\}\)\\\}\\,d\\mathbf\{t\}\\right\)\\bm\{\\xi\}=\\int\_\{B\(\{\\bf y\},\\lambda\_\{1\}\)\}\(\\bm\{\\xi\}^\{\\top\}\\bm\{\\iota\}\(\\mathbf\{t\}\)\)^\{2\}\\,d\\mathbf\{t\}\.Since𝝃⊤​𝜾​\(𝐭\)\\bm\{\\xi\}^\{\\top\}\\bm\{\\iota\}\(\\mathbf\{t\}\)is a nonzero affine function of𝐭\\mathbf\{t\}, it cannot vanish identically on the open ballB⁡\(𝐲,λ1\)B\(\{\\bf y\},\\lambda\_\{1\}\)\. Therefore the above integral is strictly positive\. Moreover, the mapping

\(𝐲,𝝃\)↦∫B⁡\(𝐲,λ1\)\(𝝃⊤​𝜾​\(𝐭\)\)2​𝑑𝐭\(\{\\bf y\},\\bm\{\\xi\}\)\\mapsto\\int\_\{B\(\{\\bf y\},\\lambda\_\{1\}\)\}\(\\bm\{\\xi\}^\{\\top\}\\bm\{\\iota\}\(\\mathbf\{t\}\)\)^\{2\}\\,d\\mathbf\{t\}is continuous, and the set

\{\(𝐲,𝝃\):𝐲∈B¯\(0,1−λ1\),∥𝝃∥=1\}\\\{\(\{\\bf y\},\\bm\{\\xi\}\):\\ \{\\bf y\}\\in\\bar\{B\}\(0,1\-\\lambda\_\{1\}\),\\ \\\|\\bm\{\\xi\}\\\|=1\\\}is compact\. The reduction fromB⁡\(0,1\)B\(0,1\)toB¯​\(0,1−λ1\)\\bar\{B\}\(0,1\-\\lambda\_\{1\}\)is sufficient for the surrounding argument\. Hence, the infimum of this mapping over the compact set is strictly positive\. This yields a uniform positive lower bound for the eigenvalues ofΩ⁡\(𝒙,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)\.

Next consider𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\. By definition,

𝚵⁡\(𝒙,wx\)=∫𝜾⁡\(𝐭\)​𝜾​\(𝐭\)⊤​𝜿​\(𝐭\)​f​\(𝒙\+wx​𝐭\)​𝑑𝐭\.\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)=\\int\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)f\(\{\\bm\{x\}\}\+w\_\{x\}\\mathbf\{t\}\)\\,d\\mathbf\{t\}\.By Assumption[1](https://arxiv.org/html/2610.33403#Thmass1), there exist constants0<cf≤Cf<∞0<c\_\{f\}\\leq C\_\{f\}<\\inftysuch that

cf≤f⁡\(𝒙\)≤Cf,∀𝒙∈𝒳\.c\_\{f\}\\leq f\(\{\\bm\{x\}\}\)\\leq C\_\{f\},\\qquad\\forall\\,\{\\bm\{x\}\}\\in\\mathcal\{X\}\.Therefore,

cf​Ω​\(𝒙,wx\)⪯𝚵⁡\(𝒙,wx\)⪯Cf​Ω​\(𝒙,wx\),c\_\{f\}\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)\\preceq\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)\\preceq C\_\{f\}\\,\\Omega\(\{\\bm\{x\}\},w\_\{x\}\),which implies that the eigenvalues of𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)are bounded above and below uniformly in𝒙∈𝒳\{\\bm\{x\}\}\\in\\mathcal\{X\}and all sufficiently smallwx\>0w\_\{x\}\>0\.

Combining the above bounds forΩ⁡\(𝒙,wx\)\\Omega\(\{\\bm\{x\}\},w\_\{x\}\)and𝚵⁡\(𝒙,wx\)\\bm\{\\varXi\}\(\{\\bm\{x\}\},w\_\{x\}\)completes the proof\. ∎

## Appendix COther proofs in Section[5](https://arxiv.org/html/2610.33403#S5)

### C\.1Proofs for the lifted joint and conditional laws

This appendix contains the proofs of the three regularity results stated in Section[5\.3](https://arxiv.org/html/2610.33403#S5.SS3): Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)\(blockwise continuity and one\-sided limits of the lifted joint density\), Corollary[5\.2](https://arxiv.org/html/2610.33403#S5.Thmcor2)\(theCrC^\{r\}extension\), and Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3)\(the corresponding piecewise structure of the lifted conditional distribution function\)\. The three results form a logical chain in which each proof relies on the preceding one:C0C^\{0\}\-regularity provides the geometric backbone,CrC^\{r\}\-regularity follows by applying theC0C^\{0\}argument coordinatewise to each partial derivative, and the conditional\-distribution structure is then derived by combiningCrC^\{r\}\-regularity with the probability law identityℙ⁡\(Z≤z∣𝐗\(c\),𝐘∈I⁡\(𝝊\)\)=ℙ⁡\(Z≤z∣𝐗\(c\),𝐗\(d\)=𝝊\)\\mathbb\{P\}\(Z\\leq z\\mid\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\\in I\(\\bm\{\\upsilon\}\)\)=\\mathbb\{P\}\(Z\\leq z\\mid\{\\bf X\}^\{\(c\)\},\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}\)\.

Throughout this appendix we adopt the notation introduced in Sections[4\.1](https://arxiv.org/html/2610.33403#S4.SS1)and[4\.2](https://arxiv.org/html/2610.33403#S4.SS2): the discrete supportΥ=∏j=1d~Υj\\varUpsilon=\\prod\_\{j=1\}^\{\\tilde\{d\}\}\\varUpsilon\_\{j\}with blocks\{I⁡\(𝝊\)\}𝝊∈Υ\\\{I\(\\bm\{\\upsilon\}\)\\\}\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}, the continualized covariate𝐗~=\(𝐗\(c\),𝐘\)\\widetilde\{\\bf X\}=\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)on𝒳~=𝒳c×𝒴\\widetilde\{\\mathcal\{X\}\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}, the interior region𝒳~∘\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}and the covariate jumping set𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}\.

### C\.2Proof of Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)

###### Theorem C\.1\(Precise version of Theorem[5\.2](https://arxiv.org/html/2610.33403#S5.Thmthm2)\)\.

Let𝐱~=\(𝐱\(c\),𝐲\)∈𝒳c×𝒴\\tilde\{\{\\bm\{x\}\}\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}be a generic point in the support of the continualized random vector defined above, where

𝒴=∏j=1d~𝒴j,𝒴j=⋃ℓ=1τjIj,ℓ,Ij,ℓ=\(υj,ℓ−1,υj,ℓ\]\.\\mathcal\{Y\}=\\prod\_\{j=1\}^\{\\tilde\{d\}\}\\mathcal\{Y\}\_\{j\},\\qquad\\mathcal\{Y\}\_\{j\}=\\bigcup\_\{\\ell=1\}^\{\\tau\_\{j\}\}I\_\{j,\\ell\},\\qquad I\_\{j,\\ell\}=\(\\upsilon\_\{j,\\ell\-1\},\\upsilon\_\{j,\\ell\}\]\.For each𝛖=\(υ1,ℓ1,…,υd~,ℓd~\)∈Υ\\bm\{\\upsilon\}=\(\\upsilon\_\{1,\\ell\_\{1\}\},\\dots,\\upsilon\_\{\\tilde\{d\},\\ell\_\{\\tilde\{d\}\}\}\)\\in\\varUpsilon, let

I\(𝝊\)=I1,ℓ1×⋯×Id~,ℓd~⊂𝒴,I\(𝝊\)∘=∏j=1d~\(υj,ℓj−1,υj,ℓj\)\.I\(\\bm\{\\upsilon\}\)=I\_\{1,\\ell\_\{1\}\}\\times\\cdots\\times I\_\{\\tilde\{d\},\\ell\_\{\\tilde\{d\}\}\}\\subset\\mathcal\{Y\},\\qquad I\(\\bm\{\\upsilon\}\)^\{\\circ\}=\\prod\_\{j=1\}^\{\\tilde\{d\}\}\(\\upsilon\_\{j,\\ell\_\{j\}\-1\},\\upsilon\_\{j,\\ell\_\{j\}\}\)\.Assume that for every𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the function

𝒙\(c\)↦f⁡\(𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\(58\)is continuous on𝒳c\\mathcal\{X\}\_\{c\}\(in the subspace topology ofℝd−d~\\mathbb\{R\}^\{d\-\\tilde\{d\}\}\)\.

Throughout, all neighborhoods of points in𝒳~=𝒳c×𝒴\\widetilde\{\\mathcal\{X\}\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}are understood in the subspace topology of𝒳c×𝒴⊂ℝd\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\\subset\\mathbb\{R\}^\{d\}\. Then the lifted joint density

g\(𝒙\(c\),𝐲\)=∑𝝊∈Υf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|𝕀\{𝐲∈I\(𝝊\)\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\\,\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}satisfies the following properties\.

1. 1\.*Interior continuity\.*ggis jointly continuous on 𝒳~∘:=𝒳c×⋃𝝊∈ΥI​\(𝝊\)∘,\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}:=\\mathcal\{X\}\_\{c\}\\times\\bigcup\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}I\(\\bm\{\\upsilon\}\)^\{\\circ\},\(59\)
2. 2\.*Localization of the discontinuity set\.*The discontinuity set ofggon𝒳c×𝒴\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}is contained in 𝒳~∙:=𝒳c×𝒴∙\.\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}:=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}^\{\\bullet\}\.
3. 3\.*One\-sided limits at partition boundary points in𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}\.*Let\(𝒙1\(c\),𝐲1\)∈𝒳c×𝒴\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\in\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}, and let𝝊1∈Υ\\bm\{\\upsilon\}\_\{1\}\\in\\varUpsilondenote the unique block index for which𝐲1∈I⁡\(𝝊1\)\{\\bf y\}\_\{1\}\\in I\(\\bm\{\\upsilon\}\_\{1\}\)\(well\-defined by the left\-open right\-closed convention\)\. 1. \(a\)*Within\-block continuity\.* lim\(𝒙\(c\),𝐲\)→\(𝒙1\(c\),𝐲1\)\(𝒙\(c\),𝐲\)∈𝒳c×I⁡\(𝝊1\)g⁡\(𝒙\(c\),𝐲\)=g⁡\(𝒙1\(c\),𝐲1\)=f⁡\(𝒙1\(c\),𝝊1\)\|I⁡\(𝝊1\)\|\.\\lim\_\{\\begin\{subarray\}\{c\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\\\ \(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\_\{1\}\)\\end\{subarray\}\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=g\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)=\\frac\{f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{1\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{1\}\)\|\}\. 2. \(b\)*Limits from adjacent blocks \(jump structure\)\.*For every block index𝝊′∈Υ\\bm\{\\upsilon\}^\{\\prime\}\\in\\varUpsilonwith𝝊′≠𝝊1\\bm\{\\upsilon\}^\{\\prime\}\\neq\\bm\{\\upsilon\}\_\{1\}and𝐲1∈I⁡\(𝝊′\)¯\{\\bf y\}\_\{1\}\\in\\overline\{I\(\\bm\{\\upsilon\}^\{\\prime\}\)\}\(i\.e\.𝐲1\{\\bf y\}\_\{1\}lies in the closure of an adjacent block\), lim\(𝒙\(c\),𝐲\)→\(𝒙1\(c\),𝐲1\)\(𝒙\(c\),𝐲\)∈𝒳c×I⁡\(𝝊′\)g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙1\(c\),𝝊′\)\|I⁡\(𝝊′\)\|\.\\lim\_\{\\begin\{subarray\}\{c\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\\\ \(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}^\{\\prime\}\)\\end\{subarray\}\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\.In general this limit differs fromg⁡\(𝒙1\(c\),𝐲1\)g\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\), soggexhibits a jump discontinuity across the corresponding partition interface\. The set of such adjacent blocks𝝊′\\bm\{\\upsilon\}^\{\\prime\}is empty if and only if every boundary coordinatey1,j=υj,τjy\_\{1,j\}=\\upsilon\_\{j,\\tau\_\{j\}\}lies on the outermost boundary of𝒴j\\mathcal\{Y\}\_\{j\}on its right side, in which case \(3b\) is vacuous\.
4. 4\.*Piecewise continuity\.*The functionggis continuous on𝒳~∘\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}, and at every\(𝒙1\(c\),𝐲1\)∈𝒳~∙\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}admits the one\-sided continuous extension described in \(3a\)\. Equivalently,ggis piecewise continuous in the sense that, for each𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the restrictiong\|𝒳c×I⁡\(𝝊\)g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}extends continuously to𝒳c×I⁡\(𝝊\)¯\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}\.

###### Proof\.

We first verify that the lifted joint density of𝐗~=\(𝐗\(c\),𝐘\)\\widetilde\{\\bf X\}=\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)on𝒳~=𝒳c×𝒴\\widetilde\{\\mathcal\{X\}\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}equals the displayed formula\. We compute the joint distribution of\(𝐗\(c\),𝐘\)\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)by disintegrating over this partition\. For any measurable rectangleA×B⊂𝒳c×𝒴A\\times B\\subset\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\},

ℙ⁡\(𝐗\(c\)∈A,𝐘∈B\)\\displaystyle\\mathbb\{P\}\\bigl\(\{\\bf X\}^\{\(c\)\}\\in A,\\,\{\\bf Y\}\\in B\\bigr\)=∑𝝊∈Υℙ⁡\(𝐗\(c\)∈A,𝐘∈B,𝐗\(d\)=𝝊\)\\displaystyle=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\mathbb\{P\}\\bigl\(\{\\bf X\}^\{\(c\)\}\\in A,\\,\{\\bf Y\}\\in B,\\,\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}\\bigr\)=∑𝝊∈Υℙ⁡\(𝐗\(c\)∈A,𝐗\(d\)=𝝊,𝐘∈B∩I⁡\(𝝊\)\)\.\\displaystyle=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\mathbb\{P\}\\bigl\(\{\\bf X\}^\{\(c\)\}\\in A,\\,\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\},\\,\{\\bf Y\}\\in B\\cap I\(\\bm\{\\upsilon\}\)\\bigr\)\.Conditional on𝐗\(d\)=𝝊\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}, the continualization takes the formYj=υj,ℓj−UjY\_\{j\}=\\upsilon\_\{j,\\ell\_\{j\}\}\-U\_\{j\}for eachjj, and theUjU\_\{j\}’s are jointly independent of𝐗\(c\)\{\\bf X\}^\{\(c\)\}; the conditional distribution of𝐘\{\\bf Y\}given\(𝐗\(c\),𝐗\(d\)=𝝊\)\(\{\\bf X\}^\{\(c\)\},\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}\)is therefore the uniform distribution onI⁡\(𝝊\)I\(\\bm\{\\upsilon\}\), with constant density1/\|I⁡\(𝝊\)\|1/\|I\(\\bm\{\\upsilon\}\)\|on this block\. Hence

ℙ⁡\(𝐗\(c\)∈A,𝐗\(d\)=𝝊,𝐘∈B∩I⁡\(𝝊\)\)=∫Af⁡\(𝒙\(c\),𝝊\)​d​𝒙\(c\)⋅λ⁡\(B∩I⁡\(𝝊\)\)\|I⁡\(𝝊\)\|,\\mathbb\{P\}\\bigl\(\{\\bf X\}^\{\(c\)\}\\in A,\\,\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\},\\,\{\\bf Y\}\\in B\\cap I\(\\bm\{\\upsilon\}\)\\bigr\)=\\int\_\{A\}f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\,d\{\\bm\{x\}\}^\{\(c\)\}\\cdot\\frac\{\\lambda\\bigl\(B\\cap I\(\\bm\{\\upsilon\}\)\\bigr\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},whereλ⁡\(⋅\)\\lambda\(\\cdot\)denotes Lebesgue measure onℝd~\\mathbb\{R\}^\{\\tilde\{d\}\}\. Writingλ\(B∩I\(𝝊\)\)=∫B𝕀\{𝐲∈I\(𝝊\)\}d𝐲\\lambda\(B\\cap I\(\\bm\{\\upsilon\}\)\)=\\int\_\{B\}\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\\,d\{\\bf y\}, we obtain

ℙ\(𝐗\(c\)∈A,𝐘∈B\)=∫A×B\{∑𝝊∈Υf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|𝕀\{𝐲∈I\(𝝊\)\}\}d𝒙\(c\)d𝐲\.\\mathbb\{P\}\\bigl\(\{\\bf X\}^\{\(c\)\}\\in A,\\,\{\\bf Y\}\\in B\\bigr\)=\\int\_\{A\\times B\}\\biggl\\\{\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\\,\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\\biggr\\\}d\{\\bm\{x\}\}^\{\(c\)\}\\,d\{\\bf y\}\.Since this holds for every measurable rectangle in𝒳c×𝒴\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}, the integrand is a version of the joint density of\(𝐗\(c\),𝐘\)\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)with respect to Lebesgue measure on𝒳c×𝒴\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\. This verifies the displayed formula

g\(𝒙\(c\),𝐲\)=∑𝝊∈Υf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|𝕀\{𝐲∈I\(𝝊\)\}\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\\,\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\.We further verify thatggintegrates to one:

∫𝒳c×𝒴g⁡\(𝒙\(c\),𝐲\)​d​𝒙\(c\)​𝑑𝐲=∑𝝊∈Υ∫𝒳cf⁡\(𝒙\(c\),𝝊\)​d​𝒙\(c\)⋅\|I⁡\(𝝊\)\|\|I⁡\(𝝊\)\|=∑𝝊∈Υℙ⁡\(𝐗\(d\)=𝝊\)=1,\\int\_\{\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\,d\{\\bm\{x\}\}^\{\(c\)\}\\,d\{\\bf y\}=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\int\_\{\\mathcal\{X\}\_\{c\}\}f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\,d\{\\bm\{x\}\}^\{\(c\)\}\\cdot\\frac\{\|I\(\\bm\{\\upsilon\}\)\|\}\{\|I\(\\bm\{\\upsilon\}\)\|\}=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\mathbb\{P\}\(\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}\)=1,where the second equality uses∫I⁡\(𝝊\)𝕀\{𝐲∈I\(𝝊\)\}d𝐲=\|I\(𝝊\)\|\\int\_\{I\(\\bm\{\\upsilon\}\)\}\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}d\{\\bf y\}=\|I\(\\bm\{\\upsilon\}\)\|and the third equality uses the partition property of𝐗\(d\)\{\\bf X\}^\{\(d\)\}\. This confirms thatggis a probability density on𝒳c×𝒴\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\.

We now turn to the regularity properties ofggstated in the theorem\. We writeB¯𝒳c​\(𝒙0\(c\),δ\):=B⁡\(𝒙0\(c\),δ\)∩𝒳c\\overline\{B\}\_\{\\mathcal\{X\}\_\{c\}\}\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\};\\delta\):=B\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\};\\delta\)\\cap\\mathcal\{X\}\_\{c\}for the relative open ball in𝒳c\\mathcal\{X\}\_\{c\}, and similarly for𝒴\\mathcal\{Y\}\.

Property \(1\)\.Fix𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilonand\(𝒙0\(c\),𝐲0\)∈𝒳~∘\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\circ\}\. Since𝐲0∈I​\(𝝊\)∘\{\\bf y\}\_\{0\}\\in I\(\\bm\{\\upsilon\}\)^\{\\circ\}, there existsδ0\>0\\delta\_\{0\}\>0such thatB⁡\(𝐲0,δ0\)⊂I​\(𝝊\)∘B\(\{\\bf y\}\_\{0\};\\delta\_\{0\}\)\\subset I\(\\bm\{\\upsilon\}\)^\{\\circ\}\.

For all0<δ<δ00<\\delta<\\delta\_\{0\}and every\(𝒙\(c\),𝐲\)∈B¯𝒳c​\(𝒙0\(c\),δ\)×B⁡\(𝐲0,δ\)\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\overline\{B\}\_\{\\mathcal\{X\}\_\{c\}\}\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\};\\delta\)\\times B\(\{\\bf y\}\_\{0\};\\delta\), we have𝐲∈I​\(𝝊\)∘\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)^\{\\circ\}, hence

g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,g⁡\(𝒙0\(c\),𝐲0\)=f⁡\(𝒙0\(c\),𝝊\)\|I⁡\(𝝊\)\|\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},\\qquad g\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)=\\frac\{f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\.In particular, on this relative neighborhoodggdoes not depend on𝐲\{\\bf y\}, so the joint continuity at\(𝒙0\(c\),𝐲0\)\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)reduces to the continuity of𝒙\(c\)↦f⁡\(𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)on𝒳c\\mathcal\{X\}\_\{c\}, which is guaranteed by \([58](https://arxiv.org/html/2610.33403#A3.E58)\)\. Thus

\|g⁡\(𝒙\(c\),𝐲\)−g⁡\(𝒙0\(c\),𝐲0\)\|=1\|I⁡\(𝝊\)\|\|f⁡\(𝒙\(c\),𝝊\)−f⁡\(𝒙0\(c\),𝝊\)\|⟶0\\bigl\|g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\-g\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\bigr\|=\\frac\{1\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\\bigl\|f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\-f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\bigr\|\\longrightarrow 0as\(𝒙\(c\),𝐲\)→\(𝒙0\(c\),𝐲0\)\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\.

Property \(2\)\.If\(𝒙0\(c\),𝐲0\)∉𝒳~∙\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\notin\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}, then for everyj∈\[d~\]j\\in\[\\tilde\{d\}\]we havey0,j≠υj,ℓy\_\{0,j\}\\neq\\upsilon\_\{j,\\ell\}for allℓ\\ell, so𝐲0\{\\bf y\}\_\{0\}lies in the interior of some unique blockI⁡\(𝝊0\)I\(\\bm\{\\upsilon\}\_\{0\}\), i\.e\.𝐲0∈I​\(𝝊0\)∘\{\\bf y\}\_\{0\}\\in I\(\\bm\{\\upsilon\}\_\{0\}\)^\{\\circ\}\. By Property \(1\),ggis continuous at\(𝒙0\(c\),𝐲0\)\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\.

Property \(3a\)\.Let\(𝒙1\(c\),𝐲1\)∈𝒳~\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\in\\widetilde\{\\mathcal\{X\}\}and let𝝊1\\bm\{\\upsilon\}\_\{1\}be the unique block index with𝐲1∈I⁡\(𝝊1\)\{\\bf y\}\_\{1\}\\in I\(\\bm\{\\upsilon\}\_\{1\}\)\. Decompose the discrete coordinate index set as

𝒟~1:=\{j∈\[d~\]:y1,j=υj,ℓj\},𝒟~2:=\[d~\]∖𝒟~1,\\tilde\{\\mathcal\{D\}\}\_\{1\}:=\\bigl\\\{j\\in\[\\tilde\{d\}\]:y\_\{1,j\}=\\upsilon\_\{j,\\ell\_\{j\}\}\\bigr\\\},\\qquad\\tilde\{\\mathcal\{D\}\}\_\{2\}:=\[\\tilde\{d\}\]\\setminus\\tilde\{\\mathcal\{D\}\}\_\{1\},whereℓj\\ell\_\{j\}is determined by𝝊1=\(υ1,ℓ1,…,υd~,ℓd~\)\\bm\{\\upsilon\}\_\{1\}=\(\\upsilon\_\{1,\\ell\_\{1\}\},\\dots,\\upsilon\_\{\\tilde\{d\},\\ell\_\{\\tilde\{d\}\}\}\)\. Coordinates in𝒟~1\\tilde\{\\mathcal\{D\}\}\_\{1\}are the boundary coordinates \(right endpoints of their respective intervals\), and coordinates in𝒟~2\\tilde\{\\mathcal\{D\}\}\_\{2\}lie strictly inside\. Write𝐲1=\(𝐲1∙,𝐲1∘\)\{\\bf y\}\_\{1\}=\(\{\\bf y\}\_\{1\}^\{\\bullet\},\{\\bf y\}\_\{1\}^\{\\circ\}\)accordingly\.

Define

δ1:=min⁡\{minj∈𝒟~1⁡\(υj,ℓj−υj,ℓj−1\),minj∈𝒟~2⁡dist⁡\(y1,j,\{υj,ℓj−1,υj,ℓj\}\)\}\>0,\\delta\_\{1\}:=\\min\\biggl\\\{\\min\_\{j\\in\\tilde\{\\mathcal\{D\}\}\_\{1\}\}\\bigl\(\\upsilon\_\{j,\\ell\_\{j\}\}\-\\upsilon\_\{j,\\ell\_\{j\}\-1\}\\bigr\),\\ \\min\_\{j\\in\\tilde\{\\mathcal\{D\}\}\_\{2\}\}\\operatorname\{dist\}\\bigl\(y\_\{1,j\},\\\{\\upsilon\_\{j,\\ell\_\{j\}\-1\},\\upsilon\_\{j,\\ell\_\{j\}\}\\\}\\bigr\)\\biggr\\\}\>0,with the convention that the minimum over an empty set is\+∞\+\\infty\. Construct the one\-sided relative neighborhood

B−\(𝒙1\(c\),𝐲1;δ\):=B¯𝒳c\(𝒙1\(c\);δ\)×B\(𝐲1∘;δ\)×∏j∈𝒟~1\(υj,ℓj−δ,υj,ℓj\],B\_\{\-\}\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\};\\delta\):=\\overline\{B\}\_\{\\mathcal\{X\}\_\{c\}\}\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\};\\delta\)\\times B\(\{\\bf y\}\_\{1\}^\{\\circ\};\\delta\)\\times\\prod\_\{j\\in\\tilde\{\\mathcal\{D\}\}\_\{1\}\}\\bigl\(\\upsilon\_\{j,\\ell\_\{j\}\}\-\\delta,\\upsilon\_\{j,\\ell\_\{j\}\}\\bigr\],which is the trace on𝒳~\\widetilde\{\\mathcal\{X\}\}of the set of points approaching𝐲1\{\\bf y\}\_\{1\}from insideI⁡\(𝝊1\)I\(\\bm\{\\upsilon\}\_\{1\}\)along all boundary coordinates\.

For any0<δ<δ10<\\delta<\\delta\_\{1\}and any\(𝒙\(c\),𝐲\)∈B−​\(𝒙1\(c\),𝐲1,δ\)\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in B\_\{\-\}\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\};\\delta\), both boundary and interior coordinates satisfyyj∈\(υj,ℓj−1,υj,ℓj\]y\_\{j\}\\in\(\\upsilon\_\{j,\\ell\_\{j\}\-1\},\\upsilon\_\{j,\\ell\_\{j\}\}\], so𝐲∈I⁡\(𝝊1\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\_\{1\}\), and

g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊1\)\|I⁡\(𝝊1\)\|\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{1\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{1\}\)\|\}\.Hence the limit along𝒳c×I⁡\(𝝊1\)\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\_\{1\}\)reduces to the limit in𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}alone, and \([58](https://arxiv.org/html/2610.33403#A3.E58)\) yields

lim\(𝒙\(c\),𝐲\)→\(𝒙1\(c\),𝐲1\)\(𝒙\(c\),𝐲\)∈𝒳c×I⁡\(𝝊1\)g⁡\(𝒙\(c\),𝐲\)=g⁡\(𝒙1\(c\),𝐲1\)=f⁡\(𝒙1\(c\),𝝊1\)\|I⁡\(𝝊1\)\|\.\\lim\_\{\\begin\{subarray\}\{c\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\\\ \(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\_\{1\}\)\\end\{subarray\}\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=g\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)=\\frac\{f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{1\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{1\}\)\|\}\.
Property \(3b\)\.Let𝝊′≠𝝊1\\bm\{\\upsilon\}^\{\\prime\}\\neq\\bm\{\\upsilon\}\_\{1\}be such that𝐲1∈I⁡\(𝝊′\)¯\{\\bf y\}\_\{1\}\\in\\overline\{I\(\\bm\{\\upsilon\}^\{\\prime\}\)\}\. Such𝝊′\\bm\{\\upsilon\}^\{\\prime\}exists if and only if some coordinatej∈𝒟~1j\\in\\tilde\{\\mathcal\{D\}\}\_\{1\}satisfiesℓj<τj\\ell\_\{j\}<\\tau\_\{j\}\(an internal partition hyperplane\); if everyj∈𝒟~1j\\in\\tilde\{\\mathcal\{D\}\}\_\{1\}hasℓj=τj\\ell\_\{j\}=\\tau\_\{j\}, the boundary coordinates lie on the outermost right boundary of𝒴\\mathcal\{Y\}and no adjacent block exists, so \(3b\) is vacuous\. Assume hereafter that such𝝊′\\bm\{\\upsilon\}^\{\\prime\}exists\.

Write𝝊′=\(υ1,ℓ1′,…,υd~,ℓd~′\)\\bm\{\\upsilon\}^\{\\prime\}=\(\\upsilon\_\{1,\\ell^\{\\prime\}\_\{1\}\},\\ldots,\\upsilon\_\{\\tilde\{d\},\\ell^\{\\prime\}\_\{\\tilde\{d\}\}\}\)\. Define the one\-sided relative neighborhood approaching𝐲1\{\\bf y\}\_\{1\}from insideI⁡\(𝝊′\)I\(\\bm\{\\upsilon\}^\{\\prime\}\),

B′​\(𝒙1\(c\),𝐲1,δ\):=B¯𝒳c​\(𝒙1\(c\),δ\)×B⁡\(𝐲1∘,δ\)×∏j∈𝒟~1Tj​\(δ\),B^\{\\prime\}\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\};\\delta\):=\\overline\{B\}\_\{\\mathcal\{X\}\_\{c\}\}\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\};\\delta\)\\times B\(\{\\bf y\}\_\{1\}^\{\\circ\};\\delta\)\\times\\prod\_\{j\\in\\tilde\{\\mathcal\{D\}\}\_\{1\}\}T\_\{j\}\(\\delta\),whereTj​\(δ\)⊂\(υj,ℓj′−1,υj,ℓj′\]T\_\{j\}\(\\delta\)\\subset\(\\upsilon\_\{j,\\ell^\{\\prime\}\_\{j\}\-1\},\\upsilon\_\{j,\\ell^\{\\prime\}\_\{j\}\}\]is the appropriate one\-sided sliver insideIj,ℓj′I\_\{j,\\ell^\{\\prime\}\_\{j\}\}withy1,j∈Tj​\(δ\)¯y\_\{1,j\}\\in\\overline\{T\_\{j\}\(\\delta\)\}\(e\.g\.\(υj,ℓj,υj,ℓj\+δ\]\(\\upsilon\_\{j,\\ell\_\{j\}\},\\upsilon\_\{j,\\ell\_\{j\}\}\+\\delta\]ifℓj′=ℓj\+1\\ell^\{\\prime\}\_\{j\}=\\ell\_\{j\}\+1\)\. For all sufficiently smallδ\\delta,B′​\(𝒙1\(c\),𝐲1,δ\)∩\(𝒳c×𝒴\)⊂𝒳c×I⁡\(𝝊′\)B^\{\\prime\}\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\};\\delta\)\\cap\(\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\)\\subset\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}^\{\\prime\}\), and on this set

g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊′\)\|I⁡\(𝝊′\)\|\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\.Then \([58](https://arxiv.org/html/2610.33403#A3.E58)\) gives

lim\(𝒙\(c\),𝐲\)→\(𝒙1\(c\),𝐲1\)\(𝒙\(c\),𝐲\)∈𝒳c×I⁡\(𝝊′\)g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙1\(c\),𝝊′\)\|I⁡\(𝝊′\)\|\.\\lim\_\{\\begin\{subarray\}\{c\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)\\\\ \(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}^\{\\prime\}\)\\end\{subarray\}\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\.This limit equalsg⁡\(𝒙1\(c\),𝐲1\)=f⁡\(𝒙1\(c\),𝝊1\)/\|I⁡\(𝝊1\)\|g\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\{\\bf y\}\_\{1\}\)=f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{1\}\)/\|I\(\\bm\{\\upsilon\}\_\{1\}\)\|if and only if

f⁡\(𝒙1\(c\),𝝊1\)\|I⁡\(𝝊1\)\|=f⁡\(𝒙1\(c\),𝝊′\)\|I⁡\(𝝊′\)\|,\\frac\{f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{1\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{1\}\)\|\}=\\frac\{f\(\{\\bm\{x\}\}\_\{1\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\},which fails generically\. Henceggadmits a jump discontinuity across every partition interface separating two blocks with distinct values off⁡\(⋅,𝝊\)/\|I⁡\(𝝊\)\|f\(\\cdot,\\bm\{\\upsilon\}\)/\|I\(\\bm\{\\upsilon\}\)\|\.

Property \(4\)\.On𝒳~∘\\tilde\{\\mathcal\{X\}\}^\{\\circ\}, every point has the form\(𝒙0\(c\),𝐲0\)\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)with𝐲0∈I​\(𝝊0\)∘\{\\bf y\}\_\{0\}\\in I\(\\bm\{\\upsilon\}\_\{0\}\)^\{\\circ\}for a unique𝝊0\\bm\{\\upsilon\}\_\{0\}; by Property \(1\),ggis continuous there\. Hence the discontinuity set ofggis contained in𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}, proving Property \(2\) and the first claim of Property \(4\)\. For the second claim, Property \(3a\) shows that for each𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the restrictiong\|𝒳c×I⁡\(𝝊\)g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}admits a continuous extension to𝒳c×I⁡\(𝝊\)¯\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}, given on the closure byf⁡\(𝒙\(c\),𝝊\)/\|I⁡\(𝝊\)\|f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)/\|I\(\\bm\{\\upsilon\}\)\|\. This is precisely the piecewise continuity statement\. ∎

### C\.3Proof of Corollary[5\.2](https://arxiv.org/html/2610.33403#S5.Thmcor2)

###### Corollary C\.1\(Precise version of Corollary[5\.2](https://arxiv.org/html/2610.33403#S5.Thmcor2)\)\.

Letr≥0r\\geq 0be an integer\. Throughout,Cr​\(𝒳c\)C^\{r\}\(\\mathcal\{X\}\_\{c\}\)denotes the class of functions whose partial derivatives up to orderrrexist on𝒳c\\mathcal\{X\}\_\{c\}\. Assume that, for every𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon,

𝒙\(c\)↦f⁡\(𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)belongs toCr​\(𝒳c\)C^\{r\}\(\\mathcal\{X\}\_\{c\}\), and that all partial derivatives of order up torrare continuous on𝒳c\\mathcal\{X\}\_\{c\}, uniformly over𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon\.

Letggdenote the lifted density of𝐗~=\(𝐗\(c\),𝐘\)\\widetilde\{\\bf X\}=\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\), namely

g\(𝒙\(c\),𝐲\)=∑𝝊∈Υf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|𝕀\{𝐲∈I\(𝝊\)\}\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\.Then the following hold\.

1. 1\.*InteriorCrC^\{r\}regularity and explicit derivatives\.*For each𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, on𝒳c×I​\(𝝊\)∘\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\}\.Henceg∈Cr​\(𝒳c×I​\(𝝊\)∘\)g\\in C^\{r\}\(\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\)in the relative product sense\. Moreover, for every multi\-indexα=\(αc,αy\)\\alpha=\(\\alpha\_\{c\},\\alpha\_\{y\}\)with\|α\|≤r\|\\alpha\|\\leq r, ∂αg⁡\(𝒙\(c\),𝐲\)=\{∂𝒙\(c\)αcf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,αy=0,0,αy≠0,\(𝒙\(c\),𝐲\)∈𝒳c×I​\(𝝊\)∘\.\\partial^\{\\alpha\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\begin\{cases\}\\dfrac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},&\\alpha\_\{y\}=0,\\\\\[6\.0pt\] 0,&\\alpha\_\{y\}\\neq 0,\\end\{cases\}\\qquad\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\.
2. 2\.*BlockwiseCrC^\{r\}extension\.*For each𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the restrictiong\|𝒳c×I⁡\(𝝊\)g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}admits aCrC^\{r\}extension to𝒳c×I⁡\(𝝊\)¯\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}, given by g𝝊ext​\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,\(𝒙\(c\),𝐲\)∈𝒳c×I⁡\(𝝊\)¯\.g\_\{\\bm\{\\upsilon\}\}^\{\\rm ext\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},\\qquad\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}\.Its derivatives satisfy ∂αg𝝊ext​\(𝒙\(c\),𝐲\)=\{∂𝒙\(c\)αcf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,αy=0,0,αy≠0,\\partial^\{\\alpha\}g\_\{\\bm\{\\upsilon\}\}^\{\\rm ext\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\begin\{cases\}\\dfrac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},&\\alpha\_\{y\}=0,\\\\\[6\.0pt\] 0,&\\alpha\_\{y\}\\neq 0,\\end\{cases\}for every\|α\|≤r\|\\alpha\|\\leq r\.
3. 3\.*One\-sided derivative limits across block boundaries\.*Let𝐲0∈𝒴∙\{\\bf y\}\_\{0\}\\in\\mathcal\{Y\}^\{\\bullet\}, and let𝝊0∈Υ\\bm\{\\upsilon\}\_\{0\}\\in\\varUpsilonbe the unique block index such that𝐲0∈I⁡\(𝝊0\)\{\\bf y\}\_\{0\}\\in I\(\\bm\{\\upsilon\}\_\{0\}\)\. If𝝊′≠𝝊0\\bm\{\\upsilon\}^\{\\prime\}\\neq\\bm\{\\upsilon\}\_\{0\}satisfies𝐲0∈I⁡\(𝝊′\)¯\{\\bf y\}\_\{0\}\\in\\overline\{I\(\\bm\{\\upsilon\}^\{\\prime\}\)\}, then for every𝒙0\(c\)∈𝒳c\{\\bm\{x\}\}\_\{0\}^\{\(c\)\}\\in\\mathcal\{X\}\_\{c\}and every multi\-indexα=\(αc,0\)\\alpha=\(\\alpha\_\{c\},0\)with\|α\|≤r\|\\alpha\|\\leq r, lim\(𝒙\(c\),𝐲\)→\(𝒙0\(c\),𝐲0\)𝐲∈I⁡\(𝝊0\)∂αg⁡\(𝒙\(c\),𝐲\)=∂𝒙\(c\)αcf⁡\(𝒙0\(c\),𝝊0\)\|I⁡\(𝝊0\)\|,\\lim\_\{\\begin\{subarray\}\{c\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\\\ \{\\bf y\}\\in I\(\\bm\{\\upsilon\}\_\{0\}\)\\end\{subarray\}\}\\partial^\{\\alpha\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{0\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{0\}\)\|\},whereas lim\(𝒙\(c\),𝐲\)→\(𝒙0\(c\),𝐲0\)𝐲∈I⁡\(𝝊′\)∂αg⁡\(𝒙\(c\),𝐲\)=∂𝒙\(c\)αcf⁡\(𝒙0\(c\),𝝊′\)\|I⁡\(𝝊′\)\|\.\\lim\_\{\\begin\{subarray\}\{c\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\to\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\\\ \{\\bf y\}\\in I\(\\bm\{\\upsilon\}^\{\\prime\}\)\\end\{subarray\}\}\\partial^\{\\alpha\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\.Therefore the one\-sided limits of the𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivative∂αg\\partial^\{\\alpha\}gcoincide if and only if ∂𝒙\(c\)αcf⁡\(𝒙0\(c\),𝝊0\)\|I⁡\(𝝊0\)\|=∂𝒙\(c\)αcf⁡\(𝒙0\(c\),𝝊′\)\|I⁡\(𝝊′\)\|\.\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{0\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{0\}\)\|\}=\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\.If no such adjacent block index𝝊′\\bm\{\\upsilon\}^\{\\prime\}exists, the corresponding cross\-block statement is vacuous\.
4. 4\.*PiecewiseCrC^\{r\}structure\.*The collection \{g\|𝒳c×I⁡\(𝝊\):𝝊∈Υ\}\\left\\\{g\|\_\{\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\}:\\bm\{\\upsilon\}\\in\\varUpsilon\\right\\\}constitutes a piecewiseCrC^\{r\}structure on𝒳~=𝒳c×𝒴\\widetilde\{\\mathcal\{X\}\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\. The blockwiseCrC^\{r\}extensions need not agree on shared block boundaries\.

###### Proof\.

Fix𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon\. On the open block𝒳c×I​\(𝝊\)∘\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, the block indicators satisfy

𝕀\{𝐲∈I\(𝝊\)\}=𝕀\{cat\(𝐲\)=𝝊\}\.\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}=\\mathbb\{I\}\\\{cat\(\{\\bf y\}\)=\\bm\{\\upsilon\}\\\}\.Therefore

g⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,\(𝒙\(c\),𝐲\)∈𝒳c×I​\(𝝊\)∘\.g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},\\qquad\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\.The right\-hand side isCrC^\{r\}in𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}by assumption and is constant in𝐲\{\\bf y\}\. Henceg∈Crg\\in C^\{r\}on the open block, and for every multi\-indexα=\(αc,αy\)\\alpha=\(\\alpha\_\{c\},\\alpha\_\{y\}\)with\|α\|≤r\|\\alpha\|\\leq r,

∂αg⁡\(𝒙\(c\),𝐲\)=\{∂𝒙\(c\)αcf⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,αy=0,0,αy≠0\.\\partial^\{\\alpha\}g\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\begin\{cases\}\\dfrac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},&\\alpha\_\{y\}=0,\\\\\[6\.0pt\] 0,&\\alpha\_\{y\}\\neq 0\.\\end\{cases\}This proves the interior derivative formula\.

Next define, for the same block,

g𝝊ext​\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,\(𝒙\(c\),𝐲\)∈𝒳c×I⁡\(𝝊\)¯\.g\_\{\\bm\{\\upsilon\}\}^\{\\rm ext\}\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},\\qquad\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}\.Sincef⁡\(⋅,𝝊\)∈Cr​\(𝒳c\)f\(\\cdot,\\bm\{\\upsilon\}\)\\in C^\{r\}\(\\mathcal\{X\}\_\{c\}\)and the right\-hand side is independent of𝐲\{\\bf y\},g𝝊ext∈Cr​\(𝒳c×I⁡\(𝝊\)¯\)g\_\{\\bm\{\\upsilon\}\}^\{\\rm ext\}\\in C^\{r\}\(\\mathcal\{X\}\_\{c\}\\times\\overline\{I\(\\bm\{\\upsilon\}\)\}\)\. It agrees withggon𝒳c×I⁡\(𝝊\)\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)\. This proves the blockwiseCrC^\{r\}extension statement\.

We now verify the one\-sided derivative limits\. Let𝐲0∈𝒴∙\{\\bf y\}\_\{0\}\\in\\mathcal\{Y\}^\{\\bullet\}, and let𝝊0\\bm\{\\upsilon\}\_\{0\}be the unique block index satisfying𝐲0∈I⁡\(𝝊0\)\{\\bf y\}\_\{0\}\\in I\(\\bm\{\\upsilon\}\_\{0\}\)\. Consider any sequence

\(𝒙n\(c\),𝐲n\)→\(𝒙0\(c\),𝐲0\),𝐲n∈I⁡\(𝝊0\)\.\(\{\\bm\{x\}\}\_\{n\}^\{\(c\)\},\{\\bf y\}\_\{n\}\)\\to\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\),\\qquad\{\\bf y\}\_\{n\}\\in I\(\\bm\{\\upsilon\}\_\{0\}\)\.For any multi\-indexα=\(αc,0\)\\alpha=\(\\alpha\_\{c\},0\)with\|α\|≤r\|\\alpha\|\\leq r, the interior formula gives

∂αg⁡\(𝒙n\(c\),𝐲n\)=∂𝒙\(c\)αcf⁡\(𝒙n\(c\),𝝊0\)\|I⁡\(𝝊0\)\|\.\\partial^\{\\alpha\}g\(\{\\bm\{x\}\}\_\{n\}^\{\(c\)\},\{\\bf y\}\_\{n\}\)=\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{n\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{0\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{0\}\)\|\}\.By continuity of∂𝒙\(c\)αcf⁡\(⋅,𝝊0\)\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\\cdot,\\bm\{\\upsilon\}\_\{0\}\)on𝒳c\\mathcal\{X\}\_\{c\}, this converges to

∂𝒙\(c\)αcf⁡\(𝒙0\(c\),𝝊0\)\|I⁡\(𝝊0\)\|\.\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}\_\{0\}\)\}\{\|I\(\\bm\{\\upsilon\}\_\{0\}\)\|\}\.
Similarly, if𝝊′≠𝝊0\\bm\{\\upsilon\}^\{\\prime\}\\neq\\bm\{\\upsilon\}\_\{0\}satisfies𝐲0∈I⁡\(𝝊′\)¯\{\\bf y\}\_\{0\}\\in\\overline\{I\(\\bm\{\\upsilon\}^\{\\prime\}\)\}, then along any sequence

\(𝒙n\(c\),𝐲n\)→\(𝒙0\(c\),𝐲0\),𝐲n∈I⁡\(𝝊′\),\(\{\\bm\{x\}\}\_\{n\}^\{\(c\)\},\{\\bf y\}\_\{n\}\)\\to\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\),\\qquad\{\\bf y\}\_\{n\}\\in I\(\\bm\{\\upsilon\}^\{\\prime\}\),we have

∂αg⁡\(𝒙n\(c\),𝐲n\)=∂𝒙\(c\)αcf⁡\(𝒙n\(c\),𝝊′\)\|I⁡\(𝝊′\)\|→∂𝒙\(c\)αcf⁡\(𝒙0\(c\),𝝊′\)\|I⁡\(𝝊′\)\|\.\\partial^\{\\alpha\}g\(\{\\bm\{x\}\}\_\{n\}^\{\(c\)\},\{\\bf y\}\_\{n\}\)=\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{n\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\\to\\frac\{\\partial^\{\\alpha\_\{c\}\}\_\{\{\\bm\{x\}\}^\{\(c\)\}\}f\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\\bm\{\\upsilon\}^\{\\prime\}\)\}\{\|I\(\\bm\{\\upsilon\}^\{\\prime\}\)\|\}\.The two one\-sided limits coincide exactly when the two displayed blockwise derivative values are equal\. This proves the jump statement for𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivatives\.

Finally, because each block restriction admits aCrC^\{r\}extension to its closed block, the family of block restrictions defines a piecewiseCrC^\{r\}structure on𝒳~=𝒳c×𝒴\\widetilde\{\\mathcal\{X\}\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\. The preceding one\-sided limit calculation shows that these extensions need not match on shared block boundaries\. ∎

### C\.4Proof of Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3)

###### Theorem C\.2\(Precise version of Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3)\)\.

Let𝐗~=\(𝐗\(c\),𝐘\)\\widetilde\{\\bf X\}=\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)be the continualized covariate vector, and write a generic point as

𝒙~=\(𝒙\(c\),𝐲\)∈𝒳c×𝒴,𝒴=∏j=1d~𝒴j\.\\tilde\{\\bm\{x\}\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\},\\qquad\\mathcal\{Y\}=\\prod\_\{j=1\}^\{\\tilde\{d\}\}\\mathcal\{Y\}\_\{j\}\.For each𝛖∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, letI⁡\(𝛖\)I\(\\bm\{\\upsilon\}\)be the associated covariate block\. Under the left\-open right\-closed convention, there is a unique block index

cat⁡\(𝐲\)=𝝊∈Υ\\mathrm\{cat\}\(\{\\bf y\}\)=\\bm\{\\upsilon\}\\in\\varUpsilonsuch that𝐲∈I⁡\(cat⁡\(𝐲\)\)\{\\bf y\}\\in I\(\\mathrm\{cat\}\(\{\\bf y\}\)\)\. Then the following statements hold\.

1. 1\.Continuous response variable\.SupposeZZis continuous on𝒵⊂ℝ\\mathcal\{Z\}\\subset\\mathbb\{R\}\. Assume that for every𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the original conditional distribution function \(z,𝒙\(c\)\)↦F⁡\(z∣𝒙\(c\),𝝊\)\(z,\{\\bm\{x\}\}^\{\(c\)\}\)\\mapsto F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)belongs toC2​\(𝒵×𝒳c\)C^\{2\}\(\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}\), with second\-order partial derivatives uniformly continuous and bounded on𝒵×𝒳c\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}, uniformly in𝝊\\bm\{\\upsilon\}\. Then the lifted conditional distribution F⁡\(z∣𝒙\(c\),𝐲\):=ℙ⁡\(Z≤z∣𝐗\(c\)=𝒙\(c\),𝐘=𝐲\)F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\):=\\mathbb\{P\}\(Z\\leq z\\mid\{\\bf X\}^\{\(c\)\}=\{\\bm\{x\}\}^\{\(c\)\},\{\\bf Y\}=\{\\bf y\}\)satisfies F\(z∣𝒙\(c\),𝐲\)=∑𝝊∈ΥF\(z∣𝒙\(c\),𝝊\)𝕀\{𝐲∈I\(𝝊\)\}\.F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\.Consequently, for each𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, \(z,𝒙\(c\),𝐲\)↦F⁡\(z∣𝒙\(c\),𝐲\)∈C2​\(𝒵×𝒳c×I​\(𝝊\)∘\)\(z,\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\mapsto F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in C^\{2\}\\bigl\(\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\\bigr\)in the relative product sense\. On this block,FFis independent of𝐲\{\\bf y\}, and hence all derivatives involving at least one derivative with respect to𝐲\{\\bf y\}vanish identically\.
2. 2\.Discrete response variable after continualization\.SupposeZZis ordered discrete on 𝒵=\{z1,…,zL\},z1<⋯<zL\.\\mathcal\{Z\}=\\\{z\_\{1\},\\ldots,z\_\{L\}\\\},\\qquad z\_\{1\}<\\cdots<z\_\{L\}\.Let z0:=z1−1,Ii=\(zi−1,zi\],𝒵~=⋃i=1LIi,z\_\{0\}:=z\_\{1\}\-1,\\qquad I\_\{i\}=\(z\_\{i\-1\},z\_\{i\}\],\\qquad\\widetilde\{\\mathcal\{Z\}\}=\\bigcup\_\{i=1\}^\{L\}I\_\{i\},and letZ~=Z−V\\tilde\{Z\}=Z\-V, where, conditional onZ=ziZ=z\_\{i\},V∼Unif⁡\(0,\|Ii\|\)V\\sim\\mathrm\{Unif\}\(0,\|I\_\{i\}\|\)independently of𝐗~\\widetilde\{\\bf X\}\. Assume that, for everyi∈\[L\]i\\in\[L\]and𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, the conditional mass 𝒙\(c\)↦P⁡\(zi∣𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)belongs toC2​\(𝒳c\)C^\{2\}\(\\mathcal\{X\}\_\{c\}\), with second\-order partial derivatives uniformly continuous and bounded on𝒳c\\mathcal\{X\}\_\{c\}, uniformly iniiand𝝊\\bm\{\\upsilon\}\. Then the lifted conditional density ofZ~\\tilde\{Z\}given\(𝒙\(c\),𝐲\)\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)is f\(z~∣𝒙\(c\),𝐲\)=∑𝝊∈Υ∑i=1LP⁡\(zi∣𝒙\(c\),𝝊\)\|Ii\|𝕀\{z~∈Ii\}𝕀\{𝐲∈I\(𝝊\)\}\.f\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}\\sum\_\{i=1\}^\{L\}\\frac\{P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\}\|\}\\mathbb\{I\}\\\{\\tilde\{z\}\\in I\_\{i\}\\\}\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\.This is a valid conditional density on𝒵~\\widetilde\{\\mathcal\{Z\}\}\. Moreover, forz~∈Ii\\tilde\{z\}\\in I\_\{i\}and𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\), F⁡\(z~∣𝒙\(c\),𝐲\)=∑k=1i−1P⁡\(zk∣𝒙\(c\),𝝊\)\+z~−zi−1\|Ii\|​P​\(zi∣𝒙\(c\),𝝊\)\.F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{k=1\}^\{i\-1\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\+\\frac\{\\tilde\{z\}\-z\_\{i\-1\}\}\{\|I\_\{i\}\|\}P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\.Consequently, for everyi∈\[L\]i\\in\[L\]and𝝊∈Υ\\bm\{\\upsilon\}\\in\\varUpsilon, \(z~,𝒙\(c\),𝐲\)↦F⁡\(z~∣𝒙\(c\),𝐲\)∈C2​\(Ii∘×𝒳c×I​\(𝝊\)∘\)\.\(\\tilde\{z\},\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\mapsto F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in C^\{2\}\\bigl\(I\_\{i\}^\{\\circ\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\\bigr\)\.On each such block,FFis independent of𝐲\{\\bf y\}, and is exactly affine inz~\\tilde\{z\}\. In particular, ∂2F∂z~2≡0,∂2F∂z~​∂xa\(c\)=1\|Ii\|​∂∂xa\(c\)​P​\(zi∣𝒙\(c\),𝝊\)\\frac\{\\partial^\{2\}F\}\{\\partial\\tilde\{z\}^\{2\}\}\\equiv 0,\\qquad\\frac\{\\partial^\{2\}F\}\{\\partial\\tilde\{z\}\\,\\partial x\_\{a\}^\{\(c\)\}\}=\\frac\{1\}\{\|I\_\{i\}\|\}\\frac\{\\partial\}\{\\partial x\_\{a\}^\{\(c\)\}\}P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)onIi∘×𝒳c×I​\(𝝊\)∘I\_\{i\}^\{\\circ\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\. Across a response interfacez~=zi\\tilde\{z\}=z\_\{i\},1≤i<L1\\leq i<L, the lifted CDF is continuous inz~\\tilde\{z\}, while itsz~\\tilde\{z\}\-derivative may jump from P⁡\(zi∣𝒙\(c\),𝝊\)\|Ii\|toP⁡\(zi\+1∣𝒙\(c\),𝝊\)\|Ii\+1\|\.\\frac\{P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\}\|\}\\quad\\text\{to\}\\quad\\frac\{P\(z\_\{i\+1\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\+1\}\|\}\.
3. 3\.Covariate jumping set\.In both cases, at every covariate boundary point \(𝒙0\(c\),𝐲0\)∈𝒳~∙=𝒳c×𝒴∙,\(\{\\bm\{x\}\}\_\{0\}^\{\(c\)\},\{\\bf y\}\_\{0\}\)\\in\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}^\{\\bullet\},the lifted conditional CDFFFadmits blockwise one\-sidedC2C^\{2\}limits from every covariate blockI⁡\(𝝊\)I\(\\bm\{\\upsilon\}\)whose closure contains𝐲0\{\\bf y\}\_\{0\}\. The values ofFFand its blockwise𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivatives on adjacent covariate blocks need not agree\.

###### Proof\.

We first record the basic conditioning identity induced by the continualization\. Since𝐗\(d\)=cat⁡\(𝐘\)\{\\bf X\}^\{\(d\)\}=\\mathrm\{cat\}\(\{\\bf Y\}\)is a measurable function of𝐘\{\\bf Y\}, and since the uniform variables used to generate𝐘\{\\bf Y\}are conditionally independent ofZZgiven\(𝐗\(c\),𝐗\(d\)\)\(\{\\bf X\}^\{\(c\)\},\{\\bf X\}^\{\(d\)\}\), the conditional law ofZZgiven\(𝐗\(c\),𝐘\)\(\{\\bf X\}^\{\(c\)\},\{\\bf Y\}\)depends on𝐘\{\\bf Y\}only through its block label\. Thus, for𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\),

ℒ⁡\(Z∣𝐗\(c\)=𝒙\(c\),𝐘=𝐲\)=ℒ⁡\(Z∣𝐗\(c\)=𝒙\(c\),𝐗\(d\)=𝝊\)\.\\mathcal\{L\}\(Z\\mid\{\\bf X\}^\{\(c\)\}=\{\\bm\{x\}\}^\{\(c\)\},\{\\bf Y\}=\{\\bf y\}\)=\\mathcal\{L\}\(Z\\mid\{\\bf X\}^\{\(c\)\}=\{\\bm\{x\}\}^\{\(c\)\},\{\\bf X\}^\{\(d\)\}=\\bm\{\\upsilon\}\)\.Equivalently,

ℙ⁡\(Z≤z∣𝐗\(c\)=𝒙\(c\),𝐘=𝐲\)=F⁡\(z∣𝒙\(c\),𝝊\),𝐲∈I⁡\(𝝊\)\.\\mathbb\{P\}\(Z\\leq z\\mid\{\\bf X\}^\{\(c\)\}=\{\\bm\{x\}\}^\{\(c\)\},\{\\bf Y\}=\{\\bf y\}\)=F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\),\\qquad\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\.
\(i\) Continuous response\.The preceding identity immediately gives

F\(z∣𝒙\(c\),𝐲\)=∑𝝊∈ΥF\(z∣𝒙\(c\),𝝊\)𝕀\{𝐲∈I\(𝝊\)\}\.F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\sum\_\{\\bm\{\\upsilon\}\\in\\varUpsilon\}F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\\mathbb\{I\}\\\{\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\\\}\.Fix𝝊\\bm\{\\upsilon\}\. On𝒵×𝒳c×I​\(𝝊\)∘\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, this function equalsF⁡\(z∣𝒙\(c\),𝝊\)F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\. By assumption this isC2C^\{2\}in\(z,𝒙\(c\)\)\(z,\{\\bm\{x\}\}^\{\(c\)\}\), with uniformly continuous and bounded second derivatives\. Since it is independent of𝐲\{\\bf y\}on this block, viewed as a function of\(z,𝒙\(c\),𝐲\)\(z,\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\), it belongs to

C2​\(𝒵×𝒳c×I​\(𝝊\)∘\),C^\{2\}\\bigl\(\\mathcal\{Z\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\\bigr\),with all𝐲\{\\bf y\}\-derivatives equal to zero\.

\(ii\) Discrete response\.For𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\), the same conditioning identity gives

ℙ⁡\(Z=zi∣𝐗\(c\)=𝒙\(c\),𝐘=𝐲\)=P⁡\(zi∣𝒙\(c\),𝝊\)\.\\mathbb\{P\}\(Z=z\_\{i\}\\mid\{\\bf X\}^\{\(c\)\}=\{\\bm\{x\}\}^\{\(c\)\},\{\\bf Y\}=\{\\bf y\}\)=P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\.GivenZ=ziZ=z\_\{i\}, the response continualization placesZ~\\tilde\{Z\}uniformly onIiI\_\{i\}\. Therefore, forz~∈Ii\\tilde\{z\}\\in I\_\{i\}and𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\),

f⁡\(z~∣𝒙\(c\),𝐲\)=P⁡\(zi∣𝒙\(c\),𝝊\)\|Ii\|\.f\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\}\|\}\.Summing overiiand𝝊\\bm\{\\upsilon\}yields the displayed density\. It integrates to one because

∫𝒵~f⁡\(z~∣𝒙\(c\),𝐲\)​𝑑z~=∑i=1L∫IiP⁡\(zi∣𝒙\(c\),𝝊\)\|Ii\|​𝑑z~=∑i=1LP⁡\(zi∣𝒙\(c\),𝝊\)=1\\int\_\{\\widetilde\{\\mathcal\{Z\}\}\}f\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\,d\\tilde\{z\}=\\sum\_\{i=1\}^\{L\}\\int\_\{I\_\{i\}\}\\frac\{P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\}\|\}\\,d\\tilde\{z\}=\\sum\_\{i=1\}^\{L\}P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)=1whenever𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\)\.

Forz~∈Ii\\tilde\{z\}\\in I\_\{i\}and𝐲∈I⁡\(𝝊\)\{\\bf y\}\\in I\(\\bm\{\\upsilon\}\),

F⁡\(z~∣𝒙\(c\),𝐲\)\\displaystyle F\(\\tilde\{z\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=∫z0z~f⁡\(s∣𝒙\(c\),𝐲\)​𝑑s\\displaystyle=\\int\_\{z\_\{0\}\}^\{\\tilde\{z\}\}f\(s\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\,ds=∑k=1i−1∫IkP⁡\(zk∣𝒙\(c\),𝝊\)\|Ik\|​𝑑s\+∫zi−1z~P⁡\(zi∣𝒙\(c\),𝝊\)\|Ii\|​𝑑s\\displaystyle=\\sum\_\{k=1\}^\{i\-1\}\\int\_\{I\_\{k\}\}\\frac\{P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{k\}\|\}\\,ds\+\\int\_\{z\_\{i\-1\}\}^\{\\tilde\{z\}\}\\frac\{P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\_\{i\}\|\}\\,ds=∑k=1i−1P⁡\(zk∣𝒙\(c\),𝝊\)\+z~−zi−1\|Ii\|​P​\(zi∣𝒙\(c\),𝝊\)\.\\displaystyle=\\sum\_\{k=1\}^\{i\-1\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\+\\frac\{\\tilde\{z\}\-z\_\{i\-1\}\}\{\|I\_\{i\}\|\}P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\.This proves the blockwise CDF formula\.

OnIi∘×𝒳c×I​\(𝝊\)∘I\_\{i\}^\{\\circ\}\\times\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}, the right\-hand side is affine inz~\\tilde\{z\},C2C^\{2\}in𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}, and independent of𝐲\{\\bf y\}\. Hence it isC2C^\{2\}on the block, with all𝐲\{\\bf y\}\-derivatives equal to zero and with

∂2F∂z~2=0,∂2F∂z~​∂xa\(c\)=1\|Ii\|​∂∂xa\(c\)​P​\(zi∣𝒙\(c\),𝝊\)\.\\frac\{\\partial^\{2\}F\}\{\\partial\\tilde\{z\}^\{2\}\}=0,\\qquad\\frac\{\\partial^\{2\}F\}\{\\partial\\tilde\{z\}\\,\\partial x\_\{a\}^\{\(c\)\}\}=\\frac\{1\}\{\|I\_\{i\}\|\}\\frac\{\\partial\}\{\\partial x\_\{a\}^\{\(c\)\}\}P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\.
At a response interfacez~=zi\\tilde\{z\}=z\_\{i\}, with1≤i<L1\\leq i<L, the left limit fromIiI\_\{i\}is

∑k=1i−1P⁡\(zk∣𝒙\(c\),𝝊\)\+P⁡\(zi∣𝒙\(c\),𝝊\)=∑k=1iP⁡\(zk∣𝒙\(c\),𝝊\),\\sum\_\{k=1\}^\{i\-1\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\+P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)=\\sum\_\{k=1\}^\{i\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\),while the right limit fromIi\+1I\_\{i\+1\}is

∑k=1iP⁡\(zk∣𝒙\(c\),𝝊\)\+0=∑k=1iP⁡\(zk∣𝒙\(c\),𝝊\)\.\\sum\_\{k=1\}^\{i\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\+0=\\sum\_\{k=1\}^\{i\}P\(z\_\{k\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\.ThusFFis continuous inz~\\tilde\{z\}across the response interface\. Itsz~\\tilde\{z\}\-derivative equalsP⁡\(zi∣𝒙\(c\),𝝊\)/\|Ii\|P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)/\|I\_\{i\}\|onIi∘I\_\{i\}^\{\\circ\}andP⁡\(zi\+1∣𝒙\(c\),𝝊\)/\|Ii\+1\|P\(z\_\{i\+1\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)/\|I\_\{i\+1\}\|onIi\+1∘I\_\{i\+1\}^\{\\circ\}, so this derivative may jump atziz\_\{i\}\.

\(iii\) Covariate jumping set\.In case \(i\), the blockwise representation shows that, on each covariate blockI​\(𝝊\)∘I\(\\bm\{\\upsilon\}\)^\{\\circ\},FFis theC2C^\{2\}functionF⁡\(z∣𝒙\(c\),𝝊\)F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\), independent of𝐲\{\\bf y\}\. In case \(ii\), for each fixed response intervalIi∘I\_\{i\}^\{\\circ\}, the displayed formula expressesFFas aC2C^\{2\}function of𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}, independent of𝐲\{\\bf y\}, on every covariate block\.

Therefore the blockwise one\-sided limits at points of𝒳~∙\\widetilde\{\\mathcal\{X\}\}^\{\\bullet\}exist by the same argument as in Corollary[5\.2](https://arxiv.org/html/2610.33403#S5.Thmcor2), applied respectively to the maps

𝒙\(c\)↦F⁡\(z∣𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)in the continuous\-response case and to the maps

𝒙\(c\)↦P⁡\(zi∣𝒙\(c\),𝝊\)\{\\bm\{x\}\}^\{\(c\)\}\\mapsto P\(z\_\{i\}\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)in the discrete\-response case\. Since these blockwise functions need not agree for adjacent𝝊\\bm\{\\upsilon\}’s, the values ofFFand its blockwise𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\-derivatives may jump across the covariate jumping set\. ∎

### C\.5Proof of Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4)

###### Proof\.

\(i\) Continuous response\.For the mixed covariate case, the only additional issue is that the Gaussian kernel has unbounded support and may assign positive weight to observations lying in blocks adjacent to the block containing the query point\. We therefore require only the continualized coordinates to stay away from the internal partition interfaces\. Denote

𝒴ϵ∘:=\{𝐲∈𝒴∘:minj∈\[d~\]1≤ℓ≤τj−1⁡\|yj−υj,ℓ\|≥ϵ\},𝒳~ϵ∘:=𝒳c×𝒴ϵ∘,\\mathcal\{Y\}\_\{\\epsilon\}^\{\\circ\}:=\\left\\\{\{\\bf y\}\\in\\mathcal\{Y\}^\{\\circ\}:\\min\_\{\\begin\{subarray\}\{c\}j\\in\[\\tilde\{d\}\]\\\\ 1\\leq\\ell\\leq\\tau\_\{j\}\-1\\end\{subarray\}\}\|y\_\{j\}\-\\upsilon\_\{j,\\ell\}\|\\geq\\epsilon\\right\\\},\\qquad\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}:=\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\_\{\\epsilon\}^\{\\circ\},whereϵ≥Cϵ​wx​\|log⁡wx\|\\epsilon\\geq C\_\{\\epsilon\}w\_\{x\}\\sqrt\{\|\\log w\_\{x\}\|\}, for some constantCϵ\>2C\_\{\\epsilon\}\>2\. Notice that no additional distance restriction is imposed on the original continuous coordinates𝒙\(c\)\{\\bm\{x\}\}^\{\(c\)\}\.

Fix𝝊=\(υ1,ℓ1,…,υd~,ℓd~\)∈Υ\\bm\{\\upsilon\}=\(\\upsilon\_\{1,\\ell\_\{1\}\},\\ldots,\\upsilon\_\{\\tilde\{d\},\\ell\_\{\\tilde\{d\}\}\}\)\\in\\varUpsilonand

𝒙~=\(𝒙\(c\),𝐲\)∈\(𝒳c×I​\(𝝊\)∘\)∩𝒳~ϵ∘\.\\tilde\{\{\\bm\{x\}\}\}=\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)\\in\\bigl\(\\mathcal\{X\}\_\{c\}\\times I\(\\bm\{\\upsilon\}\)^\{\\circ\}\\bigr\)\\cap\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\.Write𝐭=\(𝐭\(c\),𝐭\(d\)\)∈ℝd−d~×ℝd~\\mathbf\{t\}=\(\\mathbf\{t\}^\{\(c\)\},\\mathbf\{t\}^\{\(d\)\}\)\\in\\mathbb\{R\}^\{d\-\\tilde\{d\}\}\\times\\mathbb\{R\}^\{\\tilde\{d\}\}, and define

𝒜𝝊​\(𝒙~\):=\{𝐭:𝐲\+wx​𝐭\(d\)∈I⁡\(𝝊\)\}\.\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\):=\\left\\\{\\mathbf\{t\}:\{\\bf y\}\+w\_\{x\}\\mathbf\{t\}^\{\(d\)\}\\in I\(\\bm\{\\upsilon\}\)\\right\\\}\.If𝒙~\+wx​𝐭∈𝒳c×𝒴\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}and𝐭∉𝒜𝝊​\(𝒙~\)\\mathbf\{t\}\\notin\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\), then the continualized component𝐲\+wx​𝐭\(d\)\{\\bf y\}\+w\_\{x\}\\mathbf\{t\}^\{\(d\)\}lies in a block different fromI⁡\(𝝊\)I\(\\bm\{\\upsilon\}\)\. Hence at least one internal partition interface must be crossed\. By the definition of𝒳~ϵ∘\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}, this implies

maxj∈\[d~\]⁡\|tj\(d\)\|≥ϵwx\.\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\frac\{\\epsilon\}\{w\_\{x\}\}\.Consequently, defining𝒮⁡\(𝒙~\):=\{𝐭:𝒙~\+wx​𝐭∈𝒳c×𝒴\}\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\):=\\\{\\mathbf\{t\}:\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\\in\\mathcal\{X\}\_\{c\}\\times\\mathcal\{Y\}\\\}, we obtain

𝒮⁡\(𝒙~\)∩𝒜𝝊c​\(𝒙~\)⊂\{𝐭:maxj∈\[d~\]⁡\|tj\(d\)\|≥ϵwx\}\.\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}^\{c\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\subset\\left\\\{\\mathbf\{t\}:\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\frac\{\\epsilon\}\{w\_\{x\}\}\\right\\\}\.Then, for𝝁=𝝁⁡\(z,𝒙~,wz,wx\)\\bm\{\\mu\}=\\bm\{\\mu\}\(z,\\tilde\{\{\\bm\{x\}\}\},w\_\{z\},w\_\{x\}\)and𝚵=𝚵⁡\(𝒙~,wx\)\\bm\{\\varXi\}=\\bm\{\\varXi\}\(\\tilde\{\{\\bm\{x\}\}\},w\_\{x\}\), we have decompositions of the integrals over𝒮​\(𝒙~\)∩𝒜𝝊​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\)and𝒮⁡\(𝒙~\)∩𝒜𝝊c​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}^\{c\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\)as follows:

𝝁=𝝁i​n\+𝝁o​u​t,𝚵=𝚵i​n\+𝚵o​u​t,\\bm\{\\mu\}=\\bm\{\\mu\}\_\{in\}\+\\bm\{\\mu\}\_\{out\},\\qquad\\bm\{\\varXi\}=\\bm\{\\varXi\}\_\{in\}\+\\bm\{\\varXi\}\_\{out\},where the subscripts "in" and "out" denote the integrals over the sets𝒮​\(𝒙~\)∩𝒜𝝊​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\)and𝒮⁡\(𝒙~\)∩𝒜𝝊c​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}^\{c\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\), respectively\.

On𝒮​\(𝒙~\)∩𝒜𝝊​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\), both𝐲\{\\bf y\}and𝐲\+wx​𝐭\(d\)\{\\bf y\}\+w\_\{x\}\\mathbf\{t\}^\{\(d\)\}belong to the same continualization block\. By Corollary[5\.2](https://arxiv.org/html/2610.33403#S5.Thmcor2)and Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3),

f⁡\(𝒙\(c\),𝐲\)=f⁡\(𝒙\(c\),𝝊\)\|I⁡\(𝝊\)\|,F⁡\(z∣𝒙\(c\),𝐲\)=F⁡\(z∣𝒙\(c\),𝝊\),f\(\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=\\frac\{f\(\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\)\}\{\|I\(\\bm\{\\upsilon\}\)\|\},\\qquad F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\{\\bf y\}\)=F\(z\\mid\{\\bm\{x\}\}^\{\(c\)\},\\bm\{\\upsilon\}\),throughout this block\. HenceffandFFhave the requiredC2C^\{2\}regularity there, and all derivatives ofFFwith respect to𝐲\{\\bf y\}vanish\. Therefore, for𝐭∈𝒮⁡\(𝒙~\)∩𝒜𝝊​\(𝒙~\)\\mathbf\{t\}\\in\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\), the covariate Taylor expansion in Eq\. \([48](https://arxiv.org/html/2610.33403#A2.E48)\) used in the proof of Lemma[B\.1](https://arxiv.org/html/2610.33403#A2.Thmlem1)remains valid:

F⁡\(z∣𝒙~\+wx​𝐭\)=F⁡\(z∣𝒙~\)\+wx​∇𝒙~⊤F​\(z∣𝒙~\)​𝐭\+wx22​𝐭⊤​∇𝒙~⊤​∇𝒙~F​\(z∣𝒙~\)​𝐭\+Rx​\(z,𝒙~,𝐭\)\.F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)=F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\+w\_\{x\}\\nabla\_\{\\tilde\{\{\\bm\{x\}\}\}\}^\{\\top\}F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\,\\mathbf\{t\}\+\\frac\{w\_\{x\}^\{2\}\}\{2\}\\mathbf\{t\}^\{\\top\}\\nabla\_\{\\tilde\{\{\\bm\{x\}\}\}\}^\{\\top\}\\nabla\_\{\\tilde\{\{\\bm\{x\}\}\}\}F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\,\\mathbf\{t\}\+R\_\{x\}\(z,\\tilde\{\{\\bm\{x\}\}\},\\mathbf\{t\}\)\.Thus the argument leading to Eq\. \([49](https://arxiv.org/html/2610.33403#A2.E49)\) applies without modification to𝝁i​n\\bm\{\\mu\}\_\{in\}\.

It remains to control𝝁o​u​t\\bm\{\\mu\}\_\{out\}via the contribution from the cross\-block region\{𝐭:maxj∈\[d~\]⁡\|tj\(d\)\|≥ϵwx\}\\left\\\{\\mathbf\{t\}:\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\frac\{\\epsilon\}\{w\_\{x\}\}\\right\\\}\. Since0≤F~≤10\\leq\\widetilde\{F\}\\leq 1, the continualized densityffis uniformly bounded over the finitely many blocks, and‖𝜾⁡\(𝐭\)‖≲1\+‖𝐭‖\\\|\\bm\{\\iota\}\(\\mathbf\{t\}\)\\\|\\lesssim 1\+\\\|\\mathbf\{t\}\\\|, the contribution to𝝁⁡\(z,𝒙~,wz,wx\)\\bm\{\\mu\}\(z,\\tilde\{\{\\bm\{x\}\}\},w\_\{z\},w\_\{x\}\)from observations outside the query block is bounded uniformly by

Cf​∫maxj∈\[d~\]⁡\|tj\(d\)\|≥ϵ/wx\(1\+‖𝐭‖\)​𝜿​\(𝐭\)​𝑑𝐭,C\_\{f\}\\int\_\{\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\epsilon/w\_\{x\}\}\(1\+\\\|\\mathbf\{t\}\\\|\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\},whereCfC\_\{f\}is an upper bound on the density function\. Similarly, the corresponding contribution of𝚵o​u​t\\bm\{\\varXi\}\_\{out\}to𝚵⁡\(𝒙~,wx\)\\bm\{\\varXi\}\(\\tilde\{\{\\bm\{x\}\}\},w\_\{x\}\)is bounded by

Cf​∫maxj∈\[d~\]⁡\|tj\(d\)\|≥ϵ/wx\(1\+‖𝐭‖\)2​𝜿​\(𝐭\)​𝑑𝐭\.C\_\{f\}\\int\_\{\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\epsilon/w\_\{x\}\}\(1\+\\\|\\mathbf\{t\}\\\|\)^\{2\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}\.
Since𝜿\\bm\{\\kappa\}is the product Gaussian kernel andddis fixed, form=1,2m=1,2,

∫maxj∈\[d~\]⁡\|tj\(d\)\|≥a\(1\+‖𝐭‖\)m​𝜿​\(𝐭\)​𝑑𝐭\\displaystyle\\int\_\{\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq a\}\(1\+\\\|\\mathbf\{t\}\\\|\)^\{m\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}≤2∑j=1d~𝔼\[\(1\+∥𝐓∥2\)𝕀\{\|Tj\(d\)\|≥a\}\]\\displaystyle\\leq 2\\sum\_\{j=1\}^\{\\tilde\{d\}\}\\mathbb\{E\}\[\(1\+\\\|\\mathbf\{T\}\\\|^\{2\}\)\\mathbb\{I\}\\\{\|T^\{\(d\)\}\_\{j\}\|\\geq a\\\}\]≤4​d~​\(d\+2\)2​π\(1\+a\)e−a2/2,∀a≥1,\\displaystyle\\leq\\frac\{4\\tilde\{d\}\(d\+2\)\}\{\\sqrt\{2\\pi\}\}\(1\+a\)e^\{\-a^\{2\}/2\},\\ \\forall a\\geq 1,where𝐓=\(𝐓\(c\),𝐓\(d\)\)∼N⁡\(0,𝐈d\)\\mathbf\{T\}=\(\\mathbf\{T\}^\{\(c\)\},\\mathbf\{T\}^\{\(d\)\}\)\\sim N\(0,\\mathbf\{I\}\_\{d\}\)and𝐓\(d\)=\(T1\(d\),…,Td~\(d\)\)\\mathbf\{T\}^\{\(d\)\}=\(T\_\{1\}^\{\(d\)\},\\ldots,T\_\{\\tilde\{d\}\}^\{\(d\)\}\); the first inequality follows from2​x≤1\+x22x\\leq 1\+x^\{2\}; the last inequality follows from𝔼\[V2𝕀\{\|V\|≥a\}\]=2\{aϕ\(a\)\+1−Φ\(a\)\}\\mathbb\{E\}\\\!\\left\[V^\{2\}\\mathbb\{I\}\\\{\|V\|\\geq a\\\}\\right\]=2\\\{a\\phi\(a\)\+1\-\\varPhi\(a\)\\\}and the Gaussian Mills bound1−Φ⁡\(a\)≤ϕ⁡\(a\)/a1\-\\varPhi\(a\)\\leq\\phi\(a\)/a\.

Takinga=ϵ/wxa=\\epsilon/w\_\{x\}and usingϵ≥Cϵ​wx​\|log⁡wx\|\\epsilon\\geq C\_\{\\epsilon\}w\_\{x\}\\sqrt\{\|\\log w\_\{x\}\|\}, we have

∫maxj⁡\|tj\(d\)\|≥ϵ/wx\(1\+‖𝐭‖\)m​𝜿​\(𝐭\)​𝑑𝐭\\displaystyle\\int\_\{\\max\_\{j\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\epsilon/w\_\{x\}\}\(1\+\\\|\\mathbf\{t\}\\\|\)^\{m\}\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}≤4​d~​\(d\+2\)2​π​\(1\+Cϵ​\|log⁡wx\|\)​wxCϵ2/2\\displaystyle\\leq\\frac\{4\\tilde\{d\}\(d\+2\)\}\{\\sqrt\{2\\pi\}\}\(1\+C\_\{\\epsilon\}\\sqrt\{\|\\log w\_\{x\}\|\}\)w\_\{x\}^\{C\_\{\\epsilon\}^\{2\}/2\}=Cϵ\>2​o​\(wx2\),m=1,2,\\displaystyle\\overset\{C\_\{\\epsilon\}\>2\}\{=\}o\(w\_\{x\}^\{2\}\),\\qquad m=1,2,where the first inequality follows from the fact thatg\(a\)=\(1\+a\)exp\(−a2/2\)g\(a\)=\(1\+a\)\\exp\(\-a^\{2\}/2\)is monotonically decreasing fora≥1a\\geq 1\. Whenwxw\_\{x\}is sufficiently small,a≥a0:=Cϵ​\|log⁡wx\|≥1a\\geq a\_\{0\}:=C\_\{\\epsilon\}\\sqrt\{\|\\log w\_\{x\}\|\}\\geq 1, and henceg⁡\(a\)≤g⁡\(a0\)g\(a\)\\leq g\(a\_\{0\}\)\. Hence, the total cross\-block contribution iso⁡\(wx2\)o\(w\_\{x\}^\{2\}\), uniformly over𝒙~∈𝒳~ϵ∘\\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\.

Accordingly, the expansion in Eq\. \([49](https://arxiv.org/html/2610.33403#A2.E49)\) applied to the within\-block integral, together with the above cross\-block bounds, gives

𝝁=𝝁i​n\+𝝁o​u​t=𝚵i​n​𝐃wx​𝜷⋆​\(z,𝒙~\)\+O⁡\(wx2\+wz2\)\+o⁡\(wx2\)=𝚵​𝐃wx​𝜷⋆​\(z,𝒙~\)\+O⁡\(wx2\+wz2\),\\bm\{\\mu\}=\\bm\{\\mu\}\_\{in\}\+\\bm\{\\mu\}\_\{out\}=\\bm\{\\varXi\}\_\{in\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\\tilde\{\{\\bm\{x\}\}\}\)\+O\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\+o\(w\_\{x\}^\{2\}\)=\\bm\{\\varXi\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\\tilde\{\{\\bm\{x\}\}\}\)\+O\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\),uniformly overz∈ℝz\\in\\mathbb\{R\}and𝒙~∈𝒳ϵ∘\\tilde\{\{\\bm\{x\}\}\}\\in\\mathcal\{X\}\_\{\\epsilon\}^\{\\circ\}, where the last equality follows from𝚵o​u​t=o⁡\(wx2\)\\bm\{\\varXi\}\_\{out\}=o\(w\_\{x\}^\{2\}\)and\|𝐃wx​𝜷⋆​\(z,𝒙~\)\|=O⁡\(1\)\|\\mathbf\{D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(z,\\tilde\{\{\\bm\{x\}\}\}\)\|=O\(1\)uniformly\.

Moreover, since the continualized density is uniformly bounded away from zero over the finitely many blocks, the eigenvalue argument of Lemma[A\.5](https://arxiv.org/html/2610.33403#A1.Thmlem5), applied to the full continualized support, yields

sup𝒙~∈𝒳~ϵ∘\|𝚵−1​\(𝒙~,wx\)\|=O⁡\(1\)\.\\sup\_\{\\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\}\|\\bm\{\\varXi\}^\{\-1\}\(\\tilde\{\{\\bm\{x\}\}\},w\_\{x\}\)\|=O\(1\)\.Therefore,

supz∈ℝ𝒙~∈𝒳~ϵ∘\|β0⋄​\(z,𝒙~,wz,wx\)−β0⋆​\(z,𝒙~\)\|=O⁡\(wx2\+wz2\)\.\\sup\_\{\\begin\{subarray\}\{c\}z\\in\\mathbb\{R\}\\\\ \\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\\end\{subarray\}\}\\left\|\\beta\_\{0\}^\{\\diamond\}\(z,\\tilde\{\{\\bm\{x\}\}\},w\_\{z\},w\_\{x\}\)\-\\beta\_\{0\}^\{\\star\}\(z,\\tilde\{\{\\bm\{x\}\}\}\)\\right\|=O\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\)\.
Finally, Lemma[B\.2](https://arxiv.org/html/2610.33403#A2.Thmlem2)remains valid on𝒳ϵ∘\\mathcal\{X\}\_\{\\epsilon\}^\{\\circ\}, since this set is a subset of the original index set and the stochastic argument does not require differentiation across the partition interfaces\. Therefore,

supz∈ℝ𝒙~∈𝒳~ϵ∘\|F^​\(z∣𝒙~\)−F⁡\(z∣𝒙~\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{\\begin\{subarray\}\{c\}z\\in\\mathbb\{R\}\\\\ \\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\\end\{subarray\}\}\\bigl\|\\hat\{F\}\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(z\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.
\(ii\) Discrete response after continualization\.Similarly, we need to avoid the interface points in\{z0,…,zL\}\\\{z\_\{0\},\\ldots,z\_\{L\}\\\}\. Define

𝒵~ϵ′∘:=\{z~∈𝒵~∘:min0≤i≤L⁡\|z~−zi\|≥ϵ′\}\.\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}:=\\left\\\{\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}^\{\\circ\}:\\min\_\{0\\leq i\\leq L\}\|\\tilde\{z\}\-z\_\{i\}\|\\geq\\epsilon^\{\\prime\}\\right\\\}\.Suppose

ϵ′≥Cϵ′​wz​\|log⁡wz\|,Cϵ′\>2\.\\epsilon^\{\\prime\}\\geq C\_\{\\epsilon^\{\\prime\}\}w\_\{z\}\\sqrt\{\|\\log w\_\{z\}\|\},\\qquad C\_\{\\epsilon^\{\\prime\}\}\>2\.For anyz~∈𝒵~ϵ′∘\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}, letIiI\_\{i\}denote the continualization interval containingz~\\tilde\{z\}, and seta=ϵ′/wza=\\epsilon^\{\\prime\}/w\_\{z\}\. By the definition of𝒵~ϵ′∘\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}, whenever\|v\|≤a\|v\|\\leq a, bothz~\\tilde\{z\}andz~−wz​v\\tilde\{z\}\-w\_\{z\}vbelong toIiI\_\{i\}\. By Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3),F\(⋅∣𝒙~\)F\(\\cdot\\mid\\tilde\{\{\\bm\{x\}\}\}\)is affine onIiI\_\{i\}, with slopepi​\(𝒙~\)/\|Ii\|p\_\{i\}\(\\tilde\{\{\\bm\{x\}\}\}\)/\|I\_\{i\}\|\. Hence

F⁡\(z~−wz​v∣𝒙~\)−F⁡\(z~∣𝒙~\)=−wz​v​pi​\(𝒙~\)\|Ii\|,\|v\|≤a\.F\(\\tilde\{z\}\-w\_\{z\}v\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)=\-w\_\{z\}v\\frac\{p\_\{i\}\(\\tilde\{\{\\bm\{x\}\}\}\)\}\{\|I\_\{i\}\|\},\\qquad\|v\|\\leq a\.By the symmetry of the Gaussian kernel,

∫\|v\|≤a\{F\(z~−wzv∣𝒙~\)−F\(z~∣𝒙~\)\}κ\(v\)dv=−wzpi​\(𝒙~\)\|Ii\|∫−aavκ\(v\)dv=0\.\\int\_\{\|v\|\\leq a\}\\bigl\\\{F\(\\tilde\{z\}\-w\_\{z\}v\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\bigr\\\}\\kappa\(v\)\\,dv=\-w\_\{z\}\\frac\{p\_\{i\}\(\\tilde\{\{\\bm\{x\}\}\}\)\}\{\|I\_\{i\}\|\}\\int\_\{\-a\}^\{a\}v\\kappa\(v\)\\,dv=0\.Thus, the bias from the symmetric region\|v\|≤a\|v\|\\leq avanishes exactly\. Using0≤F≤10\\leq F\\leq 1and referring to Eq\. \([46](https://arxiv.org/html/2610.33403#A2.E46)\), we bound the remaining contribution as follows:

\|F~​\(z~∣𝒙~\)−F​\(z~∣𝒙~\)\|\\displaystyle\\left\|\\widetilde\{F\}\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\right\|=\|∫\|v\|\>a\{F⁡\(z~−wz​v∣𝒙~\)−F⁡\(z~∣𝒙~\)\}​κ​\(v\)​dv\|\\displaystyle=\\left\|\\int\_\{\|v\|\>a\}\\bigl\\\{F\(\\tilde\{z\}\-w\_\{z\}v\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\bigr\\\}\\kappa\(v\)\\,dv\\right\|≤2\{1−Φ\(a\)\}≤2a​2​πe−a2/2≤2Cϵ′​2​π​\|log⁡wz\|wzCϵ′2/2=o\(wz2\),\\displaystyle\\leq 2\\\{1\-\\varPhi\(a\)\\\}\\leq\\frac\{2\}\{a\\sqrt\{2\\pi\}\}e^\{\-a^\{2\}/2\}\\leq\\frac\{2\}\{C\_\{\\epsilon^\{\\prime\}\}\\sqrt\{2\\pi\|\\log w\_\{z\}\|\}\}w\_\{z\}^\{C\_\{\\epsilon^\{\\prime\}\}^\{2\}/2\}=o\(w\_\{z\}^\{2\}\),where the last inequality follows fromϵ′≥Cϵ′​wz​\|log⁡wz\|\\epsilon^\{\\prime\}\\geq C\_\{\\epsilon^\{\\prime\}\}w\_\{z\}\\sqrt\{\|\\log w\_\{z\}\|\}withCϵ′\>2C\_\{\\epsilon^\{\\prime\}\}\>2\. The response\-smoothing argument in fact holds uniformly over all continualized covariate blocks, since it only uses the piecewise\-affine structure ofFFin the response direction and0≤F≤10\\leq F\\leq 1\. Hence,

supz~∈𝒵~ϵ′∘𝒙~∈𝒳~\|F~​\(z~∣𝒙~\)−F⁡\(z~∣𝒙~\)\|=o⁡\(wz2\)\.\\sup\_\{\\begin\{subarray\}\{c\}\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}\\\\ \\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\\end\{subarray\}\}\\left\|\\widetilde\{F\}\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\right\|=o\(w\_\{z\}^\{2\}\)\.
Substituting the preceding response\-smoothing bound into the integral expression for𝝁\\bm\{\\mu\}, we write

𝝁−𝚵​𝐃wx​𝜷⋆​\(z~,𝒙~\)\\displaystyle\\bm\{\\mu\}\-\\bm\{\\varXi\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\}\)=∫𝒮⁡\(𝒙~\)𝜾⁡\(𝐭\)​𝜿​\(𝐭\)​f​\(𝒙~\+wx​𝐭\)​\{F~​\(z~∣𝒙~\+wx​𝐭\)−F⁡\(z~∣𝒙~\+wx​𝐭\)\}​d𝐭⏟=:Rz\\displaystyle=\\underbrace\{\\int\_\{\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\}\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)f\(\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)\\Bigl\\\{\\widetilde\{F\}\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)\\Bigr\\\}\\,d\\mathbf\{t\}\}\_\{=:R\_\{z\}\}\+∫𝒮⁡\(𝒙~\)𝜾⁡\(𝐭\)​𝜿​\(𝐭\)​f​\(𝒙~\+wx​𝐭\)​\{F⁡\(z~∣𝒙~\+wx​𝐭\)−𝜾​\(𝐭\)⊤​𝐃wx​𝜷⋆​\(z~,𝒙~\)\}​d𝐭⏟=:Rx\.\\displaystyle\+\\underbrace\{\\int\_\{\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\}\\bm\{\\iota\}\(\\mathbf\{t\}\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)f\(\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)\\Bigl\\\{F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\}\)\\Bigr\\\}\\,d\\mathbf\{t\}\}\_\{=:R\_\{x\}\}\.Sinceffis uniformly bounded and‖𝜾⁡\(𝐭\)‖≲1\+‖𝐭‖\\\|\\bm\{\\iota\}\(\\mathbf\{t\}\)\\\|\\lesssim 1\+\\\|\\mathbf\{t\}\\\|, it follows that

‖Rz‖≤C​∫ℝd\(1\+‖𝐭‖\)​𝜿​\(𝐭\)​𝑑𝐭⋅o⁡\(wz2\)=o⁡\(wz2\)\.\\\|R\_\{z\}\\\|\\leq C\\int\_\{\\mathbb\{R\}^\{d\}\}\(1\+\\\|\\mathbf\{t\}\\\|\)\\bm\{\\kappa\}\(\\mathbf\{t\}\)\\,d\\mathbf\{t\}\\cdot o\(w\_\{z\}^\{2\}\)=o\(w\_\{z\}^\{2\}\)\.
ForRxR\_\{x\}, decompose the integral over𝒮​\(𝒙~\)∩𝒜𝝊​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}\(\\tilde\{\{\\bm\{x\}\}\}\)and𝒮⁡\(𝒙~\)∩𝒜𝝊c​\(𝒙~\)\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}^\{c\}\(\\tilde\{\{\\bm\{x\}\}\}\)\. On the within\-block region, the covariate Taylor expansion gives

F⁡\(z~∣𝒙~\+wx​𝐭\)−𝜾​\(𝐭\)⊤​𝐃wx​𝜷⋆​\(z~,𝒙~\)=O⁡\(wx2​‖𝐭‖2\),F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\+w\_\{x\}\\mathbf\{t\}\)\-\\bm\{\\iota\}\(\\mathbf\{t\}\)^\{\\top\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\}\)=O\(w\_\{x\}^\{2\}\\\|\\mathbf\{t\}\\\|^\{2\}\),uniformly, and therefore its contribution toRxR\_\{x\}isO⁡\(wx2\)O\(w\_\{x\}^\{2\}\)\. On the cross\-block region, the integrand is uniformly bounded by a polynomial in‖𝐭‖\\\|\\mathbf\{t\}\\\|, while

𝒮⁡\(𝒙~\)∩𝒜𝝊c​\(𝒙~\)⊂\{𝐭:maxj∈\[d~\]⁡\|tj\(d\)\|≥ϵwx\}\.\\mathcal\{S\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\cap\\mathcal\{A\}\_\{\\bm\{\\upsilon\}\}^\{c\}\(\\tilde\{\{\\bm\{x\}\}\}\)\\subset\\left\\\{\\mathbf\{t\}:\\max\_\{j\\in\[\\tilde\{d\}\]\}\|t\_\{j\}^\{\(d\)\}\|\\geq\\frac\{\\epsilon\}\{w\_\{x\}\}\\right\\\}\.The Gaussian\-tail bound established in part \(i\) therefore gives ano⁡\(wx2\)o\(w\_\{x\}^\{2\}\)contribution\. Consequently,

Rx=O⁡\(wx2\)\+o⁡\(wx2\)=O⁡\(wx2\),R\_\{x\}=O\(w\_\{x\}^\{2\}\)\+o\(w\_\{x\}^\{2\}\)=O\(w\_\{x\}^\{2\}\),and hence

𝝁−𝚵​𝐃wx​𝜷⋆​\(z~,𝒙~\)=O⁡\(wx2\)\+o⁡\(wz2\)\\bm\{\\mu\}\-\\bm\{\\varXi\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\}\)=O\(w\_\{x\}^\{2\}\)\+o\(w\_\{z\}^\{2\}\)uniformly overz~∈𝒵~ϵ′∘\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}and𝒙~∈𝒳~ϵ∘\\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\.

By the population normal equation,

𝐃wx​\{𝜷⋄​\(z~,𝒙~,wz,wx\)−𝜷⋆​\(z~,𝒙~\)\}=𝚵−1​\{𝝁−𝚵​𝐃wx​𝜷⋆​\(z~,𝒙~\)\}\.\{\\bf D\}\_\{w\_\{x\}\}\\bigl\\\{\\bm\{\\beta\}^\{\\diamond\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\},w\_\{z\},w\_\{x\}\)\-\\bm\{\\beta\}^\{\\star\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\}\)\\bigr\\\}=\\bm\{\\varXi\}^\{\-1\}\\bigl\\\{\\bm\{\\mu\}\-\\bm\{\\varXi\}\{\\bf D\}\_\{w\_\{x\}\}\\bm\{\\beta\}^\{\\star\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\}\)\\bigr\\\}\.Since𝚵−1\\bm\{\\varXi\}^\{\-1\}is uniformly bounded and the first diagonal entry of𝐃wx\{\\bf D\}\_\{w\_\{x\}\}equals one,

supz~∈𝒵~ϵ′∘𝒙~∈𝒳~ϵ∘\|β0⋄​\(z~,𝒙~,wz,wx\)−F⁡\(z~∣𝒙~\)\|=O⁡\(wx2\)\+o⁡\(wz2\)\.\\sup\_\{\\begin\{subarray\}\{c\}\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}\\\\ \\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\\end\{subarray\}\}\\left\|\\beta\_\{0\}^\{\\diamond\}\(\\tilde\{z\},\\tilde\{\{\\bm\{x\}\}\},w\_\{z\},w\_\{x\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\right\|=O\(w\_\{x\}^\{2\}\)\+o\(w\_\{z\}^\{2\}\)\.Combining this oracle bias bound with the same stochastic bound as in part \(i\) yields

supz~∈𝒵~ϵ′∘𝒙~∈𝒳~ϵ∘\|F^​\(z~∣𝒙~\)−F⁡\(z~∣𝒙~\)\|=Op​\(wx2\+wz2\+\|log⁡wx\|M​wxd\)\.\\sup\_\{\\begin\{subarray\}\{c\}\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}\\\\ \\tilde\{\{\\bm\{x\}\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\}\\end\{subarray\}\}\\left\|\\hat\{F\}\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\{\\bm\{x\}\}\}\)\\right\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+w\_\{z\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\.∎

### C\.6Proof of Corollary[5\.4](https://arxiv.org/html/2610.33403#S5.Thmcor4)

###### Proof\.

Fixiiand𝒙~∈𝒳ϵ∘\\tilde\{\\bm\{x\}\}\\in\\mathcal\{X\}\_\{\\epsilon\}^\{\\circ\}\. By Theorem[5\.3](https://arxiv.org/html/2610.33403#S5.Thmthm3)\(ii\),F\(⋅∣𝒙~\)F\(\\cdot\\mid\\tilde\{\\bm\{x\}\}\)is affine onIi∩𝒵~ϵ′∘I\_\{i\}\\cap\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}with slopepi​\(𝒙~\)/\|Ii\|p\_\{i\}\(\\tilde\{\\bm\{x\}\}\)/\|I\_\{i\}\|\. Hence

F⁡\(z~i,\+∘∣𝒙~\)−F⁡\(z~i,−∘∣𝒙~\)=z~i,\+∘−z~i,−∘\|Ii\|​pi​\(𝒙~\),F\(\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\-F\(\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)=\\frac\{\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\-\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\}\{\|I\_\{i\}\|\}p\_\{i\}\(\\tilde\{\\bm\{x\}\}\),giving the exact identity

pi​\(𝒙~\)=\|Ii\|z~i,\+∘−z~i,−∘​\(F⁡\(z~i,\+∘∣𝒙~\)−F⁡\(z~i,−∘∣𝒙~\)\)\.p\_\{i\}\(\\tilde\{\\bm\{x\}\}\)=\\frac\{\|I\_\{i\}\|\}\{\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\-\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\}\\Bigl\(F\(\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\-F\(\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\\Bigr\)\.Subtracting,

p^i​\(𝒙~\)−pi​\(𝒙~\)=\|Ii\|z~i,\+∘−z~i,−∘​\[\(F^−F\)​\(z~i,\+∘∣𝒙~\)−\(F^−F\)​\(z~i,−∘∣𝒙~\)\]\.\\widehat\{p\}\_\{i\}\(\\tilde\{\\bm\{x\}\}\)\-p\_\{i\}\(\\tilde\{\\bm\{x\}\}\)=\\frac\{\|I\_\{i\}\|\}\{\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\-\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\}\\Bigl\[\(\\widehat\{F\}\-F\)\(\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\-\(\\widehat\{F\}\-F\)\(\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\\Bigr\]\.\(60\)By the separation condition \([36](https://arxiv.org/html/2610.33403#S5.E36)\),\|Ii\|/\(z~i,\+∘−z~i,−∘\)≤c−1\|I\_\{i\}\|/\(\\tilde\{z\}\_\{i,\+\}^\{\\circ\}\-\\tilde\{z\}\_\{i,\-\}^\{\\circ\}\)\\leq c^\{\-1\}is uniformly bounded inii\.

By the refined bias bound established in the proof of Theorem[5\.4](https://arxiv.org/html/2610.33403#S5.Thmthm4)\(ii\), uniformly overz~∈𝒵~ϵ′∘\\tilde\{z\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}and𝒙~∈𝒳~ϵ∘\\tilde\{\\bm\{x\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\circ\},

\|F^​\(z~∣𝒙~\)−F⁡\(z~∣𝒙~\)\|=Op​\(wx2\+\|log⁡wx\|M​wxd\)\+o⁡\(wz2\)\.\\bigl\|\\widehat\{F\}\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\)\-F\(\\tilde\{z\}\\mid\\tilde\{\\bm\{x\}\}\)\\bigr\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\+o\(w\_\{z\}^\{2\}\)\.Sincez~i,±∘∈𝒵~ϵ′∘\\tilde\{z\}\_\{i,\\pm\}^\{\\circ\}\\in\\widetilde\{\\mathcal\{Z\}\}\_\{\\epsilon^\{\\prime\}\}^\{\\circ\}andLLis fixed,

max⁡sup𝒙~∈𝒳ϵ∘i∈\[L\],z~i,±∘⁡\|\(F^−F\)​\(z~i,±∘∣𝒙~\)\|=Op​\(wx2\+\|log⁡wx\|M​wxd\)\+o⁡\(wz2\)\.\\max\_\{i\\in\[L\],\\,\\tilde\{z\}\_\{i,\\pm\}^\{\\circ\}\}\\sup\_\{\\tilde\{\\bm\{x\}\}\\in\\mathcal\{X\}\_\{\\epsilon\}^\{\\circ\}\}\\left\|\(\\widehat\{F\}\-F\)\(\\tilde\{z\}\_\{i,\\pm\}^\{\\circ\}\\mid\\tilde\{\\bm\{x\}\}\)\\right\|=O\_\{p\}\\\!\\left\(w\_\{x\}^\{2\}\+\\sqrt\{\\frac\{\|\\log w\_\{x\}\|\}\{Mw\_\{x\}^\{d\}\}\}\\right\)\+o\(w\_\{z\}^\{2\}\)\.Combining this bound with \([60](https://arxiv.org/html/2610.33403#A3.E60)\) and \([36](https://arxiv.org/html/2610.33403#S5.E36)\) proves \([37](https://arxiv.org/html/2610.33403#S5.E37)\)\.

For𝒙~∈𝒳~ϵ∙\\tilde\{\\bm\{x\}\}\\in\\widetilde\{\\mathcal\{X\}\}\_\{\\epsilon\}^\{\\bullet\}, the same result follows by evaluatingp^i\\widehat\{p\}\_\{i\}at the corresponding interior representative𝒙~ϵ∘\\tilde\{\\bm\{x\}\}\_\{\\epsilon\}^\{\\circ\}, since the conditional probability masses are invariant in𝐲\{\\bf y\}within each continualization block\. ∎

## Appendix DExperimental settings

### D\.1Data usage modes for downstream learning

A federated synthetic dataset is most useful when it can be combined with the original local data to support downstream learning tasks at each client\. This subsection formalizes four data usage modes for downstream training that differ in how synthetic and local data is weighted in the empirical loss\. These four modes are not algorithmic variants of the synthetic data generator itself but four well\-defined ways of*using*the generated data in downstream learning; they constitute the comparison framework that Section[6](https://arxiv.org/html/2610.33403#S6)evaluates empirically\.

Throughout this subsection, letℬ=\{𝒙^b\}b=1B\\mathcal\{B\}=\\\{\\hat\{\\bm\{x\}\}\_\{b\}\\\}\_\{b=1\}^\{B\}denote the synthetic dataset generated by Algorithm[2](https://arxiv.org/html/2610.33403#alg2)and shared across all clients, and letℋ\(k\)\\mathcal\{H\}^\{\(k\)\}denote clientkk’s local dataset\. We consider a generic downstream lossℓ\\ellindexed by a parameterθ\\theta\(for example, a squared\-error loss for linear regression or a negative log\-likelihood loss for logistic regression\), and define the four data usage modes as follows\.

M1: synthetic\-only learning\.Clientkkfits its downstream model entirely on the shared synthetic dataset:

θ^syn\(k\)=arg⁡min⁡∑𝒙^b∈ℬθ⁡ℓ⁡\(θ,𝒙^b\)\.\\widehat\{\\theta\}^\{\(k\)\}\_\{\\rm syn\}=\\arg\\min\_\{\\theta\}\\sum\_\{\\hat\{\{\\bm\{x\}\}\}\_\{b\}\\in\\mathcal\{B\}\}\\ell\(\\theta;\\hat\{\{\\bm\{x\}\}\}\_\{b\}\)\.\(61\)This mode is the natural baseline for clients with no or extremely limited local data\. It captures all the information that the federated framework has aggregated about the joint law, but it does not exploit any client\-specific signal\.

M2: local\-only learning\.Clientkkfits the downstream model on its local data alone:

θ^loc\(k\)=arg⁡min⁡∑𝐡i∈ℋ\(k\)θ⁡ℓ⁡\(θ,𝐡i\)\.\\widehat\{\\theta\}^\{\(k\)\}\_\{\\rm loc\}=\\arg\\min\_\{\\theta\}\\sum\_\{\{\\bf h\}\_\{i\}\\in\\mathcal\{H\}^\{\(k\)\}\}\\ell\(\\theta;\{\\bf h\}\_\{i\}\)\.\(62\)This is the standard non\-federated baseline: no benefit is drawn from the synthetic data, and each client’s estimator is governed by its local sample sizemkm\_\{k\}\.

M3: mixed\-data learning\.Clientkkpools a subset of the synthetic observationsℬp​a​r​t\{\\mathcal\{B\}\}\_\{part\}and\|ℬp​a​r​t\|=r​\|ℋ\(k\)\|,r∈\[0,∞\]\|\{\\mathcal\{B\}\}\_\{part\}\|=r\|\\mathcal\{H\}^\{\(k\)\}\|,r\\in\[0,\\infty\], whose size is controlled byrrrelative to the local sample size, with its local dataset to form a single training set with uniform weight per sample\.

θ^mix\(k\)=arg⁡minθ​\{∑𝐡i∈ℋ\(k\)ℓ⁡\(θ,𝐡i\)\+∑𝒙^b∈ℬp​a​r​tℓ⁡\(θ,𝒙^b\)\}\.\\widehat\{\\theta\}^\{\(k\)\}\_\{\\rm mix\}=\\arg\\min\_\{\\theta\}\\biggl\\\{\\sum\_\{\{\\bf h\}\_\{i\}\\in\\mathcal\{H\}^\{\(k\)\}\}\\ell\(\\theta;\{\\bf h\}\_\{i\}\)\+\\sum\_\{\\hat\{\{\\bm\{x\}\}\}\_\{b\}\\in\{\\mathcal\{B\}\}\_\{part\}\}\\ell\(\\theta;\\hat\{\{\\bm\{x\}\}\}\_\{b\}\)\\biggr\\\}\.\(63\)Each synthetic and local sample contributes equally to the empirical loss, but the number of synthetic observations is controlled\. Note that the parameterr=∞r=\\inftymeans that all synthetic data is used\.

M4: loss\-reweighted synthetic\-local learning\.Clientkkcombines synthetic and local data with a tunable weightλ∈\[0,\+∞\)\\lambda\\in\[0,\+\\infty\)on the synthetic loss:

θ^rew\(k\)​\(λ\)=arg⁡minθ​\{∑𝐡i∈ℋ\(k\)ℓ⁡\(θ,𝐡i\)\+λ​∑𝒙^b∈ℬℓ⁡\(θ,𝒙^b\)\}\.\\widehat\{\\theta\}^\{\(k\)\}\_\{\\rm rew\}\(\\lambda\)=\\arg\\min\_\{\\theta\}\\biggl\\\{\\sum\_\{\{\\bf h\}\_\{i\}\\in\\mathcal\{H\}^\{\(k\)\}\}\\\!\\\!\\ell\(\\theta;\{\\bf h\}\_\{i\}\)\+\\lambda\\\!\\\!\\sum\_\{\\hat\{\{\\bm\{x\}\}\}\_\{b\}\\in\{\\mathcal\{B\}\}\}\\\!\\\!\\ell\(\\theta;\\hat\{\{\\bm\{x\}\}\}\_\{b\}\)\\biggr\\\}\.\(64\)The weightλ\\lambdacontrols the relative influence of the two data sources\. The special caseλ=0\\lambda=0recovers Mode 2, whereasλ=1\\lambda=1recovers Mode 3 with the full synthetic dataset\.

The four modes differ in how synthetic data replaces or complements local data\. Mode 1 completely replaces local data with synthetic data; Mode 2 ignores the synthetic data entirely; and Mode 3 controls the relative contribution of synthetic data through the number of synthetic observations selected byrrrelative tomkm\_\{k\}\. Mode 4 controls the trade\-off explicitly at the loss\-function level throughλ\\lambda\.

### D\.2Simulation data

The simulated covariates are generated as follows\. First,

XB,i∼Bernoulli⁡\(π\),π=0\.45\.X\_\{B,i\}\\sim\\mathrm\{Bernoulli\}\(\\pi\),\\qquad\\pi=0\.45\.Conditional onXB,iX\_\{B,i\}, the pair\(XG1,i,XG2,i\)\(X\_\{G\_\{1\},i\},X\_\{G\_\{2\},i\}\)is sampled from a bivariate Gaussian distribution:

\(XG1,i,XG2,i\)⊤\|XB,i=0∼N⁡\(\(−1\.00\.5\),\(1\.00\.60\.61\.5\)\),\(X\_\{G\_\{1\},i\},X\_\{G\_\{2\},i\}\)^\{\\top\}\\mid X\_\{B,i\}=0\\sim N\\left\(\\begin\{pmatrix\}\-1\.0\\\\ 0\.5\\end\{pmatrix\},\\begin\{pmatrix\}1\.0&0\.6\\\\ 0\.6&1\.5\\end\{pmatrix\}\\right\),\(XG1,i,XG2,i\)⊤\|XB,i=1∼N⁡\(\(1\.2−0\.7\),\(1\.6−0\.4−0\.40\.8\)\)\.\(X\_\{G\_\{1\},i\},X\_\{G\_\{2\},i\}\)^\{\\top\}\\mid X\_\{B,i\}=1\\sim N\\left\(\\begin\{pmatrix\}1\.2\\\\ \-0\.7\\end\{pmatrix\},\\begin\{pmatrix\}1\.6&\-0\.4\\\\ \-0\.4&0\.8\\end\{pmatrix\}\\right\)\.The independent continuous covariate is generated as

XG3,i∼N⁡\(0,1\)\.X\_\{G\_\{3\},i\}\\sim N\(0,1\)\.The ordinal covariateXO,i∈\{1,2,3,4\}X\_\{O,i\}\\in\\\{1,2,3,4\\\}is obtained by discretizing the latent variable

Zi=0\.65​XB,i\+0\.45​XG1,i−0\.35​XG2,i\+εi,εi∼N⁡\(0,1\),Z\_\{i\}=0\.65X\_\{B,i\}\+0\.45X\_\{G\_\{1\},i\}\-0\.35X\_\{G\_\{2\},i\}\+\\varepsilon\_\{i\},\\qquad\\varepsilon\_\{i\}\\sim N\(0,1\),using cut points \(\-0\.8,0\.25,1\.1\):

XO,i=1\+∑c∈\{−0\.8,0\.25,1\.1\}𝟏\{Zi\>c\}\.X\_\{O,i\}=1\+\\sum\_\{c\\in\\\{\-0\.8,0\.25,1\.1\\\}\}\\mathbf\{1\}\\\{Z\_\{i\}\>c\\\}\.Finally, withXO,ic=XO,i−2\.5X\_\{O,i\}^\{c\}=X\_\{O,i\}\-2\.5, the Laplace covariate is generated as

XL,i∼Laplace⁡\(μi,si\),X\_\{L,i\}\\sim\\mathrm\{Laplace\}\(\\mu\_\{i\},s\_\{i\}\),whereμi=0\.40​XB,i\+0\.25​XG1,i−0\.15​XG2,i\+0\.20​XO,ic\\mu\_\{i\}=0\.40X\_\{B,i\}\+0\.25X\_\{G\_\{1\},i\}\-0\.15X\_\{G\_\{2\},i\}\+0\.20X\_\{O,i\}^\{c\}, andsi=exp⁡\(−0\.10\+0\.25​XB,i\+0\.08​XO,ic\)\.s\_\{i\}=\\exp\(\-0\.10\+0\.25X\_\{B,i\}\+0\.08X\_\{O,i\}^\{c\}\)\.ThusXBX\_\{B\}drives the distribution of\(XG1,XG2\)\(X\_\{G\_\{1\}\},X\_\{G\_\{2\}\}\), the ordinal variableXOX\_\{O\}depends on\(XB,XG1,XG2\)\(X\_\{B\},X\_\{G\_\{1\}\},X\_\{G\_\{2\}\}\), andXLX\_\{L\}depends on\(XB,XG1,XG2,XO\)\(X\_\{B\},X\_\{G\_\{1\}\},X\_\{G\_\{2\}\},X\_\{O\}\), whileXG3X\_\{G\_\{3\}\}is generated independently\.

Table S1:Chain order, task\-specific parameters, and evaluation metrics\.
### D\.3Real data

Table S2:Preprocessing and variable\-type assignments for the UCI datasets\.

## Appendix EBaseline methods

We compare FITS with HeteroFedSyn\[[Li et al\., 2026](https://arxiv.org/html/2610.33403#bib.bib22)\]and FedTabDiff\[[Sattarov et al\., 2024](https://arxiv.org/html/2610.33403#bib.bib9)\]\. Both baselines generate a shared synthetic tabular datasetℬ=\{𝒙^b\}b=1B\\mathcal\{B\}=\\\{\\hat\{\\bm\{x\}\}\_\{b\}\\\}\_\{b=1\}^\{B\}from the same client datasets\{ℋ\(k\)\}k=1K\\\{\\mathcal\{H\}^\{\(k\)\}\\\}\_\{k=1\}^\{K\}, wheremk=\|ℋ\(k\)\|m\_\{k\}=\|\\mathcal\{H\}^\{\(k\)\}\|andM=∑k=1KmkM=\\sum\_\{k=1\}^\{K\}m\_\{k\}\. FT\-Transformer provides an additional downstream evaluation model\. The descriptions below summarize the baseline implementations used in our experiments\.

### E\.1HeteroFedSyn

We use the adaptive marginal\-selection variant, AdaFedPrivSyn\. Continuous columns are discretized into at most100100equal\-width bins, while discrete columns retain their observed categories\. For a table withqqcolumns, clientkkcomputes one\-way count vectors𝒄a\(k\)\\bm\{c\}\_\{a\}^\{\(k\)\}and two\-way count vectors𝒄a​b\(k\)\\bm\{c\}\_\{ab\}^\{\(k\)\}\. Each attribute pair uses a shared Gaussian projection matrixΠa​b∈ℝca​cb×10\\Pi\_\{ab\}\\in\\mathbb\{R\}^\{c\_\{a\}c\_\{b\}\\times 10\}, wherecac\_\{a\}is the encoded domain size of columnaa\. The server aggregates locally perturbed statistics as

𝒑^a=1M​∑k=1K\(𝒄a\(k\)\+𝝃a\(k\)\),𝒔^a​b=1M​∑k=1K\(Πa​b⊤​𝒄a​b\(k\)\+𝜻a​b\(k\)\)\.\\hat\{\\bm\{p\}\}\_\{a\}=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{K\}\\bigl\(\\bm\{c\}\_\{a\}^\{\(k\)\}\+\\bm\{\\xi\}\_\{a\}^\{\(k\)\}\\bigr\),\\qquad\\hat\{\\bm\{s\}\}\_\{ab\}=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{K\}\\bigl\(\\Pi\_\{ab\}^\{\\top\}\\bm\{c\}\_\{ab\}^\{\(k\)\}\+\\bm\{\\zeta\}\_\{ab\}^\{\(k\)\}\\bigr\)\.Here𝝃a\(k\)\\bm\{\\xi\}\_\{a\}^\{\(k\)\}and𝜻a​b\(k\)\\bm\{\\zeta\}\_\{ab\}^\{\(k\)\}are Gaussian noise vectors\. Dependency scores compare𝒔^a​b\\hat\{\\bm\{s\}\}\_\{ab\}with the projected productΠa​b⊤​\(𝒑^a⊗𝒑^b\)\\Pi\_\{ab\}^\{\\top\}\(\\hat\{\\bm\{p\}\}\_\{a\}\\otimes\\hat\{\\bm\{p\}\}\_\{b\}\), with a correction for the aggregated noise variance\. A pair is selected when its dependency score exceeds its expected release\-noise error\. We select at most⌊\(q2\)/3⌋\\lfloor\\binom\{q\}\{2\}/3\\rfloorpairs\. After each selection, marginal consistency adjustment and3030iterations of the gradual update method \(GUM,[Zhang et al\. \[2021\]](https://arxiv.org/html/2610.33403#bib.bib27)\) produce a synthetic tabular dataset; projected residuals update the scores of remaining pairs involving covered attributes\. Previously released one\-way marginals are reused\. Continuous values are recovered as bin midpoints, and discrete values are mapped back to their original categories\.

We setϵ=108\\epsilon=10^\{8\}andδ=10−5\\delta=10^\{\-5\}\. WritingL=log⁡\(1/δ\)L=\\log\(1/\\delta\), the implementation uses the zCDP budget

ρ=\(L\+ϵ−L\)2,\(ρ1,ρ2,ρ3\)=\(0\.1​ρ,0\.1​ρ,0\.8​ρ\),\\rho=\\bigl\(\\sqrt\{L\+\\epsilon\}\-\\sqrt\{L\}\\bigr\)^\{2\},\\qquad\(\\rho\_\{1\},\\rho\_\{2\},\\rho\_\{3\}\)=\(0\.1\\rho,0\.1\\rho,0\.8\\rho\),for one\-way statistics, projected two\-way statistics, and selected two\-way marginals, respectively\. This deliberately weak privacy setting emphasizes synthesis utility rather than a privacy–utility trade\-off\.

### E\.2FedTabDiff

FedTabDiff learns a denoising diffusion model on an encoded mixed\-type tabular dataset\. Continuous columns are transformed to approximately standard normal marginals using a training\-data quantile transform with at most1,0001\{,\}000quantiles\. Each discrete column uses a two\-dimensional frozen embedding; generated embeddings are decoded to the nearest category embedding within that column\. The encoded record𝒛0\\bm\{z\}\_\{0\}is perturbed according to

𝒛t=α¯t​𝒛0\+1−α¯t​𝜺,𝜺∼N⁡\(𝟎,I\),α¯t=∏u=0t\(1−βu\)\.\\bm\{z\}\_\{t\}=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\bm\{z\}\_\{0\}\+\\sqrt\{1\-\\bar\{\\alpha\}\_\{t\}\}\\bm\{\\varepsilon\},\\qquad\\bm\{\\varepsilon\}\\sim N\(\\bm\{0\},I\),\\qquad\\bar\{\\alpha\}\_\{t\}=\\prod\_\{u=0\}^\{t\}\(1\-\\beta\_\{u\}\)\.Clients minimize the noise\-prediction loss𝔼​‖𝜺−𝜺ψ​\(𝒛t,t\)‖22\\mathbb\{E\}\\\|\\bm\{\\varepsilon\}\-\\bm\{\\varepsilon\}\_\{\\psi\}\(\\bm\{z\}\_\{t\},t\)\\\|\_\{2\}^\{2\}\. All clients participate in every communication round, and the server uses sample\-size\-weighted FedAvg:

ψ\(s\+1\)=∑k=1KmkM​ψk\(s\+1\),\\psi^\{\(s\+1\)\}=\\sum\_\{k=1\}^\{K\}\\frac\{m\_\{k\}\}\{M\}\\psi\_\{k\}^\{\(s\+1\)\},whereψk\(s\+1\)\\psi\_\{k\}^\{\(s\+1\)\}is thekk\-th client’s model at the\(s\+1\)\(s\+1\)\-th iteration andψ\(s\+1\)\\psi^\{\(s\+1\)\}is the corresponding central model\.

The noise predictor uses four hidden layers of width1,0241\{,\}024\. We use500500diffusion steps with a linear noise schedule from2×10−42\\times 10^\{\-4\}to0\.040\.04, and1,0001\{,\}000FedAvg rounds with Adam \(learning rate10−410^\{\-4\}\), batch size512512, and at most2020local updates per client per round\. Covariates and responses are synthesized jointly\.

### E\.3FT\-Transformer for downstream evaluation

We use FT\-Transformer\[[Gorishniy et al\., 2021](https://arxiv.org/html/2610.33403#bib.bib28)\]to evaluate nonlinear regression and classification on the generated data\. For each M1–M4 fit, only covariates enter the model\. Continuous features are standardized using the weighted mean and standard deviation of that fit’s training data, while nominal and ordinal predictors are mapped to categorical indices\.

Each standardized continuous featurexj\(c\)x^\{\(c\)\}\_\{j\}receives an affine token, and each discrete featurexj\(d\)x^\{\(d\)\}\_\{j\}receives a learned category embedding:

𝒕j=\{xj\(c\)​𝒂j\+𝒃j,xj\(c\)​continuous,Ej​\[xj\(d\)\]\+𝒃j,xj\(d\)​discrete,𝒕j∈ℝ32,\\bm\{t\}\_\{j\}=\\begin\{cases\}x^\{\(c\)\}\_\{j\}\\bm\{a\}\_\{j\}\+\\bm\{b\}\_\{j\},&x^\{\(c\)\}\_\{j\}\\text\{ continuous\},\\\\ E\_\{j\}\[x^\{\(d\)\}\_\{j\}\]\+\\bm\{b\}\_\{j\},&x^\{\(d\)\}\_\{j\}\\text\{ discrete\},\\end\{cases\}\\qquad\\bm\{t\}\_\{j\}\\in\\mathbb\{R\}^\{32\},where𝒂j,𝒃j∈ℝ32\\bm\{a\}\_\{j\},\\ \\bm\{b\}\_\{j\}\\in\\mathbb\{R\}^\{32\}are learnable parameter vectors andEjE\_\{j\}is a learnable embedding matrix for discrete featurejj\.

The feature tokens and a learned\[CLS\]token have dimension3232and pass through two Transformer blocks\. Each block contains residual self\-attention with four heads and a feed\-forward network of width6464with ReGLU activation\. Attention, feed\-forward, and residual dropout are all set to zero\. The final\[CLS\]representation passes through LayerNorm, ReLU, and a linear head to produce a scalar outputfθ​\(𝒙\)f\_\{\\theta\}\(\\bm\{x\}\), used for regression or as a binary\-classification logit\.

For a training table𝒯=\{\(𝒙i,yi,wi\)\}i=1n\\mathcal\{T\}=\\\{\(\\bm\{x\}\_\{i\},y\_\{i\},w\_\{i\}\)\\\}\_\{i=1\}^\{n\}specified by the data usage mode, the objective is

ℒ⁡\(θ\)=∑i=1nwi​ℓ​\(θ,𝒙i,yi\)∑i=1nwi,ℓ⁡\(θ,𝒙,y\)=\{\(fθ​\(𝒙\)−y\)2,regression,log⁡\(1\+efθ​\(𝒙\)\)−y​fθ​\(𝒙\),classification\.\\mathcal\{L\}\(\\theta\)=\\frac\{\\sum\_\{i=1\}^\{n\}w\_\{i\}\\ell\(\\theta;\\bm\{x\}\_\{i\},y\_\{i\}\)\}\{\\sum\_\{i=1\}^\{n\}w\_\{i\}\},\\qquad\\ell\(\\theta;\\bm\{x\},y\)=\\begin\{cases\}\(f\_\{\\theta\}\(\\bm\{x\}\)\-y\)^\{2\},&\\text\{regression\},\\\\ \\log\(1\+e^\{f\_\{\\theta\}\(\\bm\{x\}\)\}\)\-yf\_\{\\theta\}\(\\bm\{x\}\),&\\text\{classification\}\.\\end\{cases\}Regression responses are additionally standardized using training weights, and predictions are mapped back to the evaluation scale\. Binary responses retain their0/10/1encoding; the sigmoid of the model output gives the class probability, with threshold0\.50\.5\.

Each model uses500500AdamW updates with learning rate10−310^\{\-3\}, batch size256256, and weight decay10−510^\{\-5\}\. Minibatches are sampled with replacement with probabilitieswi/∑jwjw\_\{i\}/\\sum\_\{j\}w\_\{j\}\. Training settings and seed rules are identical across methods\. We report regression test MSE on the original response scale and classification test accuracy\.

## Appendix FSupplementary experimental results

Table S3:Simulation: linear\-regression parameter estimation error \(FITS\-G vs FITS\-K\)\.Table S4:Simulation: logistic\-regression parameter estimation error \(FITS\-G vs FITS\-K\)\.Table S5:Simulation: FT\-Transformer classification accuracy on the test data \(FITS\-G vs FITS\-K\)\.Table S6:Simulation: linear\-regression parameter estimation error \(FITS\-G: all modes\)\.Table S7:Simulation: logistic\-regression parameter estimation error \(FITS\-G: all modes\)\.Table S8:Simulation: FT\-Transformer classification accuracy on the test data \(FITS\-G: all modes\)\.Table S9:Bike: linear\-regression parameter estimation error \(FITS\-G vs FITS\-K\)\.Table S10:Bike: linear\-regression test MSE \(FITS\-G vs FITS\-K\)\.Table S11:Bike: FT\-Transformer regression MSE on the test data \(FITS\-G vs FITS\-K\)\.Table S12:Bike: linear\-regression parameter estimation error \(FITS\-G: all modes\)\.Table S13:Bike: linear\-regression test MSE \(FITS\-G: all modes\)\.Table S14:Bike: FT\-Transformer regression test MSE \(FITS\-G: all modes\)\.Table S15:Adult: logistic\-regression parameter estimation error \(FITS\-G vs FITS\-K\)\.Table S16:Adult: logistic\-regression test accuracy \(FITS\-G vs FITS\-K\)\.Table S17:Adult: FT\-Transformer classification test accuracy \(FITS\-G vs FITS\-K\)\.Table S18:Adult: logistic\-regression parameter estimation error \(FITS\-G: all modes\)\.Table S19:Adult: logistic\-regression test accuracy \(FITS\-G: all modes\)\.Table S20:Adult: FT\-Transformer classification test accuracy \(FITS\-G: all modes\)\.Table S21:Simulation: linear\-regression test MSE\.Table S22:Simulation: logistic\-regression test accuracy\.Table S23:Simulation: FT\-Transformer regression test MSE\.Table S24:Simulation: linear\-regression test MSE \(FITS\-G vs FITS\-K\)\.Table S25:Simulation: logistic\-regression test accuracy \(FITS\-G vs FITS\-K\)\.Table S26:Simulation: FT\-Transformer regression test MSE \(FITS\-G vs FITS\-K\)\.Table S27:Simulation: linear\-regression test MSE \(FITS\-G: all modes\)\.Table S28:Simulation: logistic\-regression test accuracy \(FITS\-G: all modes\)\.Table S29:Simulation: FT\-Transformer regression test MSE \(FITS\-G: all modes\)\.