Minimax-Optimal Predictive Flow Matching under Distribution Drift

# Minimax-Optimal Predictive Flow Matching under Distribution Drift
Source: [https://arxiv.org/html/2610.28898](https://arxiv.org/html/2610.28898)
Aadithya Srikanth††thanks:Correspondence to:srikan18@purdue\.edu\.Vaneet AggarwalAffiliation:\{srikan18, vaneet\}@purdue\.edu

###### Abstract

Flow matching learns a generative velocity field by regression, but under distribution drift a field must be deployed before the current data arrive\. We study this predictive problem for a fixed, possibly nonconvex, neural class, with dynamic regret against the best represented field of each round\. We propose DyProd\-FM, which refits flow matching on dyadic historical prefixes and selects among the fits with a sleeping Prod master\. Without knowing any drift budget, it attains regretO~​\(min⁡\{Nω\+1,1\+T​Ω\}\)\\widetilde\{O\}\(\\min\\\{N\_\{\\omega\}\+1,1\+\\sqrt\{T\\Omega\}\\\}\)in the relative velocity risk variationΩ\\OmegaandO~​\(min⁡\{Nμ\+1,1\+T1/3​Vμ2/3\}\)\\widetilde\{O\}\(\\min\\\{N\_\{\\mu\}\+1,1\+T^\{1/3\}V\_\{\\mu\}^\{2/3\}\\\}\)in the target\-law variationVμ=∑tW2​\(μt\+1,μt\)V\_\{\\mu\}=\\sum\_\{t\}W\_\{2\}\(\\mu\_\{t\+1\},\\mu\_\{t\}\), up to approximation and fitting costs\. Matching lower bounds for fixed classes show that both rates are optimal inTTand in the budgets up to logarithmic factors\. Prior optimal dynamic\-regret guarantees rely on a convex decision set with strongly convex or exp\-concave losses, whereas our class may be nonconvex and no curvature in its parameters is assumed\. The target\-law rate rests on a second\-order link between the two variations, which the base noise of flow matching provides\.

## 1Introduction

Flow matching constructs a generative model by regressing a velocity field along a path from a base distribution to a target distribution\([Lipman et al\., 2023](https://arxiv.org/html/2610.28898#bib.bib22);[Liu et al\., 2023](https://arxiv.org/html/2610.28898#bib.bib24);[Albergo et al\., 2025](https://arxiv.org/html/2610.28898#bib.bib2)\)\. Its statistical analysis usually fixes the target while observations accumulate\. In a changing environment, this separation between training and deployment is insufficient: a field learned from a large historical sample may accurately solve an obsolete regression problem\. Conversely, discarding history after every observation forfeits the statistical benefit of repeated sampling\. A predictive theory must quantify both effects under the information available when a field is deployed\.

We study nonstationary stochastic flow matching with independent, fresh batches\. A learner selects one represented field using past batches, then receives the current batch and scores every active candidate\. Its benchmark is the best represented field for the current population, which may change every round\. Learning rounds, along which the environment changes, are distinct from flow time, along which a selected field generates a sample\.

How should drift be measured? The*relative velocity risk variation*Ω\\Omegameasures how drift changes the comparison between represented fields\. It is class\-relative and ignores changes the class cannot see\. The*target\-law variation*Vμ=∑tW2​\(μt\+1,μt\)V\_\{\\mu\}=\\sum\_\{t\}W\_\{2\}\(\\mu\_\{t\+1\},\\mu\_\{t\}\)measures drift of the generative target itself\. It is class\-free and estimable from samples\. We determine the minimax rate under both measures, up to logarithmic factors, for a fixed class and fixed problem scales\.

Three difficulties arise\. Historical batches are independent but heterogeneous, so one common comparator need not fit all past velocities\. Historical fitting attainsO⁡\(1/n\)O\(1/n\)withnnexamples, but standard online selection adds orderT\\sqrt\{T\}regret and erases this gain\. First\-order conversion of target drift into risk drift discards squared\-loss curvature and yields onlyT​Vμ\\sqrt\{TV\_\{\\mu\}\}\. We must resolve all three for a nonconvex class without parameter\-curvature assumptions\. Section[5](https://arxiv.org/html/2610.28898#S5)resolves them in turn through historical estimation \(Section[5\.1](https://arxiv.org/html/2610.28898#S5.SS1)\), fast selection \(Section[5\.2](https://arxiv.org/html/2610.28898#S5.SS2)\), and drift transfer followed by horizon partitions \(Sections[5\.3](https://arxiv.org/html/2610.28898#S5.SS3)–[5\.5](https://arxiv.org/html/2610.28898#S5.SS5)\)\.

##### Contributions\.

1. 1\.DyProd\-FM and optimal rates in both variations\.DyProd\-FM refits flow matching on deterministic, geometrically growing historical prefixes and properly selects one fit per round with a sleeping Prod master at a single fast rate\([Gaillard et al\., 2014](https://arxiv.org/html/2610.28898#bib.bib16)\)\. It receives no budget\. Its dynamic velocity regret isO~​\(min⁡\{Nω\+1,1\+T​Ω\}\)\\widetilde\{O\}\(\\min\\\{N\_\{\\omega\}\+1,1\+\\sqrt\{T\\Omega\}\\\}\)in the relative velocity risk variation \(Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)\) andO~​\(min⁡\{Nμ\+1,1\+T1/3​Vμ2/3\}\)\\widetilde\{O\}\(\\min\\\{N\_\{\\mu\}\+1,1\+T^\{1/3\}V\_\{\\mu\}^\{2/3\}\\\}\)in the target\-law variation \(Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)\), up to approximation and fitting costs\. We complement these with matching lower bounds for fixed classes at fixed problem scales \(Appendix[F](https://arxiv.org/html/2610.28898#A6)\), showing that both rates are optimal inTTand in the budgets up to logarithmic factors\. The target\-law exponents are those of total\-variation forecasting\([Baby and Wang, 2019](https://arxiv.org/html/2610.28898#bib.bib6)\), here for a nonconvex neural class and a proper learner\.
2. 2\.A second\-order link between the two variations\.A target move is exactly a translation of the base noise plus a translation of the label at the same input \(Equation \([35](https://arxiv.org/html/2610.28898#S5.E35)\)\)\. Base smoothing controls how this change affects comparisons between fields \(Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1)\)\. For a field optimal under an earlier target, the drift contribution to excess risk is bounded by a constant times the squared Wasserstein distance to the new target, with additional approximation and smoothing remainders\. The same estimate also bounds relative\-risk variationΩ\\Omegain terms of target\-law variationVμV\_\{\\mu\}\(Proposition[2](https://arxiv.org/html/2610.28898#Thmproposition2)\)\. The smoothing modulus is sharp for Gaussian bases, and the moment condition it requires excludes uniform flow time and admits the truncated uniform, Beta, and logit\-normal laws used in practice \(Proposition[3](https://arxiv.org/html/2610.28898#Thmproposition3)\)\.
3. 3\.Analysis beyond fixed\-target flow matching\.The fast historical rate uses the Bernstein\-type variance control of squared\-loss flow matching\([Sahoo et al\., 2026](https://arxiv.org/html/2610.28898#bib.bib25)\), in the approximate form implied by the regression identity, and our lower bounds build on the rare\-mixture construction of the same work\. What is new is the nonstationary part\. \(i\) Histories are independent but not identically distributed, and their common comparator carries a drift\-induced residual even when every round is realizable \(Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)\)\. \(ii\) The same squared\-loss structure bounds the second moment of the master’s score differences by velocity risks, so selecting among the fits costs a polylogarithmic factor inTTper lifetime, rather than the square root of the lifetime’s length\. \(iii\) Relative\-risk oscillation gives first\-order historical transfer inΩ\\Omega, while base smoothing and approximate quadratic growth give second\-order transfer inVμV\_\{\\mu\}\. \(iv\) The two\-point constructions become block constructions for dynamic regret that keep every class constant fixed\.

## 2Related Work

##### Flow matching\.

Flow matching\([Lipman et al\., 2023](https://arxiv.org/html/2610.28898#bib.bib22);[Lipman et al\., 2024](https://arxiv.org/html/2610.28898#bib.bib23)\), rectified flow\([Liu et al\., 2023](https://arxiv.org/html/2610.28898#bib.bib24)\), and stochastic interpolants\([Albergo and Vanden\-Eijnden, 2023](https://arxiv.org/html/2610.28898#bib.bib1);[Albergo et al\., 2025](https://arxiv.org/html/2610.28898#bib.bib2)\)learn transport velocities by regression\. Fixed\-target theory relates velocity accuracy to sampling error\([Benton et al\., 2024](https://arxiv.org/html/2610.28898#bib.bib10)\)and bounds statistical errors and sample complexity\([Zhou and Liu, 2025](https://arxiv.org/html/2610.28898#bib.bib31);[Gaur et al\., 2025](https://arxiv.org/html/2610.28898#bib.bib17)\)\.[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)obtain order\-optimal fixed\-target rates from Bernstein\-type variance control, with a rare\-mixture lower bound\. Nonuniform flow\-time laws are standard in large\-scale training\([Esser et al\., 2024](https://arxiv.org/html/2610.28898#bib.bib15)\)\. We ask instead how much excess current velocity error accumulates when deployment precedes each fresh batch and the target drifts\.

##### Dynamic regret\.

Tracking changing comparators is classical\([Herbster and Warmuth, 1998](https://arxiv.org/html/2610.28898#bib.bib18);[Zinkevich, 2003](https://arxiv.org/html/2610.28898#bib.bib32);[Besbes et al\., 2015](https://arxiv.org/html/2610.28898#bib.bib11)\)\. Our analysis builds on strongly adaptive interval guarantees\([Daniely et al\., 2015](https://arxiv.org/html/2610.28898#bib.bib14);[Jun et al\., 2017](https://arxiv.org/html/2610.28898#bib.bib19);[Zhang et al\., 2018](https://arxiv.org/html/2610.28898#bib.bib30);[Wei et al\., 2016](https://arxiv.org/html/2610.28898#bib.bib29)\)and on Prod with sleeping experts\([Gaillard et al\., 2014](https://arxiv.org/html/2610.28898#bib.bib16)\)\. Fornnobservations with total variationCnC\_\{n\}under squared loss,[Baby and Wang \(2019\)](https://arxiv.org/html/2610.28898#bib.bib6)establish the minimax raten1/3​Cn2/3n^\{1/3\}C\_\{n\}^\{2/3\}, and[Baby and Wang \(2021\)](https://arxiv.org/html/2610.28898#bib.bib7);[Baby and Wang \(2022\)](https://arxiv.org/html/2610.28898#bib.bib8)attain such rates with proper learners for exp\-concave and strongly convex losses over convex decision sets\. Our target\-law rate has the same exponents for a nonconvex neural class, where the curvature comes from the flow\-matching regression identity and base smoothing rather than from the loss geometry in parameter space\. We also use fast\-rate arguments\([Bartlett et al\., 2005](https://arxiv.org/html/2610.28898#bib.bib9);[Srebro et al\., 2010](https://arxiv.org/html/2610.28898#bib.bib26);[Boucheron et al\., 2013](https://arxiv.org/html/2610.28898#bib.bib13)\)and two\-point lower bounds\([Tsybakov, 2009](https://arxiv.org/html/2610.28898#bib.bib28);[Sahoo et al\., 2026](https://arxiv.org/html/2610.28898#bib.bib25)\)\. Appendix[A](https://arxiv.org/html/2610.28898#A1)gives the extended discussion\.

## 3Problem Setting

### 3\.1Sampling and Velocity Risk

Letddbe the data dimension,T≥1T\\geq 1the number of learning rounds, andm≥1m\\geq 1the batch size\. Learning rounds are indexed bytt, and flow time isr∈\[0,1\]r\\in\[0,1\]\. Throughout,∥⋅∥\\lVert\\cdot\\rVertis the Euclidean norm, and matrix operator and Frobenius norms are denoted by∥⋅∥op\\lVert\\cdot\\rVert\_\{\\mathrm\{op\}\}and∥⋅∥F\\lVert\\cdot\\rVert\_\{F\}\. A fixed base law is denoted byν\\nuand the round\-tttarget byμt\\mu\_\{t\}, both onℝd\\mathbb\{R\}^\{d\}\. An exogenous standard Borel environmentEEfixes the target sequence through jointly measurable kernels\. Conditional onEE, all endpoint\-time triples are independent, and themmtriples within roundtthave common law

\(X0,X1,r\)∼ν⊗μt⊗λ,\(X\_\{0\},X\_\{1\},r\)\\sim\\nu\\otimes\\mu\_\{t\}\\otimes\\lambda,\(1\)whereλ\\lambdais a fixed flow\-time law on\[0,1\]\[0,1\]\. The relative\-risk results useλ=Unif⁡\[0,1\]\\lambda=\\mathrm\{Unif\}\[0,1\]by default, and the target\-law results use a law satisfying Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\. The environment does not respond to the learner\. All envelopes and budget caps below are deterministic and uniform over environments\. We write𝔼t\\mathbb\{E\}\_\{t\}for expectation under one fresh triple at roundtt, conditional onEE\. The outer𝔼\\mathbb\{E\}also averages over data, independent algorithmic seeds, and, if random, the environment\.

LetIr​\(x0,x1\)I\_\{r\}\(x\_\{0\},x\_\{1\}\)be the fixed interpolation, and letZr​\(x0,x1\)Z\_\{r\}\(x\_\{0\},x\_\{1\}\)denote its flow\-time derivative\. The regression input and observed label are, respectively,

U=\(Ir​\(X0,X1\),r\),Z=Zr​\(X0,X1\)\.U=\(I\_\{r\}\(X\_\{0\},X\_\{1\}\),r\),\\qquad Z=Z\_\{r\}\(X\_\{0\},X\_\{1\}\)\.\(2\)For a fixed represented class𝒱\\mathcal\{V\}, define the ideal regression fieldbtb\_\{t\}, population lossLtL\_\{t\}, and absolute velocity riskQtQ\_\{t\}by

bt​\(U\)\\displaystyle b\_\{t\}\(U\)=𝔼t​\[Z∣U\],\\displaystyle=\\mathbb\{E\}\_\{t\}\[Z\\mid U\],\(3\)Lt​\(v\)\\displaystyle L\_\{t\}\(v\)=𝔼t​∥v⁡\(U\)−Z∥2,\\displaystyle=\\mathbb\{E\}\_\{t\}\\lVert v\(U\)\-Z\\rVert^\{2\},Qt​\(v\)\\displaystyle Q\_\{t\}\(v\)=𝔼t​∥v⁡\(U\)−bt​\(U\)∥2\.\\displaystyle=\\mathbb\{E\}\_\{t\}\\lVert v\(U\)\-b\_\{t\}\(U\)\\rVert^\{2\}\.\(4\)All conditional means are formed after fixingEE\. The sampling law ofUUmay change withtt, so these are population risks under the current interpolation law\. We write∥f∥t2=𝔼t​∥f⁡\(U\)∥2\\lVert f\\rVert\_\{t\}^\{2\}=\\mathbb\{E\}\_\{t\}\\lVert f\(U\)\\rVert^\{2\}for the field norm under the round\-ttinput law\.

The best represented approximation error at roundttisata\_\{t\}, andFt​\(v\)F\_\{t\}\(v\)denotes excess velocity risk:

at=minv∈𝒱⁡Qt​\(v\),Ft​\(v\)=Qt​\(v\)−at\.a\_\{t\}=\\min\_\{v\\in\\mathcal\{V\}\}Q\_\{t\}\(v\),\\qquad F\_\{t\}\(v\)=Q\_\{t\}\(v\)\-a\_\{t\}\.\(5\)The assumptions below ensure that the minimum exists\. Conditional\-expectation orthogonality gives

Lt​\(v\)=Qt​\(v\)\+𝔼t​∥Z−bt​\(U\)∥2\.L\_\{t\}\(v\)=Q\_\{t\}\(v\)\+\\mathbb\{E\}\_\{t\}\\lVert Z\-b\_\{t\}\(U\)\\rVert^\{2\}\.\(6\)ConsequentlyFt​\(v\)=Lt​\(v\)−minu∈𝒱⁡Lt​\(u\)F\_\{t\}\(v\)=L\_\{t\}\(v\)\-\\min\_\{u\\in\\mathcal\{V\}\}L\_\{t\}\(u\)\. Ifvt∈𝒱v\_\{t\}\\in\\mathcal\{V\}is deployed at roundtt, define cumulative dynamic velocity regret and cumulative approximation error by

ℛTvel=∑t=1TFt​\(vt\),𝒜T=∑t=1Tat\.\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}=\\sum\_\{t=1\}^\{T\}F\_\{t\}\(v\_\{t\}\),\\qquad\\mathcal\{A\}\_\{T\}=\\sum\_\{t=1\}^\{T\}a\_\{t\}\.\(7\)Thus∑tQt​\(vt\)=ℛTvel\+𝒜T\\sum\_\{t\}Q\_\{t\}\(v\_\{t\}\)=\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\+\\mathcal\{A\}\_\{T\}exactly, and sinceLt−FtL\_\{t\}\-F\_\{t\}does not depend on the field, minimizing∑tFt​\(vt\)\\sum\_\{t\}F\_\{t\}\(v\_\{t\}\)is minimizing∑tLt​\(vt\)\\sum\_\{t\}L\_\{t\}\(v\_\{t\}\)\. Regret measures competition with the current class optimum, and approximation measures the class’s remaining distance to the ideal velocity\.

### 3\.2Two Measures of Nonstationarity

##### Relative velocity risk variation\.

We measure class\-relative nonstationarity by the half\-oscillation of a one\-round population\-risk increment\. For1≤t<T1\\leq t<T, define

ωt=12supu,v∈𝒱\|\\displaystyle\\omega\_\{t\}=\\frac\{1\}\{2\}\\sup\_\{u,v\\in\\mathcal\{V\}\}\\big\|\[Lt\+1​\(v\)−Lt\+1​\(u\)\]\\displaystyle\[L\_\{t\+1\}\(v\)\-L\_\{t\+1\}\(u\)\]−\[Lt\(v\)−Lt\(u\)\]\|\.\\displaystyle\-\[L\_\{t\}\(v\)\-L\_\{t\}\(u\)\]\\big\|\.\(8\)Replacing everyLtL\_\{t\}byQtQ\_\{t\}leaves this quantity unchanged\. LetΩT\\Omega\_\{T\}be total relative\-risk variation andNTωN\_\{T\}^\{\\omega\}its number of nonzero increments\. We use almost\-sure capsΩ\\OmegaandNωN\_\{\\omega\}:

ΩT=∑t<Tωt≤Ω,NTω=∑t<T𝟏\{ωt\>0\}≤Nω\.\\Omega\_\{T\}=\\sum\_\{t<T\}\\omega\_\{t\}\\leq\\Omega,\\quad N\_\{T\}^\{\\omega\}=\\sum\_\{t<T\}\\mathbf\{1\}\\\{\\omega\_\{t\}\>0\\\}\\leq N\_\{\\omega\}\.\(9\)These caps appear only in the analysis\. They are not supplied to the learner\. Common additive shifts of all fields’ risks contribute zero variation\.

The budget compares complete current risks under their respective input laws while removing the field\-independent noise term in Equation \([6](https://arxiv.org/html/2610.28898#S3.E6)\), and it is class dependent\. Its half\-oscillation form givessupv∈𝒱\|Ft\+1​\(v\)−Ft​\(v\)\|≤2​ωt\\sup\_\{v\\in\\mathcal\{V\}\}\|F\_\{t\+1\}\(v\)\-F\_\{t\}\(v\)\|\\leq 2\\omega\_\{t\}\.

##### Target\-law variation\.

LetW2W\_\{2\}be the quadratic Wasserstein distance onℝd\\mathbb\{R\}^\{d\}\. Define the target\-law variation, its change count, and their caps by

VTμ\\displaystyle V\_\{T\}^\{\\mu\}=∑t<TW2​\(μt\+1,μt\)≤Vμ,\\displaystyle=\\sum\_\{t<T\}W\_\{2\}\(\\mu\_\{t\+1\},\\mu\_\{t\}\)\\leq V\_\{\\mu\},NTμ\\displaystyle N\_\{T\}^\{\\mu\}=∑t<T𝟏\{μt\+1≠μt\}≤Nμ\.\\displaystyle=\\sum\_\{t<T\}\\mathbf\{1\}\\\{\\mu\_\{t\+1\}\\neq\\mu\_\{t\}\\\}\\leq N\_\{\\mu\}\.\(10\)This budget does not depend on the class\. With a fixed base, flow\-time law, and independent pairing,μt\+1=μt\\mu\_\{t\+1\}=\\mu\_\{t\}forcesLt\+1=LtL\_\{t\+1\}=L\_\{t\}, soNTω≤NTμN\_\{T\}^\{\\omega\}\\leq N\_\{T\}^\{\\mu\}\. Under Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5), Proposition[2](https://arxiv.org/html/2610.28898#Thmproposition2)boundsΩT\\Omega\_\{T\}by a multiple ofVTμV\_\{T\}^\{\\mu\}up to small remainders, but no class\-independent equivalence holds in either direction\.

### 3\.3Assumptions

Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[4](https://arxiv.org/html/2610.28898#Thmassumption4)follow the standard setting of fixed\-target flow\-matching sample\-complexity analysis\([Sahoo et al\., 2026](https://arxiv.org/html/2610.28898#bib.bib25)\): sub\-Gaussian endpoints, a label envelope for the path, a bounded feed\-forward network, and access to an empirical minimizer, which Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)relaxes to an approximate one\.

###### Assumption 1\(Endpoint envelope\)\.

A knownκ\>0\\kappa\>0bounds both endpoints, uniformly inttand conditionally onEE:

𝔼t​exp⁡\(∥Xj∥2/κ2\)≤2,j∈\{0,1\}\.\\mathbb\{E\}\_\{t\}\\exp\(\\lVert X\_\{j\}\\rVert^\{2\}/\\kappa^\{2\}\)\\leq 2,\\qquad j\\in\\\{0,1\\\}\.\(11\)

###### Assumption 2\(Regular compatible path\)\.

The known interpolation is jointly Borel and absolutely continuous inrrfor every endpoint pair, withI0​\(x0,x1\)=x0I\_\{0\}\(x\_\{0\},x\_\{1\}\)=x\_\{0\}andI1​\(x0,x1\)=x1I\_\{1\}\(x\_\{0\},x\_\{1\}\)=x\_\{1\}\. It has a jointly Borel derivative versionZrZ\_\{r\}and a knowncZ\>0c\_\{Z\}\>0such that

∥Zr​\(x0,x1\)∥≤cZ​\(∥x0∥\+∥x1∥\+1\)\\lVert Z\_\{r\}\(x\_\{0\},x\_\{1\}\)\\rVert\\leq c\_\{Z\}\(\\lVert x\_\{0\}\\rVert\+\\lVert x\_\{1\}\\rVert\+1\)\(12\)for almost everyrrfor each pair\. Use the version set to zero where this envelope fails\. Flow time in \([1](https://arxiv.org/html/2610.28898#S3.E1)\) has a fixed lawλ\\lambdaon\[0,1\]\[0,1\], chosen by the learner\.

###### Assumption 3\(Constrained neural class\)\.

FixD≥2D\\geq 2affine layers with widthsn0=d\+1,n1,…,nD−1,nD=dn\_\{0\}=d\+1,n\_\{1\},\\ldots,n\_\{D\-1\},n\_\{D\}=d\. For inputh0=\(x,r\)h\_\{0\}=\(x,r\), hidden stateshjh\_\{j\}, weight matricesWjW\_\{j\}, bias vectorsbjb\_\{j\}, fixed coordinatewise activationsϕj\\phi\_\{j\}, and parameter vectorθ\\theta, the field is

hj=ϕj​\(Wj​hj−1\+bj\)​\(j<D\),vθ=WD​hD−1\+bD\.h\_\{j\}=\\phi\_\{j\}\(W\_\{j\}h\_\{j\-1\}\+b\_\{j\}\)\\ \(j<D\),\\quad v\_\{\\theta\}=W\_\{D\}h\_\{D\-1\}\+b\_\{D\}\.\(13\)Fixed coordinatewise activations are differentiable,LϕL\_\{\\phi\}\-Lipschitz, and bounded in\[−b,b\]\[\-b,b\]for finiteLϕ,bL\_\{\\phi\},b\. The first hidden activation is nondecreasing\. The nonempty parameter setΘ⊂ℝP\\Theta\\subset\\mathbb\{R\}^\{P\}is closed, with finite boundsVj,cjV\_\{j\},c\_\{j\}satisfying∥Wj∥op≤Vj\\lVert W\_\{j\}\\rVert\_\{\\mathrm\{op\}\}\\leq V\_\{j\}and∥bj∥≤cj\\lVert b\_\{j\}\\rVert\\leq c\_\{j\}\. HerePPcounts the trainable scalar entries, each once\. Fixed entries are allowed\. Set𝒱=\{vθ:θ∈Θ\}\\mathcal\{V\}=\\\{v\_\{\\theta\}:\\theta\\in\\Theta\\\}, fixed across rounds\.

Proposition[B\.1](https://arxiv.org/html/2610.28898#Thmproposition1a)derives compactness, parameter covers, a global output boundBvB\_\{v\}, and a global spatial Lipschitz constantLxL\_\{x\}:

Bv=VD​b​nD−1\+cD,Lx=VD​∏j<D\(Lϕ​Vj\)\.B\_\{v\}=V\_\{D\}b\\sqrt\{n\_\{D\-1\}\}\+c\_\{D\},\\quad L\_\{x\}=V\_\{D\}\\prod\_\{j<D\}\(L\_\{\\phi\}V\_\{j\}\)\.\(14\)
To state the fitting requirement, letM\>0M\>0be a deterministic threshold andGMG\_\{M\}its endpoint gate\. Define the proxy lossℓM\\ell\_\{M\}and its boundBBby

GM\\displaystyle G\_\{M\}=𝟏​\{∥X0∥≤M,∥X1∥≤M\},\\displaystyle=\\mathbf\{1\}\\\{\\lVert X\_\{0\}\\rVert\\leq M,\\lVert X\_\{1\}\\rVert\\leq M\\\},ℓM​\(v\)\\displaystyle\\ell\_\{M\}\(v\)=GM​∥v⁡\(U\)−Z∥2,\\displaystyle=G\_\{M\}\\lVert v\(U\)\-Z\\rVert^\{2\},\(15\)B\\displaystyle B=\[Bv\+cZ​\(2​M\+1\)\]2\>0\.\\displaystyle=\[B\_\{v\}\+c\_\{Z\}\(2M\+1\)\]^\{2\}\>0\.For a deterministic historyHHcontainingqqrounds, its empirical proxy objectiveL^H,M\\widehat\{L\}\_\{H,M\}averagesℓM\\ell\_\{M\}over allm​qmqexamples, including rejected examples as zero summands\.

###### Assumption 4\(Empirical fitting interface\)\.

The fitting routine has an accuracyϵopt≥0\\epsilon\_\{\\mathrm\{opt\}\}\\geq 0: every call returns a measurablev^∈𝒱\\widehat\{v\}\\in\\mathcal\{V\}using only its historical data and independent randomness, with

L^H,M​\(v^\)≤infv∈𝒱L^H,M​\(v\)\+ϵopt\\widehat\{L\}\_\{H,M\}\(\\widehat\{v\}\)\\leq\\inf\_\{v\\in\\mathcal\{V\}\}\\widehat\{L\}\_\{H,M\}\(v\)\+\\epsilon\_\{\\mathrm\{opt\}\}\(16\)almost surely\. The learner knowsT,mT,mand validκ,cZ,Bv\\kappa,c\_\{Z\},B\_\{v\}envelopes for gating and normalization\.

The target\-law results add a design condition on the path, the base, and the flow\-time law\. For affine schedulesα,β\\alpha,\\betawriteJ=α​β′−α′​βJ=\\alpha\\beta^\{\\prime\}\-\\alpha^\{\\prime\}\\betafor their Wronskian, and for a base densitypνp\_\{\\nu\}withpν∈H1​\(ℝd\)\\sqrt\{p\_\{\\nu\}\}\\in H^\{1\}\(\\mathbb\{R\}^\{d\}\)writeIν=4∫∇pν∇pν⊤I\_\{\\nu\}=4\\int\\nabla\\sqrt\{p\_\{\\nu\}\}\\,\\nabla\\sqrt\{p\_\{\\nu\}\}^\{\\top\}for its Fisher information\.

###### Assumption 5\(Smoothing design\)\.

\(a\) The interpolation is affine,Ir​\(x0,x1\)=α⁡\(r\)​x0\+β⁡\(r\)​x1I\_\{r\}\(x\_\{0\},x\_\{1\}\)=\\alpha\(r\)x\_\{0\}\+\\beta\(r\)x\_\{1\}withα,β∈C1​\(\[0,1\]\)\\alpha,\\beta\\in C^\{1\}\(\[0,1\]\),α⁡\(0\)=β⁡\(1\)=1\\alpha\(0\)=\\beta\(1\)=1,α⁡\(1\)=β⁡\(0\)=0\\alpha\(1\)=\\beta\(0\)=0, soZr=α′​\(r\)​x0\+β′​\(r\)​x1Z\_\{r\}=\\alpha^\{\\prime\}\(r\)x\_\{0\}\+\\beta^\{\\prime\}\(r\)x\_\{1\}and \([12](https://arxiv.org/html/2610.28898#S3.E12)\) holds withcZ=max⁡\{∥α′∥∞,∥β′∥∞\}c\_\{Z\}=\\max\\\{\\lVert\\alpha^\{\\prime\}\\rVert\_\{\\infty\},\\lVert\\beta^\{\\prime\}\\rVert\_\{\\infty\}\\\}\. \(b\) The base has a density withpν∈H1​\(ℝd\)\\sqrt\{p\_\{\\nu\}\}\\in H^\{1\}\(\\mathbb\{R\}^\{d\}\)andι:=λmax​\(Iν\)<∞\\iota:=\\lambda\_\{\\max\}\(I\_\{\\nu\}\)<\\infty\. \(c\) Flow time has a fixed lawλ\\lambdawithα⁡\(r\)\>0\\alpha\(r\)\>0forλ\\lambda\-almost everyrrand

mh:=𝔼λ​β​\(r\)2α​\(r\)2<∞,mk:=𝔼λ​J​\(r\)2α​\(r\)2<∞\.m\_\{h\}:=\\mathbb\{E\}\_\{\\lambda\}\\frac\{\\beta\(r\)^\{2\}\}\{\\alpha\(r\)^\{2\}\}<\\infty,\\qquad m\_\{k\}:=\\mathbb\{E\}\_\{\\lambda\}\\frac\{J\(r\)^\{2\}\}\{\\alpha\(r\)^\{2\}\}<\\infty\.\(17\)

##### Predictive information\.

Before roundtt, candidate fields and selection probabilities depend only on earlier batches and seeds\. The learner observes labelsZZ, notbtb\_\{t\}\.

## 4Main Results

### 4\.1The Learner: DyProd\-FM

DyProd\-FM keeps fits with different starting times to balance estimation error against drift\. The dyadic family𝒟T\\mathcal\{D\}\_\{T\}consists of the distinct nonempty intersections of binary\-tree nodes on\[1,2⌈log2⁡T⌉\]\[1,2^\{\\lceil\\log\_\{2\}T\\rceil\}\]with\[1,T\]\[1,T\]\. Its root is\[1,T\]\[1,T\]\. At most1\+⌈log2⁡T⌉1\+\\lceil\\log\_\{2\}T\\rceillifetimes are active per round\. Each fits the largest power\-of\-two prefix of its past and refits when that prefix doubles\. Sleeping Prod\([Gaillard et al\., 2014](https://arxiv.org/html/2610.28898#bib.bib16)\)selects one field before the fresh batch arrives, so every deployed field belongs to𝒱\\mathcal\{V\}\.

Algorithm 1:DyProd\-FM

Given:horizonTTand batch sizemm\. Design parameters:flow\-time lawλ\\lambda, gate radiusMM\(default in Equation \([24](https://arxiv.org/html/2610.28898#S4.E24)\)\), normalizationB=\[Bv\+cZ​\(2​M\+1\)\]2B=\[B\_\{v\}\+c\_\{Z\}\(2M\+1\)\]^\{2\}, master rateη=1/32\\eta=1/32, and a training routinePrefixFMfor the network of Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3)\(Algorithm[B\.2](https://arxiv.org/html/2610.28898#A2.SS2)\)\.

1. 1:Form the dyadic lifetimes𝒟T\\mathcal\{D\}\_\{T\}, and setMe=\|𝒟T\|M\_\{e\}=\|\\mathcal\{D\}\_\{T\}\|andwI=1/Mew\_\{I\}=1/M\_\{e\}for allI∈𝒟TI\\in\\mathcal\{D\}\_\{T\}\. For each startaa, drawθa\(0\)=ΠΘ​\(ga\)\\theta\_\{a\}^\{\(0\)\}=\\Pi\_\{\\Theta\}\(g\_\{a\}\),ga∼N⁡\(0,P−1​IP\)g\_\{a\}\\sim N\(0,P^\{\-1\}I\_\{P\}\), whereIPI\_\{P\}is the identity matrix andΠΘ\\Pi\_\{\\Theta\}is a measurable Euclidean projection \(Appendix[B\.2\.1](https://arxiv.org/html/2610.28898#A2.SS2.SSS1)\)\.
2. 2:fort=1,…,Tt=1,\\ldots,Tdo
3. 3:Process the root lifetime first\. For each activeI=\[a,b\]∋tI=\[a,b\]\\ni t: ift=at=a, letvI,tv\_\{I,t\}be the most recent completed root fit, orvθ1\(0\)v\_\{\\theta\_\{1\}^\{\(0\)\}\}whent=1t=1\. Otherwiseq=2⌊log2⁡\(t−a\)⌋q=2^\{\\lfloor\\log\_\{2\}\(t\-a\)\\rfloor\}andvI,t=PrefixFM​\(L^\[a,a\+q−1\],M\)v\_\{I,t\}=\\textsc\{PrefixFM\}\(\\widehat\{L\}\_\{\[a,a\+q\-1\],M\}\), started fromθa\(0\)\\theta\_\{a\}^\{\(0\)\}ifq=1q=1and from the fit forq/2q/2otherwise, refitted only whenqqdoubles\.
4. 4:DrawItI\_\{t\}withpI,t=wI/∑K∋twKp\_\{I,t\}=w\_\{I\}/\\sum\_\{K\\ni t\}w\_\{K\}and deployvt=vIt,tv\_\{t\}=v\_\{I\_\{t\},t\}\.
5. 5:ReceiveX1,t,i∼μtX\_\{1,t,i\}\\sim\\mu\_\{t\}, drawX0,t,i∼νX\_\{0,t,i\}\\sim\\nuandrt,i∼λr\_\{t,i\}\\sim\\lambda, form triplesξt,i=\(X0,t,i,X1,t,i,rt,i\)\\xi\_\{t,i\}=\(X\_\{0,t,i\},X\_\{1,t,i\},r\_\{t,i\}\)\(i≤mi\\leq m\), and scorexI,t=1B​m​∑iℓM​\(vI,t,ξt,i\)x\_\{I,t\}=\\frac\{1\}\{Bm\}\\sum\_\{i\}\\ell\_\{M\}\(v\_\{I,t\};\\xi\_\{t,i\}\)andx¯t=∑I∋tpI,t​xI,t\\bar\{x\}\_\{t\}=\\sum\_\{I\\ni t\}p\_\{I,t\}x\_\{I,t\}\.
6. 6:UpdatewI←wI​\[1\+η⁡\(x¯t−xI,t\)\]w\_\{I\}\\leftarrow w\_\{I\}\[1\+\\eta\(\\bar\{x\}\_\{t\}\-x\_\{I,t\}\)\]for activeII\. Inactive weights are unchanged\.
7. 7:end for

Appendix[B\.2](https://arxiv.org/html/2610.28898#A2.SS2)specifies initialization and prefix training\. Appendix[C\.2\.4](https://arxiv.org/html/2610.28898#A3.SS2.SSS4)gives fitting, scoring, memory, and per\-round costs conditional on the fitting interface\.

### 4\.2Theoretical Guarantees

Expected dynamic velocity regret𝔼​ℛTvel=𝔼​∑t=1TFt​\(vt\)\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}=\\mathbb\{E\}\\sum\_\{t=1\}^\{T\}F\_\{t\}\(v\_\{t\}\)compares DyProd\-FM with each round’s best represented field\. The bounds𝖥T\\mathsf\{F\}\_\{T\}and𝖳T\\mathsf\{T\}\_\{T\}use the drift measures of Section[3\.2](https://arxiv.org/html/2610.28898#S3.SS2)\. Dividing byTTgives expected average excess velocity risk\. Use Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)’s rateη=1/32\\eta=1/32and anyM\>0M\>0\. Drift budgets enter only the analysis\.

LetMe=\|𝒟T\|<4​TM\_\{e\}=\|\\mathcal\{D\}\_\{T\}\|<4T,Λ=log⁡\(e​Me\)\\Lambda=\\log\(eM\_\{e\}\),ζ=B/\(m​T\)2\\zeta=B/\(mT\)^\{2\}, andΓ=1\+log⁡\(2​KM\)\\Gamma=1\+\\log\(2K\_\{M\}\)\. HereKMK\_\{M\}is the size of a deterministic uniformζ\\zeta\-cover of proxy losses \(Appendix[B\.4](https://arxiv.org/html/2610.28898#A2.SS4)\), used only in the analysis\. Proposition[B\.2](https://arxiv.org/html/2610.28898#Thmproposition2a)bounds the uniform original\-to\-proxy risk discrepancy byδ=δM\\delta=\\delta\_\{M\}, where

Cℓ\\displaystyle C\_\{\\ell\}=2​\[\(Bv\+cZ\)2\+cZ2\],\\displaystyle=2\[\(B\_\{v\}\+c\_\{Z\}\)^\{2\}\+c\_\{Z\}^\{2\}\],δM\\displaystyle\\delta\_\{M\}=2Cℓ\(1\+M2\+4κ2\)e−M2/\(4κ2\)\.\\displaystyle=2C\_\{\\ell\}\(1\+M^\{2\}\+4\\kappa^\{2\}\)e^\{\-M^\{2\}/\(4\\kappa^\{2\}\)\}\.\(18\)
###### Theorem 1\(Relative\-risk tracking\)\.

Under the predictive protocol, Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[4](https://arxiv.org/html/2610.28898#Thmassumption4), and the caps in Equation \([9](https://arxiv.org/html/2610.28898#S3.E9)\), Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)satisfies𝔼​ℛTvel≤𝖥T\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq\\mathsf\{F\}\_\{T\}, where

𝖥T=\\displaystyle\\mathsf\{F\}\_\{T\}=\{\}min⁡\{KT​\(Nω\+1\),KT\+2​10​KT​T​Ω\}\\displaystyle\\min\\big\\\{K\_\{T\}\(N\_\{\\omega\}\+1\),\\ K\_\{T\}\+2\\sqrt\{10K\_\{T\}T\\Omega\}\\big\\\}\+HT​𝔼​𝒜T\+T​ef\.\\displaystyle\+H\_\{T\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}\.\(19\)The interval costKTK\_\{T\}, approximation multiplierHTH\_\{T\}, residualefe\_\{f\}, and dyadic\-cover size boundDTD\_\{T\}are

DT\\displaystyle D\_\{T\}=2​\(1\+⌈log2⁡T⌉\),HT=8​log⁡\(e​T\),\\displaystyle=2\(1\+\\lceil\\log\_\{2\}T\\rceil\),\\quad H\_\{T\}=8\\log\(eT\),KT\\displaystyle K\_\{T\}=DT​\[44​B​Λ\+120​B​Γm​log⁡\(e​T\)\],\\displaystyle=D\_\{T\}\\left\[44B\\Lambda\+\\frac\{120B\\Gamma\}\{m\}\\log\(eT\)\\right\],\(20\)ef\\displaystyle e\_\{f\}=4​ϵopt\+9​ζ\+20​δ\.\\displaystyle=4\\epsilon\_\{\\mathrm\{opt\}\}\+9\\zeta\+20\\delta\.

The learner adapts to the smaller cost of countNωN\_\{\\omega\}and variationΩ\\Omega\. Approximation costsHT​𝔼​𝒜TH\_\{T\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}, whileT​efTe\_\{f\}collects fitting, covering, and gating errors\. Section[5\.3](https://arxiv.org/html/2610.28898#S5.SS3)outlines the proof, detailed in Appendix[D](https://arxiv.org/html/2610.28898#A4)\. For the variation branch with fixed positiveΩ\\Omega, takeϵopt=O\(T−1/2\)\\epsilon\_\{\\mathrm\{opt\}\}=O\(T^\{\-1/2\}\)\.

For target\-law tracking, base smoothing gives sharper risk transfer\. Define the base\-translation and label\-shift modulih⁡\(r\)=ι/8​\|β⁡\(r\)\|/α⁡\(r\)h\(r\)=\\sqrt\{\\iota/8\}\\,\|\\beta\(r\)\|/\\alpha\(r\)andk⁡\(r\)=\|J⁡\(r\)\|/α⁡\(r\)k\(r\)=\|J\(r\)\|/\\alpha\(r\), and the label thresholdMz=cZ​\(1\+M\)M\_\{z\}=c\_\{Z\}\(1\+M\)\. Using the momentsmh,mkm\_\{h\},m\_\{k\}of Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5), set

Λsm\\displaystyle\\Lambda\_\{\\mathrm\{sm\}\}=2​mk\+4​3​\(Bv\+Mz\)​ι​mh/8,\\displaystyle=2\\sqrt\{m\_\{k\}\}\+4\\sqrt\{3\}\\,\(B\_\{v\}\+M\_\{z\}\)\\sqrt\{\\iota m\_\{h\}/8\},ϵsm\\displaystyle\\epsilon\_\{\\mathrm\{sm\}\}=16​3​Bv​ι​mh/8​δM/2,\\displaystyle=16\\sqrt\{3\}\\,B\_\{v\}\\sqrt\{\\iota m\_\{h\}/8\}\\,\\sqrt\{\\delta\_\{M\}/2\},c2\\displaystyle c\_\{2\}=8​3​Bv​𝔼λ​\[h​k\],c¯=99​Λsm2\+37​c2\.\\displaystyle=8\\sqrt\{3\}\\,B\_\{v\}\\mathbb\{E\}\_\{\\lambda\}\[hk\],\\quad\\bar\{c\}=99\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\+37c\_\{2\}\.\(21\)In Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1),Λsm\\Lambda\_\{\\mathrm\{sm\}\}controls smoothing,ϵsm\\epsilon\_\{\\mathrm\{sm\}\}label tails, andc2c\_\{2\}the quadratic remainder\. Their interval coefficient isc¯\\bar\{c\}\.

###### Theorem 2\(Target\-law tracking\)\.

Under the predictive protocol, Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1),[3](https://arxiv.org/html/2610.28898#Thmassumption3),[4](https://arxiv.org/html/2610.28898#Thmassumption4),[5](https://arxiv.org/html/2610.28898#Thmassumption5), and the caps in Equation \([10](https://arxiv.org/html/2610.28898#S3.E10)\), Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)satisfies𝔼​ℛTvel≤𝖳T\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq\\mathsf\{T\}\_\{T\}, where

𝖳T=\\displaystyle\\mathsf\{T\}\_\{T\}=\{\}min\{KT\(Nμ\+1\)\+HT𝔼𝒜T\+Tef,\\displaystyle\\min\\Big\\\{K\_\{T\}\(N\_\{\\mu\}\+1\)\+H\_\{T\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\},2​KT\+\+2​\(2​c¯\)1/3​\(KT\+\)2/3​T1/3​Vμ2/3\\displaystyle\\quad 2K\_\{T\}^\{\+\}\+2\(2\\bar\{c\}\)^\{1/3\}\(K\_\{T\}^\{\+\}\)^\{2/3\}T^\{1/3\}V\_\{\\mu\}^\{2/3\}\+37ϵsmTVμ\+HT\+𝔼𝒜T\+Tef\+\}\.\\displaystyle\\quad\+37\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\}\+H\_\{T\}^\{\+\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}^\{\+\}\\Big\\\}\.\(22\)The variation branch uses the interval cost, approximation multiplier, and per\-round residual

HT\+\\displaystyle H\_\{T\}^\{\+\}=300​log⁡\(e​T\)\+138,\\displaystyle=300\\log\(eT\)\+138,KT\+\\displaystyle K\_\{T\}^\{\+\}=DT​\[44​B​Λ\+360​B​Γm​log⁡\(e​T\)\],\\displaystyle=D\_\{T\}\\left\[44B\\Lambda\+\\frac\{360B\\Gamma\}\{m\}\\log\(eT\)\\right\],\(23\)ef\+\\displaystyle e\_\{f\}^\{\+\}=10​ϵopt\+25​ζ\+55​δ\.\\displaystyle=10\\epsilon\_\{\\mathrm\{opt\}\}\+25\\zeta\+55\\delta\.

The bound𝖳T\\mathsf\{T\}\_\{T\}adapts to countNμN\_\{\\mu\}or Wasserstein movementVμV\_\{\\mu\}\. Quadratic transfer gives theT1/3​Vμ2/3T^\{1/3\}V\_\{\\mu\}^\{2/3\}cost and remainder37​ϵsm​T​Vμ37\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\}, alongside approximation and fitting errors\. Section[5\.5](https://arxiv.org/html/2610.28898#S5.SS5)outlines the proof, detailed in Appendix[E](https://arxiv.org/html/2610.28898#A5)\. For the variation branch with fixed positiveVμV\_\{\\mu\}, takeϵopt=O\(T−2/3\)\\epsilon\_\{\\mathrm\{opt\}\}=O\(T^\{\-2/3\}\)\.

The constants are valid atT=m=1T=m=1, and both theorems apply to one run under their joint assumptions\. Choosing

M=4​κ​log⁡\(e​m​T\)M=4\\kappa\\sqrt\{\\log\(emT\)\}\(24\)gives the cumulative proxy bias

T​δM≤2​Cℓ​T​\[1\+16​κ2​log⁡\(e​m​T\)\+4​κ2\]\(e​m​T\)4,T\\delta\_\{M\}\\leq\\frac\{2C\_\{\\ell\}T\[1\+16\\kappa^\{2\}\\log\(emT\)\+4\\kappa^\{2\}\]\}\{\(emT\)^\{4\}\},\(25\)andϵsm​T​Vμ=O⁡\(Vμ​log⁡\(e​m​T\)/\(m2​T\)\)\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\}=O\(V\_\{\\mu\}\\sqrt\{\\log\(emT\)\}/\(m^\{2\}T\)\)\. At fixed dimension, batch size, architecture, and regularity scales,B,Γ,Λ,ΛsmB,\\Gamma,\\Lambda,\\Lambda\_\{\\mathrm\{sm\}\}grow only polylogarithmically inTT, giving the dynamic rates in the abstract\. A persistent approximation errorat=a\>0a\_\{t\}=a\>0contributesHT​a​TH\_\{T\}aTorHT\+​a​TH\_\{T\}^\{\+\}aT, depending on the branch\. Appendix[F](https://arxiv.org/html/2610.28898#A6)accounts for approximation when matching the lower bounds\.

##### Lower bounds\.

Appendix[F](https://arxiv.org/html/2610.28898#A6)complements Theorems[1](https://arxiv.org/html/2610.28898#Thmtheorem1)and[2](https://arxiv.org/html/2610.28898#Thmtheorem2)with lower bounds for fixed classes and problem constants\. Only the target sequence varies\. Gaussian targets with at mostN<TN<Tmean changes force regret of orderN\+1N\+1in both variations \(Proposition[F\.1](https://arxiv.org/html/2610.28898#Thmproposition1b)\)\. A rare\-mixture family, the block version of the two\-point construction of[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25), forcesT​Ω/m\\sqrt\{T\\Omega/m\}under uniform flow time for one fixed class, whose trainable slope lets its cutoff follow the far target component \(Proposition[F\.2](https://arxiv.org/html/2610.28898#Thmproposition2b)\)\. Gaussian targets with slowly moving means forceT1/3Vμ2/3m−2/3T^\{1/3\}V\_\{\\mu\}^\{2/3\}m^\{\-2/3\}for every flow\-time law of Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(Proposition[F\.3](https://arxiv.org/html/2610.28898#Thmproposition3a)\)\. Theorems[1](https://arxiv.org/html/2610.28898#Thmtheorem1)and[2](https://arxiv.org/html/2610.28898#Thmtheorem2)match these bounds up to polylogarithmic factors inm​TmT, factors ofmm, and additive fitting costs\.

## 5Proof Outline of the Main Results

Our flow\-matching analysis contributes fast estimation from heterogeneous histories, a risk\-dependent second moment for proper selection, and quadratic transfer under target\-law drift\. Standard dyadic covers and Prod potentials assemble these estimates\. Following Appendix[C](https://arxiv.org/html/2610.28898#A3), we establish the shared results, then complete Theorems[1](https://arxiv.org/html/2610.28898#Thmtheorem1)and[2](https://arxiv.org/html/2610.28898#Thmtheorem2)as in Appendices[D](https://arxiv.org/html/2610.28898#A4)and[E](https://arxiv.org/html/2610.28898#A5)\.

### 5\.1Historical Estimation for Changing Targets

For a deterministic historical intervalHHofqqrounds, letn=m​qn=mqbe its number of examples\. Define its average population lossL¯H\\bar\{L\}\_\{H\}, best common fieldvH⋆v\_\{H\}^\{\\star\}, common\-field approximation errora¯H\\bar\{a\}\_\{H\}, and excess lossrHr\_\{H\}by

L¯H​\(v\)\\displaystyle\\bar\{L\}\_\{H\}\(v\)=1q​∑s∈HLs​\(v\),vH⋆∈arg​minv∈𝒱⁡L¯H​\(v\),\\displaystyle=\\frac\{1\}\{q\}\\sum\_\{s\\in H\}L\_\{s\}\(v\),\\quad v\_\{H\}^\{\\star\}\\in\\argmin\_\{v\\in\\mathcal\{V\}\}\\bar\{L\}\_\{H\}\(v\),a¯H\\displaystyle\\bar\{a\}\_\{H\}=1q​∑s∈HQs​\(vH⋆\),rH​\(v\)=L¯H​\(v\)−L¯H​\(vH⋆\)\.\\displaystyle=\\frac\{1\}\{q\}\\sum\_\{s\\in H\}Q\_\{s\}\(v\_\{H\}^\{\\star\}\),\\quad r\_\{H\}\(v\)=\\bar\{L\}\_\{H\}\(v\)\-\\bar\{L\}\_\{H\}\(v\_\{H\}^\{\\star\}\)\.\(26\)One field must serve all historical laws, soa¯H\\bar\{a\}\_\{H\}can exceed the average of theasa\_\{s\}and remain positive even when every round is realizable\. This drift\-induced residual extends fixed\-target estimation: pooling reduces sampling error but can increase the common comparator’s cost\.

Index the independent historical triples byχi\\chi\_\{i\},i=1,…,ni=1,\\ldots,n, and definegi​\(v\)=ℓM​\(v,χi\)−ℓM​\(vH⋆,χi\)g\_\{i\}\(v\)=\\ell\_\{M\}\(v;\\chi\_\{i\}\)\-\\ell\_\{M\}\(v\_\{H\}^\{\\star\};\\chi\_\{i\}\)\. Lemma[C\.1](https://arxiv.org/html/2610.28898#Thmlemma1a)in Appendix[C\.1](https://arxiv.org/html/2610.28898#A3.SS1)gives the key second\-moment bound

1n​∑iVar⁡\(gi​\(v\)\)≤8​B​\{rH​\(v\)\+2​a¯H\}\.\\frac\{1\}\{n\}\\sum\_\{i\}\\operatorname\{Var\}\(g\_\{i\}\(v\)\)\\leq 8B\\\{r\_\{H\}\(v\)\+2\\bar\{a\}\_\{H\}\\\}\.\(27\)Foru,v∈𝒱u,v\\in\\mathcal\{V\}, the absolute difference of their gated squared residuals is at most2​B​∥u⁡\(U\)−v⁡\(U\)∥2\\sqrt\{B\}\\lVert u\(U\)\-v\(U\)\\rVert\. Squaring and comparing both fields with the round\-specific ideal velocity proves \([27](https://arxiv.org/html/2610.28898#S5.E27)\)\. This approximate Bernstein condition in function space comes from the regression identity in Equation \([6](https://arxiv.org/html/2610.28898#S3.E6)\), not from parameter\-space quadratic growth or a Polyak–Łojasiewicz inequality\. Its approximation term cannot be deleted for nonconvex classes \(Appendix[C](https://arxiv.org/html/2610.28898#A3)\)\.

###### Proposition 1\(Historical estimation\)\.

Under Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[4](https://arxiv.org/html/2610.28898#Thmassumption4), fix a deterministic history ofq≥1q\\geq 1rounds, condition onEE, and letv^\\widehat\{v\}satisfy \([16](https://arxiv.org/html/2610.28898#S3.E16)\)\. Then, withn=m​qn=mq,

𝔼⁡\[rH​\(v^\)∣E\]≤36​B​Γm​q\+2​a¯H\+2​ϵopt\+5​ζ\+10​δ,\\mathbb\{E\}\[r\_\{H\}\(\\widehat\{v\}\)\\mid E\]\\leq\\frac\{36B\\Gamma\}\{mq\}\+2\\bar\{a\}\_\{H\}\+2\\epsilon\_\{\\mathrm\{opt\}\}\+5\\zeta\+10\\delta,\(28\)and the high\-probability bound \([70](https://arxiv.org/html/2610.28898#A3.E70)\) of Appendix[C](https://arxiv.org/html/2610.28898#A3)holds\.

##### Proof of Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)\.

Lemma[C\.1](https://arxiv.org/html/2610.28898#Thmlemma1a)makes Bernstein’s deviation depend on excess risk\. A union bound over the loss cover gives a uniform inequality that applies to the data\-dependent fit\. Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)bounds its empirical excess byϵopt\\epsilon\_\{\\mathrm\{opt\}\}\. Absorbing the square\-root risk term and integrating the tail yields Equation \([28](https://arxiv.org/html/2610.28898#S5.E28)\)\. Appendix[C\.1\.3](https://arxiv.org/html/2610.28898#A3.SS1.SSS3)proves this extension of[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)to heterogeneous histories\. The common comparator’s residuala¯H\\bar\{a\}\_\{H\}remains explicit while estimation scales as1/\(m​q\)1/\(mq\)\. We next show that selection preserves this fast rate\.

### 5\.2Aggregation over Dyadic Lifetimes

To control selection among historical fits, we use Appendix[C\.2](https://arxiv.org/html/2610.28898#A3.SS2)\. Index lifetimes byjj, with current fieldvj,tv\_\{j,t\}, normalized batch scorexj,t∈\[0,1\]x\_\{j,t\}\\in\[0,1\], and pre\-batch selection probabilitypj,tp\_\{j,t\}\. Sums overjjorkkbelow run over active lifetimes\. Setx¯t=∑jpj,t​xj,t\\bar\{x\}\_\{t\}=\\sum\_\{j\}p\_\{j,t\}x\_\{j,t\}and fix a lifetimej=Ij=I\. Withdj,t=x¯t−xj,td\_\{j,t\}=\\bar\{x\}\_\{t\}\-x\_\{j,t\}, weight conservation andlog⁡\(1\+u\)≥u−u2\\log\(1\+u\)\\geq u\-u^\{2\}for\|u\|≤1/2\|u\|\\leq 1/2yield the pathwise inequality

∑t∈Idj,t≤log⁡Meη\+η​∑t∈Idj,t2,\\sum\_\{t\\in I\}d\_\{j,t\}\\leq\\frac\{\\log M\_\{e\}\}\{\\eta\}\+\\eta\\sum\_\{t\\in I\}d\_\{j,t\}^\{2\},\(29\)proved pathwise in Lemma[C\.4](https://arxiv.org/html/2610.28898#Thmlemma4)by the second\-order Prod potential argument\([Gaillard et al\., 2014](https://arxiv.org/html/2610.28898#bib.bib16)\)\. The fast rate needs a population calculation\. Conditional on the pre\-round information andEE, denote expectation over the new batch and selection by𝔼new\\mathbb\{E\}\_\{\\mathrm\{new\}\}\. Jensen and the gated squared\-loss identity give

𝔼new​dj,t2≤8B​\[∑kpk,t​Qt​\(vk,t\)\+Qt​\(vj,t\)\]\.\\displaystyle\\mathbb\{E\}\_\{\\mathrm\{new\}\}d\_\{j,t\}^\{2\}\\leq\\frac\{8\}\{B\}\\bigg\[\\sum\_\{k\}p\_\{k,t\}Q\_\{t\}\(v\_\{k,t\}\)\+Q\_\{t\}\(v\_\{j,t\}\)\\bigg\]\.\(30\)The first term is the selected field’s expected absolute velocity risk\. This flow\-matching\-specific estimate extends the standard Prod argument: one fixed rate preserves fast estimation while selecting a represented field\. LetSI=∑t∈I𝔼⁡\[Ft​\(vt\)∣E\]S\_\{I\}=\\sum\_\{t\\in I\}\\mathbb\{E\}\[F\_\{t\}\(v\_\{t\}\)\\mid E\],JI=∑t∈I𝔼⁡\[Ft​\(vI,t\)∣E\]J\_\{I\}=\\sum\_\{t\\in I\}\\mathbb\{E\}\[F\_\{t\}\(v\_\{I,t\}\)\\mid E\], and𝒜I=∑t∈Iat\\mathcal\{A\}\_\{I\}=\\sum\_\{t\\in I\}a\_\{t\}\. Inserting \([30](https://arxiv.org/html/2610.28898#S5.E30)\) into \([29](https://arxiv.org/html/2610.28898#S5.E29)\) at rate1/321/32gives

SI−JI≤32​B​log⁡Me\+14​\(SI\+JI\+2​𝒜I\)\+2​\|I\|​δ,S\_\{I\}\-J\_\{I\}\\leq 32B\\log M\_\{e\}\+\\tfrac\{1\}\{4\}\(S\_\{I\}\+J\_\{I\}\+2\\mathcal\{A\}\_\{I\}\)\+2\|I\|\\delta,\(31\)and absorbingSI/4S\_\{I\}/4givesSI≤53​JI\+1283​B​log⁡Me\+23​𝒜I\+83​\|I\|​δS\_\{I\}\\leq\\frac\{5\}\{3\}J\_\{I\}\+\\frac\{128\}\{3\}B\\log M\_\{e\}\+\\frac\{2\}\{3\}\\mathcal\{A\}\_\{I\}\+\\frac\{8\}\{3\}\|I\|\\delta\(Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5)\), without averaging represented fields\. Appendix[C\.2](https://arxiv.org/html/2610.28898#A3.SS2)gives both detailed proofs\. The selection overhead is thusO⁡\(B​log⁡Me\)O\(B\\log M\_\{e\}\)per lifetime\.

It remains to boundJIJ\_\{I\}, the current risks of a lifetime’s historical fits, under each drift measure\.

### 5\.3Relative\-Risk Transfer and Proof of Theorem 1

We now transfer Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)’s historical excess to the current law\. For a historyH=\[a,a\+q−1\]H=\[a,a\+q\-1\]withq≥1q\\geq 1anda\+q≤ta\+q\\leq t, letΩa:t=∑s=at−1ωs\\Omega\_\{a:t\}=\\sum\_\{s=a\}^\{t\-1\}\\omega\_\{s\}\. Lemma[C\.2](https://arxiv.org/html/2610.28898#Thmlemma2), proved in Appendix[C\.1](https://arxiv.org/html/2610.28898#A3.SS1), gives

a¯H\\displaystyle\\bar\{a\}\_\{H\}≤1q∑s∈Has\+2Ωa:t,\\displaystyle\\leq\\frac\{1\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+2\\Omega\_\{a:t\},\(32\)Ft​\(v^\)\\displaystyle F\_\{t\}\(\\widehat\{v\}\)≤rH\(v^\)\+2Ωa:t\.\\displaystyle\\leq r\_\{H\}\(\\widehat\{v\}\)\+2\\Omega\_\{a:t\}\.\(33\)Both are first order in drift\. Insert them into Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)\. At agek=t−a≥1k=t\-a\\geq 1, the prefix lengthq≥k/2q\\geq k/2makes the1/q1/qerrors sum harmonically\. Lemma[C\.6](https://arxiv.org/html/2610.28898#Thmlemma6)\(Appendix[C\.2](https://arxiv.org/html/2610.28898#A3.SS2)\) gives the resulting lifetime bound onJIJ\_\{I\}, including approximation, fitting, and drift\.

For a consecutive intervalJJ, writen=\|J\|n=\|J\|, letΩJ\\Omega\_\{J\}sum its internal relative\-risk increments, and put𝒜J=∑t∈Jat\\mathcal\{A\}\_\{J\}=\\sum\_\{t\\in J\}a\_\{t\}\. Lemma[C\.7](https://arxiv.org/html/2610.28898#Thmlemma7)partitionsJJinto at mostDTD\_\{T\}disjoint lifetimes\. Combining their base bounds with Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5)gives

∑t∈J𝔼⁡\[Ft​\(vt\)∣E\]≤KT\+HT​𝒜J\+n​ef\+10​\(n−1\)​ΩJ\.\\sum\_\{t\\in J\}\\mathbb\{E\}\[F\_\{t\}\(v\_\{t\}\)\\mid E\]\\leq K\_\{T\}\+H\_\{T\}\\mathcal\{A\}\_\{J\}\+ne\_\{f\}\+10\(n\-1\)\\Omega\_\{J\}\.\(34\)
##### Complete Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)\.

Splitting at relative\-risk changes eliminates internal variation, givingKT​\(Nω\+1\)K\_\{T\}\(N\_\{\\omega\}\+1\)\. Alternatively, blocks of length at mosth∈\{1,…,T\}h\\in\\\{1,\\ldots,T\\\}costKT​⌈T/h⌉\+10​\(h−1\)​ΩK\_\{T\}\\lceil T/h\\rceil\+10\(h\-1\)\\Omega\. Appendix[D](https://arxiv.org/html/2610.28898#A4)chooses integerhhto bound this byKT\+2​10​KT​T​ΩK\_\{T\}\+2\\sqrt\{10K\_\{T\}T\\Omega\}\. These are proof\-only partitions\. Taking the smaller cost, addingHT​𝔼​𝒜T\+T​efH\_\{T\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}, and integrating overEEgives𝔼​ℛTvel≤𝖥T\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq\\mathsf\{F\}\_\{T\}, proving Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)\. Target\-law tracking uses the same estimates with the sharper transfer below\.

### 5\.4Base Smoothing and Target\-Law Transfer

A target\-law bound must convert Wasserstein movement into changes in velocity risk\. Foru∈𝒱u\\in\\mathcal\{V\}, writeDs,t​\(u\)=Lt​\(u\)−Ls​\(u\)D\_\{s,t\}\(u\)=L\_\{t\}\(u\)\-L\_\{s\}\(u\)and defineosc⁡\(f\)=supv∈𝒱f⁡\(v\)−infv∈𝒱f⁡\(v\)\\operatorname\{osc\}\(f\)=\\sup\_\{v\\in\\mathcal\{V\}\}f\(v\)\-\\inf\_\{v\\in\\mathcal\{V\}\}f\(v\)\. Thenωt=12​osc⁡\(Dt,t\+1\)\\omega\_\{t\}=\\frac\{1\}\{2\}\\operatorname\{osc\}\(D\_\{t,t\+1\}\)\. We must control howDs,tD\_\{s,t\}separates an earlier fit from the current optimum at second order\.

##### A target move is a base\-noise shift plus a label shift\.

Let\(X1,X1′\)\(X\_\{1\},X\_\{1\}^\{\\prime\}\)be aW2W\_\{2\}\-optimal coupling of\(μs,μt\)\(\\mu\_\{s\},\\mu\_\{t\}\), let both rounds use the sameX0∼νX\_\{0\}\\sim\\nuandr∼λr\\sim\\lambda, and writeΔ​X=X1′−X1\\Delta X=X\_\{1\}^\{\\prime\}\-X\_\{1\}\. Writingα=α⁡\(r\)\\alpha=\\alpha\(r\),β=β⁡\(r\)\\beta=\\beta\(r\), andJ=J⁡\(r\)=α​β′−α′​βJ=J\(r\)=\\alpha\\beta^\{\\prime\}\-\\alpha^\{\\prime\}\\beta, the substitutionX~0=X0\+\(β/α\)​Δ​X\\tilde\{X\}\_\{0\}=X\_\{0\}\+\(\\beta/\\alpha\)\\Delta Xgives, forα\>0\\alpha\>0, the exact identity

\(Ir​\(X0,X1′\),Zr​\(X0,X1′\)\)=\(Ir​\(X~0,X1\),Zr​\(X~0,X1\)\+Jα​Δ​X\)\.\\begin\{split\}&\\big\(I\_\{r\}\(X\_\{0\},X\_\{1\}^\{\\prime\}\),Z\_\{r\}\(X\_\{0\},X\_\{1\}^\{\\prime\}\)\\big\)\\\\ &\\quad=\\Big\(I\_\{r\}\(\\tilde\{X\}\_\{0\},X\_\{1\}\),\\,Z\_\{r\}\(\\tilde\{X\}\_\{0\},X\_\{1\}\)\+\\frac\{J\}\{\\alpha\}\\Delta X\\Big\)\.\\end\{split\}\(35\)Roundttis roundsswith its base noise translated by\(β/α\)​Δ​X\(\\beta/\\alpha\)\\Delta Xand its label translated by\(J/α\)​Δ​X\(J/\\alpha\)\\Delta Xat the same input\. The key target\-law ingredient is that base regularity controls risk comparisons without spatially differencing fields or adding Lipschitz or norm\-equivalence assumptions\.

###### Lemma 1\(Second\-order smoothing transfer\)\.

Under Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1),[3](https://arxiv.org/html/2610.28898#Thmassumption3)and[5](https://arxiv.org/html/2610.28898#Thmassumption5), for all roundss,ts,tand allu,w∈𝒱u,w\\in\\mathcal\{V\}, withW=W2​\(μs,μt\)W=W\_\{2\}\(\\mu\_\{s\},\\mu\_\{t\}\),

\|Ds,t​\(u\)−Ds,t​\(w\)\|≤\\displaystyle\|D\_\{s,t\}\(u\)\-D\_\{s,t\}\(w\)\|\\leq\{\}Λsm​W​\(∥u−w∥s\+∥u−w∥t\)\\displaystyle\\Lambda\_\{\\mathrm\{sm\}\}W\\big\(\\lVert u\-w\\rVert\_\{s\}\+\\lVert u\-w\\rVert\_\{t\}\\big\)\+ϵsm​W\+c2​W2\.\\displaystyle\+\\epsilon\_\{\\mathrm\{sm\}\}W\+c\_\{2\}W^\{2\}\.With bounded endpoints andMzM\_\{z\}an almost\-sure label bound, the same holds withϵsm=0\\epsilon\_\{\\mathrm\{sm\}\}=0\.

Appendix[E\.1\.4](https://arxiv.org/html/2610.28898#A5.SS1.SSS4)proves this using Hellinger change of measure for the base, Cauchy–Schwarz for the label shift, and label tails forϵsm\\epsilon\_\{\\mathrm\{sm\}\}\. Lemma[E\.5](https://arxiv.org/html/2610.28898#Thmlemma5a)bounds squared prediction distance by excess risk plus approximation error\. Lemma[E\.6](https://arxiv.org/html/2610.28898#Thmlemma6a)absorbs the resulting square\-root risk terms to give the next proposition\.

###### Proposition 2\(Link between the two variations\)\.

Under the assumptions of Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1), letωs,t=12​osc⁡\(Ds,t\)\\omega\_\{s,t\}=\\frac\{1\}\{2\}\\operatorname\{osc\}\(D\_\{s,t\}\)and letvs⋆v\_\{s\}^\{\\star\}minimizeLsL\_\{s\}over𝒱\\mathcal\{V\}\. For all roundss,ts,t, withW=W2​\(μs,μt\)W=W\_\{2\}\(\\mu\_\{s\},\\mu\_\{t\}\),

ωs,t\\displaystyle\\omega\_\{s,t\}≤2​Bv​Λsm​W\+12​\(ϵsm​W\+c2​W2\),\\displaystyle\\leq 2B\_\{v\}\\Lambda\_\{\\mathrm\{sm\}\}W\+\\tfrac\{1\}\{2\}\(\\epsilon\_\{\\mathrm\{sm\}\}W\+c\_\{2\}W^\{2\}\),\(36\)Ft​\(vs⋆\)\\displaystyle F\_\{t\}\(v^\{\\star\}\_\{s\}\)≤5​Λsm2​W2\+8​\(as\+at\)\+2​\(ϵsm​W\+c2​W2\)\.\\displaystyle\\leq 5\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}W^\{2\}\+8\(a\_\{s\}\+a\_\{t\}\)\+2\(\\epsilon\_\{\\mathrm\{sm\}\}W\+c\_\{2\}W^\{2\}\)\.\(37\)

Appendix[E\.1\.7](https://arxiv.org/html/2610.28898#A5.SS1.SSS7)gives the detailed proof\. Equation \([36](https://arxiv.org/html/2610.28898#S5.E36)\) boundsΩT\\Omega\_\{T\}by a multiple ofVTμV\_\{T\}^\{\\mu\}for bounded fields, yielding onlyT​Vμ\\sqrt\{TV\_\{\\mu\}\}through Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)\. Equation \([37](https://arxiv.org/html/2610.28898#S5.E37)\) bounds an earlier optimum’s excess risk byW2W^\{2\}terms plus approximation and smoothing remainders\. Section[5\.5](https://arxiv.org/html/2610.28898#S5.SS5)uses this quadratic dependence to obtainT1/3​Vμ2/3T^\{1/3\}V\_\{\\mu\}^\{2/3\}\.

###### Proposition 3\(Sharpness of the smoothing modulus\)\.

Letν=N⁡\(0,σ02​Id\)\\nu=N\(0,\\sigma\_\{0\}^\{2\}I\_\{d\}\)withσ0\>0\\sigma\_\{0\}\>0andIdI\_\{d\}the identity matrix, let Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(a\) hold, and letqr,x1q\_\{r,x\_\{1\}\}be the law ofIr​\(X0,x1\)I\_\{r\}\(X\_\{0\},x\_\{1\}\)\. Then forα⁡\(r\)\>0\\alpha\(r\)\>0,

supx1≠x1′𝖧⁡\(qr,x1,qr,x1′\)∥x1−x1′∥=\|β⁡\(r\)\|2​2​α​\(r\)​σ0=h⁡\(r\),\\sup\_\{x\_\{1\}\\neq x\_\{1\}^\{\\prime\}\}\\frac\{\\mathsf\{H\}\(q\_\{r,x\_\{1\}\},q\_\{r,x\_\{1\}^\{\\prime\}\}\)\}\{\\lVert x\_\{1\}\-x\_\{1\}^\{\\prime\}\\rVert\}=\\frac\{\|\\beta\(r\)\|\}\{2\\sqrt\{2\}\\,\\alpha\(r\)\\sigma\_\{0\}\}=h\(r\),where𝖧⁡\(P,Q\)=\[12​∫\(p−q\)2\]1/2\\mathsf\{H\}\(P,Q\)=\[\\frac\{1\}\{2\}\\int\(\\sqrt\{p\}\-\\sqrt\{q\}\)^\{2\}\]^\{1/2\}is the Hellinger distance for laws with densitiesp,qp,q\. For every affine schedule,𝔼Unif⁡\[0,1\]​\[β2/α2\]=∞\\mathbb\{E\}\_\{\\mathrm\{Unif\}\[0,1\]\}\[\\beta^\{2\}/\\alpha^\{2\}\]=\\infty\.

Smoothing vanishes near the target endpoint, excluding uniform flow time from this argument\. Appendix[E\.1\.8](https://arxiv.org/html/2610.28898#A5.SS1.SSS8)proves Proposition[3](https://arxiv.org/html/2610.28898#Thmproposition3)and motivates Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)’s flow\-time condition\. We now apply the quadratic transfer to historical fits\.

### 5\.5Proof of Theorem 2 from Target\-Law Transfer

Under Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5), Lemma[E\.7](https://arxiv.org/html/2610.28898#Thmlemma7a)applies Lemma[E\.6](https://arxiv.org/html/2610.28898#Thmlemma6a)across a historical prefix \(Appendix[E\.1](https://arxiv.org/html/2610.28898#A5.SS1)\)\. ForH=\[a,a\+q−1\]H=\[a,a\+q\-1\]withq≥1q\\geq 1anda\+q≤ta\+q\\leq t, setVa:tμ=∑s=at−1W2\(μs\+1,μs\)V^\{\\mu\}\_\{a:t\}=\\sum\_\{s=a\}^\{t\-1\}W\_\{2\}\(\\mu\_\{s\+1\},\\mu\_\{s\}\)andρ¯a:t=ϵsmVa:tμ\+c2\(Va:tμ\)2\\bar\{\\rho\}\_\{a:t\}=\\epsilon\_\{\\mathrm\{sm\}\}V^\{\\mu\}\_\{a:t\}\+c\_\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\. For everyv∈𝒱v\\in\\mathcal\{V\}, the resulting transfer is

a¯H≤\\displaystyle\\bar\{a\}\_\{H\}\\leq\{\}9q∑s∈Has\+8at\+5Λsm2\(Va:tμ\)2\+2ρ¯a:t,\\displaystyle\\frac\{9\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+8a\_\{t\}\+5\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+2\\bar\{\\rho\}\_\{a:t\},\(38\)Ft​\(v\)≤\\displaystyle F\_\{t\}\(v\)\\leq\{\}3rH\(v\)\+29Λsm2\(Va:tμ\)2\+36q∑s∈Has\\displaystyle 3r\_\{H\}\(v\)\+29\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+\\frac\{36\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+34at\+10ρ¯a:t\.\\displaystyle\+34a\_\{t\}\+10\\bar\{\\rho\}\_\{a:t\}\.\(39\)Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)and harmonic summation give the lifetime bound of Lemma[E\.8](https://arxiv.org/html/2610.28898#Thmlemma8)\. Combining it with Lemmas[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5)and[C\.7](https://arxiv.org/html/2610.28898#Thmlemma7)gives Lemma[E\.9](https://arxiv.org/html/2610.28898#Thmlemma9)\. For a consecutive intervalJJ, setn=\|J\|n=\|J\|,𝒜J=∑t∈Jat\\mathcal\{A\}\_\{J\}=\\sum\_\{t\\in J\}a\_\{t\}, andVJμ=∑s,s\+1∈JW2​\(μs\+1,μs\)V\_\{J\}^\{\\mu\}=\\sum\_\{s,s\+1\\in J\}W\_\{2\}\(\\mu\_\{s\+1\},\\mu\_\{s\}\)\. The result is

∑t∈J𝔼⁡\[Ft​\(vt\)∣E\]≤\\displaystyle\\sum\_\{t\\in J\}\\mathbb\{E\}\[F\_\{t\}\(v\_\{t\}\)\\mid E\]\\leq\{\}KT\+\+HT\+​𝒜J\+n​ef\+\\displaystyle K\_\{T\}^\{\+\}\+H\_\{T\}^\{\+\}\\mathcal\{A\}\_\{J\}\+ne\_\{f\}^\{\+\}\+\(n−1\)​\[c¯​\(VJμ\)2\+37​ϵsm​VJμ\]\.\\displaystyle\+\(n\-1\)\\big\[\\bar\{c\}\(V\_\{J\}^\{\\mu\}\)^\{2\}\+37\\epsilon\_\{\\mathrm\{sm\}\}V\_\{J\}^\{\\mu\}\\big\]\.\(40\)SinceNTω≤NTμ≤NμN\_\{T\}^\{\\omega\}\\leq N\_\{T\}^\{\\mu\}\\leq N\_\{\\mu\}, Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)gives the count branch under the same flow\-time law \(Remark[6](https://arxiv.org/html/2610.28898#Thmremark6)\)\.

For the variation branch, maximal pieces satisfyingn​\(VJμ\)2≤KT\+/c¯n\(V\_\{J\}^\{\\mu\}\)^\{2\}\\leq K\_\{T\}^\{\+\}/\\bar\{c\}each cost at most2​KT\+2K\_\{T\}^\{\+\}plus approximation, fitting, and smoothing terms\. Appendix[E\.2](https://arxiv.org/html/2610.28898#A5.SS2), following[Baby and Wang \(2019\)](https://arxiv.org/html/2610.28898#bib.bib6), bounds their number by1\+\(2​c¯​T​Vμ2/KT\+\)1/31\+\(2\\bar\{c\}TV\_\{\\mu\}^\{2\}/K\_\{T\}^\{\+\}\)^\{1/3\}\. Summing \([40](https://arxiv.org/html/2610.28898#S5.Ex21)\) gives the variation branch, including37​ϵsm​T​Vμ37\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\}\. Taking the smaller branch and integrating overEEgives𝔼​ℛTvel≤𝖳T\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq\\mathsf\{T\}\_\{T\}, proving Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)\.

## 6Conclusion

One proper, budget\-free learner attains the ratesT​Ω\\sqrt\{T\\Omega\}andT1/3​Vμ2/3T^\{1/3\}V\_\{\\mu\}^\{2/3\}for predictive flow matching, which are minimax for fixed classes up to logarithmic factors\. Flow matching supplies the curvature that convexity would otherwise provide\. Nonaffine interpolations, target\-law rates under uniform flow time, the dependence onmm, and computational training guarantees remain open\.

## AI Use Statement

The authors sought the help of large language models to improve the grammar and presentation of the manuscript\. Additionally, LLMs were used for literature survey and formatting\. The authors take complete responsibility for this work\.

## References

- Albergo and Vanden\-Eijnden \(2023\)Albergo, M\. S\. and Vanden\-Eijnden, E\. \(2023\)\.Building normalizing flows with stochastic interpolants\.In*The Eleventh International Conference on Learning Representations*\.
- Albergo et al\. \(2025\)Albergo, M\. S\., Boffi, N\. M\., and Vanden\-Eijnden, E\. \(2025\)\.Stochastic interpolants: A unifying framework for flows and diffusions\.*Journal of Machine Learning Research*, 26\(209\):1–80\.
- Aliprantis and Border \(2006\)Aliprantis, C\. D\. and Border, K\. C\. \(2006\)\.*Infinite Dimensional Analysis: A Hitchhiker’s Guide*\.Springer, 3rd edition\.
- Ambrosio et al\. \(2008\)Ambrosio, L\., Gigli, N\., and Savaré, G\. \(2008\)\.*Gradient Flows in Metric Spaces and in the Space of Probability Measures*\.Birkhäuser Basel, 2nd edition\.
- Ash and Adams \(2020\)Ash, J\. T\. and Adams, R\. P\. \(2020\)\.On warm\-starting neural network training\.In*Advances in Neural Information Processing Systems*, volume 33, pages 3884–3894\.
- Baby and Wang \(2019\)Baby, D\. and Wang, Y\.\-X\. \(2019\)\.Online forecasting of total\-variation\-bounded sequences\.In*Advances in Neural Information Processing Systems*, volume 32\.
- Baby and Wang \(2021\)Baby, D\. and Wang, Y\.\-X\. \(2021\)\.Optimal dynamic regret in exp\-concave online learning\.In*Proceedings of the 34th Conference on Learning Theory*, pages 359–409\. PMLR\.
- Baby and Wang \(2022\)Baby, D\. and Wang, Y\.\-X\. \(2022\)\.Optimal dynamic regret in proper online learning with strongly convex losses and beyond\.In*Proceedings of the 25th International Conference on Artificial Intelligence and Statistics*, pages 1805–1845\. PMLR\.
- Bartlett et al\. \(2005\)Bartlett, P\. L\., Bousquet, O\., and Mendelson, S\. \(2005\)\.Local Rademacher complexities\.*The Annals of Statistics*, 33\(4\):1497–1537\.
- Benton et al\. \(2024\)Benton, J\., Deligiannidis, G\., and Doucet, A\. \(2024\)\.Error bounds for flow matching methods\.*Transactions on Machine Learning Research*\.
- Besbes et al\. \(2015\)Besbes, O\., Gur, Y\., and Zeevi, A\. \(2015\)\.Non\-stationary stochastic optimization\.*Operations Research*, 63\(5\):1227–1244\.
- Bottou and Bousquet \(2008\)Bottou, L\. and Bousquet, O\. \(2008\)\.The tradeoffs of large scale learning\.In*Advances in Neural Information Processing Systems*, volume 20, pages 161–168\.
- Boucheron et al\. \(2013\)Boucheron, S\., Lugosi, G\., and Massart, P\. \(2013\)\.*Concentration Inequalities: A Nonasymptotic Theory of Independence*\.Oxford University Press\.
- Daniely et al\. \(2015\)Daniely, A\., Gonen, A\., and Shalev\-Shwartz, S\. \(2015\)\.Strongly adaptive online learning\.In*Proceedings of the 32nd International Conference on Machine Learning*, pages 1405–1411\. PMLR\.
- Esser et al\. \(2024\)Esser, P\., Kulal, S\., Blattmann, A\., Entezari, R\., Müller, J\., Saini, H\., Levi, Y\., Lorenz, D\., Sauer, A\., Boesel, F\., Podell, D\., Dockhorn, T\., English, Z\., and Rombach, R\. \(2024\)\.Scaling rectified flow transformers for high\-resolution image synthesis\.In*Proceedings of the 41st International Conference on Machine Learning*, pages 12606–12633\. PMLR\.
- Gaillard et al\. \(2014\)Gaillard, P\., Stoltz, G\., and van Erven, T\. \(2014\)\.A second\-order bound with excess losses\.In*Proceedings of the 27th Conference on Learning Theory*, pages 176–196\. PMLR\.
- Gaur et al\. \(2025\)Gaur, M\., Trivedi, P\., Aeron, S\., Bedi, A\. S\., Atia, G\. K\., and Aggarwal, V\. \(2025\)\.Generative modeling with continuous flows: Sample complexity of flow matching\.*arXiv preprint arXiv:2512\.01286*\.
- Herbster and Warmuth \(1998\)Herbster, M\. and Warmuth, M\. K\. \(1998\)\.Tracking the best expert\.*Machine Learning*, 32\(2\):151–178\.
- Jun et al\. \(2017\)Jun, K\.\-S\., Orabona, F\., Wright, S\., and Willett, R\. \(2017\)\.Improved strongly adaptive online learning using coin betting\.In*Proceedings of the 20th International Conference on Artificial Intelligence and Statistics*, pages 943–951\. PMLR\.
- Klein and Rio \(2005\)Klein, T\. and Rio, E\. \(2005\)\.Concentration around the mean for maxima of empirical processes\.*The Annals of Probability*, 33\(3\):1060–1077\.
- Ledoux and Talagrand \(1991\)Ledoux, M\. and Talagrand, M\. \(1991\)\.*Probability in Banach Spaces: Isoperimetry and Processes*\.Springer\.
- Lipman et al\. \(2023\)Lipman, Y\., Chen, R\. T\. Q\., Ben\-Hamu, H\., Nickel, M\., and Le, M\. \(2023\)\.Flow matching for generative modeling\.In*The Eleventh International Conference on Learning Representations*\.
- Lipman et al\. \(2024\)Lipman, Y\., Havasi, M\., Holderrieth, P\., Shaul, N\., Le, M\., Karrer, B\., Chen, R\. T\. Q\., Lopez\-Paz, D\., Ben\-Hamu, H\., and Gat, I\. \(2024\)\.Flow matching guide and code\.*arXiv preprint arXiv:2412\.06264*\.
- Liu et al\. \(2023\)Liu, X\., Gong, C\., and Liu, Q\. \(2023\)\.Flow straight and fast: Learning to generate and transfer data with rectified flow\.In*The Eleventh International Conference on Learning Representations*\.
- Sahoo et al\. \(2026\)Sahoo, H\. K\., Aggarwal, V\., and Gaur, M\. \(2026\)\.Order\-optimal sample complexity for distribution learning via flow matching\.*arXiv preprint arXiv:2601\.20250v2*\.[https://arxiv\.org/abs/2601\.20250v2](https://arxiv.org/abs/2601.20250v2)\.
- Srebro et al\. \(2010\)Srebro, N\., Sridharan, K\., and Tewari, A\. \(2010\)\.Smoothness, low noise and fast rates\.In*Advances in Neural Information Processing Systems*, volume 23, pages 2199–2207\.
- Stepanov and Trevisan \(2017\)Stepanov, E\. and Trevisan, D\. \(2017\)\.Three superposition principles: Currents, continuity equations and curves of measures\.*Journal of Functional Analysis*, 272\(3\):1044–1103\.
- Tsybakov \(2009\)Tsybakov, A\. B\. \(2009\)\.*Introduction to Nonparametric Estimation*\.Springer Series in Statistics\. Springer, New York, NY\.Translated by Vladimir Zaiats\.
- Wei et al\. \(2016\)Wei, C\.\-Y\., Hong, Y\.\-T\., and Lu, C\.\-J\. \(2016\)\.Tracking the best expert in non\-stationary stochastic environments\.In*Advances in Neural Information Processing Systems*, volume 29, pages 3972–3980\.
- Zhang et al\. \(2018\)Zhang, L\., Yang, T\., Jin, R\., and Zhou, Z\.\-H\. \(2018\)\.Dynamic regret of strongly adaptive methods\.In*Proceedings of the 35th International Conference on Machine Learning*, pages 5882–5891\. PMLR\.
- Zhou and Liu \(2025\)Zhou, Z\. and Liu, W\. \(2025\)\.An error analysis of flow matching for deep generative modeling\.In*Proceedings of the 42nd International Conference on Machine Learning*, pages 78903–78932\. PMLR\.
- Zinkevich \(2003\)Zinkevich, M\. \(2003\)\.Online convex programming and generalized infinitesimal gradient ascent\.In*Proceedings of the 20th International Conference on Machine Learning*, pages 928–936\.

Minimax\-Optimal Predictive Flow Matching under Distribution Drift Supplementary Material

## Appendix AExtended Related Work

This appendix expands Section[2](https://arxiv.org/html/2610.28898#S2)\. It places each line of work relative to the three ingredients of this paper: drifting targets in flow matching, budget\-free dynamic regret for a nonconvex class, and the two measures of nonstationarity\.

### A\.1Flow Matching, Rectified Flow, and Stochastic Interpolants

[Lipman et al\. \(2023\)](https://arxiv.org/html/2610.28898#bib.bib22)introduce flow matching, which learns a transport velocity by regressing on conditional velocities along a prescribed probability path\. Rectified flow\([Liu et al\., 2023](https://arxiv.org/html/2610.28898#bib.bib24)\)uses linear interpolation between base and target samples and favors nearly straight transport paths\. Stochastic interpolants\([Albergo and Vanden\-Eijnden, 2023](https://arxiv.org/html/2610.28898#bib.bib1);[Albergo et al\., 2025](https://arxiv.org/html/2610.28898#bib.bib2)\)formulate flows and diffusions through interpolation maps, possibly with additional latent noise, and characterize the drift by the same kind of conditional\-mean regression\. The guide of[Lipman et al\. \(2024\)](https://arxiv.org/html/2610.28898#bib.bib23)surveys paths, schedules, and implementations\.

All of these frameworks use the regression identity that underlies Equation \([6](https://arxiv.org/html/2610.28898#S3.E6)\)\. Assumption[2](https://arxiv.org/html/2610.28898#Thmassumption2)covers the linear and sine–cosine paths, and the affine schedules of Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)include both\. Interpolants with additional latent noise, nonaffine interpolations, and couplings other than independent pairing are outside the present target\-law analysis\. The flow\-time law is also a modeling choice in practice\.[Esser et al\. \(2024\)](https://arxiv.org/html/2610.28898#bib.bib15)sample flow time from a logit\-normal law when training rectified\-flow transformers at scale, and such laws satisfy Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(c\) \(Appendix[E\.1\.9](https://arxiv.org/html/2610.28898#A5.SS1.SSS9)\)\.

### A\.2Statistical Theory of Flow Matching

[Benton et al\. \(2024\)](https://arxiv.org/html/2610.28898#bib.bib10)relate the sampling error of flow\-matching methods to the accuracy of the learned velocity, under regularity of the learned flow\.[Zhou and Liu \(2025\)](https://arxiv.org/html/2610.28898#bib.bib31)give an error analysis of flow matching for deep generative modeling with fixed targets, including bounded\-support distributions and stronger score\-regularity regimes\.[Gaur et al\. \(2025\)](https://arxiv.org/html/2610.28898#bib.bib17)study the sample complexity of flow matching and separate approximation, estimation, and optimization errors\. These results motivate the error decomposition used here, with regret, approximation error, and fitting error kept separate\. We retain an explicit approximate\-global fitting interface \(Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)\) rather than proving convergence of a generic neural optimizer\.

[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)obtain order\-optimal sample complexity for rectified flows\. Their analysis rests on Bernstein\-type variance control for the squared flow\-matching loss, a localized Rademacher bound, and a rare\-mixture lower bound\. Our Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[4](https://arxiv.org/html/2610.28898#Thmassumption4)follow the same setting\. The difference is the protocol\. These works study one fixed target and i\.i\.d\. samples, whereas we study a drifting target, predictive deployment before each fresh batch, heterogeneous historical pools, and dynamic regret against the best represented field of every round\. Our question is therefore different: how much excess current velocity error accumulates when deployment precedes each fresh batch and the target changes? Appendix[C\.1\.5](https://arxiv.org/html/2610.28898#A3.SS1.SSS5)compares our direct historical estimation with the localized analysis\.

### A\.3Nonstationary Online Learning and Dynamic Regret

Tracking the best expert under switches\([Herbster and Warmuth, 1998](https://arxiv.org/html/2610.28898#bib.bib18)\)and dynamic regret for online convex optimization against moving comparators\([Zinkevich, 2003](https://arxiv.org/html/2610.28898#bib.bib32)\)are the classical starting points\.[Besbes et al\. \(2015\)](https://arxiv.org/html/2610.28898#bib.bib11)study non\-stationary stochastic optimization under a budget on the temporal variation of the loss functions\. Our relative velocity risk variation is a class\-relative budget of the same functional type, measured by the half\-oscillation of risk increments so that changes common to all represented fields are free\. The target\-law variation is different in kind: it is a budget on the targets themselves, not on the losses or on the comparators\.

Strongly adaptive methods control regret on every interval\.[Daniely et al\. \(2015\)](https://arxiv.org/html/2610.28898#bib.bib14)use geometric covering intervals, and[Jun et al\. \(2017\)](https://arxiv.org/html/2610.28898#bib.bib19)improve strongly adaptive rates through coin betting\.[Zhang et al\. \(2018\)](https://arxiv.org/html/2610.28898#bib.bib30)show how interval guarantees yield dynamic regret by partitioning the horizon according to functional variation\. Our dyadic lifetimes follow the geometric\-covering idea, and our analysis also partitions the horizon only in the proof\.[Wei et al\. \(2016\)](https://arxiv.org/html/2610.28898#bib.bib29)track the best expert in non\-stationary stochastic environments and show how statistical variance interacts with switching and variation budgets\. This stochastic\-expert view is close to our fresh\-batch protocol, in which one batch scores every active candidate\. We adapt these ingredients to stochastic population risks of data\-dependent historical fits, and we select a single field at every round so that the deployed field stays in the represented class\.

The master of DyProd\-FM is the Prod algorithm with sleeping experts\.[Gaillard et al\. \(2014\)](https://arxiv.org/html/2610.28898#bib.bib16)prove second\-order regret bounds in terms of excess losses for multiple\-rate Prod and give the sleeping\-expert reduction\. With one rate, the selection probabilities are normalized weights \(Appendix[B\.2\.1](https://arxiv.org/html/2610.28898#A2.SS2.SSS1)\)\. Our contribution at this step is the population analysis: the squared flow\-matching loss bounds the second moment of the score differences by velocity risks, which turns the potential inequality into the fast master bound of Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5)\.

### A\.4Fast Dynamic Regret and Total\-Variation Forecasting

[Baby and Wang \(2019\)](https://arxiv.org/html/2610.28898#bib.bib6)study online forecasting of total\-variation\-bounded sequences under squared loss and establish the minimax raten1/3​Cn2/3n^\{1/3\}C\_\{n\}^\{2/3\}, with a matching lower bound and a bin\-counting argument\.[Baby and Wang \(2021\)](https://arxiv.org/html/2610.28898#bib.bib7)obtain optimal dynamic regret for exp\-concave losses, and[Baby and Wang \(2022\)](https://arxiv.org/html/2610.28898#bib.bib8)obtain such rates with proper learners for strongly convex losses and beyond\. Their budgets are path variations of comparator sequences in convex decision sets\. Their optimality conclusions therefore do not directly identify the minimax rate for our relative\-risk variation and endpoint\-feedback model\.

Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)attains the exponentsT1/3​Vμ2/3T^\{1/3\}V\_\{\\mu\}^\{2/3\}of total\-variation forecasting, and its proof uses the bin\-counting partition \(Appendix[E](https://arxiv.org/html/2610.28898#A5)\)\. The decision set, however, is a nonconvex neural class, and the learner is proper\. The curvature that the convex theory obtains from strong convexity or exp\-concavity comes here from the flow\-matching regression identity and from base smoothing \(Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1)\)\.

### A\.5Fast Statistical Rates and Localization

Variance control by excess risk underlies fast rates and localized complexity methods\([Bartlett et al\., 2005](https://arxiv.org/html/2610.28898#bib.bib9);[Srebro et al\., 2010](https://arxiv.org/html/2610.28898#bib.bib26);[Boucheron et al\., 2013](https://arxiv.org/html/2610.28898#bib.bib13)\)\. In a convex class, squared loss gives an exact Bernstein condition\. In a nonconvex represented class it does not, because the approximation error remains in the comparison between risk and distance \(Lemma[C\.1](https://arxiv.org/html/2610.28898#Thmlemma1a)\)\. We use one\-sided Bernstein inequalities for independent, non\-identically distributed observations together with a deterministic cover of the class, which gives explicit constants for the expectation bounds used online \(Appendix[C](https://arxiv.org/html/2610.28898#A3)\)\. For this parametric class the localized route gives the same order \(Appendix[C\.1\.5](https://arxiv.org/html/2610.28898#A3.SS1.SSS5)\)\.

### A\.6Minimax Lower Bounds

Two\-point and multiple\-hypothesis reductions are the standard tools of nonparametric minimax theory\([Tsybakov, 2009](https://arxiv.org/html/2610.28898#bib.bib28)\)\.[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)use a rare\-mixture pair to show that their fixed\-target sample complexity cannot be improved\. Our lower bounds apply such reductions block by block in time \(Appendix[F](https://arxiv.org/html/2610.28898#A6)\)\. Every hard instance keeps the class, the envelope, and the flow\-time law fixed, and only the target sequence varies withTTand the budget\. In the rare\-mixture construction a trainable slope lets one fixed class follow a far target component, whose location grows likelog⁡\(1/η\)\\sqrt\{\\log\(1/\\eta\)\}as in[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)\. This rules out constructions that shrink the hard class as the budget changes, which would change the constants that the upper bounds depend on\.

## Appendix BAlgorithm and Consequences of Assumptions

### B\.1Notation

SymbolMeaning in Theorems[1](https://arxiv.org/html/2610.28898#Thmtheorem1)–[2](https://arxiv.org/html/2610.28898#Thmtheorem2)T,m,dT,m,dHorizon, triples per round, dimension\.ν,μt,λ\\nu,\\mu\_\{t\},\\lambdaBase law, round\-tttarget law, flow\-time law\.𝒱,Bv,Lx\\mathcal\{V\},B\_\{v\},L\_\{x\}Represented class, its output bound, and its spatial Lipschitz constant \(Equation \([14](https://arxiv.org/html/2610.28898#S3.E14)\)\)\.Qt,at,FtQ\_\{t\},a\_\{t\},F\_\{t\}Absolute velocity risk, class approximation error, excess velocity risk \(Equation \([5](https://arxiv.org/html/2610.28898#S3.E5)\)\)\.ℛTvel,𝒜T\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\},\\mathcal\{A\}\_\{T\}Dynamic velocity regret and cumulative approximation error \(Equation \([7](https://arxiv.org/html/2610.28898#S3.E7)\)\)\.Ω,Nω\\Omega,N\_\{\\omega\}Caps on the relative velocity risk variation and its change count \(Equation \([9](https://arxiv.org/html/2610.28898#S3.E9)\)\)\.Vμ,NμV\_\{\\mu\},N\_\{\\mu\}Caps on the target\-law variation and its change count \(Equation \([10](https://arxiv.org/html/2610.28898#S3.E10)\)\)\.κ,cZ\\kappa,c\_\{Z\}Endpoint and label envelopes \(Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[2](https://arxiv.org/html/2610.28898#Thmassumption2)\)\.ϵopt\\epsilon\_\{\\mathrm\{opt\}\}Accuracy of the fitting routine \(Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)\)\.ι,mh,mk\\iota,m\_\{h\},m\_\{k\}Base Fisher\-information bound and flow\-time moments \(Equation \([17](https://arxiv.org/html/2610.28898#S3.E17)\)\)\.M,B,δM,B,\\deltaGate radius, proxy loss bound, original\-to\-proxy discrepancy \(Equations \([15](https://arxiv.org/html/2610.28898#S3.E15)\), \([18](https://arxiv.org/html/2610.28898#S4.E18)\)\)\.ζ,KM,Γ\\zeta,K\_\{M\},\\GammaCover accuracyB/\(m​T\)2B/\(mT\)^\{2\}, cover size, andΓ=1\+log⁡\(2​KM\)\\Gamma=1\+\\log\(2K\_\{M\}\)\.Me,ΛM\_\{e\},\\LambdaNumber of lifetimes andlog⁡\(e​Me\)\\log\(eM\_\{e\}\)\.DT,KT,HT,efD\_\{T\},K\_\{T\},H\_\{T\},e\_\{f\}Constants of Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)\(Equation \([20](https://arxiv.org/html/2610.28898#S4.E20)\)\)\.Λsm,ϵsm,c2,c¯\\Lambda\_\{\\mathrm\{sm\}\},\\epsilon\_\{\\mathrm\{sm\}\},c\_\{2\},\\bar\{c\}Smoothing constants of Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)\(Equation \([21](https://arxiv.org/html/2610.28898#S4.E21)\)\)\.KT\+,HT\+,ef\+K\_\{T\}^\{\+\},H\_\{T\}^\{\+\},e\_\{f\}^\{\+\}Remaining constants of Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)\(Equation \([23](https://arxiv.org/html/2610.28898#S4.E23)\)\)\.𝖥T,𝖳T\\mathsf\{F\}\_\{T\},\\mathsf\{T\}\_\{T\}The bounds in Equations \([19](https://arxiv.org/html/2610.28898#S4.E19)\) and \([22](https://arxiv.org/html/2610.28898#S4.E22)\)\.The same letter can carry a distinct role only with its specified index and type: for example,bt​\(x,r\)b\_\{t\}\(x,r\)is a vector field whereasbjb\_\{j\}in a network is a finite\-dimensional layer bias\. Empty sums equal zero and empty products equal one\. All logarithms without a subscript are natural logarithms\. A singleton interval has no internal variation edges\.

### B\.2Algorithm and Implementation Details

Algorithm 2:PrefixFM, gated flow\-matching training on a historical prefixGiven:the stored triplesξs,i=\(X0,s,i,X1,s,i,rs,i\)\\xi\_\{s,i\}=\(X\_\{0,s,i\},X\_\{1,s,i\},r\_\{s,i\}\)of the prefixH=\[a,a\+q−1\]H=\[a,a\+q\-1\],i≤mi\\leq m, formed in line 5 of Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1), and the gate radiusMM\.Design parameters:the network of Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3), an optimizer such as stochastic gradient descent or Adam, a number of iterationsNN, and the initial parametersθstart\\theta\_\{\\mathrm\{start\}\}:θa\(0\)\\theta\_\{a\}^\{\(0\)\}of Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)for the prefix\[a,a\]\[a,a\], and the completed fit for\[a,a\+q/2−1\]\[a,a\+q/2\-1\]for the prefix\[a,a\+q−1\]\[a,a\+q\-1\]withq≥2q\\geq 2\(Appendix[B\.2\.1](https://arxiv.org/html/2610.28898#A2.SS2.SSS1)\)\.1:For every stored triple, formUs,i=\(Irs,i​\(X0,s,i,X1,s,i\),rs,i\)U\_\{s,i\}=\(I\_\{r\_\{s,i\}\}\(X\_\{0,s,i\},X\_\{1,s,i\}\),r\_\{s,i\}\),Zs,i=Zrs,i​\(X0,s,i,X1,s,i\)Z\_\{s,i\}=Z\_\{r\_\{s,i\}\}\(X\_\{0,s,i\},X\_\{1,s,i\}\), andGs,i=𝟏​\{∥X0,s,i∥≤M,∥X1,s,i∥≤M\}G\_\{s,i\}=\\mathbf\{1\}\\\{\\lVert X\_\{0,s,i\}\\rVert\\leq M,\\lVert X\_\{1,s,i\}\\rVert\\leq M\\\}\.2:Setθ←θstart\\theta\\leftarrow\\theta\_\{\\mathrm\{start\}\}and reset the optimizer state\.3:forj=1,…,Nj=1,\\ldots,Ndo4:Updateθ\\thetato decreaseL^H,M​\(θ\)=1m​q​∑s∈H∑i≤mGs,i​∥vθ​\(Us,i\)−Zs,i∥2\\widehat\{L\}\_\{H,M\}\(\\theta\)=\\frac\{1\}\{mq\}\\sum\_\{s\\in H\}\\sum\_\{i\\leq m\}G\_\{s,i\}\\lVert v\_\{\\theta\}\(U\_\{s,i\}\)\-Z\_\{s,i\}\\rVert^\{2\}, and projectθ\\thetaontoΘ\\Theta\.5:end for6:returnvθv\_\{\\theta\}\.

Algorithm[B\.2](https://arxiv.org/html/2610.28898#A2.SS2)is the standard flow\-matching training loop, as in[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25), applied to the gated loss of one historical prefix\. Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)asks only that the returned network be withinϵopt\\epsilon\_\{\\mathrm\{opt\}\}of the minimum ofL^H,M\\widehat\{L\}\_\{H,M\}over𝒱\\mathcal\{V\}\. The optimizer andNNdetermineϵopt\\epsilon\_\{\\mathrm\{opt\}\}, which the analysis treats as given\. Accuracy guarantees for particular training routines are outside the scope of this work\.

##### Flow\-time laws\.

The flow time of every triple is drawn fromλ\\lambdawhen the triple is formed\. Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)usesλ=Unif⁡\[0,1\]\\lambda=\\mathrm\{Unif\}\[0,1\], the standard choice\. Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)uses any law satisfying Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(c\), for exampleUnif⁡\[0,1−η0\]\\mathrm\{Unif\}\[0,1\-\\eta\_\{0\}\],Beta⁡\(a,b\)\\mathrm\{Beta\}\(a,b\)withb\>2b\>2, or a logit\-normal law\([Esser et al\., 2024](https://arxiv.org/html/2610.28898#bib.bib15)\)\. By Remark[6](https://arxiv.org/html/2610.28898#Thmremark6), Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)also holds under the laws of Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5), so with such a law one run satisfies both theorems\.

#### B\.2\.1Initialization and Predictability

Each expert is a dyadic lifetime with its current fitted field\. Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)uses the design parameters of Section[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)\. The fitting accuracyϵopt\\epsilon\_\{\\mathrm\{opt\}\}is determined by the training routine and is not an input\.

##### Initialization\.

For the compact feasible parameter setΘ⊂ℝP\\Theta\\subset\\mathbb\{R\}^\{P\}of Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3), draw independently across starts and independently of the observations and selection seeds

ga∼N⁡\(0,P−1​IP\),θa\(0\)=ΠΘ​\(ga\),ΠΘ​\(g\)=lexmin⁡arg​minθ∈Θ⁡∥θ−g∥2\.g\_\{a\}\\sim N\(0,P^\{\-1\}I\_\{P\}\),\\qquad\\theta\_\{a\}^\{\(0\)\}=\\Pi\_\{\\Theta\}\(g\_\{a\}\),\\qquad\\Pi\_\{\\Theta\}\(g\)=\\operatorname\{lexmin\}\\argmin\_\{\\theta\\in\\Theta\}\\lVert\\theta\-g\\rVert^\{2\}\.\(41\)Order trainable coordinates by layer, with row\-major weights followed by biases, and retain prescribed fixed entries\. Compactness and the lexicographic tie rule define a deterministic measurable projection, so every initial field belongs to𝒱\\mathcal\{V\}\.

##### Prefixes and warm starts\.

Process the root first\. At age zero, a lifetime uses the latest completed root fit, or the fixed startup fieldvθ1\(0\)v\_\{\\theta\_\{1\}^\{\(0\)\}\}if none exists\. At agek≥1k\\geq 1, it uses its own prefix of lengthq=2⌊log2⁡k⌋q=2^\{\\lfloor\\log\_\{2\}k\\rfloor\}: theq=1q=1fit starts fromθa\(0\)\\theta\_\{a\}^\{\(0\)\}, and each longer fit starts from the completedq/2q/2fit with optimizer buffers and step counter reset\. Lifetimes with the same start share initialization and fits\. Nested prefixes use only their own data\. The root fallback never initializes a fit\. The analysis depends only on the empirical gap in Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4), so projected shrink\-and\-perturb or fresh starts are also valid when they use only prefix data and independent randomness\([Ash and Adams, 2020](https://arxiv.org/html/2610.28898#bib.bib5)\)\.

##### Predictability and weights\.

Every prefix ends byt−1t\-1, refits occur only when its length doubles, and scoring and weight updates use the pre\-batch fields\. The root is always active\. All conceptual weights start at1/Me1/M\_\{e\}and receive no new mass on waking\. Lemma[C\.4](https://arxiv.org/html/2610.28898#Thmlemma4)ensures positivity, and the common rateη=1/32\\eta=1/32makes Prod probabilities normalized weights\([Gaillard et al\., 2014](https://arxiv.org/html/2610.28898#bib.bib16)\)\. The predictable fallback satisfies0≤Fa​\(v\)≤B\+δ0\\leq F\_\{a\}\(v\)\\leq B\+\\deltaby Equation \([54](https://arxiv.org/html/2610.28898#A2.E54)\), which is all the lifetime proof needs at age zero\. These rules specify initialization and data use\. Optimizer convergence is covered only by Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)\.

### B\.3Probability Model and Regression Identity

#### B\.3\.1Conditioning and Predictability

The environment takes values in a standard Borel space, and the target distribution is a jointly measurable probability kernel of\(E,t\)\(E,t\)\. The base is fixed\. Given the environment, generate all triples with the product law in \([1](https://arxiv.org/html/2610.28898#S3.E1)\)\. Independent fitting and selection seeds may be sampled on an additional product probability space\. This construction permits a random environment while ensuring that the targets cannot adapt to sample noise or the learner’s choices\.

For analysis, fixEE\. Letℱt−1\\mathcal\{F\}\_\{t\-1\}contain all earlier batches, earlier selection seeds, and the fitting randomness used to form the round\-ttcandidates\. Conditional onEEandℱt−1\\mathcal\{F\}\_\{t\-1\}, candidate fields and probabilities are fixed\. The new selection seed and new batch are independent of this information and of each other\. Ifft​\(v\)f\_\{t\}\(v\)is any integrable population functional, proper random selection consequently satisfies

𝔼\[ft\(vt\)∣E,ℱt−1\]=∑j​activepj,tft\(vj,t\)\.\\mathbb\{E\}\[f\_\{t\}\(v\_\{t\}\)\\mid E,\\mathcal\{F\}\_\{t\-1\}\]=\\sum\_\{j\\ \\mathrm\{active\}\}p\_\{j,t\}f\_\{t\}\(v\_\{j,t\}\)\.\(42\)For a current empirical proxy score, the same conditioning gives its population meanLt,M​\(vj,t\)/BL\_\{t,M\}\(v\_\{j,t\}\)/B\. The fields may depend arbitrarily on their earlier training observations, while the evaluation batch is fresh\. Conditioning additionally on the selected index is unnecessary and would obscure why every active model can be compared on one batch\.

A historical prefix attached to a deterministic lifetime and a deterministic age has deterministic endpoints\. Its fit is random, but the concentration theorem is uniform in the candidate parameter after the history is fixed\. We do not apply a fixed\-sample theorem to a stopping\-time history\. Population\-dependent partitions used later are deterministic onceEEis fixed\. These are the exact reasons that the historical and online arguments are compatible\.

#### B\.3\.2Finiteness, Measurability, and the Regression Identity

Assumption[1](https://arxiv.org/html/2610.28898#Thmassumption1)and Jensen imply finite endpoint second moments\. WriteS=∥X0∥\+∥X1∥S=\\lVert X\_\{0\}\\rVert\+\\lVert X\_\{1\}\\rVertin this paragraph\. Assumption[2](https://arxiv.org/html/2610.28898#Thmassumption2), or Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(a\), yields∥Z∥≤cZ​\(S\+1\)\\lVert Z\\rVert\\leq c\_\{Z\}\(S\+1\), and Proposition[B\.1](https://arxiv.org/html/2610.28898#Thmproposition1a)gives∥vθ∥≤Bv\\lVert v\_\{\\theta\}\\rVert\\leq B\_\{v\}\. Hence\[Bv\+cZ​\(S\+1\)\]2\[B\_\{v\}\+c\_\{Z\}\(S\+1\)\]^\{2\}is an integrable common envelope for every original loss\. Conditional expectation is well defined inL2L^\{2\}, and Jensen gives𝔼t​∥bt​\(U\)∥2≤𝔼t​∥Z∥2\\mathbb\{E\}\_\{t\}\\lVert b\_\{t\}\(U\)\\rVert^\{2\}\\leq\\mathbb\{E\}\_\{t\}\\lVert Z\\rVert^\{2\}\.

The standard Borel construction permits disintegration to obtain a jointly measurable version of the conditional mean as a function of the environment and input\. To verify Equation \([6](https://arxiv.org/html/2610.28898#S3.E6)\), write

v⁡\(U\)−Z=\[v⁡\(U\)−bt​\(U\)\]\+\[bt​\(U\)−Z\]\.v\(U\)\-Z=\[v\(U\)\-b\_\{t\}\(U\)\]\+\[b\_\{t\}\(U\)\-Z\]\.\(43\)Expand its squared norm\. The cross term has conditional expectation zero becausev​\(U\)−bt​\(U\)v\(U\)\-b\_\{t\}\(U\)is measurable inUUand𝔼t​\[bt​\(U\)−Z∣U\]=0\\mathbb\{E\}\_\{t\}\[b\_\{t\}\(U\)\-Z\\mid U\]=0\. Both factors are square integrable, so this orthogonality is legitimate\. Taking expectations gives Equation \([6](https://arxiv.org/html/2610.28898#S3.E6)\), and subtracting the minimum gives Equation \([5](https://arxiv.org/html/2610.28898#S3.E5)\) and summing gives Equation \([7](https://arxiv.org/html/2610.28898#S3.E7)\)\.

For fixed data, the empirical proxy objective is continuous inθ\\theta\. For a fixed environment, dominated convergence gives continuity ofLt​\(vθ\)L\_\{t\}\(v\_\{\\theta\}\)andQt​\(vθ\)Q\_\{t\}\(v\_\{\\theta\}\)\. Compactness therefore gives all stated minima\. A countable dense subset ofΘ\\Thetayields the same extrema of continuous functions, proving measurability of minimum risks and the suprema definingωt\\omega\_\{t\}\. Measurable minimizers follow from the measurable maximum theorem applied to the compact\-valued argmin correspondence\([Aliprantis and Border, 2006](https://arxiv.org/html/2610.28898#bib.bib3)\)\. The existence of a measurable optimizer is a mathematical statement, not a computational complexity bound\.

#### B\.3\.3Risk Oscillation and an Offset Example

For a bounded real functionffon𝒱\\mathcal\{V\}, defineosc⁡\(f\)=supv∈𝒱f⁡\(v\)−infv∈𝒱f⁡\(v\)\\operatorname\{osc\}\(f\)=\\sup\_\{v\\in\\mathcal\{V\}\}f\(v\)\-\\inf\_\{v\\in\\mathcal\{V\}\}f\(v\)\. Then2​ωt=osc⁡\(Lt\+1−Lt\)2\\omega\_\{t\}=\\operatorname\{osc\}\(L\_\{t\+1\}\-L\_\{t\}\)\. The elementary inequalityosc⁡\(f\+g\)≤osc⁡\(f\)\+osc⁡\(g\)\\operatorname\{osc\}\(f\+g\)\\leq\\operatorname\{osc\}\(f\)\+\\operatorname\{osc\}\(g\)proves that fors<ts<t,

osc⁡\(Lt−Ls\)≤2​∑j=st−1ωj\.\\operatorname\{osc\}\(L\_\{t\}\-L\_\{s\}\)\\leq 2\\sum\_\{j=s\}^\{t\-1\}\\omega\_\{j\}\.\(44\)Moreover, adding any constant toffleaves its oscillation unchanged\. In fact,12​osc⁡\(f\)=infc∈ℝsupv∈𝒱\|f⁡\(v\)−c\|\\frac\{1\}\{2\}\\operatorname\{osc\}\(f\)=\\inf\_\{c\\in\\mathbb\{R\}\}\\sup\_\{v\\in\\mathcal\{V\}\}\|f\(v\)\-c\|, since the midpoint of its range attains the infimum\. Thus the budget ignores precisely the common component of a risk increment\. This quotient interpretation is elementary and is not a new seminorm construction\.

For a concrete flow\-matching example, letd=1d=1, take a centered Gaussian base, and use linear interpolation\. Let the represented class be constant fieldsvc​\(x,r\)=cv\_\{c\}\(x,r\)=cfor\|c\|≤R\|c\|\\leq RwithR\>0R\>0\. This is an admissible constrained network subclass with zero output weights and trainable bounded output bias\. Let the target alternate betweenN⁡\(0,1\)N\(0,1\)andN⁡\(0,4\)N\(0,4\)\. The observed label isZ=X1−X0Z=X\_\{1\}\-X\_\{0\}and has mean zero\. ConsequentlyLt​\(vc\)=c2\+𝔼t​Z2L\_\{t\}\(v\_\{c\}\)=c^\{2\}\+\\mathbb\{E\}\_\{t\}Z^\{2\}, so every relative risk is unchanged andωt=0\\omega\_\{t\}=0, whereassupc\|Lt\+1​\(vc\)−Lt​\(vc\)\|=3\\sup\_\{c\}\|L\_\{t\+1\}\(v\_\{c\}\)\-L\_\{t\}\(v\_\{c\}\)\|=3at each alternation\. The endpoints satisfy a common finite sub\-Gaussian envelope\. This example illustrates what the regret budget measures\. It does not assert that the constant\-field class accurately generates both targets\.

### B\.4Neural\-Class Consequences and Covering Numbers

#### B\.4\.1Compactness and Network Bounds

###### Proposition B\.1\.

Under Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3), the parameter set is compact and the field is globally bounded byBvB\_\{v\}and globally spatiallyLxL\_\{x\}\-Lipschitz, with the constants in Equation \([14](https://arxiv.org/html/2610.28898#S3.E14)\)\. It is differentiable in its input\. For each finite radiusRR, definekj=Lϕk\_\{j\}=L\_\{\\phi\}forj<Dj<DandkD=1k\_\{D\}=1, and define hidden\-state envelopes byH0​\(R\)=R2\+1H\_\{0\}\(R\)=\\sqrt\{R^\{2\}\+1\}andHj​\(R\)=b​njH\_\{j\}\(R\)=b\\sqrt\{n\_\{j\}\}for1≤j<D1\\leq j<D\. Then a parameter\-sensitivity constant is

CR=∑j=1Dkj​\[Hj−1​\(R\)\+1\]​∏i=j\+1D\(ki​Vi\),C\_\{R\}=\\sum\_\{j=1\}^\{D\}k\_\{j\}\[H\_\{j\-1\}\(R\)\+1\]\\prod\_\{i=j\+1\}^\{D\}\(k\_\{i\}V\_\{i\}\),\(45\)and it satisfies

sup∥x∥≤R,r∈\[0,1\]∥vθ​\(x,r\)−vθ′​\(x,r\)∥≤CR​∥θ−θ′∥\.\\sup\_\{\\lVert x\\rVert\\leq R,\\ r\\in\[0,1\]\}\\lVert v\_\{\\theta\}\(x,r\)\-v\_\{\\theta^\{\\prime\}\}\(x,r\)\\rVert\\leq C\_\{R\}\\lVert\\theta\-\\theta^\{\\prime\}\\rVert\.\(46\)A valid radius containing the parameter set is

RΘ2=∑j=1D\[min⁡\(nj,nj−1\)​Vj2\+cj2\]\.R\_\{\\Theta\}^\{2\}=\\sum\_\{j=1\}^\{D\}\\big\[\\min\(n\_\{j\},n\_\{j\-1\}\)V\_\{j\}^\{2\}\+c\_\{j\}^\{2\}\\big\]\.\(47\)

###### Proof\.

Every matrix entry is bounded by the operator\-norm envelope and every bias coordinate by its Euclidean envelope\. Closedness and finite dimension therefore imply compactness\. More quantitatively, the number of nonzero singular values ofWjW\_\{j\}is at mostmin⁡\(nj,nj−1\)\\min\(n\_\{j\},n\_\{j\-1\}\), so∥Wj∥F2≤min⁡\(nj,nj−1\)​Vj2\\lVert W\_\{j\}\\rVert\_\{F\}^\{2\}\\leq\\min\(n\_\{j\},n\_\{j\-1\}\)V\_\{j\}^\{2\}\. Sum these bounds and the squared bias bounds\. Removing fixed entries can only reduce the trainable\-vector norm, which proves Equation \([47](https://arxiv.org/html/2610.28898#A2.E47)\)\.

Each coordinate of the final hidden vector is bounded bybb, hence∥hD−1∥≤b​nD−1\\lVert h\_\{D\-1\}\\rVert\\leq b\\sqrt\{n\_\{D\-1\}\}\. Applying the output affine map gives∥vθ∥≤VD​b​nD−1\+cD=Bv\\lVert v\_\{\\theta\}\\rVert\\leq V\_\{D\}b\\sqrt\{n\_\{D\-1\}\}\+c\_\{D\}=B\_\{v\}\. For two inputs with the same flow time, every hidden layer multiplies their distance by at mostLϕ​VjL\_\{\\phi\}V\_\{j\}and the output layer by at mostVDV\_\{D\}\. Composing givesLxL\_\{x\}\. The same reasoning applies to the full input\(x,r\)\(x,r\)\. Composition of differentiable maps gives differentiability, and continuous derivatives are not needed for these estimates\.

To prove parameter sensitivity, compare two networks at the same input\. Letdj=∥hj−hj′∥d\_\{j\}=\\lVert h\_\{j\}\-h^\{\\prime\}\_\{j\}\\rVertfor hidden layers and letdDd\_\{D\}denote the final output difference\. Adding and subtractingWj​hj−1′\+bjW\_\{j\}h^\{\\prime\}\_\{j\-1\}\+b\_\{j\}inside the layer difference gives

dj\\displaystyle d\_\{j\}≤kj​∥Wj​\(hj−1−hj−1′\)\+\(Wj−Wj′\)​hj−1′\+bj−bj′∥\\displaystyle\\leq k\_\{j\}\\lVert W\_\{j\}\(h\_\{j\-1\}\-h^\{\\prime\}\_\{j\-1\}\)\+\(W\_\{j\}\-W^\{\\prime\}\_\{j\}\)h^\{\\prime\}\_\{j\-1\}\+b\_\{j\}\-b^\{\\prime\}\_\{j\}\\rVert≤kj​Vj​dj−1\+kj​\[∥Wj−Wj′∥op​Hj−1​\(R\)\+∥bj−bj′∥\]\.\\displaystyle\\leq k\_\{j\}V\_\{j\}d\_\{j\-1\}\+k\_\{j\}\\big\[\\lVert W\_\{j\}\-W^\{\\prime\}\_\{j\}\\rVert\_\{\\mathrm\{op\}\}H\_\{j\-1\}\(R\)\+\\lVert b\_\{j\}\-b^\{\\prime\}\_\{j\}\\rVert\\big\]\.\(48\)The first step forj=Dj=DuseskD=1k\_\{D\}=1\. Each matrix operator\-norm difference is at most its Frobenius difference, which in turn is at most the norm of the vector of all changed entries, and the same is true of each bias block\. Both are at most∥θ−θ′∥\\lVert\\theta\-\\theta^\{\\prime\}\\rVert\. Sinced0=0d\_\{0\}=0, expand the recursion: the perturbation introduced at layerjjis multiplied by the Lipschitz constants of layersj\+1,…,Dj\+1,\\ldots,D\. Summing these contributions is exactly Equation \([45](https://arxiv.org/html/2610.28898#A2.E45)\)\. This proves \([46](https://arxiv.org/html/2610.28898#A2.E46)\)\. ∎

The product\-sum expression is valid if a weight bound is zero, if a product equals one, and if different layers have different bounds\. It never divides byLϕ​V−1L\_\{\\phi\}V\-1\. For a fixed architecture,CR=O⁡\(1\+R\)C\_\{R\}=O\(1\+R\)\. The first\-layer monotonicity retained in Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3)is not used in this proof\.

#### B\.4\.2Gated State Radius and Joint Loss Cover

Absolute continuity of the interpolation and Assumption[2](https://arxiv.org/html/2610.28898#Thmassumption2)imply, for every endpoint pair,

∥Ir​\(x0,x1\)∥≤∥x0∥\+∫0r∥Zs​\(x0,x1\)∥​𝑑s≤∥x0∥\+cZ​\(∥x0∥\+∥x1∥\+1\)\.\\lVert I\_\{r\}\(x\_\{0\},x\_\{1\}\)\\rVert\\leq\\lVert x\_\{0\}\\rVert\+\\int\_\{0\}^\{r\}\\lVert Z\_\{s\}\(x\_\{0\},x\_\{1\}\)\\rVert\\,ds\\leq\\lVert x\_\{0\}\\rVert\+c\_\{Z\}\(\\lVert x\_\{0\}\\rVert\+\\lVert x\_\{1\}\\rVert\+1\)\.\(49\)On the gate, define the resulting spatial radius asRM=M\+cZ​\(2​M\+1\)R\_\{M\}=M\+c\_\{Z\}\(2M\+1\)\. A linear\-endpoint interpolationIr=α⁡\(r\)​x0\+β⁡\(r\)​x1I\_\{r\}=\\alpha\(r\)x\_\{0\}\+\\beta\(r\)x\_\{1\}permits the sharper radiusM​supr\(\|α⁡\(r\)\|\+\|β⁡\(r\)\|\)M\\sup\_\{r\}\(\|\\alpha\(r\)\|\+\|\\beta\(r\)\|\), whereα,β\\alpha,\\betaare its scalar schedules\. The radius need not equalMM: the sine–cosine path can have radius2​M\\sqrt\{2\}M\.

###### Corollary B\.1\.

LetLM=2​CRM​BL\_\{M\}=2C\_\{R\_\{M\}\}\\sqrt\{B\}be the parameter\-Lipschitz constant for the proxy loss\. WriteN⁡\(ε,𝒞,∥⋅∥∞\)N\(\\varepsilon,\\mathcal\{C\},\\lVert\\cdot\\rVert\_\{\\infty\}\)for the covering number of a function class𝒞\\mathcal\{C\}at accuracyε\\varepsilonin supremum norm over all endpoint\-time triples\. For any accuracyε\>0\\varepsilon\>0, the proxy\-loss class has a uniform cover with centers in𝒱\\mathcal\{V\}and log cardinality at most

log⁡N⁡\(ε,\{ℓM​\(vθ\):θ∈Θ\},∥⋅∥∞\)≤P​log⁡\(1\+2​RΘ​LMε\)\.\\log N\(\\varepsilon,\\\{\\ell\_\{M\}\(v\_\{\\theta\}\):\\theta\\in\\Theta\\\},\\lVert\\cdot\\rVert\_\{\\infty\}\)\\leq P\\log\\left\(1\+\\frac\{2R\_\{\\Theta\}L\_\{M\}\}\{\\varepsilon\}\\right\)\.\(50\)If the loss class is constant, one center suffices\. The same cover approximates proxy excess losses against any one fixed field\.

###### Proof\.

For represented predictionsu,vu,v, the difference\-of\-squares identity on the gate gives

\|∥v−Z∥2−∥u−Z∥2\|\\displaystyle\\lvert\\lVert v\-Z\\rVert^\{2\}\-\\lVert u\-Z\\rVert^\{2\}\\rvert=\|⟨v−u,v\+u−2​Z⟩\|\\displaystyle=\\lvert\\langle v\-u,v\+u\-2Z\\rangle\\rvert≤∥v−u∥​\(∥v−Z∥\+∥u−Z∥\)≤2​B​∥v−u∥\.\\displaystyle\\leq\\lVert v\-u\\rVert\(\\lVert v\-Z\\rVert\+\\lVert u\-Z\\rVert\)\\leq 2\\sqrt\{B\}\\lVert v\-u\\rVert\.\(51\)Outside the gate both losses are zero\. Inequality \([46](https://arxiv.org/html/2610.28898#A2.E46)\), with radiusRMR\_\{M\}, makes this difference at mostLM​∥θ−θ′∥L\_\{M\}\\lVert\\theta\-\\theta^\{\\prime\}\\rVertfor every triple\. The Borel derivative version specified in Assumption[2](https://arxiv.org/html/2610.28898#Thmassumption2)satisfies its envelope everywhere, so the supremum assertion is literal and not restricted to a high\-probability set of samples\.

For a parameter\-net radiusγ\>0\\gamma\>0, choose a maximalγ\\gamma\-separated set of points inΘ\\Theta\. The open radius\-γ/2\\gamma/2balls around them are disjoint and contained in a radius\-\(RΘ\+γ/2\)\(R\_\{\\Theta\}\+\\gamma/2\)ball inℝP\\mathbb\{R\}^\{P\}\. Comparing volumes bounds their number by\(1\+2​RΘ/γ\)P\(1\+2R\_\{\\Theta\}/\\gamma\)^\{P\}\. Maximality makes this aγ\\gamma\-net\. Takeγ=ε/LM\\gamma=\\varepsilon/L\_\{M\}ifLM\>0L\_\{M\}\>0\. IfLM=0L\_\{M\}=0, every parameter has the same loss\. This proves \([50](https://arxiv.org/html/2610.28898#A2.E50)\)\. Subtracting a fixed comparator does not change the covering distances\. A supremum\-norm cover is also a cover in every empiricalL2L^\{2\}norm, under heterogeneous as well as identically distributed samples\. ∎

The cover is for the complete vector\-output network, not separate scalar\-coordinate networks\. The parameter count already contains the output dimension\. Dimension may nevertheless enterBvB\_\{v\},RΘR\_\{\\Theta\},LxL\_\{x\}, and the endpoint envelope\. These constants are not all logarithmic in architecture size\.

#### B\.4\.3Learned Dynamics and Approximation Boundaries

For a represented field, global input Lipschitzness and continuity imply existence and uniqueness of the learned ODE on\[0,1\]\[0,1\]\. Its bounded speed gives

supr∈\[0,1\]∥Yr∥≤∥Y0∥\+Bv\.\\sup\_\{r\\in\[0,1\]\}\\lVert Y\_\{r\}\\rVert\\leq\\lVert Y\_\{0\}\\rVert\+B\_\{v\}\.\(52\)There is thus no finite\-time escape\. A measurable solution map follows from Picard iteration, and \([52](https://arxiv.org/html/2610.28898#A2.E52)\) gives finite second moments when the base has finite second moment\. None of these statements grants uniqueness for the ideal field\.

Bounded outputs do impose an approximation restriction\. Writex\+=max⁡\{x,0\}x\_\{\+\}=\\max\\\{x,0\\\}for the positive part\. The reverse triangle inequality gives, for every representedvv,

Qt​\(v\)≥𝔼t​\(∥bt​\(U\)∥−Bv\)\+2,Q\_\{t\}\(v\)\\geq\\mathbb\{E\}\_\{t\}\(\\lVert b\_\{t\}\(U\)\\rVert\-B\_\{v\}\)\_\{\+\}^\{2\},\(53\)The right side may be positive even for infinite data\. Thus Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3)alone does not imply realizability\. Two further limits of the class are worth stating\. The cover complexity in \([50](https://arxiv.org/html/2610.28898#A2.E50)\) is linear in the numberPPof trainable parameters, as is the localized Rademacher bound of[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)\(Appendix[C\.1\.5](https://arxiv.org/html/2610.28898#A3.SS1.SSS5)\), so the guarantees can be loose for heavily overparameterized networks\. Bounded hidden activations exclude unclipped ReLU, GELU, and SiLU networks, a restriction shared with[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)\. If widths or norm bounds increase withTTto reduce this floor, their effects on the cover and the flow\-stability constant must be included\. Parameter sensitivity is local in the input radius: the familyvw​\(x\)=tanh⁡\(w​x\)v\_\{w\}\(x\)=\\tanh\(wx\),w∈\[0,1\]w\\in\[0,1\], hassupx\|vε​\(x\)−v0​\(x\)\|=1\\sup\_\{x\}\|v\_\{\\varepsilon\}\(x\)\-v\_\{0\}\(x\)\|=1for everyε\>0\\varepsilon\>0, precluding a uniform global parameter\-Lipschitz bound\. The gated radius is therefore necessary in the entropy calculation\.

### B\.5Endpoint Gating and Original\-Risk Transfer

###### Proposition B\.2\.

Under Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[3](https://arxiv.org/html/2610.28898#Thmassumption3), the proxy in Equation \([15](https://arxiv.org/html/2610.28898#S3.E15)\) satisfies0≤ℓM​\(v\)≤B0\\leq\\ell\_\{M\}\(v\)\\leq B\. Its population versionLt,M​\(v\)=𝔼t​ℓM​\(v\)L\_\{t,M\}\(v\)=\\mathbb\{E\}\_\{t\}\\ell\_\{M\}\(v\)obeys, for every round and represented field,

0≤Lt​\(v\)−Lt,M​\(v\)≤δM,0\\leq L\_\{t\}\(v\)\-L\_\{t,M\}\(v\)\\leq\\delta\_\{M\},\(54\)withδM\\delta\_\{M\}given by Equation \([18](https://arxiv.org/html/2610.28898#S4.E18)\)\. The calibration in Equation \([24](https://arxiv.org/html/2610.28898#S4.E24)\) gives \([25](https://arxiv.org/html/2610.28898#S4.E25)\)\.

###### Proof\.

Fix the environment and round\. LetS=∥X0∥\+∥X1∥S=\\lVert X\_\{0\}\\rVert\+\\lVert X\_\{1\}\\rVert\. Since the represented output is bounded,

∥v⁡\(U\)−Z∥2\\displaystyle\\lVert v\(U\)\-Z\\rVert^\{2\}≤\[Bv\+cZ\+cZ​S\]2\\displaystyle\\leq\[B\_\{v\}\+c\_\{Z\}\+c\_\{Z\}S\]^\{2\}≤2​\(Bv\+cZ\)2\+2​cZ2​S2≤Cℓ​\(1\+S2\)\.\\displaystyle\\leq 2\(B\_\{v\}\+c\_\{Z\}\)^\{2\}\+2c\_\{Z\}^\{2\}S^\{2\}\\leq C\_\{\\ell\}\(1\+S^\{2\}\)\.\(55\)On the gateS≤2​MS\\leq 2M, giving the proxy boundBB\. To bound the omitted expected loss, useS2≤2​\(∥X0∥2\+∥X1∥2\)S^\{2\}\\leq 2\(\\lVert X\_\{0\}\\rVert^\{2\}\+\\lVert X\_\{1\}\\rVert^\{2\}\)and Cauchy–Schwarz:

𝔼t​eS2/\(4​κ2\)\\displaystyle\\mathbb\{E\}\_\{t\}e^\{S^\{2\}/\(4\\kappa^\{2\}\)\}≤𝔼t​e\(∥X0∥2\+∥X1∥2\)/\(2​κ2\)\\displaystyle\\leq\\mathbb\{E\}\_\{t\}e^\{\(\\lVert X\_\{0\}\\rVert^\{2\}\+\\lVert X\_\{1\}\\rVert^\{2\}\)/\(2\\kappa^\{2\}\)\}≤\(𝔼t​e∥X0∥2/κ2\)1/2​\(𝔼t​e∥X1∥2/κ2\)1/2≤2\.\\displaystyle\\leq\\big\(\\mathbb\{E\}\_\{t\}e^\{\\lVert X\_\{0\}\\rVert^\{2\}/\\kappa^\{2\}\}\\big\)^\{1/2\}\\big\(\\mathbb\{E\}\_\{t\}e^\{\\lVert X\_\{1\}\\rVert^\{2\}/\\kappa^\{2\}\}\\big\)^\{1/2\}\\leq 2\.\(56\)This step does not require independence of the endpoints\. Markov’s inequality givesℙ\(S2\>u∣E\)≤2e−u/\(4κ2\)\\mathbb\{P\}\(S^\{2\}\>u\\mid E\)\\leq 2e^\{\-u/\(4\\kappa^\{2\}\)\}\. Rejection impliesS\>MS\>M\. For any nonnegative random variableYYand thresholdaa, tail integration gives

𝔼\[\(1\+Y\)𝟏\{Y\>a\}\]=\(1\+a\)ℙ\(Y\>a\)\+∫a∞ℙ\(Y\>u\)du\.\\mathbb\{E\}\[\(1\+Y\)\\mathbf\{1\}\\\{Y\>a\\\}\]=\(1\+a\)\\mathbb\{P\}\(Y\>a\)\+\\int\_\{a\}^\{\\infty\}\\mathbb\{P\}\(Y\>u\)\\,du\.\(57\)Apply it withY=S2Y=S^\{2\}anda=M2a=M^\{2\}\. The omitted loss is at most

Cℓ𝔼t\[\(1\+S2\)𝟏\{GM=0\}\]\\displaystyle C\_\{\\ell\}\\mathbb\{E\}\_\{t\}\[\(1\+S^\{2\}\)\\mathbf\{1\}\\\{G\_\{M\}=0\\\}\]≤2Cℓ\[\(1\+M2\)e−M2/\(4κ2\)\+∫M2∞e−u/\(4κ2\)du\]\\displaystyle\\leq 2C\_\{\\ell\}\\left\[\(1\+M^\{2\}\)e^\{\-M^\{2\}/\(4\\kappa^\{2\}\)\}\+\\int\_\{M^\{2\}\}^\{\\infty\}e^\{\-u/\(4\\kappa^\{2\}\)\}du\\right\]=2Cℓ\(1\+M2\+4κ2\)e−M2/\(4κ2\)\.\\displaystyle=2C\_\{\\ell\}\(1\+M^\{2\}\+4\\kappa^\{2\}\)e^\{\-M^\{2\}/\(4\\kappa^\{2\}\)\}\.\(58\)The loss is nonnegative, so the discrepancy also has the correct lower sign\. SubstitutingM2=16​κ2​log⁡\(e​m​T\)M^\{2\}=16\\kappa^\{2\}\\log\(emT\)makes the exponential\(e​m​T\)−4\(emT\)^\{\-4\}, which proves \([25](https://arxiv.org/html/2610.28898#S4.E25)\)\. ∎

#### B\.5\.1Consequences Used by the Statistical Proof

Define the nonnegative bias functionβt​\(v\)=Lt​\(v\)−Lt,M​\(v\)\\beta\_\{t\}\(v\)=L\_\{t\}\(v\)\-L\_\{t,M\}\(v\)\. Inequality \([54](https://arxiv.org/html/2610.28898#A2.E54)\) statesβt​\(v\)∈\[0,δ\]\\beta\_\{t\}\(v\)\\in\[0,\\delta\]\. Thus a proxy difference and its original counterpart differ by at mostδ\\delta, and the later estimates conservatively use2​δ2\\delta\. If proxy losses of two fields differ uniformly by at mostζ\\zeta, their original risks differ by at mostζ\+2​δ\\zeta\+2\\delta\. These inequalities also hold after averaging over any deterministic historical interval\. They are uniform overvv, so they apply to data\-dependent fits\.

The gate is shared by all models, independent of their predictions\. A rejected example remains in the denominator of the empirical average\. This makes the objective exactly the empirical mean ofℓM\\ell\_\{M\}and the score expectation exactlyLt,ML\_\{t,M\}\. Dividing instead by the number accepted would define a different random\-denominator objective\. Resampling until acceptance would change the data law\. Neither operation is part of the learner\.

The gated conditional regression target need not equalbtb\_\{t\}: wherever𝔼t​\[GM∣U\]\>0\\mathbb\{E\}\_\{t\}\[G\_\{M\}\\mid U\]\>0, the gated minimizer among arbitrary measurable fields is𝔼t​\[GM​Z∣U\]/𝔼t​\[GM∣U\]\\mathbb\{E\}\_\{t\}\[G\_\{M\}Z\\mid U\]/\\mathbb\{E\}\_\{t\}\[G\_\{M\}\\mid U\]\. We do not substitute it for the original conditional mean\. Instead, we bound gated second moments by ungated velocity risks and use \([54](https://arxiv.org/html/2610.28898#A2.E54)\) only for mean\-risk transfer\. This avoids a hidden assumption that truncation preserves the regression target\.

#### B\.5\.2Tail Formulations and Bounded\-Support Recovery

To relate Assumption[1](https://arxiv.org/html/2610.28898#Thmassumption1)to directional tails, suppose for a finiteK\>0K\>0thatℙ\(\|⟨u,X⟩\|≥z∣E\)≤2e−z2/K\\mathbb\{P\}\(\|\\langle u,X\\rangle\|\\geq z\\mid E\)\\leq 2e^\{\-z^\{2\}/K\}for every unit vectoruuandz≥0z\\geq 0\. Each coordinate then satisfies

𝔼\[eXi2/\(3​K\)∣E\]=1\+13​K∫0∞es/\(3​K\)ℙ\(Xi2\>s∣E\)ds≤1\+23​K∫0∞e−2s/\(3K\)ds=2\.\\mathbb\{E\}\[e^\{X\_\{i\}^\{2\}/\(3K\)\}\\mid E\]=1\+\\frac\{1\}\{3K\}\\int\_\{0\}^\{\\infty\}e^\{s/\(3K\)\}\\mathbb\{P\}\(X\_\{i\}^\{2\}\>s\\mid E\)\\,ds\\leq 1\+\\frac\{2\}\{3K\}\\int\_\{0\}^\{\\infty\}e^\{\-2s/\(3K\)\}ds=2\.\(59\)Convexity of the exponential gives𝔼⁡\[e∥X∥2/\(3​K​d\)∣E\]≤d−1​∑i𝔼⁡\[eXi2/\(3​K\)∣E\]≤2\\mathbb\{E\}\[e^\{\\lVert X\\rVert^\{2\}/\(3Kd\)\}\\mid E\]\\leq d^\{\-1\}\\sum\_\{i\}\\mathbb\{E\}\[e^\{X\_\{i\}^\{2\}/\(3K\)\}\\mid E\]\\leq 2\. Thusκ2=3​K​d\\kappa^\{2\}=3Kdis sufficient without coordinate independence\. Conversely, Assumption[1](https://arxiv.org/html/2610.28898#Thmassumption1)givesℙ\(∥X∥≥R∣E\)≤2e−R2/κ2\\mathbb\{P\}\(\\lVert X\\rVert\\geq R\\mid E\)\\leq 2e^\{\-R^\{2\}/\\kappa^\{2\}\}and hence directional control\. This comparison retains dimension dependence and requires a uniform envelope across rounds\.

If endpoints already have known bounded support, choose the proxy to be the original loss, use its positive envelopeBB, and setδ=0\\delta=0\. The remaining proofs apply unchanged\. If the loss is identically zero, regret vanishes and normalization can be omitted\. Under the stated sub\-Gaussian model,cZ\>0c\_\{Z\}\>0and the chosenMMensureB\>0B\>0without such a special case\.

## Appendix CUseful Results to Prove Theorems 1 and 2

The historical\-estimation and aggregation arguments below apply to both theorems\. The first\-order transfer and lifetime bounds also supply the change\-count branch of Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)through Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)\. The final lemma provides the dyadic cover used in both proofs\.

### C\.1Historical Statistical Estimation

Fix the environment throughout this appendix\. LetHHbe a deterministic interval ofq≥1q\\geq 1rounds\. It containsn=m​qn=mqobservations, denoted byξ1,…,ξn\\xi\_\{1\},\\ldots,\\xi\_\{n\}in an arbitrary deterministic order preserving each observation’s round label\. Writes⁡\(i\)s\(i\)for observationii’s round, andPiP\_\{i\}for its conditional law\. Independence holds acrossii, but the lawsPiP\_\{i\}may differ\. The comparatorvH⋆v\_\{H\}^\{\\star\}, errora¯H\\bar\{a\}\_\{H\}, and excessrHr\_\{H\}are defined in Equation \([26](https://arxiv.org/html/2610.28898#S5.E26)\)\. The population comparator is fixed under the conditioned environment and is not fitted to these observations\.

#### C\.1\.1A Derived Approximate Bernstein Inequality

###### Lemma C\.1\.

For a fixed represented fieldvv, letgi​\(v\)=ℓM​\(v,ξi\)−ℓM​\(vH⋆,ξi\)g\_\{i\}\(v\)=\\ell\_\{M\}\(v;\\xi\_\{i\}\)\-\\ell\_\{M\}\(v\_\{H\}^\{\\star\};\\xi\_\{i\}\)\. Then\|gi​\(v\)\|≤B\|g\_\{i\}\(v\)\|\\leq Band

1n​∑i𝔼​gi​\(v\)2≤8​B​\{rH​\(v\)\+2​a¯H\}\.\\frac\{1\}\{n\}\\sum\_\{i\}\\mathbb\{E\}g\_\{i\}\(v\)^\{2\}\\leq 8B\\\{r\_\{H\}\(v\)\+2\\bar\{a\}\_\{H\}\\\}\.\(60\)Consequently the same right\-hand side bounds the average of their variances\.

For a single target law this is the Bernstein\-type variance control of squared\-loss flow matching used by[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25), in the approximate form that follows from the regression identity\. For a heterogeneous history the comparator is the best common fieldvH⋆v\_\{H\}^\{\\star\}, and the residuala¯H\\bar\{a\}\_\{H\}can be positive even when every round is realizable\.

###### Proof\.

Both proxy losses lie in\[0,B\]\[0,B\]\. On an accepted observation both ungated residuals have norm at mostB\\sqrt\{B\}, so the difference\-of\-squares identity yields

gi​\(v\)2≤4​B​GM​\(ξi\)​∥v⁡\(Ui\)−vH⋆​\(Ui\)∥2\.g\_\{i\}\(v\)^\{2\}\\leq 4BG\_\{M\}\(\\xi\_\{i\}\)\\lVert v\(U\_\{i\}\)\-v\_\{H\}^\{\\star\}\(U\_\{i\}\)\\rVert^\{2\}\.\(61\)Insert the ideal fieldbs⁡\(i\)b\_\{s\(i\)\}appropriate to this observation’s law and use∥x\+y∥2≤2​∥x∥2\+2​∥y∥2\\lVert x\+y\\rVert^\{2\}\\leq 2\\lVert x\\rVert^\{2\}\+2\\lVert y\\rVert^\{2\}:

𝔼​gi​\(v\)2≤8​B​\{Qs⁡\(i\)​\(v\)\+Qs⁡\(i\)​\(vH⋆\)\}\.\\mathbb\{E\}g\_\{i\}\(v\)^\{2\}\\leq 8B\\\{Q\_\{s\(i\)\}\(v\)\+Q\_\{s\(i\)\}\(v\_\{H\}^\{\\star\}\)\\\}\.\(62\)Dropping the gate is valid because the squared quantities are nonnegative\. Each round occurs exactlymmtimes\. Averaging overiitherefore gives8​B​\[q−1​∑sQs​\(v\)\+a¯H\]8B\[q^\{\-1\}\\sum\_\{s\}Q\_\{s\}\(v\)\+\\bar\{a\}\_\{H\}\]\. The regression identity impliesq−1​∑sQs​\(v\)=rH​\(v\)\+a¯Hq^\{\-1\}\\sum\_\{s\}Q\_\{s\}\(v\)=r\_\{H\}\(v\)\+\\bar\{a\}\_\{H\}, since the irreducible regression\-noise term cancels separately at each round\. Substituting proves \([60](https://arxiv.org/html/2610.28898#A3.E60)\)\. FinallyVar⁡\(gi\)≤𝔼​gi2\\operatorname\{Var\}\(g\_\{i\}\)\\leq\\mathbb\{E\}g\_\{i\}^\{2\}for each individual law\. ∎

The intermediate function\-space inequality is

1q​∑s∈H𝔼s​∥v⁡\(U\)−vH⋆​\(U\)∥2≤2​rH​\(v\)\+4​a¯H\.\\frac\{1\}\{q\}\\sum\_\{s\\in H\}\\mathbb\{E\}\_\{s\}\\lVert v\(U\)\-v\_\{H\}^\{\\star\}\(U\)\\rVert^\{2\}\\leq 2r\_\{H\}\(v\)\+4\\bar\{a\}\_\{H\}\.\(63\)It follows from the same insertion ofbsb\_\{s\}\. This is not a parameter\-space curvature assumption, anda¯H\\bar\{a\}\_\{H\}cannot in general be removed\. At a tie between two separated class minimizers, excess risk may be zero while their prediction separation is positive\. For example, take a scalar base concentrated at zero, a target uniform on\{−1,1\}\\\{\-1,1\\\}, linear interpolation, and only the two constant fieldsv±=±cv\_\{\\pm\}=\\pm c\. Both minimize the loss, yet their sampled loss differenceg=−4​c​X1g=\-4cX\_\{1\}has

L⁡\(v\+\)−L⁡\(v−\)=0,Var⁡\(g\)=16​c2\.L\(v\_\{\+\}\)\-L\(v\_\{\-\}\)=0,\\qquad\\operatorname\{Var\}\(g\)=16c^\{2\}\.\(64\)No constant bounds this variance by their zero excess risk, so the approximation term in \([60](https://arxiv.org/html/2610.28898#A3.E60)\) is substantive for nonconvex classes\. The example explains a proof boundary and is not a lower bound on achievable regret\.

#### C\.1\.2Uniform Concentration for a Fixed History

For a fixedvv, center each observation separately by definingYi​\(v\)=𝔼​gi​\(v\)−gi​\(v\)Y\_\{i\}\(v\)=\\mathbb\{E\}g\_\{i\}\(v\)\-g\_\{i\}\(v\)\. The variables are independent and mean zero, with\|Yi​\(v\)\|≤2​B\|Y\_\{i\}\(v\)\|\\leq 2B\. Their total variance is at most8​B​n​\[rH​\(v\)\+2​a¯H\]8Bn\[r\_\{H\}\(v\)\+2\\bar\{a\}\_\{H\}\]\. The one\-sided Bernstein inequality for independent bounded variables\([Boucheron et al\., 2013](https://arxiv.org/html/2610.28898#bib.bib13)\)then implies, for anyu≥0u\\geq 0, except on an event of probability at moste−ue^\{\-u\},

1n​∑i𝔼​gi​\(v\)−1n​∑igi​\(v\)≤4​B⁡\[rH​\(v\)\+2​a¯H\]​un\+2​B​un\.\\frac\{1\}\{n\}\\sum\_\{i\}\\mathbb\{E\}g\_\{i\}\(v\)\-\\frac\{1\}\{n\}\\sum\_\{i\}g\_\{i\}\(v\)\\leq 4\\sqrt\{\\frac\{B\[r\_\{H\}\(v\)\+2\\bar\{a\}\_\{H\}\]u\}\{n\}\}\+\\frac\{2Bu\}\{n\}\.\(65\)The standard linear Bernstein remainder2​B​u/\(3​n\)2Bu/\(3n\)is enlarged to2​B​u/n2Bu/n\. No identical\-distribution reduction is used\.

Choose a deterministicζ\\zeta\-cover of the proxy\-loss class withKMK\_\{M\}centers, as in Corollary[B\.1](https://arxiv.org/html/2610.28898#Thmcorollary1)\. For a desiredz≥0z\\geq 0, setu=log⁡KM\+zu=\\log K\_\{M\}\+z\. Apply \([65](https://arxiv.org/html/2610.28898#A3.E65)\) to each center and sum the failure probabilities:KM​e−u=e−zK\_\{M\}e^\{\-u\}=e^\{\-z\}\. On the resulting event the inequalities hold at every center simultaneously\.

For a general fieldvv, letvkv\_\{k\}be a center whose proxy loss is withinζ\\zetauniformly\. DefineHu=B​u/nH\_\{u\}=Bu/nfor this proof\. Empirical comparator differences satisfy

\|1n​∑igi​\(v\)−1n​∑igi​\(vk\)\|≤ζ,\\lvert\\frac\{1\}\{n\}\\sum\_\{i\}g\_\{i\}\(v\)\-\\frac\{1\}\{n\}\\sum\_\{i\}g\_\{i\}\(v\_\{k\}\)\\rvert\\leq\\zeta,\(66\)and the original population excesses satisfy\|rH​\(v\)−rH​\(vk\)\|≤ζ\+2​δ\|r\_\{H\}\(v\)\-r\_\{H\}\(v\_\{k\}\)\|\\leq\\zeta\+2\\delta\. In addition, the average proxy excess againstvH⋆v\_\{H\}^\{\\star\}differs fromrH​\(vk\)r\_\{H\}\(v\_\{k\}\)by at most2​δ2\\delta\. Combining these three comparisons with \([65](https://arxiv.org/html/2610.28898#A3.E65)\) atvkv\_\{k\}gives

rH​\(v\)\\displaystyle r\_\{H\}\(v\)≤1n​∑igi​\(v\)\+2​ζ\+4​δ\+4​Hu​\[rH​\(vk\)\+2​a¯H\]\+2​Hu\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{i\}g\_\{i\}\(v\)\+2\\zeta\+4\\delta\+4\\sqrt\{H\_\{u\}\[r\_\{H\}\(v\_\{k\}\)\+2\\bar\{a\}\_\{H\}\]\}\+2H\_\{u\}≤1n​∑igi​\(v\)\+2​ζ\+4​δ\+4​Hu​\[rH​\(v\)\+2​a¯H\+ζ\+2​δ\]\+2​Hu\.\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{i\}g\_\{i\}\(v\)\+2\\zeta\+4\\delta\+4\\sqrt\{H\_\{u\}\[r\_\{H\}\(v\)\+2\\bar\{a\}\_\{H\}\+\\zeta\+2\\delta\]\}\+2H\_\{u\}\.\(67\)This holds for every field on one event\. A data\-dependent approximate minimizer may therefore be substituted without assuming it is independent of the data\. The cover centers themselves are fixed before observing the history\.

#### C\.1\.3Proof of Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)

The original population minimizervH⋆v\_\{H\}^\{\\star\}is an admissible empirical competitor\. By Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4),

1n​∑igi​\(v^\)=L^H,M​\(v^\)−L^H,M​\(vH⋆\)≤ϵopt\.\\frac\{1\}\{n\}\\sum\_\{i\}g\_\{i\}\(\\widehat\{v\}\)=\\widehat\{L\}\_\{H,M\}\(\\widehat\{v\}\)\-\\widehat\{L\}\_\{H,M\}\(v\_\{H\}^\{\\star\}\)\\leq\\epsilon\_\{\\mathrm\{opt\}\}\.\(68\)Letr=rH​\(v^\)r=r\_\{H\}\(\\widehat\{v\}\)andx=2​a¯H\+ζ\+2​δx=2\\bar\{a\}\_\{H\}\+\\zeta\+2\\delta, both nonnegative\. User\+x≤r\+x\\sqrt\{r\+x\}\\leq\\sqrt\{r\}\+\\sqrt\{x\}in \([67](https://arxiv.org/html/2610.28898#A3.E67)\), and then4​Hu​r≤r/2\+8​Hu4\\sqrt\{H\_\{u\}r\}\\leq r/2\+8H\_\{u\}\. This yields

r\\displaystyle r≤ϵopt\+2​ζ\+4​δ\+12​r\+10​Hu\+4​Hu​x,\\displaystyle\\leq\\epsilon\_\{\\mathrm\{opt\}\}\+2\\zeta\+4\\delta\+\\tfrac\{1\}\{2\}r\+10H\_\{u\}\+4\\sqrt\{H\_\{u\}x\},r\\displaystyle r≤2​ϵopt\+4​ζ\+8​δ\+20​Hu\+8​Hu​x\.\\displaystyle\\leq 2\\epsilon\_\{\\mathrm\{opt\}\}\+4\\zeta\+8\\delta\+20H\_\{u\}\+8\\sqrt\{H\_\{u\}x\}\.\(69\)SubstitutingHu=B⁡\(log⁡KM\+z\)/nH\_\{u\}=B\(\\log K\_\{M\}\+z\)/ngives, with probability at least1−e−z1\-e^\{\-z\},

rH​\(v^\)≤\\displaystyle r\_\{H\}\(\\widehat\{v\}\)\\leq\{\}2​ϵopt\+4​ζ\+8​δ\+20​B​\(log⁡KM\+z\)n\\displaystyle 2\\epsilon\_\{\\mathrm\{opt\}\}\+4\\zeta\+8\\delta\+\\frac\{20B\(\\log K\_\{M\}\+z\)\}\{n\}\+8​B⁡\(log⁡KM\+z\)n​\(2​a¯H\+ζ\+2​δ\)\.\\displaystyle\+8\\sqrt\{\\frac\{B\(\\log K\_\{M\}\+z\)\}\{n\}\(2\\bar\{a\}\_\{H\}\+\\zeta\+2\\delta\)\}\.\(70\)This is the high\-probability part of Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)\. Applying the elementary inequality8​Hu​x≤16​Hu\+x8\\sqrt\{H\_\{u\}x\}\\leq 16H\_\{u\}\+xto the final term instead gives

rH​\(v^\)≤2​ϵopt\+36​Hu\+2​a¯H\+5​ζ\+10​δ\.r\_\{H\}\(\\widehat\{v\}\)\\leq 2\\epsilon\_\{\\mathrm\{opt\}\}\+36H\_\{u\}\+2\\bar\{a\}\_\{H\}\+5\\zeta\+10\\delta\.\(71\)This inequality also fails with probability at moste−ze^\{\-z\}for everyz≥0z\\geq 0\. Integrating the upper tail is legitimate because the risk is bounded by a finite original\-loss envelope\. For clarity, if a nonnegative random variableRRobeysℙ⁡\(R\>a\+c​z\)≤e−z\\mathbb\{P\}\(R\>a\+cz\)\\leq e^\{\-z\}for everyz≥0z\\geq 0, then

𝔼​R≤a\+∫a∞ℙ⁡\(R\>s\)​𝑑s≤a\+c​∫0∞e−z​𝑑z=a\+c\.\\mathbb\{E\}R\\leq a\+\\int\_\{a\}^\{\\infty\}\\mathbb\{P\}\(R\>s\)ds\\leq a\+c\\int\_\{0\}^\{\\infty\}e^\{\-z\}dz=a\+c\.\(72\)Apply this fact to \([71](https://arxiv.org/html/2610.28898#A3.E71)\)\. It replaceslog⁡KM\+z\\log K\_\{M\}\+zbylog⁡KM\+1\\log K\_\{M\}\+1, which is at mostΓ=1\+log⁡\(2​KM\)\\Gamma=1\+\\log\(2K\_\{M\}\), and proves \([28](https://arxiv.org/html/2610.28898#S5.E28)\)\.

The square\-root interaction can also be retained in expectation\. Inequality \([69](https://arxiv.org/html/2610.28898#A3.E69)\) bounds the upper quantiles by its increasing right side evaluated at a mean\-one exponential random variable\. Integrate those quantiles and use concavity of the square root\. The resulting sharper inequality is

𝔼​rH​\(v^\)≤2​ϵopt\+4​ζ\+8​δ\+20​B​Γn\+8​B​Γn​\(2​a¯H\+ζ\+2​δ\)\.\\mathbb\{E\}r\_\{H\}\(\\widehat\{v\}\)\\leq 2\\epsilon\_\{\\mathrm\{opt\}\}\+4\\zeta\+8\\delta\+\\frac\{20B\\Gamma\}\{n\}\+8\\sqrt\{\\frac\{B\\Gamma\}\{n\}\(2\\bar\{a\}\_\{H\}\+\\zeta\+2\\delta\)\}\.\(73\)This auxiliary exponential variable is only a tail\-integration device\. It is not an additional sampling assumption\. In a stationary realizable problem,a¯H=0\\bar\{a\}\_\{H\}=0and the leading statistical contribution isB​Γ/nB\\Gamma/n\. In a stationary misspecified problem, \([73](https://arxiv.org/html/2610.28898#A3.E73)\) retains an interaction of orderB​Γ​a¯H/n\\sqrt\{B\\Gamma\\bar\{a\}\_\{H\}/n\}, precluding a blanket1/n1/nproper\-selection claim for arbitrary nonconvex classes\.

#### C\.1\.4First\-Order Historical\-to\-Current Transfer

###### Lemma C\.2\.

LetH=\[a,a\+q−1\]H=\[a,a\+q\-1\]precede roundt\>a\+q−1t\>a\+q\-1\. WithΩa:t=∑s=at−1ωs\\Omega\_\{a:t\}=\\sum\_\{s=a\}^\{t\-1\}\\omega\_\{s\}, the bounds \([32](https://arxiv.org/html/2610.28898#S5.E32)\) and \([33](https://arxiv.org/html/2610.28898#S5.E33)\) hold for every represented field used as a historical fit\. They hold pointwise, before expectation over its data\.

###### Proof\.

Choosevt⋆∈arg​minv⁡Lt​\(v\)v\_\{t\}^\{\\star\}\\in\\argmin\_\{v\}L\_\{t\}\(v\)\. For anys∈Hs\\in H, letvs⋆v\_\{s\}^\{\\star\}be a round\-ssminimizer\. The optimality ofvt⋆v\_\{t\}^\{\\star\}atttand the telescoping oscillation bound give

Fs​\(vt⋆\)\\displaystyle F\_\{s\}\(v\_\{t\}^\{\\star\}\)=Ls​\(vt⋆\)−Ls​\(vs⋆\)\\displaystyle=L\_\{s\}\(v\_\{t\}^\{\\star\}\)\-L\_\{s\}\(v\_\{s\}^\{\\star\}\)≤\[Ls​\(vt⋆\)−Lt​\(vt⋆\)\]−\[Ls​\(vs⋆\)−Lt​\(vs⋆\)\]\\displaystyle\\leq\[L\_\{s\}\(v\_\{t\}^\{\\star\}\)\-L\_\{t\}\(v\_\{t\}^\{\\star\}\)\]\-\[L\_\{s\}\(v\_\{s\}^\{\\star\}\)\-L\_\{t\}\(v\_\{s\}^\{\\star\}\)\]≤osc\(Ls−Lt\)≤2Ωa:t\.\\displaystyle\\leq\\operatorname\{osc\}\(L\_\{s\}\-L\_\{t\}\)\\leq 2\\Omega\_\{a:t\}\.\(74\)The historical minimizer also minimizes the average ofQsQ\_\{s\}, because the regression\-noise offsets are independent of the field\. Thus

a¯H≤1q∑s∈HQs\(vt⋆\)=1q∑s∈H\[as\+Fs\(vt⋆\)\]≤1q∑s∈Has\+2Ωa:t\.\\bar\{a\}\_\{H\}\\leq\\frac\{1\}\{q\}\\sum\_\{s\\in H\}Q\_\{s\}\(v\_\{t\}^\{\\star\}\)=\\frac\{1\}\{q\}\\sum\_\{s\\in H\}\[a\_\{s\}\+F\_\{s\}\(v\_\{t\}^\{\\star\}\)\]\\leq\\frac\{1\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+2\\Omega\_\{a:t\}\.\(75\)For any fieldvv, expand its current excess againstvt⋆v\_\{t\}^\{\\star\}around historical average losses:

Ft​\(v\)\\displaystyle F\_\{t\}\(v\)=L¯H​\(v\)−L¯H​\(vt⋆\)\+\(Lt−L¯H\)​\(v\)−\(Lt−L¯H\)​\(vt⋆\)\\displaystyle=\\bar\{L\}\_\{H\}\(v\)\-\\bar\{L\}\_\{H\}\(v\_\{t\}^\{\\star\}\)\+\(L\_\{t\}\-\\bar\{L\}\_\{H\}\)\(v\)\-\(L\_\{t\}\-\\bar\{L\}\_\{H\}\)\(v\_\{t\}^\{\\star\}\)≤rH​\(v\)\+osc⁡\(Lt−L¯H\)≤rH​\(v\)\+1q​∑s∈Hosc⁡\(Lt−Ls\)\\displaystyle\\leq r\_\{H\}\(v\)\+\\operatorname\{osc\}\(L\_\{t\}\-\\bar\{L\}\_\{H\}\)\\leq r\_\{H\}\(v\)\+\\frac\{1\}\{q\}\\sum\_\{s\\in H\}\\operatorname\{osc\}\(L\_\{t\}\-L\_\{s\}\)≤rH\(v\)\+2Ωa:t\.\\displaystyle\\leq r\_\{H\}\(v\)\+2\\Omega\_\{a:t\}\.\(76\)The first inequality usesL¯H​\(vH⋆\)≤L¯H​\(vt⋆\)\\bar\{L\}\_\{H\}\(v\_\{H\}^\{\\star\}\)\\leq\\bar\{L\}\_\{H\}\(v\_\{t\}^\{\\star\}\)\. This proves both claims and displays exactly where common offsets cancel\. ∎

Both bounds are first order in the oscillation budget and hold for every flow\-time law\. Lemma[E\.7](https://arxiv.org/html/2610.28898#Thmlemma7a)gives second\-order counterparts in the target\-law budget under Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\.

#### C\.1\.5A Localized Rademacher Route

[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)obtain the fixed\-target rate by localized Rademacher complexity\([Bartlett et al\., 2005](https://arxiv.org/html/2610.28898#bib.bib9)\): they cover the loss class through the parameter\-Lipschitz property of the network, bound the local Rademacher complexity by Dudley’s entropy integral, and solve a sub\-root fixed point\. This subsubsection runs the same route for a heterogeneous history and compares the result with Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)\. Both routes give a statistical term of orderB​P​log⁡\(⋅\)/nBP\\log\(\\cdot\)/n, with the same approximation residual\.

##### Setting\.

FixH=\[a,a\+q−1\]H=\[a,a\+q\-1\]andn=m​qn=mq\. The examplesξ1,…,ξn\\xi\_\{1\},\\ldots,\\xi\_\{n\}are independent, andξi\\xi\_\{i\}has the lawPiP\_\{i\}of its round\. For a functionggof one example letP¯​g=n−1​∑i𝔼Pi​g​\(ξi\)\\bar\{P\}g=n^\{\-1\}\\sum\_\{i\}\\mathbb\{E\}\_\{P\_\{i\}\}g\(\\xi\_\{i\}\),Pn​g=n−1​∑ig⁡\(ξi\)P\_\{n\}g=n^\{\-1\}\\sum\_\{i\}g\(\\xi\_\{i\}\), andT⁡\(g\)=n−1​∑i𝔼Pi​g​\(ξi\)2T\(g\)=n^\{\-1\}\\sum\_\{i\}\\mathbb\{E\}\_\{P\_\{i\}\}g\(\\xi\_\{i\}\)^\{2\}\. Letgv=ℓM​\(v,⋅\)−ℓM​\(vH⋆,⋅\)g\_\{v\}=\\ell\_\{M\}\(v;\\cdot\)\-\\ell\_\{M\}\(v\_\{H\}^\{\\star\};\\cdot\),𝒢=\{gv:v∈𝒱\}\\mathcal\{G\}=\\\{g\_\{v\}:v\\in\\mathcal\{V\}\\\}, and𝒢⁡\(r\)=\{g∈𝒢:T⁡\(g\)≤r\}\\mathcal\{G\}\(r\)=\\\{g\\in\\mathcal\{G\}:T\(g\)\\leq r\\\}\. Eachgvg\_\{v\}takes values in\[−B,B\]\[\-B,B\], andgvH⋆=0g\_\{v\_\{H\}^\{\\star\}\}=0lies in every𝒢⁡\(r\)\\mathcal\{G\}\(r\)\. For a classℱ\\mathcal\{F\}writeℛn​\(ℱ\)=𝔼​supf∈ℱn−1​∑iσi​f​\(ξi\)\\mathcal\{R\}\_\{n\}\(\\mathcal\{F\}\)=\\mathbb\{E\}\\sup\_\{f\\in\\mathcal\{F\}\}n^\{\-1\}\\sum\_\{i\}\\sigma\_\{i\}f\(\\xi\_\{i\}\)with independent Rademacher signsσi\\sigma\_\{i\}\. LetL=P​log⁡\(1\+4​RΘ​LM​n/B\)L=P\\log\(1\+4R\_\{\\Theta\}L\_\{M\}n/B\)\.

##### Step 1: Bernstein condition with a residual\.

By \([54](https://arxiv.org/html/2610.28898#A2.E54)\),rH​\(v\)≤P¯​gv\+δr\_\{H\}\(v\)\\leq\\bar\{P\}g\_\{v\}\+\\delta\. Lemma[C\.1](https://arxiv.org/html/2610.28898#Thmlemma1a)then gives, withB0=8​BB\_\{0\}=8Bandρ0=δ\+2​a¯H\\rho\_\{0\}=\\delta\+2\\bar\{a\}\_\{H\},

T⁡\(gv\)≤8​B​\{rH​\(v\)\+2​a¯H\}≤B0​\(P¯​gv\+ρ0\)\(v∈𝒱\)\.T\(g\_\{v\}\)\\leq 8B\\\{r\_\{H\}\(v\)\+2\\bar\{a\}\_\{H\}\\\}\\leq B\_\{0\}\(\\bar\{P\}g\_\{v\}\+\\rho\_\{0\}\)\\qquad\(v\\in\\mathcal\{V\}\)\.\(77\)

##### Step 2: local complexity from the cover\.

###### Lemma C\.3\.

For everyr\>0r\>0,ℛn​\(𝒢⁡\(r\)\)≤24​r​L/n\+\(576​L\+8\)​B/n\\mathcal\{R\}\_\{n\}\(\\mathcal\{G\}\(r\)\)\\leq 24\\sqrt\{rL/n\}\+\(576L\+8\)B/n\.

###### Proof\.

Subtracting the fixed functionℓM​\(vH⋆,⋅\)\\ell\_\{M\}\(v\_\{H\}^\{\\star\};\\cdot\)preserves supremum\-norm distances, so Corollary[B\.1](https://arxiv.org/html/2610.28898#Thmcorollary1)gives a supremum\-normε\\varepsilon\-cover of𝒢\\mathcal\{G\}of log size at mostP​log⁡\(1\+2​RΘ​LM/ε\)P\\log\(1\+2R\_\{\\Theta\}L\_\{M\}/\\varepsilon\)\. A supremum\-norm cover is anL2​\(Pn\)L^\{2\}\(P\_\{n\}\)cover, and an externalε/2\\varepsilon/2\-cover of the subset𝒢⁡\(r\)\\mathcal\{G\}\(r\)yields an internalε\\varepsilon\-cover\. Hencelog⁡N⁡\(ε,𝒢⁡\(r\),L2​\(Pn\)\)≤P​log⁡\(1\+4​RΘ​LM/ε\)\\log N\(\\varepsilon,\\mathcal\{G\}\(r\),L^\{2\}\(P\_\{n\}\)\)\\leq P\\log\(1\+4R\_\{\\Theta\}L\_\{M\}/\\varepsilon\)\. Conditionally on the sample, Dudley’s entropy integral\([Srebro et al\., 2010](https://arxiv.org/html/2610.28898#bib.bib26), Lemma A\.3\), withρ^2=supg∈𝒢⁡\(r\)Pn​g2\\hat\{\\rho\}^\{2\}=\\sup\_\{g\\in\\mathcal\{G\}\(r\)\}P\_\{n\}g^\{2\}and the cutoffα=B/n\\alpha=B/n, gives

𝔼σ​supg∈𝒢⁡\(r\)1n​∑iσi​g​\(ξi\)≤4​α\+12n​∫αρ^log⁡N⁡\(ε,𝒢⁡\(r\),L2​\(Pn\)\)​𝑑ε≤4​Bn\+12​ρ^​Ln,\\mathbb\{E\}\_\{\\sigma\}\\sup\_\{g\\in\\mathcal\{G\}\(r\)\}\\frac\{1\}\{n\}\\sum\_\{i\}\\sigma\_\{i\}g\(\\xi\_\{i\}\)\\leq 4\\alpha\+\\frac\{12\}\{\\sqrt\{n\}\}\\int\_\{\\alpha\}^\{\\hat\{\\rho\}\}\\sqrt\{\\log N\(\\varepsilon,\\mathcal\{G\}\(r\),L^\{2\}\(P\_\{n\}\)\)\}\\,d\\varepsilon\\leq\\frac\{4B\}\{n\}\+12\\hat\{\\rho\}\\sqrt\{\\frac\{L\}\{n\}\},because the covering number decreases inε\\varepsilon\. Ifρ^<α\\hat\{\\rho\}<\\alpha, Cauchy–Schwarz gives𝔼σ​supg∈𝒢⁡\(r\)n−1​∑iσi​g​\(ξi\)≤ρ^<α\\mathbb\{E\}\_\{\\sigma\}\\sup\_\{g\\in\\mathcal\{G\}\(r\)\}n^\{\-1\}\\sum\_\{i\}\\sigma\_\{i\}g\(\\xi\_\{i\}\)\\leq\\hat\{\\rho\}<\\alpha, so the same bound holds\. Take expectations and apply Jensen’s inequality,𝔼​ρ^≤\(𝔼​ρ^2\)1/2\\mathbb\{E\}\\hat\{\\rho\}\\leq\(\\mathbb\{E\}\\hat\{\\rho\}^\{2\}\)^\{1/2\}\. Symmetrization for independent observations\([Boucheron et al\., 2013](https://arxiv.org/html/2610.28898#bib.bib13)\)and the contraction principle\([Ledoux and Talagrand, 1991](https://arxiv.org/html/2610.28898#bib.bib21)\)forx↦x2x\\mapsto x^\{2\}, which is2​B2B\-Lipschitz on\[−B,B\]\[\-B,B\], give

𝔼​ρ^2≤supg∈𝒢⁡\(r\)T⁡\(g\)\+𝔼​supg∈𝒢⁡\(r\)\(Pn−P¯\)​g2≤r\+4​B​ℛn​\(𝒢⁡\(r\)\)\.\\mathbb\{E\}\\hat\{\\rho\}^\{2\}\\leq\\sup\_\{g\\in\\mathcal\{G\}\(r\)\}T\(g\)\+\\mathbb\{E\}\\sup\_\{g\\in\\mathcal\{G\}\(r\)\}\(P\_\{n\}\-\\bar\{P\}\)g^\{2\}\\leq r\+4B\\,\\mathcal\{R\}\_\{n\}\(\\mathcal\{G\}\(r\)\)\.WithR=ℛn​\(𝒢​\(r\)\)R=\\mathcal\{R\}\_\{n\}\(\\mathcal\{G\}\(r\)\), this yieldsR≤4​B/n\+12​L​r/n\+576​B​L​R/nR\\leq 4B/n\+12\\sqrt\{Lr/n\}\+\\sqrt\{576BLR/n\}\. The last term is at mostR/2\+288​B​L/nR/2\+288BL/n, and rearranging proves the lemma\. ∎

##### Step 3: the fixed point\.

The functionψ⁡\(r\)=B0​\[24​r​L/n\+\(576​L\+8\)​B/n\]\\psi\(r\)=B\_\{0\}\[24\\sqrt\{rL/n\}\+\(576L\+8\)B/n\]is nonnegative, nondecreasing, andψ⁡\(r\)/r\\psi\(r\)/\\sqrt\{r\}is nonincreasing, so it is sub\-root, andψ⁡\(r\)≥B0​ℛn​\(𝒢⁡\(r\)\)\\psi\(r\)\\geq B\_\{0\}\\mathcal\{R\}\_\{n\}\(\\mathcal\{G\}\(r\)\)by Lemma[C\.3](https://arxiv.org/html/2610.28898#Thmlemma3)\. Its fixed point solvesr=c​r\+br=c\\sqrt\{r\}\+bwithc=24​B0​L/nc=24B\_\{0\}\\sqrt\{L/n\}andb=B0​\(576​L\+8\)​B/nb=B\_\{0\}\(576L\+8\)B/n, sor∗≤c2\+2​br^\{\*\}\\leq c^\{2\}\+2b\. WithB0=8​BB\_\{0\}=8B,

r∗B0≤\(5760​L\+16\)​Bn\.\\frac\{r^\{\*\}\}\{B\_\{0\}\}\\leq\\frac\{\(5760L\+16\)B\}\{n\}\.\(78\)

##### Step 4: the localized inequality\.

Part 1 of Theorem 3\.3 of[Bartlett et al\. \(2005\)](https://arxiv.org/html/2610.28898#bib.bib9), which allows a general functionalTT, is stated for i\.i\.d\. samples\. Its proof uses the sampling law only through symmetrization and Talagrand’s concentration inequality, and both hold for independent, non\-identically distributed observations once the lawPPis replaced by the averageP¯\\bar\{P\}and the variance byn−1​∑iVarPin^\{\-1\}\\sum\_\{i\}\\operatorname\{Var\}\_\{P\_\{i\}\}\([Klein and Rio, 2005](https://arxiv.org/html/2610.28898#bib.bib20);[Boucheron et al\., 2013](https://arxiv.org/html/2610.28898#bib.bib13)\)\. In this form, for a classℱ\\mathcal\{F\}with ranges in an interval of lengthℓ\\elland a functional withn−1​∑iVarPi⁡f≤T⁡\(f\)≤B0​P¯​fn^\{\-1\}\\sum\_\{i\}\\operatorname\{Var\}\_\{P\_\{i\}\}f\\leq T\(f\)\\leq B\_\{0\}\\bar\{P\}f, everyK\>1K\>1andx\>0x\>0give, with probability at least1−e−x1\-e^\{\-x\}and for allf∈ℱf\\in\\mathcal\{F\},

P¯​f≤KK−1​Pn​f\+c1​K​r∗B0\+x⁡\(c2​ℓ\+c3​B0​K\)n,\\bar\{P\}f\\leq\\frac\{K\}\{K\-1\}P\_\{n\}f\+c\_\{1\}K\\frac\{r^\{\*\}\}\{B\_\{0\}\}\+\\frac\{x\(c\_\{2\}\\ell\+c\_\{3\}B\_\{0\}K\)\}\{n\},\(79\)wherec1,c2,c3c\_\{1\},c\_\{2\},c\_\{3\}are absolute constants andr∗r^\{\*\}is the fixed point of a sub\-rootψ≥B0​ℛn​\{f∈ℱ:T⁡\(f\)≤r\}\\psi\\geq B\_\{0\}\\mathcal\{R\}\_\{n\}\\\{f\\in\\mathcal\{F\}:T\(f\)\\leq r\\\}\.

##### Why the product\-sample version holds\.

The proof of Part 1 has three ingredients\. First, for a star\-shaped normalization ofℱ\\mathcal\{F\}, it boundsZ=supf∑i\{𝔼Pi​f​\(ξi\)−f⁡\(ξi\)\}Z=\\sup\_\{f\}\\sum\_\{i\}\\\{\\mathbb\{E\}\_\{P\_\{i\}\}f\(\\xi\_\{i\}\)\-f\(\\xi\_\{i\}\)\\\}above its mean by a Talagrand\-type inequality whose inputs are the rangeℓ\\elland the variance proxysupfn−1​∑iVarPi⁡f\\sup\_\{f\}n^\{\-1\}\\sum\_\{i\}\\operatorname\{Var\}\_\{P\_\{i\}\}f\. Each summand is centered under its own lawPiP\_\{i\}, and the inequality of[Klein and Rio \(2005\)](https://arxiv.org/html/2610.28898#bib.bib20)gives exactly this bound for independent, non\-identically distributed coordinates, with absolute constants\. Second, it bounds𝔼​Z\\mathbb\{E\}Zby2​n​ℛn2n\\mathcal\{R\}\_\{n\}, and symmetrization needs only independence of the coordinates\. Third, it combines these bounds over the shells\{T\(f\)≤r\}\\\{T\(f\)\\leq r\\\}through the sub\-root property and the fixed point, which is deterministic algebra and does not involve the sampling law\. ReplacingPPbyP¯\\bar\{P\}throughout therefore gives \([79](https://arxiv.org/html/2610.28898#A3.E79)\), with constantsc1,c2,c3c\_\{1\},c\_\{2\},c\_\{3\}that may differ from those of[Bartlett et al\. \(2005\)](https://arxiv.org/html/2610.28898#bib.bib9)\.

Apply \([79](https://arxiv.org/html/2610.28898#A3.E79)\) to the shifted classfv=gv\+ρ0f\_\{v\}=g\_\{v\}\+\\rho\_\{0\}with the functionalT~​\(fv\)=T⁡\(gv\)\\widetilde\{T\}\(f\_\{v\}\)=T\(g\_\{v\}\), which is the second moment ofgvg\_\{v\}rather than offvf\_\{v\}\. Its range has lengthℓ=2​B\\ell=2B, its variances equal those ofgvg\_\{v\}and are at mostT~​\(fv\)\\widetilde\{T\}\(f\_\{v\}\), and \([77](https://arxiv.org/html/2610.28898#A3.E77)\) readsT~​\(fv\)≤B0​P¯​fv\\widetilde\{T\}\(f\_\{v\}\)\\leq B\_\{0\}\\bar\{P\}f\_\{v\}\. A constant shift does not change the Rademacher complexity, since𝔼​∑iσi​ρ0=0\\mathbb\{E\}\\sum\_\{i\}\\sigma\_\{i\}\\rho\_\{0\}=0, so the sameψ\\psiandr∗r^\{\*\}apply\. Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)givesPn​gv^≤ϵoptP\_\{n\}g\_\{\\widehat\{v\}\}\\leq\\epsilon\_\{\\mathrm\{opt\}\}, hencePn​fv^≤ϵopt\+ρ0P\_\{n\}f\_\{\\widehat\{v\}\}\\leq\\epsilon\_\{\\mathrm\{opt\}\}\+\\rho\_\{0\}\. WithK=2K=2andrH​\(v^\)≤P¯​gv^\+δr\_\{H\}\(\\widehat\{v\}\)\\leq\\bar\{P\}g\_\{\\widehat\{v\}\}\+\\delta, with probability at least1−e−x1\-e^\{\-x\},

rH​\(v^\)≤2​ϵopt\+2​a¯H\+2​δ\+2​c1​\(5760​L\+16\)​Bn\+2​x​\(c2\+8​c3\)​Bn\.r\_\{H\}\(\\widehat\{v\}\)\\leq 2\\epsilon\_\{\\mathrm\{opt\}\}\+2\\bar\{a\}\_\{H\}\+2\\delta\+2c\_\{1\}\\frac\{\(5760L\+16\)B\}\{n\}\+\\frac\{2x\(c\_\{2\}\+8c\_\{3\}\)B\}\{n\}\.\(80\)Integrating the tail overxxgives the same bound in expectation withxxreplaced by11\.

##### Comparison\.

Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)gives𝔼​rH​\(v^\)≤36​B​Γ/n\+2​a¯H\+2​ϵopt\+5​ζ\+10​δ\\mathbb\{E\}r\_\{H\}\(\\widehat\{v\}\)\\leq 36B\\Gamma/n\+2\\bar\{a\}\_\{H\}\+2\\epsilon\_\{\\mathrm\{opt\}\}\+5\\zeta\+10\\deltawithΓ=1\+log⁡\(2​KM\)\\Gamma=1\+\\log\(2K\_\{M\}\)and, by \([50](https://arxiv.org/html/2610.28898#A2.E50)\),log⁡KM≤P​log⁡\(1\+2​RΘ​LM​\(m​T\)2/B\)\\log K\_\{M\}\\leq P\\log\(1\+2R\_\{\\Theta\}L\_\{M\}\(mT\)^\{2\}/B\)\. The localized bound \([80](https://arxiv.org/html/2610.28898#A3.E80)\) has statistical term of orderB​P​log⁡\(1\+4​RΘ​LM​n/B\)/nBP\\log\(1\+4R\_\{\\Theta\}L\_\{M\}n/B\)/n\. The two routes therefore share

- •the factorBBfrom the loss range,
- •the linear factorPPfrom the parametric cover,
- •the depth factor, which enters both logarithms throughCRMC\_\{R\_\{M\}\}inLML\_\{M\}, which in general is the product\-sum expression Equation \([45](https://arxiv.org/html/2610.28898#A2.E45)\) with layer\-specific boundsVjV\_\{j\}, of orderD​log⁡\(Lϕ​V\)D\\log\(L\_\{\\phi\}V\)when all layers share one boundVV,
- •the approximation residual2​a¯H2\\bar\{a\}\_\{H\}and the fitting term2​ϵopt2\\epsilon\_\{\\mathrm\{opt\}\}\.

They differ inside the logarithm, wherennreplaces\(m​T\)2\(mT\)^\{2\}, and in the constants: the cover route has explicit constants, while the localized route inherits the unspecified absolute constants of the concentration inequality\. The two routes therefore give the same rate up to logarithmic factors\. For a short history, withnnmuch smaller thanm​TmT, the cover route pays a larger logarithm\. This is the same order as the fixed\-target boundO~​\(B​P/n\)\\widetilde\{O\}\(BP/n\)of[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)\. Localization therefore does not remove the dependence onPPfor this class\. A width\-free bound would need norm\-based complexity, which neither analysis assumes\. The online analysis uses Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)because its constants are explicit\.

### C\.2Aggregation Analysis of DyProd\-FM

The algorithm and its initialization rules are given in Appendix[B\.2](https://arxiv.org/html/2610.28898#A2.SS2)\. Here we establish the potential, population, and lifetime bounds used by both main theorems\.

#### C\.2\.1Pathwise Potential Bound

###### Lemma C\.4\.

For a lifetimej=Ij=I, Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)obeys \([29](https://arxiv.org/html/2610.28898#S5.E29)\) on every sequence of bounded scores\. No stochastic hypothesis is needed\.

###### Proof\.

The normalized scores lie in\[0,1\]\[0,1\]\. Their weighted average also lies in\[0,1\]\[0,1\], so\|dj,t\|≤1\|d\_\{j,t\}\|\\leq 1and1\+η​dj,t≥1/2\>01\+\\eta d\_\{j,t\}\\geq 1/2\>0\. The total conceptual weight is initially one\. The change in total weight at a round equals

∑j∈𝒥tη​wj​\(x¯t−xj,t\)\\displaystyle\\sum\_\{j\\in\\mathcal\{J\}\_\{t\}\}\\eta w\_\{j\}\(\\bar\{x\}\_\{t\}\-x\_\{j,t\}\)=η⁡\[x¯t​∑j∈𝒥twj−∑j∈𝒥twj​xj,t\]=0,\\displaystyle=\\eta\\Big\[\\bar\{x\}\_\{t\}\\sum\_\{j\\in\\mathcal\{J\}\_\{t\}\}w\_\{j\}\-\\sum\_\{j\\in\\mathcal\{J\}\_\{t\}\}w\_\{j\}x\_\{j,t\}\\Big\]=0,\(81\)by the definition ofpj,tp\_\{j,t\}\. Sleeping weights do not change, so the total remains one, including weights for future and expired lifetimes\. Every individual weight is therefore at most one\.

The inequalitylog⁡\(1\+u\)≥u−u2\\log\(1\+u\)\\geq u\-u^\{2\}holds on\[−1/2,1/2\]\[\-1/2,1/2\]\. Apply it to each update ofjjand sum over its lifetime:

0≥log⁡wj,end≥−log⁡Me\+η​∑t∈Idj,t−η2​∑t∈Idj,t2\.0\\geq\\log w\_\{j,\\mathrm\{end\}\}\\geq\-\\log M\_\{e\}\+\\eta\\sum\_\{t\\in I\}d\_\{j,t\}\-\\eta^\{2\}\\sum\_\{t\\in I\}d\_\{j,t\}^\{2\}\.\(82\)Rearrange and divide byη\>0\\eta\>0to obtain \([29](https://arxiv.org/html/2610.28898#S5.E29)\)\. Only updates duringIIenter this product, because the weight is unchanged when the lifetime sleeps\. ∎

#### C\.2\.2Population Transfer and the Fast Master

All expectations in this subsubsection condition on the fixed environment\. For a lifetimeII, defineSI,JIS\_\{I\},J\_\{I\}and𝒜I\\mathcal\{A\}\_\{I\}as before \([31](https://arxiv.org/html/2610.28898#S5.E31)\)\.

###### Lemma C\.5\.

Every lifetime satisfies

SI≤53​JI\+1283​B​log⁡Me\+23​𝒜I\+83​\|I\|​δ\.S\_\{I\}\\leq\\frac\{5\}\{3\}J\_\{I\}\+\\frac\{128\}\{3\}B\\log M\_\{e\}\+\\frac\{2\}\{3\}\\mathcal\{A\}\_\{I\}\+\\frac\{8\}\{3\}\|I\|\\delta\.\(83\)

###### Proof\.

First condition onEEandℱt−1\\mathcal\{F\}\_\{t\-1\}\. Writej=Ij=Ifor the fixed lifetime\. For the current batch, subtract its score from the mixture score:

dj,t=∑kpk,t​1B​m​∑i=1m\[ℓM​\(vk,t,ξt,i\)−ℓM​\(vj,t,ξt,i\)\]\.d\_\{j,t\}=\\sum\_\{k\}p\_\{k,t\}\\frac\{1\}\{Bm\}\\sum\_\{i=1\}^\{m\}\[\\ell\_\{M\}\(v\_\{k,t\};\\xi\_\{t,i\}\)\-\\ell\_\{M\}\(v\_\{j,t\};\\xi\_\{t,i\}\)\]\.\(84\)Jensen’s inequality over the probabilities and over themmobservations gives

𝔼\[dj,t2∣E,ℱt−1\]≤1B2∑kpk,t𝔼t\[\(ℓM\(vk,t\)−ℓM\(vj,t\)\)2\]\.\\mathbb\{E\}\[d\_\{j,t\}^\{2\}\\mid E,\\mathcal\{F\}\_\{t\-1\}\]\\leq\\frac\{1\}\{B^\{2\}\}\\sum\_\{k\}p\_\{k,t\}\\mathbb\{E\}\_\{t\}\[\(\\ell\_\{M\}\(v\_\{k,t\}\)\-\\ell\_\{M\}\(v\_\{j,t\}\)\)^\{2\}\]\.\(85\)The difference\-of\-squares calculation in Lemma[C\.1](https://arxiv.org/html/2610.28898#Thmlemma1a)gives, for any two represented fieldsu,vu,vfixed under this conditioning,

𝔼t​\[\(ℓM​\(u\)−ℓM​\(v\)\)2\]≤4​B​𝔼t​∥u⁡\(U\)−v⁡\(U\)∥2≤8​B​\[Qt​\(u\)\+Qt​\(v\)\]\.\\mathbb\{E\}\_\{t\}\[\(\\ell\_\{M\}\(u\)\-\\ell\_\{M\}\(v\)\)^\{2\}\]\\leq 4B\\mathbb\{E\}\_\{t\}\\lVert u\(U\)\-v\(U\)\\rVert^\{2\}\\leq 8B\[Q\_\{t\}\(u\)\+Q\_\{t\}\(v\)\]\.\(86\)Substitution into \([85](https://arxiv.org/html/2610.28898#A3.E85)\) proves \([30](https://arxiv.org/html/2610.28898#S5.E30)\)\. It bounds a second moment, not just the batch\-noise variance: the squared population score difference remains even for very largemm\.

The conditional mean score difference is exactly the difference of population proxy risks, divided byBB\. Proper selection in Equation \([42](https://arxiv.org/html/2610.28898#A2.E42)\) and the uniform original\-risk transfer give

SI−JI≤B​∑t∈I𝔼​dj,t\+2​\|I\|​δ\.S\_\{I\}\-J\_\{I\}\\leq B\\sum\_\{t\\in I\}\\mathbb\{E\}d\_\{j,t\}\+2\|I\|\\delta\.\(87\)The subtraction ofata\_\{t\}cancels between the two compared risks\. Forη=1/32\\eta=1/32, take expectations in the potential bound, then substitute the second\-moment estimate\. SinceB​η⋅8/B=1/4B\\eta\\cdot 8/B=1/4andQt=Ft\+atQ\_\{t\}=F\_\{t\}\+a\_\{t\},

SI−JI≤32​B​log⁡Me\+14​\(SI\+JI\+2​𝒜I\)\+2​\|I\|​δ\.S\_\{I\}\-J\_\{I\}\\leq 32B\\log M\_\{e\}\+\\tfrac\{1\}\{4\}\(S\_\{I\}\+J\_\{I\}\+2\\mathcal\{A\}\_\{I\}\)\+2\|I\|\\delta\.\(88\)SubtractSI/4S\_\{I\}/4and moveJIJ\_\{I\}to the right\. The left becomes3​SI/43S\_\{I\}/4, and the comparator coefficient becomes5​JI/45J\_\{I\}/4\. Division by3/43/4proves \([83](https://arxiv.org/html/2610.28898#A3.E83)\)\. ∎

Only the potential inequality is pathwise\. The population statement additionally uses fresh regression feedback, predictable candidates, and squared loss\. The two statements should not be conflated\. The second moment in \([30](https://arxiv.org/html/2610.28898#S5.E30)\) does not decay with the batch size: for the per\-example normalized score differenceΔ\\Deltabetween the mixture and lifetimejj, with independent batch copiesΔ1,…,Δm\\Delta\_\{1\},\\ldots,\\Delta\_\{m\},

𝔼new​\(1m​∑iΔi\)2=\(𝔼new​Δ\)2\+Varnew⁡\(Δ\)m\.\\mathbb\{E\}\_\{\\mathrm\{new\}\}\\Big\(\\frac\{1\}\{m\}\\sum\_\{i\}\\Delta\_\{i\}\\Big\)^\{2\}=\(\\mathbb\{E\}\_\{\\mathrm\{new\}\}\\Delta\)^\{2\}\+\\frac\{\\operatorname\{Var\}\_\{\\mathrm\{new\}\}\(\\Delta\)\}\{m\}\.\(89\)The first term survives asmmgrows, which is why the constants separate the1/m1/mgain of base estimation from themm\-free aggregation cost\. A square\-root master rateη≍\|I\|−1/2\\eta\\asymp\|I\|^\{\-1/2\}is not needed: every bound in this paper is obtained from \([83](https://arxiv.org/html/2610.28898#A3.E83)\)\.

#### C\.2\.3Historical Fits over a Lifetime

DefineeM=2​ϵopt\+5​ζ\+10​δe\_\{M\}=2\\epsilon\_\{\\mathrm\{opt\}\}\+5\\zeta\+10\\deltafor this subsubsection\. For any lifetimeII, letnI=\|I\|n\_\{I\}=\|I\|, and letΩI\\Omega\_\{I\}sum only its internal variation increments\.

###### Lemma C\.6\.

The fitted model of every lifetime satisfies

JI≤B\+72​B​Γm​log⁡\(e​nI\)\+4​log⁡\(e​nI\)​𝒜I\+nI​eM\+6​\(nI−1\)​ΩI\.J\_\{I\}\\leq B\+\\frac\{72B\\Gamma\}\{m\}\\log\(en\_\{I\}\)\+4\\log\(en\_\{I\}\)\\mathcal\{A\}\_\{I\}\+n\_\{I\}e\_\{M\}\+6\(n\_\{I\}\-1\)\\Omega\_\{I\}\.\(90\)

###### Proof\.

Write the lifetime’s start asaa\. At its first round the deployed fieldvI,av\_\{I,a\}is a predictable member of𝒱\\mathcal\{V\}\. Nonnegativity of the minimum loss and \([54](https://arxiv.org/html/2610.28898#A2.E54)\) giveFa​\(vI,a\)≤La​\(vI,a\)≤B\+δF\_\{a\}\(v\_\{I,a\}\)\\leq L\_\{a\}\(v\_\{I,a\}\)\\leq B\+\\delta\.

At a later round letk=t−a≥1k=t\-a\\geq 1be the age andq=2⌊log2⁡k⌋q=2^\{\\lfloor\\log\_\{2\}k\\rfloor\}its historical prefix length\. Thenk/2≤q≤kk/2\\leq q\\leq k\. Apply \([28](https://arxiv.org/html/2610.28898#S5.E28)\) to this deterministic prefix and then both inequalities of Lemma[C\.2](https://arxiv.org/html/2610.28898#Thmlemma2)\. The resulting current excess obeys

𝔼Ft\(vI,t\)≤36​B​Γm​q\+2q∑s=aa\+q−1as\+eM\+6Ωa:t\.\\mathbb\{E\}F\_\{t\}\(v\_\{I,t\}\)\\leq\\frac\{36B\\Gamma\}\{mq\}\+\\frac\{2\}\{q\}\\sum\_\{s=a\}^\{a\+q\-1\}a\_\{s\}\+e\_\{M\}\+6\\Omega\_\{a:t\}\.\(91\)The drift coefficient is six: the factor two multiplyinga¯H\\bar\{a\}\_\{H\}in the statistical bound contributes four through \([32](https://arxiv.org/html/2610.28898#S5.E32)\), and current\-risk transfer contributes two\.

For the statistical term,1/q≤2/k1/q\\leq 2/kand the harmonic sum gives

∑k=1nI−11q≤2​∑k=1nI−11k≤2​log⁡\(e​nI\)\.\\sum\_\{k=1\}^\{n\_\{I\}\-1\}\\frac\{1\}\{q\}\\leq 2\\sum\_\{k=1\}^\{n\_\{I\}\-1\}\\frac\{1\}\{k\}\\leq 2\\log\(en\_\{I\}\)\.\(92\)For a fixed earlier rounds∈Is\\in I, its coefficient in the double approximation sum is at most∑k=1nI−12/q≤4​log⁡\(e​nI\)\\sum\_\{k=1\}^\{n\_\{I\}\-1\}2/q\\leq 4\\log\(en\_\{I\}\)\. Summing this bound overssgives the approximation term in \([90](https://arxiv.org/html/2610.28898#A3.E90)\)\. Every internal drift edge can occur in at mostnI−1n\_\{I\}\-1of the quantitiesΩa:t\\Omega\_\{a:t\}\. Thus∑t\>aΩa:t≤\(nI−1\)ΩI\\sum\_\{t\>a\}\\Omega\_\{a:t\}\\leq\(n\_\{I\}\-1\)\\Omega\_\{I\}\. There arenI−1n\_\{I\}\-1fitting residuals, and the unused one\-round allowanceeM≥δe\_\{M\}\\geq\\deltaabsorbs the initial tail term\. This proves \([90](https://arxiv.org/html/2610.28898#A3.E90)\), includingnI=1n\_\{I\}=1\. ∎

No union bound over experts is needed: each expectation inequality holds for its deterministic prefix\. Linearity of expectation permits summing these bounds even when their training data overlap\. Lemma[E\.8](https://arxiv.org/html/2610.28898#Thmlemma8)is the target\-law counterpart of this lemma and uses the same prefixes and the same fits\.

#### C\.2\.4Resource Accounting and Computational Scope

LetN=2⌈log2⁡T⌉N=2^\{\\lceil\\log\_\{2\}T\\rceil\}denote the containing power of two\. There are2​N−1<4​T2N\-1<4Tbinary\-tree nodes, soMe<4​TM\_\{e\}<4T\. At each scale, at most one original node contains a given round, and clipping and removing duplicates cannot increase this count\. Hence at most1\+⌈log2⁡T⌉1\+\\lceil\\log\_\{2\}T\\rceillifetime models are active\. Storing their parameters costsO⁡\(P​log⁡T\)O\(P\\log T\)scalar entries\. A direct implementation stores at mostm​TmTtriples, orO⁡\(m​d​T\)O\(mdT\)scalars, in addition toO⁡\(T\)O\(T\)conceptual weights\. Future weights may be initialized lazily at their preassigned value, and expired weights never need to be updated again\.

An original interval of length2k2^\{k\}triggers at mostkknontrivial fitting calls, because prefix lengths are1,2,…,2k−11,2,\\ldots,2^\{k\-1\}\. There areN/2kN/2^\{k\}original nodes of that length\. Therefore the total number of fitting calls is at mostN​∑k≥1k/2k≤2​N=O⁡\(T\)N\\sum\_\{k\\geq 1\}k/2^\{k\}\\leq 2N=O\(T\), and clipping and deduplication do not increase it\. The sum of prefix lengths within one original node is less than its length, and total original\-node lengths per scale equalNN\. Thus the total size of datasets passed to fits, counting repeated historical examples, isO⁡\(m​T​log⁡T\)O\(mT\\log T\)\. This is a data\-volume count, not an optimizer running time: a fitting routine may revisit its dataset many times\. The load is uneven in time\. A round at which a long prefix doubles retrains on up toO⁡\(m​T\)O\(mT\)examples, while the amortized load isO⁡\(m​log⁡T\)O\(m\\log T\)examples per round\.

Scoring usesO⁡\(m​T​log⁡T\)O\(mT\\log T\)field evaluations\. If a field evaluation costsCevalC\_\{\\mathrm\{eval\}\}, the corresponding cost isO⁡\(Ceval​m​T​log⁡T\)O\(C\_\{\\mathrm\{eval\}\}mT\\log T\), and master arithmetic isO⁡\(T​log⁡T\)O\(T\\log T\)\. No dimension or minibatch factor is silently removed from these counts\. The general computational guarantee remains conditional on the cost of a fitting routine satisfying Assumption[4](https://arxiv.org/html/2610.28898#Thmassumption4)\.

### C\.3Disjoint Dyadic Covers

###### Lemma C\.7\(Disjoint dyadic covers\)\.

Every consecutive intervalJ⊆\[1,T\]J\\subseteq\[1,T\]admits a partition into at mostDT=2​\(1\+⌈log2⁡T⌉\)D\_\{T\}=2\(1\+\\lceil\\log\_\{2\}T\\rceil\)disjoint lifetimes in𝒟T\\mathcal\{D\}\_\{T\}, with at most two pieces of each length\.

###### Proof\.

Fix any consecutive intervalJ⊆\[1,T\]J\\subseteq\[1,T\]\. Use the untruncated aligned dyadic intervals that are maximal among those contained inJJ\. They partitionJJ: each point belongs to a singleton and hence to a largest admissible containing dyadic interval, and two maximal dyadic intervals are either disjoint or equal\. There are at most two such intervals of any length\. Indeed, the parent of a maximal interval is not contained inJJ, so it crosses a boundary ofJJ\. At a fixed scale at most one parent crosses each boundary, and such a parent has at most one child contained inJJ\. Each chosen interval is an original tree node contained in\[1,T\]\[1,T\], so it is among the algorithm’s lifetimes\. The number of pieces is therefore at mostDT=2​\(1\+⌈log2⁡T⌉\)D\_\{T\}=2\(1\+\\lceil\\log\_\{2\}T\\rceil\)\. ∎

## Appendix DProof of Theorem 1

We prove Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)without an asymptotic optimization over real\-valued block lengths\. First fix the environment\. All expectations until the final step are conditional on it, and𝒜J\\mathcal\{A\}\_\{J\}is consequently deterministic\.

Step 1: Combine the Master and Base Bounds\.For a dyadic lifetimeII, combine the historical lifetime bound of Lemma[C\.6](https://arxiv.org/html/2610.28898#Thmlemma6), Equation \([90](https://arxiv.org/html/2610.28898#A3.E90)\), with the fast master bound of Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5), Equation \([83](https://arxiv.org/html/2610.28898#A3.E83)\), both proved in Appendix[C](https://arxiv.org/html/2610.28898#A3)\. Lemma[C\.6](https://arxiv.org/html/2610.28898#Thmlemma6)uses Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1)and the first\-order transfer of Lemma[C\.2](https://arxiv.org/html/2610.28898#Thmlemma2)\. Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5)combines the potential inequality of Lemma[C\.4](https://arxiv.org/html/2610.28898#Thmlemma4)with the second\-moment control of Lemma[C\.1](https://arxiv.org/html/2610.28898#Thmlemma1a)\. The non\-statistical constant is\(5/3\)​B\+\(128/3\)​B​log⁡Me\(5/3\)B\+\(128/3\)B\\log M\_\{e\}, which is at most44​B​Λ44B\\Lambda\. The statistical coefficient becomes\(5/3\)​72=120\(5/3\)72=120\. The approximation coefficient is

203​log⁡\(e​\|I\|\)\+23≤8​log⁡\(e​T\)=HT\.\\frac\{20\}\{3\}\\log\(e\|I\|\)\+\\frac\{2\}\{3\}\\leq 8\\log\(eT\)=H\_\{T\}\.\(93\)The coefficients ofϵopt,ζ,δ\\epsilon\_\{\\mathrm\{opt\}\},\\zeta,\\deltaare10/310/3,25/325/3, and50/3\+8/3=58/350/3\+8/3=58/3, respectively\. They are at most4,9,204,9,20\. Finally\(5/3\)​6=10\(5/3\)6=10for drift\. Thus

SI≤KT/DT\+HT​𝒜I\+\|I\|​ef\+10​\(\|I\|−1\)​ΩI\.S\_\{I\}\\leq K\_\{T\}/D\_\{T\}\+H\_\{T\}\\mathcal\{A\}\_\{I\}\+\|I\|e\_\{f\}\+10\(\|I\|\-1\)\\Omega\_\{I\}\.\(94\)The same fields and selected sequence enter every later bound, including the target\-law bound of Appendix[E](https://arxiv.org/html/2610.28898#A5)\. There is no comparison between separate learner runs\.

Step 2: Apply the Disjoint Dyadic Cover\.Apply Lemma[C\.7](https://arxiv.org/html/2610.28898#Thmlemma7)to any consecutive intervalJ⊆\[1,T\]J\\subseteq\[1,T\]\.

Approximation and residual sums over the cover have no overlap\. Internal variation edges of its pieces are a subset of those ofJJ, and\|I\|−1≤\|J\|−1\|I\|\-1\\leq\|J\|\-1\. Consequently

∑I​in​cover\(\|I\|−1\)​ΩI≤\(\|J\|−1\)​ΩJ\.\\sum\_\{I\\ \\mathrm\{in\\ cover\}\}\(\|I\|\-1\)\\Omega\_\{I\}\\leq\(\|J\|\-1\)\\Omega\_\{J\}\.\(95\)Summing \([94](https://arxiv.org/html/2610.28898#A4.E94)\) over the cover gives exactly \([34](https://arxiv.org/html/2610.28898#S5.E34)\)\.

Step 3: Change\-Count Partition\.Partition\[1,T\]\[1,T\]immediately after every edge withωt\>0\\omega\_\{t\}\>0\. Letkkbe the resulting number of intervals, and letn1,…,nkn\_\{1\},\\ldots,n\_\{k\}be their lengths\. Thenk≤Nω\+1k\\leq N\_\{\\omega\}\+1,∑jnj=T\\sum\_\{j\}n\_\{j\}=T, and each interval has zero internal variation\. Applying the interval bound established in Step 2, Equation \([34](https://arxiv.org/html/2610.28898#S5.E34)\), to each interval gives

𝔼⁡\[ℛTvel∣E\]≤KT​k\+HT​𝒜T\+T​ef≤KT​\(Nω\+1\)\+HT​𝒜T\+T​ef\.\\mathbb\{E\}\[\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\mid E\]\\leq K\_\{T\}k\+H\_\{T\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}\\leq K\_\{T\}\(N\_\{\\omega\}\+1\)\+H\_\{T\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}\.\(96\)The partition depends only on the conditioned environment, and the algorithm never needs to observe it\.

Step 4: Variation Partition\.Choose an integerh∈\[1,T\]h\\in\[1,T\]for the proof and partition\[1,T\]\[1,T\]into⌈T/h⌉\\lceil T/h\\rceilconsecutive blocks of length at mosthh\. Each block has at mosth−1h\-1internal edges, and distinct blocks have disjoint internal edges, so summing the interval bound from Step 2, Equation \([34](https://arxiv.org/html/2610.28898#S5.E34)\), gives, after removing the global approximation and residual terms, the integer optimization

minh∈\{1,…,T\}⁡\{KT​⌈T/h⌉\+10​\(h−1\)​Ω\}\.\\min\_\{h\\in\\\{1,\\ldots,T\\\}\}\\\{K\_\{T\}\\lceil T/h\\rceil\+10\(h\-1\)\\Omega\\\}\.\(97\)IfΩ=0\\Omega=0, chooseh=Th=Tand obtainKTK\_\{T\}\. IfΩ\>0\\Omega\>0, defineh∗=KT​T/\(10​Ω\)h\_\{\*\}=\\sqrt\{K\_\{T\}T/\(10\\Omega\)\}\. There are three cases\.

1. 1\.If1≤h∗≤T1\\leq h\_\{\*\}\\leq T, seth=⌈h∗⌉h=\\lceil h\_\{\*\}\\rceil\. Use⌈T/h⌉≤T/h\+1\\lceil T/h\\rceil\\leq T/h\+1,h≥h∗h\\geq h\_\{\*\}, andh−1≤h∗h\-1\\leq h\_\{\*\}to get KT​⌈T/h⌉\+10​\(h−1\)​Ω≤KT\+KT​T/h∗\+10​h∗​Ω=KT\+2​10​KT​T​Ω\.K\_\{T\}\\lceil T/h\\rceil\+10\(h\-1\)\\Omega\\leq K\_\{T\}\+K\_\{T\}T/h\_\{\*\}\+10h\_\{\*\}\\Omega=K\_\{T\}\+2\\sqrt\{10K\_\{T\}T\\Omega\}\.\(98\)
2. 2\.Ifh∗\>Th\_\{\*\}\>T, chooseh=Th=T\. Letx=10​T​Ωx=10T\\Omega\. The condition impliesx<KTx<K\_\{T\}, since10​Ω<KT/T10\\Omega<K\_\{T\}/T\. Therefore10​\(T−1\)​Ω≤x≤KT​x10\(T\-1\)\\Omega\\leq x\\leq\\sqrt\{K\_\{T\}x\}, and the cost is at mostKT\+10​KT​T​ΩK\_\{T\}\+\\sqrt\{10K\_\{T\}T\\Omega\}\.
3. 3\.Ifh∗<1h\_\{\*\}<1, chooseh=1h=1\. NowKT​T<10​ΩK\_\{T\}T<10\\Omega, which givesKT​T≤10​KT​T​ΩK\_\{T\}T\\leq\\sqrt\{10K\_\{T\}T\\Omega\}\. The cost isKT​TK\_\{T\}T, again bounded by the desired expression\.

RestoringHT​𝒜T\+T​efH\_\{T\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}and comparing with \([96](https://arxiv.org/html/2610.28898#A4.E96)\) proves Equation \([19](https://arxiv.org/html/2610.28898#S4.E19)\) conditionally onEE\. The proof handlesT=1T=1and the transitions between regimes without assuming a real\-valued window can be used by the algorithm\.

Step 5: Integrate the Environment\.The separate count and variation bounds hold for every admissible fixed environment\. The deterministic caps and constants are uniform, so integration preserves them and replaces𝒜T\\mathcal\{A\}\_\{T\}with𝔼​𝒜T\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\. Both unconditional bounds therefore hold, and their minimum also bounds𝔼​ℛTvel\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\. This is not an interchange of expectation and a random minimum\. For completeness, the separate uniform bound follows fromFt​\(v\)≤Lt​\(v\)≤B\+δF\_\{t\}\(v\)\\leq L\_\{t\}\(v\)\\leq B\+\\deltafor every represented field, giving𝔼​ℛTvel≤T⁡\(B\+δ\)\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq T\(B\+\\delta\)\. The stated theorem keeps the count and variation bounds derived above\.

At fixed architectural and endpoint scales, Equation \([24](https://arxiv.org/html/2610.28898#S4.E24)\) givesB=O⁡\(log⁡\(e​m​T\)\)B=O\(\\log\(emT\)\),RM=O⁡\(log⁡\(e​m​T\)\)R\_\{M\}=O\(\\sqrt\{\\log\(emT\)\}\), andCRM=O⁡\(1\+RM\)C\_\{R\_\{M\}\}=O\(1\+R\_\{M\}\)\. Inequality \([50](https://arxiv.org/html/2610.28898#A2.E50)\) withε=ζ\\varepsilon=\\zetathen givesΓ=O⁡\(1\+P​log⁡\(e​m​T\)\)\\Gamma=O\(1\+P\\log\(emT\)\), with fixed constants retained\. AlsoMe<4​TM\_\{e\}<4T,DT=O⁡\(log⁡\(e​T\)\)D\_\{T\}=O\(\\log\(eT\)\), andT​ζ=B/\(m2​T\)T\\zeta=B/\(m^\{2\}T\)\. These relations justify the suppressed logarithms in the abstract\. If architecture, batch size, or endpoint scale varies with the horizon, the explicit theorem is the relevant statement\.

Thus, under the predictive protocol, Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1)–[4](https://arxiv.org/html/2610.28898#Thmassumption4), and the caps in Equation \([9](https://arxiv.org/html/2610.28898#S3.E9)\), Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)satisfies

𝔼​ℛTvel≤\\displaystyle\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq\{\}𝖥T,\\displaystyle\\mathsf\{F\}\_\{T\},𝖥T=\\displaystyle\\mathsf\{F\}\_\{T\}=\{\}min⁡\{KT​\(Nω\+1\),KT\+2​10​KT​T​Ω\}\\displaystyle\\min\\big\\\{K\_\{T\}\(N\_\{\\omega\}\+1\),\\ K\_\{T\}\+2\\sqrt\{10K\_\{T\}T\\Omega\}\\big\\\}\+HT​𝔼​𝒜T\+T​ef\.\\displaystyle\+H\_\{T\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}\.\(99\)Hence Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)is proved\.□\\square

## Appendix EProof of Theorem 2

### E\.1Supporting Lemmas Needed for Theorem 2

This appendix proves Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)\. We first establish Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1)and Propositions[2](https://arxiv.org/html/2610.28898#Thmproposition2)and[3](https://arxiv.org/html/2610.28898#Thmproposition3), and compute the flow\-time moments of Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\. We then derive the historical and interval bounds and complete the proof using the total\-variation partition\. Throughout, Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1),[3](https://arxiv.org/html/2610.28898#Thmassumption3)and[5](https://arxiv.org/html/2610.28898#Thmassumption5)hold, and the environment is fixed until the final integration step\. All results of Appendices[B](https://arxiv.org/html/2610.28898#A2)–[C](https://arxiv.org/html/2610.28898#A3)used here hold under Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)’s flow\-time law by Remark[6](https://arxiv.org/html/2610.28898#Thmremark6)\. For lawsP,QP,Qwith densitiesp,qp,qwith respect to a common measure, the Hellinger distance is

𝖧2​\(P,Q\)=12​∫\(p−q\)2=1−∫p​q∈\[0,1\]\.\\mathsf\{H\}^\{2\}\(P,Q\)=\\frac\{1\}\{2\}\\int\(\\sqrt\{p\}\-\\sqrt\{q\}\)^\{2\}=1\-\\int\\sqrt\{pq\}\\in\[0,1\]\.\(100\)
#### E\.1\.1Hellinger Tools

###### Lemma E\.1\(Hellinger change of measure\)\.

For every measurableFFwith∫F2​d​\(P\+Q\)<∞\\int F^\{2\}d\(P\+Q\)<\\infty,

\|∫F​𝑑P−∫F​𝑑Q\|≤2​𝖧​\(P,Q\)​\(∫F2​𝑑P\+∫F2​𝑑Q\)1/2\.\\Big\|\\int F\\,dP\-\\int F\\,dQ\\Big\|\\leq 2\\mathsf\{H\}\(P,Q\)\\Big\(\\int F^\{2\}dP\+\\int F^\{2\}dQ\\Big\)^\{1/2\}\.\(101\)

###### Proof\.

Write∫F⁡\(p−q\)=∫F⁡\(p\+q\)​\(p−q\)\\int F\(p\-q\)=\\int F\(\\sqrt\{p\}\+\\sqrt\{q\}\)\(\\sqrt\{p\}\-\\sqrt\{q\}\)and apply Cauchy–Schwarz:

\|∫F⁡\(p−q\)\|≤\(∫F2​\(p\+q\)2\)1/2​\(∫\(p−q\)2\)1/2\.\\Big\|\\int F\(p\-q\)\\Big\|\\leq\\Big\(\\int F^\{2\}\(\\sqrt\{p\}\+\\sqrt\{q\}\)^\{2\}\\Big\)^\{1/2\}\\Big\(\\int\(\\sqrt\{p\}\-\\sqrt\{q\}\)^\{2\}\\Big\)^\{1/2\}\.The first factor is at most\(2​∫F2​\(p\+q\)\)1/2\(2\\int F^\{2\}\(p\+q\)\)^\{1/2\}because\(p\+q\)2≤2​\(p\+q\)\(\\sqrt\{p\}\+\\sqrt\{q\}\)^\{2\}\\leq 2\(p\+q\), and the second equals2​𝖧​\(P,Q\)\\sqrt\{2\}\\,\\mathsf\{H\}\(P,Q\)\. ∎

###### Lemma E\.2\(Hellinger cost of translating the base\)\.

Forσ∈ℝd\\sigma\\in\\mathbb\{R\}^\{d\}, letνσ\\nu\_\{\\sigma\}be the law ofX0\+σX\_\{0\}\+\\sigmawithX0∼νX\_\{0\}\\sim\\nu\. Then𝖧2​\(νσ,ν\)≤ι​∥σ∥2/8\\mathsf\{H\}^\{2\}\(\\nu\_\{\\sigma\},\\nu\)\\leq\\iota\\lVert\\sigma\\rVert^\{2\}/8\.

###### Proof\.

Letf=pν∈H1​\(ℝd\)f=\\sqrt\{p\_\{\\nu\}\}\\in H^\{1\}\(\\mathbb\{R\}^\{d\}\)\. For smooth compactly supportedff, the fundamental theorem of calculus and Jensen give

\|f⁡\(x−σ\)−f⁡\(x\)\|2=\|∫01⟨∇f​\(x−θ​σ\),σ⟩​𝑑θ\|2≤∫01⟨∇f​\(x−θ​σ\),σ⟩2​𝑑θ\.\|f\(x\-\\sigma\)\-f\(x\)\|^\{2\}=\\Big\|\\int\_\{0\}^\{1\}\\langle\\nabla f\(x\-\\theta\\sigma\),\\sigma\\rangle d\\theta\\Big\|^\{2\}\\leq\\int\_\{0\}^\{1\}\\langle\\nabla f\(x\-\\theta\\sigma\),\\sigma\\rangle^\{2\}d\\theta\.Integrating inxxand using translation invariance of Lebesgue measure gives∫\|f\(⋅−σ\)−f\|2≤∫⟨∇f,σ⟩2\\int\|f\(\\cdot\-\\sigma\)\-f\|^\{2\}\\leq\\int\\langle\\nabla f,\\sigma\\rangle^\{2\}\. Both sides are continuous inH1H^\{1\}, so the inequality extends to everyf∈H1​\(ℝd\)f\\in H^\{1\}\(\\mathbb\{R\}^\{d\}\)by density\. SinceIν=4∫∇f∇f⊤I\_\{\\nu\}=4\\int\\nabla f\\nabla f^\{\\top\}, the right side equalsσ⊤​Iν​σ/4≤ι​∥σ∥2/4\\sigma^\{\\top\}I\_\{\\nu\}\\sigma/4\\leq\\iota\\lVert\\sigma\\rVert^\{2\}/4\. Divide by two\. ∎

Forν=N⁡\(0,σ02​Id\)\\nu=N\(0,\\sigma\_\{0\}^\{2\}I\_\{d\}\)the exact value is𝖧2\(νσ,ν\)=1−exp\(−∥σ∥2/\(8σ02\)\)\\mathsf\{H\}^\{2\}\(\\nu\_\{\\sigma\},\\nu\)=1\-\\exp\(\-\\lVert\\sigma\\rVert^\{2\}/\(8\\sigma\_\{0\}^\{2\}\)\), which the lemma bounds by∥σ∥2/\(8​σ02\)\\lVert\\sigma\\rVert^\{2\}/\(8\\sigma\_\{0\}^\{2\}\)because1−e−x≤x1\-e^\{\-x\}\\leq x\.

#### E\.1\.2The Base\-Shift Representation

###### Lemma E\.3\(A target move is a base\-noise shift plus a label shift\)\.

Let\(X1,X1′\)\(X\_\{1\},X\_\{1\}^\{\\prime\}\)have marginals\(μs,μt\)\(\\mu\_\{s\},\\mu\_\{t\}\), and letX0∼νX\_\{0\}\\sim\\nuandr∼λr\\sim\\lambdabe independent of it and of each other\. On\{α\(r\)\>0\}\\\{\\alpha\(r\)\>0\\\}putΔ​X=X1′−X1\\Delta X=X\_\{1\}^\{\\prime\}\-X\_\{1\},σr=\(β⁡\(r\)/α⁡\(r\)\)​Δ​X\\sigma\_\{r\}=\(\\beta\(r\)/\\alpha\(r\)\)\\Delta X,er=\(J⁡\(r\)/α⁡\(r\)\)​Δ​Xe\_\{r\}=\(J\(r\)/\\alpha\(r\)\)\\Delta X, andX~0=X0\+σr\\tilde\{X\}\_\{0\}=X\_\{0\}\+\\sigma\_\{r\}\. Then Equation \([35](https://arxiv.org/html/2610.28898#S5.E35)\) holds, and conditional on\(X1,X1′,r\)\(X\_\{1\},X\_\{1\}^\{\\prime\},r\)the law ofX~0\\tilde\{X\}\_\{0\}isνσr\\nu\_\{\\sigma\_\{r\}\}\.

###### Proof\.

First,α​X~0\+β​X1=α​X0\+β​Δ​X\+β​X1=α​X0\+β​X1′\\alpha\\tilde\{X\}\_\{0\}\+\\beta X\_\{1\}=\\alpha X\_\{0\}\+\\beta\\Delta X\+\\beta X\_\{1\}=\\alpha X\_\{0\}\+\\beta X\_\{1\}^\{\\prime\}\. Second, usingJ/α=β′−α′​β/αJ/\\alpha=\\beta^\{\\prime\}\-\\alpha^\{\\prime\}\\beta/\\alpha,

α′​X~0\+β′​X1\+Jα​Δ​X=α′​X0\+α′​βα​Δ​X\+β′​X1\+β′​Δ​X−α′​βα​Δ​X=α′​X0\+β′​X1′\.\\alpha^\{\\prime\}\\tilde\{X\}\_\{0\}\+\\beta^\{\\prime\}X\_\{1\}\+\\frac\{J\}\{\\alpha\}\\Delta X=\\alpha^\{\\prime\}X\_\{0\}\+\\frac\{\\alpha^\{\\prime\}\\beta\}\{\\alpha\}\\Delta X\+\\beta^\{\\prime\}X\_\{1\}\+\\beta^\{\\prime\}\\Delta X\-\\frac\{\\alpha^\{\\prime\}\\beta\}\{\\alpha\}\\Delta X=\\alpha^\{\\prime\}X\_\{0\}\+\\beta^\{\\prime\}X\_\{1\}^\{\\prime\}\.The conditional law statement holds becauseX0X\_\{0\}is independent of\(X1,X1′,r\)\(X\_\{1\},X\_\{1\}^\{\\prime\},r\)andσr\\sigma\_\{r\}is a function of these variables\. ∎

The pair on the left of Equation \([35](https://arxiv.org/html/2610.28898#S5.E35)\), together withrr, has the round\-ttexample law, while\(Ir​\(X0,X1\),Zr​\(X0,X1\),r\)\(I\_\{r\}\(X\_\{0\},X\_\{1\}\),Z\_\{r\}\(X\_\{0\},X\_\{1\}\),r\)has the round\-sslaw\.

#### E\.1\.3Label Tails

###### Lemma E\.4\(Label tails\)\.

For every roundttandM≥0M\\geq 0, withMz=cZ​\(1\+M\)M\_\{z\}=c\_\{Z\}\(1\+M\),

𝔼t\[∥Z∥2𝟏\{∥Z∥\>Mz\}\]≤4cZ2\(1\+M2\+4κ2\)e−M2/\(4κ2\)≤δM/2\.\\mathbb\{E\}\_\{t\}\\big\[\\lVert Z\\rVert^\{2\}\\mathbf\{1\}\\\{\\lVert Z\\rVert\>M\_\{z\}\\\}\\big\]\\leq 4c\_\{Z\}^\{2\}\(1\+M^\{2\}\+4\\kappa^\{2\}\)e^\{\-M^\{2\}/\(4\\kappa^\{2\}\)\}\\leq\\delta\_\{M\}/2\.\(102\)

###### Proof\.

LetS=∥X0∥\+∥X1∥S=\\lVert X\_\{0\}\\rVert\+\\lVert X\_\{1\}\\rVert\. By Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(a\),∥Z∥≤cZ​S≤cZ​\(1\+S\)\\lVert Z\\rVert\\leq c\_\{Z\}S\\leq c\_\{Z\}\(1\+S\), so\{∥Z∥\>Mz\}⊆\{S\>M\}\\\{\\lVert Z\\rVert\>M\_\{z\}\\\}\\subseteq\\\{S\>M\\\}and∥Z∥2≤2​cZ2​\(1\+S2\)\\lVert Z\\rVert^\{2\}\\leq 2c\_\{Z\}^\{2\}\(1\+S^\{2\}\)\. The tail computation in the proof of Proposition[B\.2](https://arxiv.org/html/2610.28898#Thmproposition2a)gives𝔼t\[\(1\+S2\)𝟏\{S2\>M2\}\]≤2\(1\+M2\+4κ2\)e−M2/\(4κ2\)\\mathbb\{E\}\_\{t\}\[\(1\+S^\{2\}\)\\mathbf\{1\}\\\{S^\{2\}\>M^\{2\}\\\}\]\\leq 2\(1\+M^\{2\}\+4\\kappa^\{2\}\)e^\{\-M^\{2\}/\(4\\kappa^\{2\}\)\}\. Multiply by2​cZ22c\_\{Z\}^\{2\}\. FinallyCℓ≥4​cZ2C\_\{\\ell\}\\geq 4c\_\{Z\}^\{2\}, so the bound is at most\(2​cZ2/Cℓ\)​δM≤δM/2\(2c\_\{Z\}^\{2\}/C\_\{\\ell\}\)\\delta\_\{M\}\\leq\\delta\_\{M\}/2\. ∎

#### E\.1\.4Proof of Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1)

Fix roundss,ts,tandu,w∈𝒱u,w\\in\\mathcal\{V\}, writeΔ=u−w\\Delta=u\-w, and letW=W2​\(μs,μt\)W=W\_\{2\}\(\\mu\_\{s\},\\mu\_\{t\}\), which is finite by Assumption[1](https://arxiv.org/html/2610.28898#Thmassumption1)\. IfW=0W=0the two rounds have the same example law and both sides vanish\. Otherwise take aW2W\_\{2\}\-optimal coupling\(X1,X1′\)\(X\_\{1\},X\_\{1\}^\{\\prime\}\), which exists onℝd\\mathbb\{R\}^\{d\}, and the variables of Lemma[E\.3](https://arxiv.org/html/2610.28898#Thmlemma3a)\. WriteU=\(Ir​\(X0,X1\),r\)U=\(I\_\{r\}\(X\_\{0\},X\_\{1\}\),r\),Z=Zr​\(X0,X1\)Z=Z\_\{r\}\(X\_\{0\},X\_\{1\}\)for the round\-ssexample andU′,Z′U^\{\\prime\},Z^\{\\prime\}for the round\-ttexample built fromX1′X\_\{1\}^\{\\prime\}\. Define

φ⁡\(U,Z\)=∥u⁡\(U\)−Z∥2−∥w⁡\(U\)−Z∥2=⟨Δ⁡\(U\),u⁡\(U\)\+w⁡\(U\)−2​Z⟩\.\\varphi\(U,Z\)=\\lVert u\(U\)\-Z\\rVert^\{2\}\-\\lVert w\(U\)\-Z\\rVert^\{2\}=\\langle\\Delta\(U\),u\(U\)\+w\(U\)\-2Z\\rangle\.It is integrable because\|φ\|≤2​Bv​\(2​Bv\+2​∥Z∥\)\|\\varphi\|\\leq 2B\_\{v\}\(2B\_\{v\}\+2\\lVert Z\\rVert\)\. ThenDs,t​\(u\)−Ds,t​\(w\)=𝔼​φ​\(U′,Z′\)−𝔼​φ​\(U,Z\)D\_\{s,t\}\(u\)\-D\_\{s,t\}\(w\)=\\mathbb\{E\}\\varphi\(U^\{\\prime\},Z^\{\\prime\}\)\-\\mathbb\{E\}\\varphi\(U,Z\)\.

Step 1: separate the two shifts\.By Equation \([35](https://arxiv.org/html/2610.28898#S5.E35)\),U′=\(Ir​\(X~0,X1\),r\)U^\{\\prime\}=\(I\_\{r\}\(\\tilde\{X\}\_\{0\},X\_\{1\}\),r\)andZ′=Zr​\(X~0,X1\)\+erZ^\{\\prime\}=Z\_\{r\}\(\\tilde\{X\}\_\{0\},X\_\{1\}\)\+e\_\{r\}\. Expanding the label shift,

φ⁡\(U′,Z′\)=φ⁡\(U′,Zr​\(X~0,X1\)\)−2​⟨Δ⁡\(U′\),er⟩\.\\varphi\(U^\{\\prime\},Z^\{\\prime\}\)=\\varphi\\big\(U^\{\\prime\},Z\_\{r\}\(\\tilde\{X\}\_\{0\},X\_\{1\}\)\\big\)\-2\\langle\\Delta\(U^\{\\prime\}\),e\_\{r\}\\rangle\.LetF⁡\(x0\)=φ⁡\(\(Ir​\(x0,X1\),r\),Zr​\(x0,X1\)\)F\(x\_\{0\}\)=\\varphi\(\(I\_\{r\}\(x\_\{0\},X\_\{1\}\),r\),Z\_\{r\}\(x\_\{0\},X\_\{1\}\)\), a function ofx0x\_\{0\}given\(X1,r\)\(X\_\{1\},r\)\. Thenφ⁡\(U′,Zr​\(X~0,X1\)\)=F⁡\(X~0\)\\varphi\(U^\{\\prime\},Z\_\{r\}\(\\tilde\{X\}\_\{0\},X\_\{1\}\)\)=F\(\\tilde\{X\}\_\{0\}\)andφ⁡\(U,Z\)=F⁡\(X0\)\\varphi\(U,Z\)=F\(X\_\{0\}\), so

Ds,t​\(u\)−Ds,t​\(w\)=𝔼⁡\[F⁡\(X~0\)−F⁡\(X0\)\]−2​𝔼​⟨Δ⁡\(U′\),er⟩\.D\_\{s,t\}\(u\)\-D\_\{s,t\}\(w\)=\\mathbb\{E\}\[F\(\\tilde\{X\}\_\{0\}\)\-F\(X\_\{0\}\)\]\-2\\mathbb\{E\}\\langle\\Delta\(U^\{\\prime\}\),e\_\{r\}\\rangle\.\(103\)
Step 2: the label shift\.Since∥er∥=k⁡\(r\)​∥Δ​X∥\\lVert e\_\{r\}\\rVert=k\(r\)\\lVert\\Delta X\\rVertandrris independent of\(X1,X1′\)\(X\_\{1\},X\_\{1\}^\{\\prime\}\), Cauchy–Schwarz gives

2​\|𝔼⁡⟨Δ⁡\(U′\),er⟩\|≤2​\(𝔼​∥Δ⁡\(U′\)∥2\)1/2​\(𝔼​k​\(r\)2​∥Δ​X∥2\)1/2=2​mk​W​∥Δ∥t\.2\|\\mathbb\{E\}\\langle\\Delta\(U^\{\\prime\}\),e\_\{r\}\\rangle\|\\leq 2\(\\mathbb\{E\}\\lVert\\Delta\(U^\{\\prime\}\)\\rVert^\{2\}\)^\{1/2\}\(\\mathbb\{E\}k\(r\)^\{2\}\\lVert\\Delta X\\rVert^\{2\}\)^\{1/2\}=2\\sqrt\{m\_\{k\}\}\\,W\\lVert\\Delta\\rVert\_\{t\}\.
Step 3: the base\-noise shift\.Condition on𝒢=σ⁡\(X1,X1′,r\)\\mathcal\{G\}=\\sigma\(X\_\{1\},X\_\{1\}^\{\\prime\},r\)\. By Lemma[E\.3](https://arxiv.org/html/2610.28898#Thmlemma3a),X0∼νX\_\{0\}\\sim\\nuandX~0∼νσr\\tilde\{X\}\_\{0\}\\sim\\nu\_\{\\sigma\_\{r\}\}conditionally\. Lemmas[E\.1](https://arxiv.org/html/2610.28898#Thmlemma1b)and[E\.2](https://arxiv.org/html/2610.28898#Thmlemma2a)give

\|𝔼⁡\[F⁡\(X~0\)−F⁡\(X0\)∣𝒢\]\|≤2​h​\(r\)​∥Δ​X∥​\(𝔼⁡\[F​\(X0\)2∣𝒢\]\+𝔼⁡\[F​\(X~0\)2∣𝒢\]\)1/2\.\|\\mathbb\{E\}\[F\(\\tilde\{X\}\_\{0\}\)\-F\(X\_\{0\}\)\\mid\\mathcal\{G\}\]\|\\leq 2h\(r\)\\lVert\\Delta X\\rVert\\big\(\\mathbb\{E\}\[F\(X\_\{0\}\)^\{2\}\\mid\\mathcal\{G\}\]\+\\mathbb\{E\}\[F\(\\tilde\{X\}\_\{0\}\)^\{2\}\\mid\\mathcal\{G\}\]\\big\)^\{1/2\}\.\(104\)To bound the second moments, split the label atMzM\_\{z\}\. Since∥Δ∥≤2​Bv\\lVert\\Delta\\rVert\\leq 2B\_\{v\}and∥u\+w∥≤2​Bv\\lVert u\+w\\rVert\\leq 2B\_\{v\},

\|F⁡\(X0\)\|\\displaystyle\|F\(X\_\{0\}\)\|≤\(2Bv\+2Mz\)∥Δ\(U\)∥\+4Bv∥Z∥𝟏\{∥Z∥\>Mz\},\\displaystyle\\leq\(2B\_\{v\}\+2M\_\{z\}\)\\lVert\\Delta\(U\)\\rVert\+4B\_\{v\}\\lVert Z\\rVert\\mathbf\{1\}\\\{\\lVert Z\\rVert\>M\_\{z\}\\\},\|F⁡\(X~0\)\|\\displaystyle\|F\(\\tilde\{X\}\_\{0\}\)\|=\|φ⁡\(U′,Z′−er\)\|\\displaystyle=\|\\varphi\(U^\{\\prime\},Z^\{\\prime\}\-e\_\{r\}\)\|≤\(2Bv\+2Mz\)∥Δ\(U′\)∥\+4Bv∥Z′∥𝟏\{∥Z′∥\>Mz\}\+4Bv∥er∥\.\\displaystyle\\leq\(2B\_\{v\}\+2M\_\{z\}\)\\lVert\\Delta\(U^\{\\prime\}\)\\rVert\+4B\_\{v\}\\lVert Z^\{\\prime\}\\rVert\\mathbf\{1\}\\\{\\lVert Z^\{\\prime\}\\rVert\>M\_\{z\}\\\}\+4B\_\{v\}\\lVert e\_\{r\}\\rVert\.Apply\(a\+b\+c\)2≤3​\(a2\+b2\+c2\)\(a\+b\+c\)^\{2\}\\leq 3\(a^\{2\}\+b^\{2\}\+c^\{2\}\), then∑ixi≤∑ixi\\sqrt\{\\sum\_\{i\}x\_\{i\}\}\\leq\\sum\_\{i\}\\sqrt\{x\_\{i\}\}\. WithAs=𝔼⁡\[∥Δ⁡\(U\)∥2∣𝒢\]A\_\{s\}=\\mathbb\{E\}\[\\lVert\\Delta\(U\)\\rVert^\{2\}\\mid\\mathcal\{G\}\],At=𝔼⁡\[∥Δ⁡\(U′\)∥2∣𝒢\]A\_\{t\}=\\mathbb\{E\}\[\\lVert\\Delta\(U^\{\\prime\}\)\\rVert^\{2\}\\mid\\mathcal\{G\}\], andτs,τt\\tau\_\{s\},\\tau\_\{t\}the conditional label\-tail moments ofZZandZ′Z^\{\\prime\}, the bracket in \([104](https://arxiv.org/html/2610.28898#A5.E104)\) has square root at most

3​\(2​Bv\+2​Mz\)​\(As\+At\)\+4​3​Bv​\(τs\+τt\)\+4​3​Bv​k​\(r\)​∥Δ​X∥\.\\sqrt\{3\}\(2B\_\{v\}\+2M\_\{z\}\)\(\\sqrt\{A\_\{s\}\}\+\\sqrt\{A\_\{t\}\}\)\+4\\sqrt\{3\}B\_\{v\}\(\\sqrt\{\\tau\_\{s\}\}\+\\sqrt\{\\tau\_\{t\}\}\)\+4\\sqrt\{3\}B\_\{v\}k\(r\)\\lVert\\Delta X\\rVert\.Multiply by2​h​\(r\)​∥Δ​X∥2h\(r\)\\lVert\\Delta X\\rVertand take expectations\. By Cauchy–Schwarz and independence ofrrand\(X1,X1′\)\(X\_\{1\},X\_\{1\}^\{\\prime\}\),𝔼⁡\[h⁡\(r\)​∥Δ​X∥​As\]≤\(𝔼​h2​W2\)1/2​∥Δ∥s\\mathbb\{E\}\[h\(r\)\\lVert\\Delta X\\rVert\\sqrt\{A\_\{s\}\}\]\\leq\(\\mathbb\{E\}h^\{2\}\\,W^\{2\}\)^\{1/2\}\\lVert\\Delta\\rVert\_\{s\}, and similarly withAtA\_\{t\}and∥Δ∥t\\lVert\\Delta\\rVert\_\{t\}\. SinceZZandZ′Z^\{\\prime\}have the round\-ssand round\-ttlabel laws, Lemma[E\.4](https://arxiv.org/html/2610.28898#Thmlemma4a)gives𝔼​τs,𝔼​τt≤δM/2\\mathbb\{E\}\\tau\_\{s\},\\mathbb\{E\}\\tau\_\{t\}\\leq\\delta\_\{M\}/2\. Finally𝔼⁡\[h⁡\(r\)​k​\(r\)​∥Δ​X∥2\]=𝔼λ​\[h​k\]​W2\\mathbb\{E\}\[h\(r\)k\(r\)\\lVert\\Delta X\\rVert^\{2\}\]=\\mathbb\{E\}\_\{\\lambda\}\[hk\]W^\{2\}\. With𝔼λ​h2=ι​mh/8\\mathbb\{E\}\_\{\\lambda\}h^\{2\}=\\iota m\_\{h\}/8,

\|𝔼⁡\[F⁡\(X~0\)−F⁡\(X0\)\]\|≤\\displaystyle\|\\mathbb\{E\}\[F\(\\tilde\{X\}\_\{0\}\)\-F\(X\_\{0\}\)\]\|\\leq\{\}4​3​\(Bv\+Mz\)​ι​mh/8​W​\(∥Δ∥s\+∥Δ∥t\)\\displaystyle 4\\sqrt\{3\}\(B\_\{v\}\+M\_\{z\}\)\\sqrt\{\\iota m\_\{h\}/8\}\\,W\(\\lVert\\Delta\\rVert\_\{s\}\+\\lVert\\Delta\\rVert\_\{t\}\)\+16​3​Bv​ι​mh/8​δM/2​W\+8​3​Bv​𝔼λ​\[h​k\]​W2\.\\displaystyle\+16\\sqrt\{3\}B\_\{v\}\\sqrt\{\\iota m\_\{h\}/8\}\\sqrt\{\\delta\_\{M\}/2\}\\,W\+8\\sqrt\{3\}B\_\{v\}\\mathbb\{E\}\_\{\\lambda\}\[hk\]W^\{2\}\.
Step 4: combine\.Adding Step 2 to this bound and using the definitions in Equation \([21](https://arxiv.org/html/2610.28898#S4.E21)\) proves the lemma\. If endpoints are bounded and∥Z∥≤Mz\\lVert Z\\rVert\\leq M\_\{z\}almost surely, the tail moments vanish andϵsm\\epsilon\_\{\\mathrm\{sm\}\}can be replaced by zero\.□\\square

The proof never differences a represented field at two inputs, so the smoothing transfer needs no additional Lipschitz or norm\-equivalence assumption on the class, and only the output boundBvB\_\{v\}enters\. It also never uses the ideal field or regularity of the targets\.

#### E\.1\.5Approximate Quadratic Growth

###### Lemma E\.5\.

For every roundtt, everyu∈𝒱u\\in\\mathcal\{V\}, and every minimizervt⋆v\_\{t\}^\{\\star\}ofLtL\_\{t\}over𝒱\\mathcal\{V\},∥u−vt⋆∥t≤Ft​\(u\)\+2​at\\lVert u\-v\_\{t\}^\{\\star\}\\rVert\_\{t\}\\leq\\sqrt\{F\_\{t\}\(u\)\}\+2\\sqrt\{a\_\{t\}\}\.

###### Proof\.

The triangle inequality through the ideal field gives∥u−vt⋆∥t≤Qt​\(u\)\+Qt​\(vt⋆\)\\lVert u\-v\_\{t\}^\{\\star\}\\rVert\_\{t\}\\leq\\sqrt\{Q\_\{t\}\(u\)\}\+\\sqrt\{Q\_\{t\}\(v\_\{t\}^\{\\star\}\)\}\. By Equation \([5](https://arxiv.org/html/2610.28898#S3.E5)\),Qt​\(u\)=Ft​\(u\)\+atQ\_\{t\}\(u\)=F\_\{t\}\(u\)\+a\_\{t\}andQt​\(vt⋆\)=atQ\_\{t\}\(v\_\{t\}^\{\\star\}\)=a\_\{t\}, so the right side is at mostFt​\(u\)\+at\+at\\sqrt\{F\_\{t\}\(u\)\}\+\\sqrt\{a\_\{t\}\}\+\\sqrt\{a\_\{t\}\}\. ∎

This is the flow\-matching substitute for quadratic growth: it holds for every nonconvex class, with the approximation error as its only residual\. It is exact in the realizable case\.

#### E\.1\.6Second\-Order Drift Transfer

###### Lemma E\.6\.

Lets,ts,tbe rounds and suppose that for someℏ,ρ≥0\\hbar,\\rho\\geq 0and allu,w∈𝒱u,w\\in\\mathcal\{V\}

\|Ds,t​\(u\)−Ds,t​\(w\)\|≤ℏ​\(∥u−w∥s\+∥u−w∥t\)\+ρ\.\|D\_\{s,t\}\(u\)\-D\_\{s,t\}\(w\)\|\\leq\\hbar\(\\lVert u\-w\\rVert\_\{s\}\+\\lVert u\-w\\rVert\_\{t\}\)\+\\rho\.\(105\)Then, withvs⋆,vt⋆v\_\{s\}^\{\\star\},v\_\{t\}^\{\\star\}minimizers ofLs,LtL\_\{s\},L\_\{t\}over𝒱\\mathcal\{V\}:

1. \(i\)Ft​\(vs⋆\)≤5​ℏ2\+8​\(as\+at\)\+2​ρF\_\{t\}\(v\_\{s\}^\{\\star\}\)\\leq 5\\hbar^\{2\}\+8\(a\_\{s\}\+a\_\{t\}\)\+2\\rhoandFs​\(vt⋆\)≤5​ℏ2\+8​\(as\+at\)\+2​ρF\_\{s\}\(v\_\{t\}^\{\\star\}\)\\leq 5\\hbar^\{2\}\+8\(a\_\{s\}\+a\_\{t\}\)\+2\\rho,
2. \(ii\)Ft​\(u\)≤3​Fs​\(u\)\+14​ℏ2\+12​as\+10​at\+4​ρF\_\{t\}\(u\)\\leq 3F\_\{s\}\(u\)\+14\\hbar^\{2\}\+12a\_\{s\}\+10a\_\{t\}\+4\\rhofor everyu∈𝒱u\\in\\mathcal\{V\}\.

###### Proof\.

*Sandwich\.*Sincemin𝒱⁡Ls≤Ls​\(vt⋆\)\\min\_\{\\mathcal\{V\}\}L\_\{s\}\\leq L\_\{s\}\(v\_\{t\}^\{\\star\}\), for everyuu,

Ft​\(u\)−Fs​\(u\)=Ds,t​\(u\)−\(min𝒱⁡Lt−min𝒱⁡Ls\)≤Ds,t​\(u\)−Ds,t​\(vt⋆\)\.F\_\{t\}\(u\)\-F\_\{s\}\(u\)=D\_\{s,t\}\(u\)\-\\big\(\\min\_\{\\mathcal\{V\}\}L\_\{t\}\-\\min\_\{\\mathcal\{V\}\}L\_\{s\}\\big\)\\leq D\_\{s,t\}\(u\)\-D\_\{s,t\}\(v\_\{t\}^\{\\star\}\)\.\(106\)Exchanging the roles ofssandtt, and usingDt,s=−Ds,tD\_\{t,s\}=\-D\_\{s,t\}, givesFs​\(u\)−Ft​\(u\)≤Ds,t​\(vs⋆\)−Ds,t​\(u\)F\_\{s\}\(u\)\-F\_\{t\}\(u\)\\leq D\_\{s,t\}\(v\_\{s\}^\{\\star\}\)\-D\_\{s,t\}\(u\)\.

*Part \(i\)\.*Letx=Fs​\(vt⋆\)x=\\sqrt\{F\_\{s\}\(v\_\{t\}^\{\\star\}\)\},y=Ft​\(vs⋆\)y=\\sqrt\{F\_\{t\}\(v\_\{s\}^\{\\star\}\)\},z=max⁡\{x,y\}z=\\max\\\{x,y\\\}andA=2​as\+2​atA=2\\sqrt\{a\_\{s\}\}\+2\\sqrt\{a\_\{t\}\}\. Apply \([106](https://arxiv.org/html/2610.28898#A5.E106)\) withu=vs⋆u=v\_\{s\}^\{\\star\}and its exchanged form withu=vt⋆u=v\_\{t\}^\{\\star\}\. Both give an upper boundDs,t​\(vs⋆\)−Ds,t​\(vt⋆\)D\_\{s,t\}\(v\_\{s\}^\{\\star\}\)\-D\_\{s,t\}\(v\_\{t\}^\{\\star\}\), which by \([105](https://arxiv.org/html/2610.28898#A5.E105)\) is at mostℏ​\(∥vs⋆−vt⋆∥s\+∥vs⋆−vt⋆∥t\)\+ρ\\hbar\(\\lVert v\_\{s\}^\{\\star\}\-v\_\{t\}^\{\\star\}\\rVert\_\{s\}\+\\lVert v\_\{s\}^\{\\star\}\-v\_\{t\}^\{\\star\}\\rVert\_\{t\}\)\+\\rho\. Lemma[E\.5](https://arxiv.org/html/2610.28898#Thmlemma5a)at roundsswithu=vt⋆u=v\_\{t\}^\{\\star\}and at roundttwithu=vs⋆u=v\_\{s\}^\{\\star\}bounds the two norms byx\+2​asx\+2\\sqrt\{a\_\{s\}\}andy\+2​aty\+2\\sqrt\{a\_\{t\}\}\. Hencez2≤ℏ​\(2​z\+A\)\+ρz^\{2\}\\leq\\hbar\(2z\+A\)\+\\rho, soz≤ℏ\+ℏ2\+ℏ​A\+ρz\\leq\\hbar\+\\sqrt\{\\hbar^\{2\}\+\\hbar A\+\\rho\}and

z2≤2​ℏ2\+2​\(ℏ2\+ℏ​A\+ρ\)≤5​ℏ2\+A2\+2​ρ≤5​ℏ2\+8​\(as\+at\)\+2​ρ,z^\{2\}\\leq 2\\hbar^\{2\}\+2\(\\hbar^\{2\}\+\\hbar A\+\\rho\)\\leq 5\\hbar^\{2\}\+A^\{2\}\+2\\rho\\leq 5\\hbar^\{2\}\+8\(a\_\{s\}\+a\_\{t\}\)\+2\\rho,using2​ℏ​A≤ℏ2\+A22\\hbar A\\leq\\hbar^\{2\}\+A^\{2\}andA2≤8​\(as\+at\)A^\{2\}\\leq 8\(a\_\{s\}\+a\_\{t\}\)\.

*Part \(ii\)\.*By \([106](https://arxiv.org/html/2610.28898#A5.E106)\) and \([105](https://arxiv.org/html/2610.28898#A5.E105)\),Ft​\(u\)−Fs​\(u\)≤ℏ​\(∥u−vt⋆∥s\+∥u−vt⋆∥t\)\+ρF\_\{t\}\(u\)\-F\_\{s\}\(u\)\\leq\\hbar\(\\lVert u\-v\_\{t\}^\{\\star\}\\rVert\_\{s\}\+\\lVert u\-v\_\{t\}^\{\\star\}\\rVert\_\{t\}\)\+\\rho\. Lemma[E\.5](https://arxiv.org/html/2610.28898#Thmlemma5a)gives∥u−vt⋆∥t≤Ft​\(u\)\+2​at\\lVert u\-v\_\{t\}^\{\\star\}\\rVert\_\{t\}\\leq\\sqrt\{F\_\{t\}\(u\)\}\+2\\sqrt\{a\_\{t\}\}and

∥u−vt⋆∥s≤∥u−vs⋆∥s\+∥vs⋆−vt⋆∥s≤Fs​\(u\)\+z\+4​as\.\\lVert u\-v\_\{t\}^\{\\star\}\\rVert\_\{s\}\\leq\\lVert u\-v\_\{s\}^\{\\star\}\\rVert\_\{s\}\+\\lVert v\_\{s\}^\{\\star\}\-v\_\{t\}^\{\\star\}\\rVert\_\{s\}\\leq\\sqrt\{F\_\{s\}\(u\)\}\+z\+4\\sqrt\{a\_\{s\}\}\.Applyℏ​Ft≤Ft/2\+ℏ2/2\\hbar\\sqrt\{F\_\{t\}\}\\leq F\_\{t\}/2\+\\hbar^\{2\}/2,ℏ​Fs≤Fs/2\+ℏ2/2\\hbar\\sqrt\{F\_\{s\}\}\\leq F\_\{s\}/2\+\\hbar^\{2\}/2,ℏ​z≤\(ℏ2\+z2\)/2\\hbar z\\leq\(\\hbar^\{2\}\+z^\{2\}\)/2,2​ℏ​at≤ℏ2\+at2\\hbar\\sqrt\{a\_\{t\}\}\\leq\\hbar^\{2\}\+a\_\{t\}, and4​ℏ​as≤2​ℏ2\+2​as4\\hbar\\sqrt\{a\_\{s\}\}\\leq 2\\hbar^\{2\}\+2a\_\{s\}\. Collecting terms,

12​Ft​\(u\)≤32​Fs​\(u\)\+92​ℏ2\+12​z2\+at\+2​as\+ρ\.\\tfrac\{1\}\{2\}F\_\{t\}\(u\)\\leq\\tfrac\{3\}\{2\}F\_\{s\}\(u\)\+\\tfrac\{9\}\{2\}\\hbar^\{2\}\+\\tfrac\{1\}\{2\}z^\{2\}\+a\_\{t\}\+2a\_\{s\}\+\\rho\.Multiply by two and insert part \(i\) forz2z^\{2\}\. ∎

#### E\.1\.7Proof of Proposition[2](https://arxiv.org/html/2610.28898#Thmproposition2)

By definition2​ωs,t=supuDs,t​\(u\)−infwDs,t​\(w\)=supu,w\(Ds,t​\(u\)−Ds,t​\(w\)\)2\\omega\_\{s,t\}=\\sup\_\{u\}D\_\{s,t\}\(u\)\-\\inf\_\{w\}D\_\{s,t\}\(w\)=\\sup\_\{u,w\}\(D\_\{s,t\}\(u\)\-D\_\{s,t\}\(w\)\)\. Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1)and∥u−w∥r≤2​Bv\\lVert u\-w\\rVert\_\{r\}\\leq 2B\_\{v\}give2​ωs,t≤4​Bv​Λsm​W\+ϵsm​W\+c2​W22\\omega\_\{s,t\}\\leq 4B\_\{v\}\\Lambda\_\{\\mathrm\{sm\}\}W\+\\epsilon\_\{\\mathrm\{sm\}\}W\+c\_\{2\}W^\{2\}, which is \([36](https://arxiv.org/html/2610.28898#S5.E36)\)\. For consecutive rounds, summation givesΩT≤2​Bv​Λsm​VTμ\+12​ϵsm​VTμ\+12​c2​∑tW2​\(μt\+1,μt\)2\\Omega\_\{T\}\\leq 2B\_\{v\}\\Lambda\_\{\\mathrm\{sm\}\}V\_\{T\}^\{\\mu\}\+\\frac\{1\}\{2\}\\epsilon\_\{\\mathrm\{sm\}\}V\_\{T\}^\{\\mu\}\+\\frac\{1\}\{2\}c\_\{2\}\\sum\_\{t\}W\_\{2\}\(\\mu\_\{t\+1\},\\mu\_\{t\}\)^\{2\}\. Inequality \([37](https://arxiv.org/html/2610.28898#S5.E37)\) is part \(i\) of Lemma[E\.6](https://arxiv.org/html/2610.28898#Thmlemma6a)withℏ=Λsm​W\\hbar=\\Lambda\_\{\\mathrm\{sm\}\}Wandρ=ϵsm​W\+c2​W2\\rho=\\epsilon\_\{\\mathrm\{sm\}\}W\+c\_\{2\}W^\{2\}, whose hypothesis is Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1)\.□\\square

#### E\.1\.8Proof of Proposition[3](https://arxiv.org/html/2610.28898#Thmproposition3)

Givenx1x\_\{1\},Ir​\(X0,x1\)=α​X0\+β​x1∼N⁡\(β​x1,α2​σ02​Id\)I\_\{r\}\(X\_\{0\},x\_\{1\}\)=\\alpha X\_\{0\}\+\\beta x\_\{1\}\\sim N\(\\beta x\_\{1\},\\alpha^\{2\}\\sigma\_\{0\}^\{2\}I\_\{d\}\)\. For two such Gaussians with common covariance the Bhattacharyya coefficient isexp\(−β2∥x1−x1′∥2/\(8α2σ02\)\)\\exp\(\-\\beta^\{2\}\\lVert x\_\{1\}\-x\_\{1\}^\{\\prime\}\\rVert^\{2\}/\(8\\alpha^\{2\}\\sigma\_\{0\}^\{2\}\)\), so

𝖧2​\(qr,x1,qr,x1′\)=1−exp⁡\(−β2​∥x1−x1′∥28​α2​σ02\)\.\\mathsf\{H\}^\{2\}\(q\_\{r,x\_\{1\}\},q\_\{r,x\_\{1\}^\{\\prime\}\}\)=1\-\\exp\\Big\(\-\\frac\{\\beta^\{2\}\\lVert x\_\{1\}\-x\_\{1\}^\{\\prime\}\\rVert^\{2\}\}\{8\\alpha^\{2\}\\sigma\_\{0\}^\{2\}\}\\Big\)\.Since1−e−x≤x1\-e^\{\-x\}\\leq xwith ratio tending to one asx↓0x\\downarrow 0, the supremum of𝖧/∥x1−x1′∥\\mathsf\{H\}/\\lVert x\_\{1\}\-x\_\{1\}^\{\\prime\}\\rVertequals\|β\|/\(2​2​α​σ0\)\|\\beta\|/\(2\\sqrt\{2\}\\alpha\\sigma\_\{0\}\), and this ish⁡\(r\)h\(r\)withι=1/σ02\\iota=1/\\sigma\_\{0\}^\{2\}\.

For the second claim,α⁡\(1\)=0\\alpha\(1\)=0andα∈C1\\alpha\\in C^\{1\}giveα⁡\(r\)≤∥α′∥∞​\(1−r\)\\alpha\(r\)\\leq\\lVert\\alpha^\{\\prime\}\\rVert\_\{\\infty\}\(1\-r\), with∥α′∥∞\>0\\lVert\\alpha^\{\\prime\}\\rVert\_\{\\infty\}\>0becauseα⁡\(0\)=1\\alpha\(0\)=1\. Continuity andβ⁡\(1\)=1\\beta\(1\)=1giver1<1r\_\{1\}<1withβ≥1/2\\beta\\geq 1/2on\[r1,1\]\[r\_\{1\},1\]\. Therefore

∫01β​\(r\)2α​\(r\)2​𝑑r≥14​∥α′∥∞2​∫r11d​r\(1−r\)2=∞\.□\\int\_\{0\}^\{1\}\\frac\{\\beta\(r\)^\{2\}\}\{\\alpha\(r\)^\{2\}\}dr\\geq\\frac\{1\}\{4\\lVert\\alpha^\{\\prime\}\\rVert\_\{\\infty\}^\{2\}\}\\int\_\{r\_\{1\}\}^\{1\}\\frac\{dr\}\{\(1\-r\)^\{2\}\}=\\infty\.\\qquad\\square

#### E\.1\.9Admissible Flow\-Time Laws

##### Truncated uniform\.

For the linear path,J≡1J\\equiv 1\. Withλ=Unif⁡\[0,1−η0\]\\lambda=\\mathrm\{Unif\}\[0,1\-\\eta\_\{0\}\]and the substitutionu=1−ru=1\-r,

mk=11−η0​∫η01d​uu2=1η0,mh=11−η0​∫η01\(1−u\)2u2​𝑑u=η0−1−2​log⁡\(1/η0\)−η01−η0\.m\_\{k\}=\\frac\{1\}\{1\-\\eta\_\{0\}\}\\int\_\{\\eta\_\{0\}\}^\{1\}\\frac\{du\}\{u^\{2\}\}=\\frac\{1\}\{\\eta\_\{0\}\},\\qquad m\_\{h\}=\\frac\{1\}\{1\-\\eta\_\{0\}\}\\int\_\{\\eta\_\{0\}\}^\{1\}\\frac\{\(1\-u\)^\{2\}\}\{u^\{2\}\}du=\\frac\{\\eta\_\{0\}^\{\-1\}\-2\\log\(1/\\eta\_\{0\}\)\-\\eta\_\{0\}\}\{1\-\\eta\_\{0\}\}\.For the sine–cosine path,J≡π/2J\\equiv\\pi/2, and∫01−η0sec2⁡\(π​r/2\)​𝑑r=2π​cot⁡\(π​η0/2\)\\int\_\{0\}^\{1\-\\eta\_\{0\}\}\\sec^\{2\}\(\\pi r/2\)dr=\\frac\{2\}\{\\pi\}\\cot\(\\pi\\eta\_\{0\}/2\)\. Hence

mk=π​cot⁡\(π​η0/2\)2​\(1−η0\),mh=2π​cot⁡\(π​η0/2\)−\(1−η0\)1−η0\.m\_\{k\}=\\frac\{\\pi\\cot\(\\pi\\eta\_\{0\}/2\)\}\{2\(1\-\\eta\_\{0\}\)\},\\qquad m\_\{h\}=\\frac\{\\frac\{2\}\{\\pi\}\\cot\(\\pi\\eta\_\{0\}/2\)\-\(1\-\\eta\_\{0\}\)\}\{1\-\\eta\_\{0\}\}\.Both are of order1/η01/\\eta\_\{0\}, soΛsm=O\(η0−1/2log⁡\(e​m​T\)\)\\Lambda\_\{\\mathrm\{sm\}\}=O\(\\eta\_\{0\}^\{\-1/2\}\\sqrt\{\\log\(emT\)\}\)\.

##### Beta and logit\-normal laws\.

For the linear path andλ=Beta⁡\(a,b\)\\lambda=\\mathrm\{Beta\}\(a,b\),mk=𝔼​\(1−r\)−2=B⁡\(a,b−2\)/B⁡\(a,b\)m\_\{k\}=\\mathbb\{E\}\(1\-r\)^\{\-2\}=B\(a,b\-2\)/B\(a,b\), finite exactly whenb\>2b\>2, andmh≤mkm\_\{h\}\\leq m\_\{k\}\. Iflogit⁡\(r\)∼N⁡\(μℓ,s2\)\\mathrm\{logit\}\(r\)\\sim N\(\\mu\_\{\\ell\},s^\{2\}\), then\(1−r\)−2=\(1\+eY\)2≤2\+2​e2​Y\(1\-r\)^\{\-2\}=\(1\+e^\{Y\}\)^\{2\}\\leq 2\+2e^\{2Y\}withY=logit⁡\(r\)Y=\\mathrm\{logit\}\(r\), somk≤2\+2​e2​μℓ\+2​s2m\_\{k\}\\leq 2\+2e^\{2\\mu\_\{\\ell\}\+2s^\{2\}\}\. For the sine–cosine path,α⁡\(r\)=sin⁡\(π⁡\(1−r\)/2\)≥1−r\\alpha\(r\)=\\sin\(\\pi\(1\-r\)/2\)\\geq 1\-rby concavity, so the same bounds hold after multiplyingmkm\_\{k\}byπ2/4\\pi^\{2\}/4\.

#### E\.1\.10Historical Transfers

###### Lemma E\.7\(Target\-law transfer\)\.

LetH=\[a,a\+q−1\]H=\[a,a\+q\-1\]precede roundtt\. Then \([38](https://arxiv.org/html/2610.28898#S5.E38)\) and \([39](https://arxiv.org/html/2610.28898#S5.E39)\) hold for everyv∈𝒱v\\in\\mathcal\{V\}, pointwise before expectation over any data\.

###### Proof\.

Fors∈Hs\\in H, the triangle inequality forW2W\_\{2\}givesW2\(μs,μt\)≤Va:tμW\_\{2\}\(\\mu\_\{s\},\\mu\_\{t\}\)\\leq V^\{\\mu\}\_\{a:t\}\. By Lemma[1](https://arxiv.org/html/2610.28898#Thmlemma1), hypothesis \([105](https://arxiv.org/html/2610.28898#A5.E105)\) holds for\(s,t\)\(s,t\)withℏ≤ΛsmVμa:t\\hbar\\leq\\Lambda\_\{\\mathrm\{sm\}\}V^\{\\mu\}\_\{a:t\}andρ≤ρ¯a:t\\rho\\leq\\bar\{\\rho\}\_\{a:t\}, since both remainders increase with the distance\.

*Historical approximation\.*Because regression\-noise offsets do not depend on the field,vH⋆v\_\{H\}^\{\\star\}also minimizesq−1​∑s∈HQsq^\{\-1\}\\sum\_\{s\\in H\}Q\_\{s\}, as in Lemma[C\.2](https://arxiv.org/html/2610.28898#Thmlemma2)\. Therefore

a¯H≤1q​∑s∈HQs​\(vt⋆\)=1q​∑s∈H\[as\+Fs​\(vt⋆\)\],\\bar\{a\}\_\{H\}\\leq\\frac\{1\}\{q\}\\sum\_\{s\\in H\}Q\_\{s\}\(v\_\{t\}^\{\\star\}\)=\\frac\{1\}\{q\}\\sum\_\{s\\in H\}\[a\_\{s\}\+F\_\{s\}\(v\_\{t\}^\{\\star\}\)\],and Lemma[E\.6](https://arxiv.org/html/2610.28898#Thmlemma6a)\(i\) bounds eachFs​\(vt⋆\)F\_\{s\}\(v\_\{t\}^\{\\star\}\)by5Λsm2\(Va:tμ\)2\+8\(as\+at\)\+2ρ¯a:t5\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+8\(a\_\{s\}\+a\_\{t\}\)\+2\\bar\{\\rho\}\_\{a:t\}\. This proves \([38](https://arxiv.org/html/2610.28898#S5.E38)\)\.

*Current excess\.*Average Lemma[E\.6](https://arxiv.org/html/2610.28898#Thmlemma6a)\(ii\) overs∈Hs\\in H:

Ft\(v\)≤3q∑s∈HFs\(v\)\+14Λsm2\(Va:tμ\)2\+12q∑s∈Has\+10at\+4ρ¯a:t\.F\_\{t\}\(v\)\\leq\\frac\{3\}\{q\}\\sum\_\{s\\in H\}F\_\{s\}\(v\)\+14\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+\\frac\{12\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+10a\_\{t\}\+4\\bar\{\\rho\}\_\{a:t\}\.By definition,q−1​∑sFs​\(v\)=rH​\(v\)\+q−1​∑s\[Ls​\(vH⋆\)−Ls​\(vs⋆\)\]q^\{\-1\}\\sum\_\{s\}F\_\{s\}\(v\)=r\_\{H\}\(v\)\+q^\{\-1\}\\sum\_\{s\}\[L\_\{s\}\(v\_\{H\}^\{\\star\}\)\-L\_\{s\}\(v\_\{s\}^\{\\star\}\)\]\. SinceL¯H​\(vH⋆\)≤L¯H​\(vt⋆\)\\bar\{L\}\_\{H\}\(v\_\{H\}^\{\\star\}\)\\leq\\bar\{L\}\_\{H\}\(v\_\{t\}^\{\\star\}\), the bracket average is at mostq−1∑sFs\(vt⋆\)≤5Λsm2\(Va:tμ\)2\+8q−1∑sas\+8at\+2ρ¯a:tq^\{\-1\}\\sum\_\{s\}F\_\{s\}\(v\_\{t\}^\{\\star\}\)\\leq 5\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+8q^\{\-1\}\\sum\_\{s\}a\_\{s\}\+8a\_\{t\}\+2\\bar\{\\rho\}\_\{a:t\}\. Substituting gives \([39](https://arxiv.org/html/2610.28898#S5.E39)\):3⋅5\+14=293\\cdot 5\+14=29,3⋅8\+12=363\\cdot 8\+12=36,3⋅8\+10=343\\cdot 8\+10=34, and3⋅2\+4=103\\cdot 2\+4=10\. ∎

#### E\.1\.11Historical Fits over a Lifetime

For a lifetimeI=\[a,a\+nI−1\]I=\[a,a\+n\_\{I\}\-1\], letVIμ=∑s=aa\+nI−2W2​\(μs\+1,μs\)V\_\{I\}^\{\\mu\}=\\sum\_\{s=a\}^\{a\+n\_\{I\}\-2\}W\_\{2\}\(\\mu\_\{s\+1\},\\mu\_\{s\}\)be its internal target\-law variation, and recalleM=2​ϵopt\+5​ζ\+10​δe\_\{M\}=2\\epsilon\_\{\\mathrm\{opt\}\}\+5\\zeta\+10\\deltafrom Appendix[C](https://arxiv.org/html/2610.28898#A3)\.

###### Lemma E\.8\.

The fitted model of every lifetime satisfies

JI≤\\displaystyle J\_\{I\}\\leq\{\}B\+δ\+216​B​Γm​log⁡\(e​nI\)\+\(180​log⁡\(e​nI\)\+82\)​𝒜I\\displaystyle B\+\\delta\+\\frac\{216B\\Gamma\}\{m\}\\log\(en\_\{I\}\)\+\(180\\log\(en\_\{I\}\)\+82\)\\mathcal\{A\}\_\{I\}\+\(nI−1\)​\[59​Λsm2​\(VIμ\)2\+22​\(ϵsm​VIμ\+c2​\(VIμ\)2\)\+3​eM\]\.\\displaystyle\+\(n\_\{I\}\-1\)\\big\[59\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V\_\{I\}^\{\\mu\}\)^\{2\}\+22\(\\epsilon\_\{\\mathrm\{sm\}\}V\_\{I\}^\{\\mu\}\+c\_\{2\}\(V\_\{I\}^\{\\mu\}\)^\{2\}\)\+3e\_\{M\}\\big\]\.\(107\)

###### Proof\.

Att=at=a, the deployed fieldvI,a∈𝒱v\_\{I,a\}\\in\\mathcal\{V\}satisfiesFa​\(vI,a\)≤La​\(vI,a\)≤B\+δF\_\{a\}\(v\_\{I,a\}\)\\leq L\_\{a\}\(v\_\{I,a\}\)\\leq B\+\\delta\. At a later round, letk=t−ak=t\-aandq=2⌊log2⁡k⌋q=2^\{\\lfloor\\log\_\{2\}k\\rfloor\}, sok/2≤q≤kk/2\\leq q\\leq kand the prefixH=\[a,a\+q−1\]H=\[a,a\+q\-1\]is deterministic\. Apply the current\-risk transfer from Lemma[E\.7](https://arxiv.org/html/2610.28898#Thmlemma7a), Equation \([39](https://arxiv.org/html/2610.28898#S5.E39)\), pointwise to the fit, and then the historical estimation bound of Proposition[1](https://arxiv.org/html/2610.28898#Thmproposition1), Equation \([28](https://arxiv.org/html/2610.28898#S5.E28)\), proved in Appendix[C](https://arxiv.org/html/2610.28898#A3)\. This gives

𝔼Ft\(vI,t\)≤3\[36​B​Γm​q\+2a¯H\+eM\]\+29Λsm2\(Va:tμ\)2\+36q∑s∈Has\+34at\+10ρ¯a:t\.\\mathbb\{E\}F\_\{t\}\(v\_\{I,t\}\)\\leq 3\\Big\[\\frac\{36B\\Gamma\}\{mq\}\+2\\bar\{a\}\_\{H\}\+e\_\{M\}\\Big\]\+29\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+\\frac\{36\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+34a\_\{t\}\+10\\bar\{\\rho\}\_\{a:t\}\.Insert the historical\-approximation bound from Lemma[E\.7](https://arxiv.org/html/2610.28898#Thmlemma7a), Equation \([38](https://arxiv.org/html/2610.28898#S5.E38)\), fora¯H\\bar\{a\}\_\{H\}, which is deterministic under the fixed environment:

𝔼Ft\(vI,t\)≤108​B​Γm​q\+90q∑s∈Has\+82at\+59Λsm2\(Va:tμ\)2\+22ρ¯a:t\+3eM\.\\mathbb\{E\}F\_\{t\}\(v\_\{I,t\}\)\\leq\\frac\{108B\\Gamma\}\{mq\}\+\\frac\{90\}\{q\}\\sum\_\{s\\in H\}a\_\{s\}\+82a\_\{t\}\+59\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V^\{\\mu\}\_\{a:t\}\)^\{2\}\+22\\bar\{\\rho\}\_\{a:t\}\+3e\_\{M\}\.Now sum overk=1,…,nI−1k=1,\\ldots,n\_\{I\}\-1\. As in Lemma[C\.6](https://arxiv.org/html/2610.28898#Thmlemma6),∑k1/q≤2​log⁡\(e​nI\)\\sum\_\{k\}1/q\\leq 2\\log\(en\_\{I\}\), so the statistical terms sum to at most216​B​Γ​log⁡\(e​nI\)/m216B\\Gamma\\log\(en\_\{I\}\)/mand eachasa\_\{s\}receives total coefficient at most180​log⁡\(e​nI\)180\\log\(en\_\{I\}\)from the historical sums\. The current terms contribute82​𝒜I82\\mathcal\{A\}\_\{I\}\. Fort∈It\\in I,Vμa:t≤VIμV^\{\\mu\}\_\{a:t\}\\leq V\_\{I\}^\{\\mu\}, so the drift terms contribute at most\(nI−1\)​\[59​Λsm2​\(VIμ\)2\+22​\(ϵsm​VIμ\+c2​\(VIμ\)2\)\]\(n\_\{I\}\-1\)\[59\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V\_\{I\}^\{\\mu\}\)^\{2\}\+22\(\\epsilon\_\{\\mathrm\{sm\}\}V\_\{I\}^\{\\mu\}\+c\_\{2\}\(V\_\{I\}^\{\\mu\}\)^\{2\}\)\], and the residuals\(nI−1\)​3​eM\(n\_\{I\}\-1\)3e\_\{M\}\. ∎

#### E\.1\.12Interval Bound

###### Lemma E\.9\.

For every lifetimeII,

SI≤KT\+/DT\+HT\+​𝒜I\+nI​ef\+\+\(nI−1\)​\[99​Λsm2​\(VIμ\)2\+37​\(ϵsm​VIμ\+c2​\(VIμ\)2\)\],S\_\{I\}\\leq K\_\{T\}^\{\+\}/D\_\{T\}\+H\_\{T\}^\{\+\}\\mathcal\{A\}\_\{I\}\+n\_\{I\}e\_\{f\}^\{\+\}\+\(n\_\{I\}\-1\)\\big\[99\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\(V\_\{I\}^\{\\mu\}\)^\{2\}\+37\(\\epsilon\_\{\\mathrm\{sm\}\}V\_\{I\}^\{\\mu\}\+c\_\{2\}\(V\_\{I\}^\{\\mu\}\)^\{2\}\)\\big\],\(108\)and for every consecutive intervalJ⊆\[1,T\]J\\subseteq\[1,T\]inequality \([40](https://arxiv.org/html/2610.28898#S5.Ex21)\) holds\.

###### Proof\.

Insert the target\-law lifetime estimate of Lemma[E\.8](https://arxiv.org/html/2610.28898#Thmlemma8), Equation \([107](https://arxiv.org/html/2610.28898#A5.E107)\), into the shared fast master bound of Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5), Equation \([83](https://arxiv.org/html/2610.28898#A3.E83)\)\. The non\-statistical constant\(5/3\)​B\+\(128/3\)​B​log⁡Me\(5/3\)B\+\(128/3\)B\\log M\_\{e\}is at most44​B​Λ44B\\Lambda, and\(5/3\)​216=360\(5/3\)216=360gives the statistical term ofKT\+/DTK\_\{T\}^\{\+\}/D\_\{T\}\. The approximation coefficient is\(5/3\)​\(180​log⁡\(e​nI\)\+82\)\+2/3≤300​log⁡\(e​T\)\+138=HT\+\(5/3\)\(180\\log\(en\_\{I\}\)\+82\)\+2/3\\leq 300\\log\(eT\)\+138=H\_\{T\}^\{\+\}\. The drift coefficients are\(5/3\)​59≤99\(5/3\)59\\leq 99and\(5/3\)​22≤37\(5/3\)22\\leq 37\. The residuals are\(5/3\)​\(nI−1\)​3​eM\+\(5/3\)​δ\+\(8/3\)​nI​δ≤nI​\(5​eM\+13​δ/3\)≤nI​ef\+\(5/3\)\(n\_\{I\}\-1\)3e\_\{M\}\+\(5/3\)\\delta\+\(8/3\)n\_\{I\}\\delta\\leq n\_\{I\}\(5e\_\{M\}\+13\\delta/3\)\\leq n\_\{I\}e\_\{f\}^\{\+\}\. This proves \([108](https://arxiv.org/html/2610.28898#A5.E108)\)\.

For an intervalJJ, take the disjoint dyadic cover of Lemma[C\.7](https://arxiv.org/html/2610.28898#Thmlemma7): at mostDTD\_\{T\}lifetimes, whose internal edges are internal edges ofJJ\. Then∑I\(nI−1\)​\(VIμ\)2≤\(nJ−1\)​\(∑IVIμ\)2≤\(nJ−1\)​\(VJμ\)2\\sum\_\{I\}\(n\_\{I\}\-1\)\(V\_\{I\}^\{\\mu\}\)^\{2\}\\leq\(n\_\{J\}\-1\)\(\\sum\_\{I\}V\_\{I\}^\{\\mu\}\)^\{2\}\\leq\(n\_\{J\}\-1\)\(V\_\{J\}^\{\\mu\}\)^\{2\}and∑I\(nI−1\)​VIμ≤\(nJ−1\)​VJμ\\sum\_\{I\}\(n\_\{I\}\-1\)V\_\{I\}^\{\\mu\}\\leq\(n\_\{J\}\-1\)V\_\{J\}^\{\\mu\}\. Summing \([108](https://arxiv.org/html/2610.28898#A5.E108)\) and usingc¯=99​Λsm2\+37​c2\\bar\{c\}=99\\Lambda\_\{\\mathrm\{sm\}\}^\{2\}\+37c\_\{2\}gives \([40](https://arxiv.org/html/2610.28898#S5.Ex21)\)\. ∎

### E\.2Proof of Theorem 2

#### E\.2\.1Total\-Variation Partition

For the variation branch we use Lemma[E\.9](https://arxiv.org/html/2610.28898#Thmlemma9), proved above by combining Lemma[E\.8](https://arxiv.org/html/2610.28898#Thmlemma8)with the shared fast master bound of Lemma[C\.5](https://arxiv.org/html/2610.28898#Thmlemma5)and the dyadic cover of Lemma[C\.7](https://arxiv.org/html/2610.28898#Thmlemma7)\. The target\-law lifetime estimate itself uses Lemmas[E\.7](https://arxiv.org/html/2610.28898#Thmlemma7a)and[E\.6](https://arxiv.org/html/2610.28898#Thmlemma6a)\. For the count branch we use the first\-order lifetime bound of Lemma[C\.6](https://arxiv.org/html/2610.28898#Thmlemma6)with the same shared aggregation and covering lemmas, as established in Appendix[D](https://arxiv.org/html/2610.28898#A4)\.

Count part\.By Remark[6](https://arxiv.org/html/2610.28898#Thmremark6), the count bound of Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)holds under the flow\-time law of Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\. SinceNTω≤NTμ≤NμN\_\{T\}^\{\\omega\}\\leq N\_\{T\}^\{\\mu\}\\leq N\_\{\\mu\}\(Section[3\.2](https://arxiv.org/html/2610.28898#S3.SS2)\), it gives the first entry of Equation \([22](https://arxiv.org/html/2610.28898#S4.E22)\)\.

Variation part\.Letτ0=KT\+/c¯\\tau\_\{0\}=K\_\{T\}^\{\+\}/\\bar\{c\}\. Construct piecesJ1,J2,…,JM′J\_\{1\},J\_\{2\},\\ldots,J\_\{M^\{\\prime\}\}from left to right: each starts after the previous one ends and is the longest intervalJJwith\|J\|​\(VJμ\)2≤τ0\|J\|\(V\_\{J\}^\{\\mu\}\)^\{2\}\\leq\\tau\_\{0\}\. A singleton has zero internal variation, so every piece is nonempty and the construction terminates atTT\. Letni=\|Ji\|n\_\{i\}=\|J\_\{i\}\|\. Fori<M′i<M^\{\\prime\}, extendingJiJ\_\{i\}by the next round violates the defining inequality\. The extension has lengthni\+1≤2​nin\_\{i\}\+1\\leq 2n\_\{i\}and internal variationVi\+=VJiμ\+W2​\(μei\+1,μei\)V\_\{i\}^\{\+\}=V\_\{J\_\{i\}\}^\{\\mu\}\+W\_\{2\}\(\\mu\_\{e\_\{i\}\+1\},\\mu\_\{e\_\{i\}\}\), whereeie\_\{i\}is the last round ofJiJ\_\{i\}\. HenceVi\+\>τ0/\(2​ni\)V\_\{i\}^\{\+\}\>\\sqrt\{\\tau\_\{0\}/\(2n\_\{i\}\)\}\. The edges counted by theVi\+V\_\{i\}^\{\+\}are disjoint, so∑i<M′Vi\+≤Vμ\\sum\_\{i<M^\{\\prime\}\}V\_\{i\}^\{\+\}\\leq V\_\{\\mu\}\. By convexity ofx↦x−1/2x\\mapsto x^\{\-1/2\}and∑i<M′ni≤T\\sum\_\{i<M^\{\\prime\}\}n\_\{i\}\\leq T,

Vμ\>τ02∑i<M′ni−1/2≥τ02\(M′−1\)3/2T−1/2,V\_\{\\mu\}\>\\sqrt\{\\frac\{\\tau\_\{0\}\}\{2\}\}\\sum\_\{i<M^\{\\prime\}\}n\_\{i\}^\{\-1/2\}\\geq\\sqrt\{\\frac\{\\tau\_\{0\}\}\{2\}\}\\,\(M^\{\\prime\}\-1\)^\{3/2\}T^\{\-1/2\},soM′≤1\+\(2​c¯​T​Vμ2/KT\+\)1/3M^\{\\prime\}\\leq 1\+\(2\\bar\{c\}TV\_\{\\mu\}^\{2\}/K\_\{T\}^\{\+\}\)^\{1/3\}\. IfVμ=0V\_\{\\mu\}=0, thenM′=1M^\{\\prime\}=1\. On each piece, the interval bound of Lemma[E\.9](https://arxiv.org/html/2610.28898#Thmlemma9), Equation \([40](https://arxiv.org/html/2610.28898#S5.Ex21)\), and\(ni−1\)​c¯​\(VJiμ\)2≤c¯​τ0=KT\+\(n\_\{i\}\-1\)\\bar\{c\}\(V\_\{J\_\{i\}\}^\{\\mu\}\)^\{2\}\\leq\\bar\{c\}\\tau\_\{0\}=K\_\{T\}^\{\+\}bound the cost by2​KT\+\+HT\+​𝒜Ji\+ni​ef\+\+37​ϵsm​ni​VJiμ2K\_\{T\}^\{\+\}\+H\_\{T\}^\{\+\}\\mathcal\{A\}\_\{J\_\{i\}\}\+n\_\{i\}e\_\{f\}^\{\+\}\+37\\epsilon\_\{\\mathrm\{sm\}\}n\_\{i\}V\_\{J\_\{i\}\}^\{\\mu\}\. Summing, and using∑ini​VJiμ≤T​Vμ\\sum\_\{i\}n\_\{i\}V^\{\\mu\}\_\{J\_\{i\}\}\\leq TV\_\{\\mu\},

𝔼⁡\[ℛTvel∣E\]≤2​KT\+​M′\+HT\+​𝒜T\+T​ef\+\+37​ϵsm​T​Vμ,\\mathbb\{E\}\[\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\mid E\]\\leq 2K\_\{T\}^\{\+\}M^\{\\prime\}\+H\_\{T\}^\{\+\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}^\{\+\}\+37\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\},which is the second entry of Equation \([22](https://arxiv.org/html/2610.28898#S4.E22)\) after inserting the bound onM′M^\{\\prime\}\.

Integration\.The caps and constants are deterministic, so both conditional bounds integrate overEE, with𝒜T\\mathcal\{A\}\_\{T\}replaced by𝔼​𝒜T\\mathbb\{E\}\\mathcal\{A\}\_\{T\}, and the minimum of the two unconditional bounds then bounds𝔼​ℛTvel\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\. The supplementary uniform estimate also follows fromFt​\(v\)≤Lt​\(v\)≤B\+δF\_\{t\}\(v\)\\leq L\_\{t\}\(v\)\\leq B\+\\delta, giving𝔼​ℛTvel≤T⁡\(B\+δ\)\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq T\(B\+\\delta\)\. It is separate from the count and variation guarantee stated in the theorem\.

Thus, under the predictive protocol, Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1),[3](https://arxiv.org/html/2610.28898#Thmassumption3),[4](https://arxiv.org/html/2610.28898#Thmassumption4),[5](https://arxiv.org/html/2610.28898#Thmassumption5), and the caps in Equation \([10](https://arxiv.org/html/2610.28898#S3.E10)\), Algorithm[4\.1](https://arxiv.org/html/2610.28898#S4.SS1)satisfies

𝔼​ℛTvel≤\\displaystyle\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\leq\{\}𝖳T,\\displaystyle\\mathsf\{T\}\_\{T\},𝖳T=\\displaystyle\\mathsf\{T\}\_\{T\}=\{\}min\{KT\(Nμ\+1\)\+HT𝔼𝒜T\+Tef,\\displaystyle\\min\\Big\\\{K\_\{T\}\(N\_\{\\mu\}\+1\)\+H\_\{T\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\},2​KT\+\+2​\(2​c¯\)1/3​\(KT\+\)2/3​T1/3​Vμ2/3\\displaystyle\\quad 2K\_\{T\}^\{\+\}\+2\(2\\bar\{c\}\)^\{1/3\}\(K\_\{T\}^\{\+\}\)^\{2/3\}T^\{1/3\}V\_\{\\mu\}^\{2/3\}\+37ϵsmTVμ\+HT\+𝔼𝒜T\+Tef\+\}\.\\displaystyle\\quad\+37\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\}\+H\_\{T\}^\{\+\}\\mathbb\{E\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}^\{\+\}\\Big\\\}\.\(109\)Hence Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)is proved\.□\\square

## Appendix FMinimax Lower Bounds

This appendix proves the lower bounds summarized after Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)\. A lower bound holds*at fixed scales*if every hard instance uses one class and satisfies the assumptions with the same constantsd,m,κ,cZ,ι,λd,m,\\kappa,c\_\{Z\},\\iota,\\lambdaand the same architectural bounds, all independent ofTTand the budget, so that only the target sequence varies\. All three constructions meet this requirement\. The counting and target\-law bounds use the Gaussian class of Appendix[F\.2](https://arxiv.org/html/2610.28898#A6.SS2)\. The relative\-risk bound uses the class𝒱⋆\\mathcal\{V\}\_\{\\star\}of Appendix[F\.4](https://arxiv.org/html/2610.28898#A6.SS4), whose trainable slope lets its cutoff follow a far target component that moves out likelog⁡\(1/η\)\\sqrt\{\\log\(1/\\eta\)\}, as in the fixed\-target construction of[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)\. The bounds are for proper learners, which deploy fields in the fixed class\. Every construction draws signs at random and bounds the regret averaged over the signs, so for every learner some sign sequence attains the bound\. This is the order of quantifiers in Propositions[F\.1](https://arxiv.org/html/2610.28898#Thmproposition1b)–[F\.3](https://arxiv.org/html/2610.28898#Thmproposition3a)\. Table[2](https://arxiv.org/html/2610.28898#A6.T2)collects the resulting rates\. All constructions use the sine–cosine schedulesα⁡\(r\)=cos⁡\(π​r/2\)\\alpha\(r\)=\\cos\(\\pi r/2\),β⁡\(r\)=sin⁡\(π​r/2\)\\beta\(r\)=\\sin\(\\pi r/2\), a standard Gaussian base, independent pairing, and fixedmm\. WriteϕN\\phi\_\{N\}andΦ\\Phifor the standard normal density and distribution function\. A*learner*is any rule that deploysvt∈𝒱v\_\{t\}\\in\\mathcal\{V\}as a measurable function of earlier batches and independent seeds\.

Table 2:Leading rates at fixed scales, for one fixed class per construction, logarithms suppressed\. Upper bounds additionally payO⁡\(log⁡T\)​𝔼​𝒜TO\(\\log T\)\\mathbb\{E\}\\mathcal\{A\}\_\{T\}andTTtimes the fitting and proxy residuals\.### F\.1Information Structure and a Two\-Point Reduction

###### Lemma F\.1\(Sine–cosine triples reveal both endpoints\)\.

WithJ=α​β′−α′​β≡π/2J=\\alpha\\beta^\{\\prime\}\-\\alpha^\{\\prime\}\\beta\\equiv\\pi/2,X0=\(β′​Ir−β​Zr\)/JX\_\{0\}=\(\\beta^\{\\prime\}I\_\{r\}\-\\beta Z\_\{r\}\)/JandX1=\(α​Zr−α′​Ir\)/JX\_\{1\}=\(\\alpha Z\_\{r\}\-\\alpha^\{\\prime\}I\_\{r\}\)/J\. Consequently, for two target sequences that agree before roundt0t\_\{0\}and whose round laws differ only on rounds in\[t0,t\)\[t\_\{0\},t\), the lawsP,P′P,P^\{\\prime\}of the learner’s information before roundttsatisfyTV\(P,P′\)≤TV\(⊗iμ\(i\),⊗iμ\(i\)′\)\\mathrm\{TV\}\(P,P^\{\\prime\}\)\\leq\\mathrm\{TV\}\(\\otimes\_\{i\}\\mu\_\{\(i\)\},\\otimes\_\{i\}\\mu^\{\\prime\}\_\{\(i\)\}\), where the products run over the at mostm⁡\(t−t0\)m\(t\-t\_\{0\}\)target draws of those rounds\.

###### Proof\.

The identities follow fromβ′​I−β​Z=\(α​β′−α′​β\)​X0\\beta^\{\\prime\}I\-\\beta Z=\(\\alpha\\beta^\{\\prime\}\-\\alpha^\{\\prime\}\\beta\)X\_\{0\}andα​Z−α′​I=\(α​β′−α′​β\)​X1\\alpha Z\-\\alpha^\{\\prime\}I=\(\\alpha\\beta^\{\\prime\}\-\\alpha^\{\\prime\}\\beta\)X\_\{1\}\. Hence the information is a measurable function of the target draws together with base draws, flow times, earlier\-round draws, and seeds, all of which have the same law under both sequences and are independent of the differing draws\. Total variation does not increase under such a map\. ∎

###### Lemma F\.2\(Two\-point reduction\)\.

LetYYbe the learner’s information and seed before a round, with lawP\+P\_\{\+\}orP−P\_\{\-\}, and let the deployed field be a measurable function ofYY\. For nonnegative functionsf±f\_\{\\pm\}on𝒱\\mathcal\{V\},

12​𝔼\+​f\+​\(v\)\+12​𝔼−​f−​\(v\)≥12​\(1−TV⁡\(P\+,P−\)\)​infu∈𝒱\{f\+​\(u\)\+f−​\(u\)\}\.\\tfrac\{1\}\{2\}\\mathbb\{E\}\_\{\+\}f\_\{\+\}\(v\)\+\\tfrac\{1\}\{2\}\\mathbb\{E\}\_\{\-\}f\_\{\-\}\(v\)\\geq\\tfrac\{1\}\{2\}\\big\(1\-\\mathrm\{TV\}\(P\_\{\+\},P\_\{\-\}\)\\big\)\\inf\_\{u\\in\\mathcal\{V\}\}\\\{f\_\{\+\}\(u\)\+f\_\{\-\}\(u\)\\\}\.

###### Proof\.

With densitiesp±p\_\{\\pm\}for a dominating measure,𝔼\+​f\+\+𝔼−​f−=∫\(f\+​p\+\+f−​p−\)≥∫\(f\+\+f−\)​min⁡\(p\+,p−\)\\mathbb\{E\}\_\{\+\}f\_\{\+\}\+\\mathbb\{E\}\_\{\-\}f\_\{\-\}=\\int\(f\_\{\+\}p\_\{\+\}\+f\_\{\-\}p\_\{\-\}\)\\geq\\int\(f\_\{\+\}\+f\_\{\-\}\)\\min\(p\_\{\+\},p\_\{\-\}\), and∫min⁡\(p\+,p−\)=1−TV⁡\(P\+,P−\)\\int\\min\(p\_\{\+\},p\_\{\-\}\)=1\-\\mathrm\{TV\}\(P\_\{\+\},P\_\{\-\}\)\. ∎

All constructions below draw block signs independently and uniformly\. A lower bound on the expected regret under this prior implies the same bound for some deterministic sign sequence, because the prior has finite support\. The environment is therefore exogenous and the caps hold for every sign sequence\.

### F\.2The Gaussian Family

FixR0\>0R\_\{0\}\>0and an exogenous sequence of meansξt∈ℝd\\xi\_\{t\}\\in\\mathbb\{R\}^\{d\}with∥ξt∥≤R0\\lVert\\xi\_\{t\}\\rVert\\leq R\_\{0\}\. Letν=N⁡\(0,Id\)\\nu=N\(0,I\_\{d\}\),μt=N⁡\(ξt,Id\)\\mu\_\{t\}=N\(\\xi\_\{t\},I\_\{d\}\),Ir=α⁡\(r\)​X0\+β⁡\(r\)​X1I\_\{r\}=\\alpha\(r\)X\_\{0\}\+\\beta\(r\)X\_\{1\},Zr=α′​\(r\)​X0\+β′​\(r\)​X1Z\_\{r\}=\\alpha^\{\\prime\}\(r\)X\_\{0\}\+\\beta^\{\\prime\}\(r\)X\_\{1\}, andw⁡\(r\)=β′​\(r\)=π2​cos⁡\(π​r/2\)w\(r\)=\\beta^\{\\prime\}\(r\)=\\frac\{\\pi\}\{2\}\\cos\(\\pi r/2\)\. Forϑ∈ℝd\\vartheta\\in\\mathbb\{R\}^\{d\}with∥ϑ∥≤R0\\lVert\\vartheta\\rVert\\leq R\_\{0\}, letvϑ​\(x,r\)=w⁡\(r\)​ϑv\_\{\\vartheta\}\(x,r\)=w\(r\)\\vartheta\. For a flow\-time lawλ\\lambdaputc∗λ=𝔼λ​\[w​\(r\)2\]c\_\{\*\}^\{\\lambda\}=\\mathbb\{E\}\_\{\\lambda\}\[w\(r\)^\{2\}\], which equalsc∗=π2/8c\_\{\*\}=\\pi^\{2\}/8for the uniform law\.

###### Lemma F\.3\(Gaussian family\)\.

For every flow\-time lawλ\\lambda:

1. \(i\)Assumption[1](https://arxiv.org/html/2610.28898#Thmassumption1)holds withκ2=8​\(d\+R02\+1\)\\kappa^\{2\}=8\(d\+R\_\{0\}^\{2\}\+1\), Assumption[2](https://arxiv.org/html/2610.28898#Thmassumption2)withcZ=π/2c\_\{Z\}=\\pi/2, and Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(a\)–\(b\) withι=1\\iota=1,
2. \(ii\)the class\{vϑ\}\\\{v\_\{\\vartheta\}\\\}satisfies Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3)with one hidden unit and the saturating sine ϕ⁡\(z\)=−1​\(z<−π2\),ϕ⁡\(z\)=sin⁡z⁡\(\|z\|≤π2\),ϕ⁡\(z\)=1​\(z\>π2\);\\phi\(z\)=\-1\\ \(z<\-\\tfrac\{\\pi\}\{2\}\),\\qquad\\phi\(z\)=\\sin z\\ \(\|z\|\\leq\\tfrac\{\\pi\}\{2\}\),\\qquad\\phi\(z\)=1\\ \(z\>\\tfrac\{\\pi\}\{2\}\);\(110\)
3. \(iii\)bt​\(x,r\)=w⁡\(r\)​ξtb\_\{t\}\(x,r\)=w\(r\)\\xi\_\{t\}forλ\\lambda\-almost everyrr,Qt​\(vϑ\)=c∗λ​∥ϑ−ξt∥2Q\_\{t\}\(v\_\{\\vartheta\}\)=c\_\{\*\}^\{\\lambda\}\\lVert\\vartheta\-\\xi\_\{t\}\\rVert^\{2\}, andat=0a\_\{t\}=0,
4. \(iv\)ωt=2​c∗λ​R0​∥ξt\+1−ξt∥\\omega\_\{t\}=2c\_\{\*\}^\{\\lambda\}R\_\{0\}\\lVert\\xi\_\{t\+1\}\-\\xi\_\{t\}\\rVertandW2​\(μt\+1,μt\)=∥ξt\+1−ξt∥W\_\{2\}\(\\mu\_\{t\+1\},\\mu\_\{t\}\)=\\lVert\\xi\_\{t\+1\}\-\\xi\_\{t\}\\rVert\.

###### Proof\.

*\(i\)*ForX∼N⁡\(ξ,Id\)X\\sim N\(\\xi,I\_\{d\}\)andκ2\>2\\kappa^\{2\}\>2, completing the square gives𝔼e∥X∥2/κ2=\(1−2/κ2\)−d/2exp\(∥ξ∥2/\(κ2−2\)\)\\mathbb\{E\}e^\{\\lVert X\\rVert^\{2\}/\\kappa^\{2\}\}=\(1\-2/\\kappa^\{2\}\)^\{\-d/2\}\\exp\(\\lVert\\xi\\rVert^\{2\}/\(\\kappa^\{2\}\-2\)\)\. By−log⁡\(1−u\)≤u/\(1−u\)\-\\log\(1\-u\)\\leq u/\(1\-u\), its logarithm is at most\(d\+R02\)/\(κ2−2\)<1/8<log⁡2\(d\+R\_\{0\}^\{2\}\)/\(\\kappa^\{2\}\-2\)<1/8<\\log 2forκ2=8​\(d\+R02\+1\)\\kappa^\{2\}=8\(d\+R\_\{0\}^\{2\}\+1\), for base and targets alike\. The schedules are smooth with correct endpoints and\|α′\|,\|β′\|≤π/2\|\\alpha^\{\\prime\}\|,\|\\beta^\{\\prime\}\|\\leq\\pi/2\. The Gaussian base has Fisher informationIdI\_\{d\}\.

*\(ii\)*The functionϕ\\phiis bounded by one, nondecreasing, continuously differentiable \(its interior derivative vanishes at the junctions\), and11\-Lipschitz\. Fix the first affine map toz=π2​\(1−r\)z=\\frac\{\\pi\}\{2\}\(1\-r\)with zero spatial weights\. Forr∈\[0,1\]r\\in\[0,1\]the hidden value isϕ⁡\(z\)=sin⁡\(π2​\(1−r\)\)=cos⁡\(π​r/2\)\\phi\(z\)=\\sin\(\\frac\{\\pi\}\{2\}\(1\-r\)\)=\\cos\(\\pi r/2\)\. With output matrixπ2​ϑ\\frac\{\\pi\}\{2\}\\varthetaand zero output bias, the output isw⁡\(r\)​ϑw\(r\)\\vartheta\. The trainable parameter is the output column, a compact ball\.

*\(iii\)*WriteX0=G0X\_\{0\}=G\_\{0\}andX1=ξt\+G1X\_\{1\}=\\xi\_\{t\}\+G\_\{1\}with independent standard Gaussian vectors\. At a fixed flow time,IrI\_\{r\}has meanβ​ξt\\beta\\xi\_\{t\}and covariance\(α2\+β2\)​Id=Id\(\\alpha^\{2\}\+\\beta^\{2\}\)I\_\{d\}=I\_\{d\},ZrZ\_\{r\}has meanβ′​ξt\\beta^\{\\prime\}\\xi\_\{t\}, andCov⁡\(Zr,Ir\)=\(α​α′\+β​β′\)​Id=0\\operatorname\{Cov\}\(Z\_\{r\},I\_\{r\}\)=\(\\alpha\\alpha^\{\\prime\}\+\\beta\\beta^\{\\prime\}\)I\_\{d\}=0becauseα2\+β2≡1\\alpha^\{2\}\+\\beta^\{2\}\\equiv 1\. Joint Gaussianity makes the centered label independent of the input, sobt​\(x,r\)=w⁡\(r\)​ξtb\_\{t\}\(x,r\)=w\(r\)\\xi\_\{t\}at every fixed time, hence forλ\\lambda\-almost everyrr\. ThenQt​\(vϑ\)=𝔼λ​\[w​\(r\)2\]​∥ϑ−ξt∥2Q\_\{t\}\(v\_\{\\vartheta\}\)=\\mathbb\{E\}\_\{\\lambda\}\[w\(r\)^\{2\}\]\\lVert\\vartheta\-\\xi\_\{t\}\\rVert^\{2\}, andϑ=ξt\\vartheta=\\xi\_\{t\}is in the class\.

*\(iv\)*WithΔt=ξt\+1−ξt\\Delta\_\{t\}=\\xi\_\{t\+1\}\-\\xi\_\{t\},\[Qt\+1​\(vϑ\)−Qt\+1​\(vϑ′\)\]−\[Qt​\(vϑ\)−Qt​\(vϑ′\)\]=−2​c∗λ​⟨ϑ−ϑ′,Δt⟩\[Q\_\{t\+1\}\(v\_\{\\vartheta\}\)\-Q\_\{t\+1\}\(v\_\{\\vartheta^\{\\prime\}\}\)\]\-\[Q\_\{t\}\(v\_\{\\vartheta\}\)\-Q\_\{t\}\(v\_\{\\vartheta^\{\\prime\}\}\)\]=\-2c\_\{\*\}^\{\\lambda\}\\langle\\vartheta\-\\vartheta^\{\\prime\},\\Delta\_\{t\}\\rangle\. The difference of two radius\-R0R\_\{0\}balls is the radius\-2​R02R\_\{0\}ball, so half the supremum of the absolute value is2​c∗λ​R0​∥Δt∥2c\_\{\*\}^\{\\lambda\}R\_\{0\}\\lVert\\Delta\_\{t\}\\rVert\. Finally, translation byΔt\\Delta\_\{t\}is an optimal coupling of two Gaussians with equal covariance, and Jensen’s inequality bounds every coupling cost below by∥Δt∥2\\lVert\\Delta\_\{t\}\\rVert^\{2\}\. ∎

The relative\-risk variation of this family is thus2​c∗λ​R02c\_\{\*\}^\{\\lambda\}R\_\{0\}times its target\-law variation, and a stale optimum loses exactlyFt​\(vξs\)=c∗λ​∥ξt−ξs∥2F\_\{t\}\(v\_\{\\xi\_\{s\}\}\)=c\_\{\*\}^\{\\lambda\}\\lVert\\xi\_\{t\}\-\\xi\_\{s\}\\rVert^\{2\}\. For a historyHHofqqrounds, letξ¯H=q−1​∑s∈Hξs\\bar\{\\xi\}\_\{H\}=q^\{\-1\}\\sum\_\{s\\in H\}\\xi\_\{s\}\. This mean is admissible because the radius\-R0R\_\{0\}ball is convex\. Lemma[F\.3](https://arxiv.org/html/2610.28898#Thmlemma3b)\(iii\) makesvξ¯Hv\_\{\\bar\{\\xi\}\_\{H\}\}the best common field, so

a¯H=c∗λq​∑s∈H∥ξs−ξ¯H∥2,\\bar\{a\}\_\{H\}=\\frac\{c\_\{\*\}^\{\\lambda\}\}\{q\}\\sum\_\{s\\in H\}\\lVert\\xi\_\{s\}\-\\bar\{\\xi\}\_\{H\}\\rVert^\{2\},althoughas=0a\_\{s\}=0for every round\. Thus historical pooling can create a positive approximation residual even in a realizable family\.

### F\.3Counting Lower Bounds

###### Proposition F\.1\(Counting lower bound\)\.

For every flow\-time lawλ\\lambda, everyN<TN<T, and every learner, there is a mean sequence for the Gaussian family withd=1d=1andNTω=NTμ≤NN\_\{T\}^\{\\omega\}=N\_\{T\}^\{\\mu\}\\leq Non which𝔼​ℛTvel≥c∗λ​R02​\(N\+1\)\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq c\_\{\*\}^\{\\lambda\}R\_\{0\}^\{2\}\(N\+1\)\.

###### Proof\.

Use targetsN⁡\(R0​ζj,1\)N\(R\_\{0\}\\zeta\_\{j\},1\)onK=N\+1K=N\+1consecutive blocks of lengths at least one, where the signsζj\\zeta\_\{j\}are independent and uniform\. By Lemma[F\.3](https://arxiv.org/html/2610.28898#Thmlemma3b)\(iii\),Ft​\(vϑ\)=c∗λ​\(ϑ−ξt\)2F\_\{t\}\(v\_\{\\vartheta\}\)=c\_\{\*\}^\{\\lambda\}\(\\vartheta\-\\xi\_\{t\}\)^\{2\}\. At the first round of blockjjthe learner’s information is independent ofζj\\zeta\_\{j\}, so Lemma[F\.2](https://arxiv.org/html/2610.28898#Thmlemma2b)with zero total variation gives

𝔼​Ft​\(vt\)≥12​infϑc∗λ​\[\(ϑ−R0\)2\+\(ϑ\+R0\)2\]=c∗λ​R02\.\\mathbb\{E\}F\_\{t\}\(v\_\{t\}\)\\geq\\tfrac\{1\}\{2\}\\inf\_\{\\vartheta\}c\_\{\*\}^\{\\lambda\}\[\(\\vartheta\-R\_\{0\}\)^\{2\}\+\(\\vartheta\+R\_\{0\}\)^\{2\}\]=c\_\{\*\}^\{\\lambda\}R\_\{0\}^\{2\}\.Summing over blocks gives𝔼​ℛTvel≥c∗λ​R02​K\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq c\_\{\*\}^\{\\lambda\}R\_\{0\}^\{2\}K\. Target changes occur only at block boundaries, soNTμ≤K−1N\_\{T\}^\{\\mu\}\\leq K\-1, andNTω=NTμN\_\{T\}^\{\\omega\}=N\_\{T\}^\{\\mu\}by Lemma[F\.3](https://arxiv.org/html/2610.28898#Thmlemma3b)\(iv\)\. ∎

### F\.4The Rare\-Mixture Family for Relative\-Risk Variation

The two\-point construction of[Sahoo et al\. \(2026\)](https://arxiv.org/html/2610.28898#bib.bib25)hides the difference between two targets in a far component of small mass: total variation of orderη\\eta, velocity separation of orderR2​ηR^\{2\}\\eta\. We use it block by block, with the far mass as the drift parameter\. Regret is measured against a class, so the class must see the far component while staying fixed\. A trainable slope achieves this: shrinking the slope moves the cutoff of the field into the tails, while every weight and bias bound stays fixed\.

##### One fixed class\.

With the saturating sineϕ\\phiof Equation \([110](https://arxiv.org/html/2610.28898#A6.E110)\) andb∘=11​π/2b\_\{\\circ\}=11\\pi/2, let

ψγ\(x\)=1\+12ϕ\(γx−b∘\)\+12ϕ\(−γx−b∘\),𝒱⋆=\{vθ,γ\(x,r\)=θψγ\(x\):\|θ\|≤1,0≤γ≤4\}\.\\psi\_\{\\gamma\}\(x\)=1\+\\tfrac\{1\}\{2\}\\phi\(\\gamma x\-b\_\{\\circ\}\)\+\\tfrac\{1\}\{2\}\\phi\(\-\\gamma x\-b\_\{\\circ\}\),\\qquad\\mathcal\{V\}\_\{\\star\}=\\\{v\_\{\\theta,\\gamma\}\(x,r\)=\\theta\\psi\_\{\\gamma\}\(x\):\|\\theta\|\\leq 1,\\ 0\\leq\\gamma\\leq 4\\\}\.\(111\)One of the two arguments is always at most−b∘<−π/2\-b\_\{\\circ\}<\-\\pi/2, soψγ​\(x\)=12​\[1\+ϕ⁡\(γ​\|x\|−b∘\)\]\\psi\_\{\\gamma\}\(x\)=\\frac\{1\}\{2\}\[1\+\\phi\(\\gamma\|x\|\-b\_\{\\circ\}\)\]\. Thusψγ\\psi\_\{\\gamma\}is even,0≤ψγ≤10\\leq\\psi\_\{\\gamma\}\\leq 1,ψ0=0\\psi\_\{0\}=0, and forγ\>0\\gamma\>0

ψγ​\(x\)=0​if​\|x\|≤5​π/γ,ψγ​\(x\)=1​if​\|x\|≥6​π/γ\.\\psi\_\{\\gamma\}\(x\)=0\\ \\text\{ if \}\|x\|\\leq 5\\pi/\\gamma,\\qquad\\psi\_\{\\gamma\}\(x\)=1\\ \\text\{ if \}\|x\|\\geq 6\\pi/\\gamma\.\(112\)

##### Instance\.

Letλ=Unif⁡\[0,1\]\\lambda=\\mathrm\{Unif\}\[0,1\], fixη∈\(0,e−12\]\\eta\\in\(0,e^\{\-12\}\], putLη=log⁡\(1/η\)L\_\{\\eta\}=\\log\(1/\\eta\)andR=8​LηR=\\sqrt\{8L\_\{\\eta\}\}, and let

μs=\(1−η\)​N​\(0,1\)\+η​N​\(s​R,1\),s∈\{−1,\+1\}\.\\mu\_\{s\}=\(1\-\\eta\)N\(0,1\)\+\\eta N\(sR,1\),\\qquad s\\in\\\{\-1,\+1\\\}\.\(113\)Only the targets vary withη\\eta\. The class remains𝒱⋆\\mathcal\{V\}\_\{\\star\}\.

###### Lemma F\.4\(Fixed class and constants\)\.

For everyη\\eta, Assumption[1](https://arxiv.org/html/2610.28898#Thmassumption1)holds withκ2=18\\kappa^\{2\}=18and Assumption[2](https://arxiv.org/html/2610.28898#Thmassumption2)withcZ=π/2c\_\{Z\}=\\pi/2\. The class𝒱⋆\\mathcal\{V\}\_\{\\star\}satisfies Assumption[3](https://arxiv.org/html/2610.28898#Thmassumption3)with two affine layers of widths\(2,2,1\)\(2,2,1\),P=5P=5trainable entries in a compact set of radiusRΘ=67/2R\_\{\\Theta\}=\\sqrt\{67/2\}, layer boundsV1=4​2V\_\{1\}=4\\sqrt\{2\}, first\-layer bias norm11​π/211\\pi/\\sqrt\{2\},V2=1/2V\_\{2\}=1/\\sqrt\{2\}, output bias bound11,Bv=2B\_\{v\}=2, andLx=4L\_\{x\}=4\. On\|x\|≤S\|x\|\\leq Sits parameter sensitivity is at mostCS=\(S2\+1\+1\)/2\+2\+1C\_\{S\}=\(\\sqrt\{S^\{2\}\+1\}\+1\)/\\sqrt\{2\}\+\\sqrt\{2\}\+1\. None of these constants depends onη\\eta,TT,mm, or the budget\.

###### Proof\.

ForX∼N⁡\(ξ,1\)X\\sim N\(\\xi,1\),𝔼eX2/18=\(8/9\)−1/2eξ2/16\\mathbb\{E\}e^\{X^\{2\}/18\}=\(8/9\)^\{\-1/2\}e^\{\\xi^\{2\}/16\}\. For the base this is1\.061<21\.061<2\. For Equation \([113](https://arxiv.org/html/2610.28898#A6.E113)\),eR2/16=η−1/2e^\{R^\{2\}/16\}=\\eta^\{\-1/2\}, so𝔼eX2/18=\(8/9\)−1/2\(1−η\+η\)≤1\.061⋅1\.25<2\\mathbb\{E\}e^\{X^\{2\}/18\}=\(8/9\)^\{\-1/2\}\(1\-\\eta\+\\sqrt\{\\eta\}\)\\leq 1\.061\\cdot 1\.25<2\. The sine–cosine schedules have\|α′\|,\|β′\|≤π/2\|\\alpha^\{\\prime\}\|,\|\\beta^\{\\prime\}\|\\leq\\pi/2\. For input\(x,r\)\(x,r\)take

W1=\(γ0−γ0\),b1=\(−b∘−b∘\),W2=\(θ/2,θ/2\),b2=θ,W\_\{1\}=\\begin\{pmatrix\}\\gamma&0\\\\ \-\\gamma&0\\end\{pmatrix\},\\qquad b\_\{1\}=\\begin\{pmatrix\}\-b\_\{\\circ\}\\\\ \-b\_\{\\circ\}\\end\{pmatrix\},\\qquad W\_\{2\}=\(\\theta/2,\\theta/2\),\\qquad b\_\{2\}=\\theta,with the activationϕ\\phi, which is bounded by one,11\-Lipschitz,C1C^\{1\}, and nondecreasing\. The output isθ\+θ2​\(ϕ1\+ϕ2\)=θ​ψγ\\theta\+\\frac\{\\theta\}\{2\}\(\\phi\_\{1\}\+\\phi\_\{2\}\)=\\theta\\psi\_\{\\gamma\}\. The trainable entries formΘ⋆=\{\(γ,−γ,θ/2,θ/2,θ\):γ∈\[0,4\],θ∈\[−1,1\]\}⊂ℝ5\\Theta\_\{\\star\}=\\\{\(\\gamma,\-\\gamma,\\theta/2,\\theta/2,\\theta\):\\gamma\\in\[0,4\],\\theta\\in\[\-1,1\]\\\}\\subset\\mathbb\{R\}^\{5\}, a compact set with squared radius2⋅42\+3/2=67/22\\cdot 4^\{2\}\+3/2=67/2\. The norms giveV1=4​2V\_\{1\}=4\\sqrt\{2\},∥b1∥=11​π/2\\lVert b\_\{1\}\\rVert=11\\pi/\\sqrt\{2\},V2=1/2V\_\{2\}=1/\\sqrt\{2\}, and\|b2\|≤1\|b\_\{2\}\|\\leq 1, soBv=V2​2\+1=2B\_\{v\}=V\_\{2\}\\sqrt\{2\}\+1=2andLx=V2​V1=4L\_\{x\}=V\_\{2\}V\_\{1\}=4\. A first\-layer perturbation changes the output by at mostV2​S2\+1V\_\{2\}\\sqrt\{S^\{2\}\+1\}times its norm, and an output\-layer perturbation by at most2\+1\\sqrt\{2\}\+1times its norm, which givesCSC\_\{S\}\. It grows only with the input radius, as the gated cover of Corollary[B\.1](https://arxiv.org/html/2610.28898#Thmcorollary1)requires\. ∎

Forξ∼N⁡\(0,1\)\\xi\\sim N\(0,1\)ands∈\{−1,\+1\}s\\in\\\{\-1,\+1\\\}let

Nη​\(γ\)=𝔼s​\[ψγ​\(Ir\)2\],Gη​\(γ\)=∫01β′​\(r\)​𝔼ξ​ψγ​\(β⁡\(r\)​R\+ξ\)​𝑑r\.N\_\{\\eta\}\(\\gamma\)=\\mathbb\{E\}\_\{s\}\[\\psi\_\{\\gamma\}\(I\_\{r\}\)^\{2\}\],\\qquad G\_\{\\eta\}\(\\gamma\)=\\int\_\{0\}^\{1\}\\beta^\{\\prime\}\(r\)\\,\\mathbb\{E\}\_\{\\xi\}\\psi\_\{\\gamma\}\(\\beta\(r\)R\+\\xi\)\\,dr\.Reflectiony↦−yy\\mapsto\-ymapsμ\+\\mu\_\{\+\}toμ−\\mu\_\{\-\}and fixesψγ\\psi\_\{\\gamma\}, soNη​\(γ\)N\_\{\\eta\}\(\\gamma\)does not depend onss\. Also0≤Gη​\(γ\)≤∫01β′=10\\leq G\_\{\\eta\}\(\\gamma\)\\leq\\int\_\{0\}^\{1\}\\beta^\{\\prime\}=1\.

###### Lemma F\.5\(Risk geometry over the whole class\)\.

Letη≤e−12\\eta\\leq e^\{\-12\}, and letpsp\_\{s\}be the posterior probability of the far component\.

1. \(i\)bs​\(y,r\)=β′​\(r\)​s​R​ps​\(y,r\)b\_\{s\}\(y,r\)=\\beta^\{\\prime\}\(r\)sR\\,p\_\{s\}\(y,r\), and there is a constantCηC\_\{\\eta\}, independent ofs,θ,γs,\\theta,\\gamma, withLs​\(vθ,γ\)=Cη\+θ2​Nη​\(γ\)−2​s​θ​η​R​Gη​\(γ\)L\_\{s\}\(v\_\{\\theta,\\gamma\}\)=C\_\{\\eta\}\+\\theta^\{2\}N\_\{\\eta\}\(\\gamma\)\-2s\\theta\\eta RG\_\{\\eta\}\(\\gamma\)\.
2. \(ii\)infv∈𝒱⋆\{F\+​\(v\)\+F−​\(v\)\}≥12​η​R\\inf\_\{v\\in\\mathcal\{V\}\_\{\\star\}\}\\\{F\_\{\+\}\(v\)\+F\_\{\-\}\(v\)\\\}\\geq\\frac\{1\}\{2\}\\eta R\.
3. \(iii\)If consecutive rounds have opposite signs, thenωt=4​η​R​max0≤γ≤4​Gη​\(γ\)≤4​η​R\\omega\_\{t\}=4\\eta R\\max\_\{0\\leq\\gamma\\leq 4\}G\_\{\\eta\}\(\\gamma\)\\leq 4\\eta R\. Equal signs giveωt=0\\omega\_\{t\}=0\.
4. \(iv\)as≤π24​R2​ηa\_\{s\}\\leq\\frac\{\\pi^\{2\}\}\{4\}R^\{2\}\\eta\.

###### Proof\.

*\(i\)*For a target componentN⁡\(c,1\)N\(c,1\)and a fixedrr,\(Ir,Zr\)\(I\_\{r\},Z\_\{r\}\)is jointly Gaussian withVar⁡\(Ir\)=α2\+β2=1\\mathrm\{Var\}\(I\_\{r\}\)=\\alpha^\{2\}\+\\beta^\{2\}=1andCov⁡\(Ir,Zr\)=α​α′\+β​β′=0\\mathrm\{Cov\}\(I\_\{r\},Z\_\{r\}\)=\\alpha\\alpha^\{\\prime\}\+\\beta\\beta^\{\\prime\}=0, so its conditional mean label isβ′​c\\beta^\{\\prime\}c\. Conditioning on the component givesbs=β′​s​R​psb\_\{s\}=\\beta^\{\\prime\}sRp\_\{s\}, because the bulk hasc=0c=0\. The bulk has mean label zero, andψγ\\psi\_\{\\gamma\}is even, so the tower property gives𝔼s​\[ψγ​\(Ir\)​Zr\]=s​η​R​Gη​\(γ\)\\mathbb\{E\}\_\{s\}\[\\psi\_\{\\gamma\}\(I\_\{r\}\)Z\_\{r\}\]=s\\eta RG\_\{\\eta\}\(\\gamma\)\. Expanding the square proves \(i\), withCη=𝔼s​Zr2C\_\{\\eta\}=\\mathbb\{E\}\_\{s\}Z\_\{r\}^\{2\}, which is the same for both signs\.

*\(ii\)*By compactness and dominated convergence the minima below exist\. Letdη=min\|θ\|≤1,0≤γ≤4⁡\{θ2​Nη​\(γ\)−2​θ​η​R​Gη​\(γ\)\}d\_\{\\eta\}=\\min\_\{\|\\theta\|\\leq 1,\\,0\\leq\\gamma\\leq 4\}\\\{\\theta^\{2\}N\_\{\\eta\}\(\\gamma\)\-2\\theta\\eta RG\_\{\\eta\}\(\\gamma\)\\\}\. Replacingθ\\thetaby−θ\-\\thetaexchanges the signs, so the minimum loss for either sign isCη\+dηC\_\{\\eta\}\+d\_\{\\eta\}\. Hence, for every admissible\(θ,γ\)\(\\theta,\\gamma\),

F\+​\(vθ,γ\)\+F−​\(vθ,γ\)=2​θ2​Nη​\(γ\)−2​dη≥−2​dη\.F\_\{\+\}\(v\_\{\\theta,\\gamma\}\)\+F\_\{\-\}\(v\_\{\\theta,\\gamma\}\)=2\\theta^\{2\}N\_\{\\eta\}\(\\gamma\)\-2d\_\{\\eta\}\\geq\-2d\_\{\\eta\}\.\(114\)To bounddηd\_\{\\eta\}, use the witness slopeγη=10​π/R\\gamma\_\{\\eta\}=10\\pi/R, which lies in\[0,4\]\[0,4\]becauseR≥96R\\geq\\sqrt\{96\}\. By \([112](https://arxiv.org/html/2610.28898#A6.E112)\) its field vanishes on\|x\|≤R/2=2​Lη\|x\|\\leq R/2=\\sqrt\{2L\_\{\\eta\}\}and equals one on\|x\|≥3​R/5\|x\|\\geq 3R/5\. Under the bulk,Ir∼N⁡\(0,1\)I\_\{r\}\\sim N\(0,1\)for everyrr, so Mills’ inequality gives

Nη\(γη\)≤Pr\(\|ξ\|\>2​Lη\)\+η≤η\(1\+\(πLη\)−1/2\)≤76η\.N\_\{\\eta\}\(\\gamma\_\{\\eta\}\)\\leq\\Pr\(\|\\xi\|\>\\sqrt\{2L\_\{\\eta\}\}\)\+\\eta\\leq\\eta\\big\(1\+\(\\pi L\_\{\\eta\}\)^\{\-1/2\}\\big\)\\leq\\tfrac\{7\}\{6\}\\eta\.On\{r:β⁡\(r\)≥3/5\}\\\{r:\\beta\(r\)\\geq 3/5\\\}the eventξ≥0\\xi\\geq 0putsβ​R\+ξ\\beta R\+\\xion the unit plateau\. Sinceβ′≥0\\beta^\{\\prime\}\\geq 0andψγ≥0\\psi\_\{\\gamma\}\\geq 0,Gη\(γη\)≥12∫\{β≥3/5\}β′\(r\)dr=15G\_\{\\eta\}\(\\gamma\_\{\\eta\}\)\\geq\\frac\{1\}\{2\}\\int\_\{\\\{\\beta\\geq 3/5\\\}\}\\beta^\{\\prime\}\(r\)\\,dr=\\frac\{1\}\{5\}\. Evaluating the minimum at\(1,γη\)\(1,\\gamma\_\{\\eta\}\),

−2​dη≥4​η​R​Gη​\(γη\)−2​Nη​\(γη\)≥η⁡\(4​R5−73\)≥12​η​R,\-2d\_\{\\eta\}\\geq 4\\eta RG\_\{\\eta\}\(\\gamma\_\{\\eta\}\)\-2N\_\{\\eta\}\(\\gamma\_\{\\eta\}\)\\geq\\eta\\Big\(\\frac\{4R\}\{5\}\-\\frac\{7\}\{3\}\\Big\)\\geq\\frac\{1\}\{2\}\\eta R,where the last step usesR≥96\>8R\\geq\\sqrt\{96\}\>8\. With \([114](https://arxiv.org/html/2610.28898#A6.E114)\) this proves the separation for the whole class\. The witness need not be the global optimizer\.

*\(iii\)*By \(i\),L−​\(vθ,γ\)−L\+​\(vθ,γ\)=4​θ​η​R​Gη​\(γ\)L\_\{\-\}\(v\_\{\\theta,\\gamma\}\)\-L\_\{\+\}\(v\_\{\\theta,\\gamma\}\)=4\\theta\\eta RG\_\{\\eta\}\(\\gamma\)\. Over the class this ranges over the symmetric interval with endpoints±4​η​R​maxγ​Gη​\(γ\)\\pm 4\\eta R\\max\_\{\\gamma\}G\_\{\\eta\}\(\\gamma\), andωt\\omega\_\{t\}is half its oscillation\. Equal signs give equal risks\.

*\(iv\)*Since0∈𝒱⋆0\\in\\mathcal\{V\}\_\{\\star\},ps2≤psp\_\{s\}^\{2\}\\leq p\_\{s\}, and𝔼s​ps​\(Ir,r\)=η\\mathbb\{E\}\_\{s\}p\_\{s\}\(I\_\{r\},r\)=\\eta, we getas≤Qs​\(0\)=𝔼s​bs​\(Ir,r\)2≤π24​R2​𝔼s​ps​\(Ir,r\)2≤π24​R2​ηa\_\{s\}\\leq Q\_\{s\}\(0\)=\\mathbb\{E\}\_\{s\}b\_\{s\}\(I\_\{r\},r\)^\{2\}\\leq\\frac\{\\pi^\{2\}\}\{4\}R^\{2\}\\mathbb\{E\}\_\{s\}p\_\{s\}\(I\_\{r\},r\)^\{2\}\\leq\\frac\{\\pi^\{2\}\}\{4\}R^\{2\}\\eta\. ∎

###### Proposition F\.2\(Relative\-risk variation lower bound\)\.

Fix the class𝒱⋆\\mathcal\{V\}\_\{\\star\}, the standard Gaussian base, the sine–cosine path, andλ=Unif⁡\[0,1\]\\lambda=\\mathrm\{Unif\}\[0,1\]\. LetT≥2T\\geq 2andm​T≥e12/2mT\\geq e^\{12\}/2, and letKmax=⌊min⁡\{T/2,2​m​T​e−12\}⌋K\_\{\\max\}=\\lfloor\\min\\\{T/2,2mTe^\{\-12\}\\\}\\rfloorandRK=8​log⁡\(4​m​⌊T/K⌋\)R\_\{K\}=\\sqrt\{8\\log\(4m\\lfloor T/K\\rfloor\)\}\. For everyΩ\\Omegawith

2​R1m​T≤Ω≤2​Kmax2​RKmaxm​T,\\frac\{2R\_\{1\}\}\{mT\}\\leq\\Omega\\leq\\frac\{2K\_\{\\max\}^\{2\}R\_\{K\_\{\\max\}\}\}\{mT\},\(115\)and every proper learner in𝒱⋆\\mathcal\{V\}\_\{\\star\}, there is a sign sequence for the family in Equation \([113](https://arxiv.org/html/2610.28898#A6.E113)\) withΩT≤Ω\\Omega\_\{T\}\\leq\\Omegaon which𝔼​ℛTvel≥120​T​Ω/m\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq\\frac\{1\}\{20\}\\sqrt\{T\\Omega/m\}\. The class, the envelope, the flow\-time law, and every architectural bound are independent ofTT,mm, andΩ\\Omega\.

###### Proof\.

FixK≤KmaxK\\leq K\_\{\\max\}, letH=⌊T/K⌋≥T/\(2​K\)H=\\lfloor T/K\\rfloor\\geq T/\(2K\), and letη=1/\(4​m​H\)\\eta=1/\(4mH\)\. The choiceK≤2​m​T​e−12K\\leq 2mTe^\{\-12\}gives4​m​H≥e124mH\\geq e^\{12\}, that is,η≤e−12\\eta\\leq e^\{\-12\}\. UseKKblocks of lengthHHwith independent uniform signs, sampled independently of the learner, and let the remainingT−K​HT\-KHrounds keep the last sign\. At a round of blockjj, the two sign hypotheses for that block agree before it, so by Lemma[F\.1](https://arxiv.org/html/2610.28898#Thmlemma1c)the learner’s information differs through at mostm​HmHtarget draws, andTV≤m​H​TV​\(μ\+,μ−\)≤m​H​η=1/4\\mathrm\{TV\}\\leq mH\\,\\mathrm\{TV\}\(\\mu\_\{\+\},\\mu\_\{\-\}\)\\leq mH\\eta=1/4\. This holds even if the learner knows the block boundaries and the mixture parameters\. Lemmas[F\.2](https://arxiv.org/html/2610.28898#Thmlemma2b)and[F\.5](https://arxiv.org/html/2610.28898#Thmlemma5b)\(ii\) give𝔼​Ft​\(vt\)≥12⋅34⋅12​η​R=316​η​R\\mathbb\{E\}F\_\{t\}\(v\_\{t\}\)\\geq\\frac\{1\}\{2\}\\cdot\\frac\{3\}\{4\}\\cdot\\frac\{1\}\{2\}\\eta R=\\frac\{3\}\{16\}\\eta Rat every block round, and summing over theK​HKHblock rounds withH​η=1/\(4​m\)H\\eta=1/\(4m\),

𝔼​ℛTvel≥3​K​R64​m\.\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq\\frac\{3KR\}\{64m\}\.By Lemma[F\.5](https://arxiv.org/html/2610.28898#Thmlemma5b)\(iii\), there are at mostK−1K\-1sign changes, soΩT≤4​\(K−1\)​η​R≤K​R/\(m​H\)≤2​K2​R/\(m​T\)\\Omega\_\{T\}\\leq 4\(K\-1\)\\eta R\\leq KR/\(mH\)\\leq 2K^\{2\}R/\(mT\)almost surely\.

GivenΩ\\Omegain the range \([115](https://arxiv.org/html/2610.28898#A6.E115)\), letKKbe the largest integer in\[1,Kmax\]\[1,K\_\{\\max\}\]with2​K2​RK/\(m​T\)≤Ω2K^\{2\}R\_\{K\}/\(mT\)\\leq\\Omega\. It exists becauseK=1K=1qualifies, and the instance then hasΩT≤Ω\\Omega\_\{T\}\\leq\\Omega\. IfK<KmaxK<K\_\{\\max\}, thenΩ<2​\(K\+1\)2​RK\+1/\(m​T\)≤8​K2​RK/\(m​T\)\\Omega<2\(K\+1\)^\{2\}R\_\{K\+1\}/\(mT\)\\leq 8K^\{2\}R\_\{K\}/\(mT\), becauseHH, and henceRKR\_\{K\}, does not increase withKK\. IfK=KmaxK=K\_\{\\max\}, the upper end of the range gives the same inequality\. In both casesK≥m​T​Ω/\(8​RK\)K\\geq\\sqrt\{mT\\Omega/\(8R\_\{K\}\)\}, so

𝔼​ℛTvel≥3​K​RK64​m≥364​RK8​T​Ωm≥120​T​Ωm,\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq\\frac\{3KR\_\{K\}\}\{64m\}\\geq\\frac\{3\}\{64\}\\sqrt\{\\frac\{R\_\{K\}\}\{8\}\}\\sqrt\{\\frac\{T\\Omega\}\{m\}\}\\geq\\frac\{1\}\{20\}\\sqrt\{\\frac\{T\\Omega\}\{m\}\},usingRK≥96R\_\{K\}\\geq\\sqrt\{96\}, which makes the middle coefficient at least0\.05180\.0518\. The bound holds on average over the block signs, so for every learner some sign sequence attains it\. Lemma[F\.4](https://arxiv.org/html/2610.28898#Thmlemma4b)shows that every instance satisfies the assumptions with the stated constants\. ∎

##### Approximation and target\-law budget of the family\.

By Lemma[F\.5](https://arxiv.org/html/2610.28898#Thmlemma5b)\(iv\),𝒜T≤T​π24​R2​η≤π28​K​R2/m\\mathcal\{A\}\_\{T\}\\leq T\\frac\{\\pi^\{2\}\}\{4\}R^\{2\}\\eta\\leq\\frac\{\\pi^\{2\}\}\{8\}KR^\{2\}/m, andW2​\(μ\+,μ−\)≤2​R​ηW\_\{2\}\(\\mu\_\{\+\},\\mu\_\{\-\}\)\\leq 2R\\sqrt\{\\eta\}: couple the bulks identically and move the far massη\\etaby2​R2R\. The approximation cost of the upper bound is thereforeO⁡\(log⁡\(e​T\)​R\)O\(\\log\(eT\)R\)times the block lower bound3​K​R/\(64​m\)3KR/\(64m\), a polylogarithmic factor sinceR=O⁡\(log⁡\(e​m​T\)\)R=O\(\\sqrt\{\\log\(emT\)\}\)\. The class hasP=5P=5andCS=O⁡\(1\+S\)C\_\{S\}=O\(1\+S\), so the constants of Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)on this family are polylogarithmic inm​TmTas well\.

### F\.5The Gaussian Family for Target\-Law Variation

###### Proposition F\.3\(Target\-law variation lower bound\)\.

Letλ\\lambdasatisfy Assumption[5](https://arxiv.org/html/2610.28898#Thmassumption5)\(c\),R0\>0R\_\{0\}\>0,T≥2T\\geq 2, and

\(2mT\)−1/2≤Vμ≤min\{T/\(4m\),16R03mT\}\.\(2mT\)^\{\-1/2\}\\leq V\_\{\\mu\}\\leq\\min\\\{T/\(4\\sqrt\{m\}\),\\,16R\_\{0\}^\{3\}mT\\\}\.For every learner there is a mean sequence for the Gaussian family with∥ξt∥≤R0\\lVert\\xi\_\{t\}\\rVert\\leq R\_\{0\}andVTμ≤VμV\_\{T\}^\{\\mu\}\\leq V\_\{\\mu\}on which

𝔼ℛTvel≥c∗λ34T1/3Vμ2/3m−2/3\.\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq\\frac\{c\_\{\*\}^\{\\lambda\}\}\{34\}\\,T^\{1/3\}V\_\{\\mu\}^\{2/3\}m^\{\-2/3\}\.

###### Proof\.

Use the Gaussian family withd=1d=1\. By Lemma[F\.3](https://arxiv.org/html/2610.28898#Thmlemma3b)it satisfies Assumptions[1](https://arxiv.org/html/2610.28898#Thmassumption1),[3](https://arxiv.org/html/2610.28898#Thmassumption3)and[5](https://arxiv.org/html/2610.28898#Thmassumption5)withκ2=8​\(2\+R02\)\\kappa^\{2\}=8\(2\+R\_\{0\}^\{2\}\),ι=1\\iota=1, and constants that do not depend onλ\\lambda\. LetK=⌊\(2​m​T​Vμ2\)1/3⌋K=\\lfloor\(2mTV\_\{\\mu\}^\{2\}\)^\{1/3\}\\rfloor\. The lower end of the range givesK≥1K\\geq 1, and henceK≥12​\(2​m​T​Vμ2\)1/3K\\geq\\frac\{1\}\{2\}\(2mTV\_\{\\mu\}^\{2\}\)^\{1/3\}\. The upper end givesK≤T/2K\\leq T/2andK≤8​R02​m​TK\\leq 8R\_\{0\}^\{2\}mT\. LetH=⌊T/K⌋≥T/\(2​K\)H=\\lfloor T/K\\rfloor\\geq T/\(2K\)andδ0=1/\(4​m​H\)\\delta\_\{0\}=1/\(4\\sqrt\{mH\}\), soδ0≤R0\\delta\_\{0\}\\leq R\_\{0\}\. UseKKblocks of lengthHHwith meansξt=δ0​ζj\\xi\_\{t\}=\\delta\_\{0\}\\zeta\_\{j\}for every roundttof blockjjand independent uniform signsζj\\zeta\_\{j\}, the remainder keeping the last mean\.

At a round of blockjj, the information differs through at mostm​HmHdraws\. By Lemma[F\.1](https://arxiv.org/html/2610.28898#Thmlemma1c),

KL\(N\(δ0,1\)⊗m​H∥N\(−δ0,1\)⊗m​H\)=2mHδ02=18,\\mathrm\{KL\}\\big\(N\(\\delta\_\{0\},1\)^\{\\otimes mH\}\\,\\\|\\,N\(\-\\delta\_\{0\},1\)^\{\\otimes mH\}\\big\)=2mH\\delta\_\{0\}^\{2\}=\\tfrac\{1\}\{8\},and Pinsker’s inequality givesTV≤1/4\\mathrm\{TV\}\\leq 1/4\. SinceFt​\(vϑ\)=c∗λ​\(ϑ−ξt\)2F\_\{t\}\(v\_\{\\vartheta\}\)=c\_\{\*\}^\{\\lambda\}\(\\vartheta\-\\xi\_\{t\}\)^\{2\}and\(ϑ−δ0\)2\+\(ϑ\+δ0\)2≥2​δ02\(\\vartheta\-\\delta\_\{0\}\)^\{2\}\+\(\\vartheta\+\\delta\_\{0\}\)^\{2\}\\geq 2\\delta\_\{0\}^\{2\}, Lemma[F\.2](https://arxiv.org/html/2610.28898#Thmlemma2b)gives𝔼​Ft​\(vt\)≥34​c∗λ​δ02\\mathbb\{E\}F\_\{t\}\(v\_\{t\}\)\\geq\\frac\{3\}\{4\}c\_\{\*\}^\{\\lambda\}\\delta\_\{0\}^\{2\}\. Summing overK​HKHrounds,

𝔼​ℛTvel≥34​K​H​c∗λ​δ02=364​c∗λ​K/m\.\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq\\tfrac\{3\}\{4\}KHc\_\{\*\}^\{\\lambda\}\\delta\_\{0\}^\{2\}=\\tfrac\{3\}\{64\}c\_\{\*\}^\{\\lambda\}K/m\.The budget isVTμ≤2​δ0​\(K−1\)<K/\(2​m​H\)≤K3/2/2​m​T≤VμV\_\{T\}^\{\\mu\}\\leq 2\\delta\_\{0\}\(K\-1\)<K/\(2\\sqrt\{mH\}\)\\leq K^\{3/2\}/\\sqrt\{2mT\}\\leq V\_\{\\mu\}\. Finally,

𝔼ℛTvel≥3128c∗λ\(2mTVμ2\)1/3/m=3⋅21/3128c∗λT1/3Vμ2/3m−2/3,\\mathbb\{E\}\\mathcal\{R\}\_\{T\}^\{\\mathrm\{vel\}\}\\geq\\tfrac\{3\}\{128\}c\_\{\*\}^\{\\lambda\}\(2mTV\_\{\\mu\}^\{2\}\)^\{1/3\}/m=\\tfrac\{3\\cdot 2^\{1/3\}\}\{128\}c\_\{\*\}^\{\\lambda\}T^\{1/3\}V\_\{\\mu\}^\{2/3\}m^\{\-2/3\},and3⋅21/3/128≥1/343\\cdot 2^\{1/3\}/128\\geq 1/34\. The family is realizable,at=0a\_\{t\}=0\.∎

### F\.6Matching the Upper Bounds

###### Corollary F\.1\(Optimality up to logarithmic factors\)\.

Fix the scales and the gate choice in Equation \([24](https://arxiv.org/html/2610.28898#S4.E24)\)\. On each family above, the upper bound of Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)or[2](https://arxiv.org/html/2610.28898#Thmtheorem2)exceeds the corresponding lower bound by at most a factor polylogarithmic inm​TmTand polynomial inmm, plus the additive fitting cost4​T​ϵopt4T\\epsilon\_\{\\mathrm\{opt\}\}\(respectively10​T​ϵopt10T\\epsilon\_\{\\mathrm\{opt\}\}\)\. Hence the ratesNω\+1N\_\{\\omega\}\+1,T​Ω\\sqrt\{T\\Omega\},Nμ\+1N\_\{\\mu\}\+1, andT1/3​Vμ2/3T^\{1/3\}V\_\{\\mu\}^\{2/3\}are optimal inTTand in the budget up to logarithmic factors\.

###### Proof\.

At fixed scales, Equation \([24](https://arxiv.org/html/2610.28898#S4.E24)\) makesBB,Γ\\Gamma,Λ\\Lambda, andΛsm\\Lambda\_\{\\mathrm\{sm\}\}polylogarithmic inm​TmT, soKTK\_\{T\},KT\+K\_\{T\}^\{\+\},c¯\\bar\{c\},HTH\_\{T\}, andHT\+H\_\{T\}^\{\+\}are polylogarithmic, whileT​ζT\\zeta,T​δT\\delta, andϵsm​T​Vμ\\epsilon\_\{\\mathrm\{sm\}\}TV\_\{\\mu\}areO⁡\(1\)O\(1\)\. ThusT​ef=4​T​ϵopt\+O⁡\(1\)Te\_\{f\}=4T\\epsilon\_\{\\mathrm\{opt\}\}\+O\(1\)andT​ef\+=10​T​ϵopt\+O⁡\(1\)Te\_\{f\}^\{\+\}=10T\\epsilon\_\{\\mathrm\{opt\}\}\+O\(1\)\.

- •*Counting\.*On the Gaussian family𝒜T=0\\mathcal\{A\}\_\{T\}=0, and Theorems[1](https://arxiv.org/html/2610.28898#Thmtheorem1)and[2](https://arxiv.org/html/2610.28898#Thmtheorem2)giveKT​\(N\+1\)\+T​efK\_\{T\}\(N\+1\)\+Te\_\{f\}againstc∗λ​R02​\(N\+1\)c\_\{\*\}^\{\\lambda\}R\_\{0\}^\{2\}\(N\+1\)\.
- •*Relative\-risk variation\.*On the rare mixture withKKblocks,K≍m​T​Ω/RK\\asymp\\sqrt\{mT\\Omega/R\}, Theorem[1](https://arxiv.org/html/2610.28898#Thmtheorem1)givesKT\+2​10​KT​T​Ω\+HT​𝒜T\+T​efK\_\{T\}\+2\\sqrt\{10K\_\{T\}T\\Omega\}\+H\_\{T\}\\mathcal\{A\}\_\{T\}\+Te\_\{f\}, whereHT​𝒜T=O⁡\(log⁡\(e​T\)​R2​K/m\)H\_\{T\}\\mathcal\{A\}\_\{T\}=O\(\\log\(eT\)R^\{2\}K/m\)\. The first three terms are a polylogarithmic multiple of3​K​R/\(64​m\)3KR/\(64m\)and ofT​Ω/m\\sqrt\{T\\Omega/m\}, up to a factorm\\sqrt\{m\}\.
- •*Target\-law variation\.*On the Gaussian family𝒜T=0\\mathcal\{A\}\_\{T\}=0, and Theorem[2](https://arxiv.org/html/2610.28898#Thmtheorem2)gives2​KT\+\+2​\(2​c¯\)1/3​\(KT\+\)2/3​T1/3​Vμ2/3\+T​ef\+\+O⁡\(1\)2K\_\{T\}^\{\+\}\+2\(2\\bar\{c\}\)^\{1/3\}\(K\_\{T\}^\{\+\}\)^\{2/3\}T^\{1/3\}V\_\{\\mu\}^\{2/3\}\+Te\_\{f\}^\{\+\}\+O\(1\)against\(c∗λ/34\)T1/3Vμ2/3m−2/3\(c\_\{\*\}^\{\\lambda\}/34\)T^\{1/3\}V\_\{\\mu\}^\{2/3\}m^\{\-2/3\}\.

In all four cases the exponents ofTTand of the budget coincide\. ∎