CryptoL: Towards Scale Dominance and Physics Constraints Mitigation in Financial Multivariate Time Series Forecasting

# CryptoL: Towards Scale Dominance and Physics Constraints Mitigation in Financial Multivariate Time Series Forecasting
Source: [https://arxiv.org/html/2609.11206](https://arxiv.org/html/2609.11206)
Mohammad Hassan HeydariArmon RasooliNote:These authors contributed equally\.Maryam AmirshahkaramiNote:These authors contributed equally\.Mohammad Ebrahim MahdaviNote:These authors contributed equally\.Hossein Karshenas

###### Abstract

Cryptocurrency forecasting presents a distinctive combination of extreme cross\-asset scale heterogeneity, non\-stationary dynamics, and structural dependencies among Open–High–Low–Close \(OHLC\) variables\. We present CryptoL, a unified framework designed to address these challenges within multivariate time\-series forecasting\. CryptoL evaluates forecasting error in context\-normalized coordinates within the RevIN pipeline, preventing inverse normalization from introducing an additional squared\-scale weighting into the MSE objective\. We formally characterize this effect through the empirical risk and parameter\-gradient geometry, establishing the conditions under which large\-scale assets can disproportionately influence shared\-model optimization\. Beyond loss\-space normalization, CryptoL examines channel\-independent and channel\-dependent normalization for OHLC data, showing that a shared channel\-dependent affine transformation preserves candle\-order relations that independent channel transformations need not preserve\. The framework further incorporates scale\-adaptive numerical stabilization to reduce distortions caused by a fixed normalization constant across assets spanning many orders of magnitude, together with a soft feasibility loss that penalizes violations of the defining OHLC inequalities\. Experiments across heterogeneous cryptocurrency assets evaluate these components through controlled ablations and demonstrate improvements in forecasting accuracy, training stability, and the frequency of financially valid OHLC predictions relative to the considered baselines\. CryptoL therefore provides an integrated approach to scale\-balanced optimization, structure\-preserving normalization, numerical stabilization, and constraint\-aware cryptocurrency forecasting\.

1Faculty of Engineering, Azad University

2Faculty of Computer Engineering, University of Isfahan

3Department of Electrical Engineering, Iran University of Science and Technology

y\.jaliltaheri@iau\.ir, m\.heydari@mehr\.ui\.ac\.ir, a\.rasouli@ec\.iut\.ac\.ir

maryamamirshahkarami@mehr\.ui\.ac\.ir, mhm\.ebrh\.mahdavi@gmail\.com, h\.karshenas@eng\.ui\.ac\.ir

## Introduction

Multivariate financial forecasting using open–high–low–close \(OHLC\) data provides a detailed representation of price evolution, but predicting these highly coupled variables jointly is challenging due to abrupt regime changes and strict physical candlestick rules\([Das et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib5);[Serdar et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib6);[Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16);[Wang et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib1)\)\. This task is especially demanding in cryptocurrency markets, which exhibit extreme non\-stationarity and cross\-asset scale heterogeneity\([Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16);[Kim et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib8);[Berthelier et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib15)\)\. Models must learn simultaneously from assets ranging from fractions of a cent to tens of thousands of dollars, making joint optimization difficult when using standard forecasting backbones that struggle with temporal distribution shifts and scale differences\([Kim et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib8);[Liu et al\. 2023](https://arxiv.org/html/2609.11206#bib.bib9);[Ye et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib10);[Berthelier et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib15);[Feng et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib14)\)\.

![Refer to caption](https://arxiv.org/html/2609.11206v1/cryptol.png)Figure 1:CryptoL frameworkWhile Reversible Instance Normalization \(RevIN\) is commonly used to mitigate distribution shifts, its implementation details have significant structural and numerical consequences for OHLC data\([Kim et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib8);[Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16);[Tepelyan 2025](https://arxiv.org/html/2609.11206#bib.bib3)\)\. Channel\-independent standardization can disrupt the relative ordering of candlestick channels, whereas channel\-dependent normalization preserves these essential relationships\. Additionally, standard RevIN evaluates forecasting errors in the original physical scale, which implicitly prioritizes high\-value assets during shared model optimization\([Kim et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib8);[Berthelier et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib15);[Liu et al\. 2023](https://arxiv.org/html/2609.11206#bib.bib9);[Ye et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib10);[Feng et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib14)\)\. Training directly in normalized target space which is referred to as Two\-Phase RevIN \(TP\-RevIN\), removes this scale\-dependent weighting, while scale\-adaptive stabilization prevents fixed normalization constants from distorting low\-valued series\([Berthelier et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib15);[Feng et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib14)\)\.

Even with proper normalization, models can still produce financially inadmissible predictions that violate core candlestick inequalities\([Wang et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib1);[Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16)\)\. Previous approaches addressed this structural consistency through specialized architectures or transformations, but these are often difficult to integrate with modern time\-series foundation models\. To address these limitations, we presentCryptoL, a unified framework that combines structure\-aware normalization, scale\-balanced optimization via TP\-RevIN, and an auxiliary physics\-informed loss\. This integrated approach discourages invalid candlestick structures while maintaining numerical stability and forecasting accuracy across diverse asset scales\.

The contributions of this work are summarized as follows:

- •We systematically analyze channel\-independent and channel\-dependent RevIN for multivariate OHLC forecasting, demonstrate their different effects on candlestick structure, and investigate fixed and scale\-adaptive epsilon formulations for stable normalization across assets with extreme numerical ranges\.
- •We provide a theoretical and extensive empirical analysis of TP\-RevIN, equivalently normalized\-space MSE training, showing how the location of the training objective affects scale\-induced optimization imbalance in heterogeneous cryptocurrency forecasting\.
- •We introduce an auxiliary physics\-informed loss that penalizes violations of the intrinsic OHLC ordering constraints and compare the resulting framework with unconstrained forecasting, conventional RevIN variants, and alternative normalization approaches\.

## Related Work

Multivariate financial forecasting aims to model temporal, cross\-variable, and cross\-asset dependencies jointly\([Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16);[Das et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib5);[Serdar et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib6);[Tepelyan 2025](https://arxiv.org/html/2609.11206#bib.bib3)\)\. Classical approaches have used state\-space models, Kalman filtering, and dynamic conditional correlation models, while recent methods employ recurrent networks, autoencoders, Transformers, and MLP\-based architectures\. StockMixer models indicator, temporal, and stock\-level interactions, whereas foundation\-model studies have shown that related multivariate inputs can improve financial forecasting\([Fan and Shen 2024](https://arxiv.org/html/2609.11206#bib.bib4)\)\. DPP instead learns representations directly from decomposed candlestick charts to predict future price movements\([Hung and Chen 2021](https://arxiv.org/html/2609.11206#bib.bib17)\)\.

OHLC forecasting introduces additional structural requirements because valid candlesticks must satisfy fixed relationships among the open, high, low, and close prices\([Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16);[Tepelyan 2025](https://arxiv.org/html/2609.11206#bib.bib3);[Wang et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib1)\)\. Existing studies have addressed these constraints through invertible transformations that guarantee valid reconstructed outputs\([Wang et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib1)\), as well as hybrid autoencoder and multitask architectures designed to model channel dependencies\([Chakraborty et al\. 2022](https://arxiv.org/html/2609.11206#bib.bib2)\)\. Other work has incorporated the timestamps of OHLC events to enrich the bar representation\([Das et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib5);[Serdar et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib6);[Simtharakao 2022](https://arxiv.org/html/2609.11206#bib.bib16);[Fiszeder et al\. 2023](https://arxiv.org/html/2609.11206#bib.bib7)\)\. These studies indicate that OHLC forecasting is not a standard multivariate regression problem, since predictions must preserve the internal structure of each candlestick\.

Normalization methods have been widely studied for handling non\-stationarity and distribution shift in time\-series forecasting\. RevIN removes instance\-specific statistics before forecasting and restores them afterward\([Kim et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib8)\)\. SAN extends this principle through local temporal\-slice normalization\([Liu et al\. 2023](https://arxiv.org/html/2609.11206#bib.bib9)\), while FAN uses dominant frequency components to address both trend and seasonal non\-stationarity\([Ye et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib10)\)\. However, most normalization studies focus on temporal distribution changes and give less attention to how the normalization axis affects structured multivariate outputs such as OHLC channels\.

Recent work has further distinguished representation normalization from the coordinate system in which the training loss is evaluated\. GTT trains on targets normalized using input\-context statistics, thereby learning directly in a unified curve\-shape space\([Feng et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib14)\)\. A comparative study of normalization in foundation models showed that MSE and MAE remain scale sensitive when predictions are denormalized before loss calculation\([Ahmed et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib18)\)\. Another recent analysis of RevIN compared conventional and normalized backpropagation through normalized MSE and reported benefits for heterogeneous\-scale data\([Berthelier et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib15)\)\. Building on these directions, our work jointly studies OHLC\-aware normalization, normalized\-space optimization, adaptive numerical stabilization, and auxiliary enforcement of candlestick validity\.

## Methodology

### Preliminaries and Problem Definition

Let

𝒟\\displaystyle\\mathcal\{D\}=\{\(𝐗n,𝐘n\)\}n=1N,\\displaystyle=\\left\\\{\\left\(\\mathbf\{X\}\_\{n\},\\mathbf\{Y\}\_\{n\}\\right\)\\right\\\}\_\{n=1\}^\{N\},\(1\)𝐗n\\displaystyle\\mathbf\{X\}\_\{n\}∈ℝL×4,𝐘n∈ℝH×4,\\displaystyle\\in\\mathbb\{R\}^\{L\\times 4\},\\qquad\\mathbf\{Y\}\_\{n\}\\in\\mathbb\{R\}^\{H\\times 4\},denote a collection of cryptocurrency forecasting samples, where𝐗n\\mathbf\{X\}\_\{n\}is an observed context window of lengthLLand𝐘n\\mathbf\{Y\}\_\{n\}is the corresponding forecast horizon of lengthHH\. The four channels are ordered as

𝐗n,t,:=\(On,t,Hn,t,Ln,t,Cn,t\),\\mathbf\{X\}\_\{n,t,:\}=\\left\(O\_\{n,t\},H\_\{n,t\},L\_\{n,t\},C\_\{n,t\}\\right\),\(2\)representing the open, high, low, and close prices, respectively\. A valid OHLC observation belongs to the feasible set

ΩOHLC=\{\(O,H,L,C\)∈ℝ4:\\displaystyle\\Omega\_\{\\mathrm\{OHLC\}\}=\\bigl\\\{\(O,H,L,C\)\\in\\mathbb\{R\}^\{4\}:H≥max⁡\(O,C\),\\displaystyle H\\geq\\max\(O,C\),\(3\)L≤min\(O,C\)\}\.\\displaystyle L\\leq\\min\(O,C\)\\bigr\\\}\.Given a forecasting modelf𝜽f\_\{\\bm\{\\theta\}\}, the objective is to estimate

𝐘^n=f𝜽​\(𝐗n\)\\widehat\{\\mathbf\{Y\}\}\_\{n\}=f\_\{\\bm\{\\theta\}\}\(\\mathbf\{X\}\_\{n\}\)\(4\)across cryptocurrencies whose numerical scales may differ by several orders of magnitude, while maintaining stable optimization and reducing violations of the constraints in Equation \([3](https://arxiv.org/html/2609.11206#Sx3.E3)\)\. All normalization statistics are computed exclusively from the observed context𝐗n\\mathbf\{X\}\_\{n\}and are reused to normalize the associated target and restore the prediction to its original physical scale\.

### Channel\-wise and Dynamic\-Epsilon Normalization

We investigate two choices for the axes over which RevIN statistics are computed\. In*channel\-independent*\(CI\) normalization, each OHLC channel is normalized using its own context statistics\. For channelc∈\{O,H,L,C\}c\\in\\\{O,H,L,C\\\}, these statistics are

μn,cCI\\displaystyle\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}=1L​∑t=1LXn,t,c,\\displaystyle=\\frac\{1\}\{L\}\\sum\_\{t=1\}^\{L\}X\_\{n,t,c\},\(5\)vn,cCI\\displaystyle v\_\{n,c\}^\{\\mathrm\{CI\}\}=1L​∑t=1L\(Xn,t,c−μn,cCI\)2,\\displaystyle=\\frac\{1\}\{L\}\\sum\_\{t=1\}^\{L\}\\left\(X\_\{n,t,c\}\-\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}\\right\)^\{2\},with effective scale

sn,cCI=vn,cCI\+ϵn,c\.s\_\{n,c\}^\{\\mathrm\{CI\}\}=\\sqrt\{v\_\{n,c\}^\{\\mathrm\{CI\}\}\+\\epsilon\_\{n,c\}\}\.\(6\)The context and target are then transformed as

X~n,t,cCI\\displaystyle\\widetilde\{X\}\_\{n,t,c\}^\{\\mathrm\{CI\}\}=Xn,t,c−μn,cCIsn,cCI,\\displaystyle=\\frac\{X\_\{n,t,c\}\-\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}\}\{s\_\{n,c\}^\{\\mathrm\{CI\}\}\},\(7\)Y~n,τ,cCI\\displaystyle\\widetilde\{Y\}\_\{n,\\tau,c\}^\{\\mathrm\{CI\}\}=Yn,τ,c−μn,cCIsn,cCI\.\\displaystyle=\\frac\{Y\_\{n,\\tau,c\}\-\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}\}\{s\_\{n,c\}^\{\\mathrm\{CI\}\}\}\.CI normalization standardizes open, high, low, and close separately\. Because the four channels undergo different affine transformations, their original ordering is not mathematically guaranteed to remain unchanged in normalized space\.

In*channel\-dependent*\(CD\) normalization, one shared mean and variance are computed jointly over the temporal and channel dimensions:

μnCD\\displaystyle\\mu\_\{n\}^\{\\mathrm\{CD\}\}=14​L​∑t=1L∑c=14Xn,t,c,\\displaystyle=\\frac\{1\}\{4L\}\\sum\_\{t=1\}^\{L\}\\sum\_\{c=1\}^\{4\}X\_\{n,t,c\},\(8\)vnCD\\displaystyle v\_\{n\}^\{\\mathrm\{CD\}\}=14​L​∑t=1L∑c=14\(Xn,t,c−μnCD\)2\.\\displaystyle=\\frac\{1\}\{4L\}\\sum\_\{t=1\}^\{L\}\\sum\_\{c=1\}^\{4\}\\left\(X\_\{n,t,c\}\-\\mu\_\{n\}^\{\\mathrm\{CD\}\}\\right\)^\{2\}\.The corresponding effective scale is

snCD=vnCD\+ϵn,s\_\{n\}^\{\\mathrm\{CD\}\}=\\sqrt\{v\_\{n\}^\{\\mathrm\{CD\}\}\+\\epsilon\_\{n\}\},\(9\)and the same transformation is applied to all OHLC channels:

X~n,t,cCD\\displaystyle\\widetilde\{X\}\_\{n,t,c\}^\{\\mathrm\{CD\}\}=Xn,t,c−μnCDsnCD,\\displaystyle=\\frac\{X\_\{n,t,c\}\-\\mu\_\{n\}^\{\\mathrm\{CD\}\}\}\{s\_\{n\}^\{\\mathrm\{CD\}\}\},\(10\)Y~n,τ,cCD\\displaystyle\\widetilde\{Y\}\_\{n,\\tau,c\}^\{\\mathrm\{CD\}\}=Yn,τ,c−μnCDsnCD\.\\displaystyle=\\frac\{Y\_\{n,\\tau,c\}\-\\mu\_\{n\}^\{\\mathrm\{CD\}\}\}\{s\_\{n\}^\{\\mathrm\{CD\}\}\}\.Unlike CI, CD applies one common affine transformation to the complete OHLC sample and therefore retains the relative organization of the candlestick channels\. Further mathematical proofs and analysis of order preservation under CD and CI normalization are provided in Appendix[B](https://arxiv.org/html/2609.11206#A2)\.

We additionally examine fixed and dynamic formulations of the stabilizing term used in Equations \([6](https://arxiv.org/html/2609.11206#Sx3.E6)\) and \([9](https://arxiv.org/html/2609.11206#Sx3.E9)\)\. The fixed formulation is

ϵfix=10−5,\\epsilon^\{\\mathrm\{fix\}\}=10^\{\-5\},\(11\)whereas the dynamic formulation adapts the stabilizer to the magnitude of the context mean:

ϵdyn=10−5​\(μ2\+10−12\)\.\\epsilon^\{\\mathrm\{dyn\}\}=10^\{\-5\}\\left\(\\mu^\{2\}\+10^\{\-12\}\\right\)\.\(12\)For CI normalization,μ\\mudenotes the corresponding channelwise mean; for CD normalization, it denotes the shared sample\-level mean\. Dynamic epsilon is designed to prevent a fixed numerical constant from having a disproportionately large effect on cryptocurrencies with very small quoted values\. Additional mathematical analysis of the fixed and dynamic epsilon formulations is provided in Appendix[B](https://arxiv.org/html/2609.11206#A2)\.

Table 1:Comparison of Different RevIN techniques\. Lower numbers indicate lower errors and better performances\. All of the experiments in this table are done using training with MSE only\. Best metrics in each category are colored withRedand Second best metrics are colored withBlue\.Table 2:Comparison of Models with FAN, SAN, and TP\-RevIN Normalization Techniques\. Experiments are done under 1h time frame\. For TP\-RevIN, here we use dynamic epsilon\.
### Two\-Phase RevIN and Scale\-Balanced Optimization

Let the normalization statistics selected in the previous subsection be denoted by𝝁n∈ℝ4\\bm\{\\mu\}\_\{n\}\\in\\mathbb\{R\}^\{4\}and𝐒n∈ℝ4×4\\mathbf\{S\}\_\{n\}\\in\\mathbb\{R\}^\{4\\times 4\}, where𝐒n\\mathbf\{S\}\_\{n\}is a positive diagonal channel\-scale matrix\. Under CD normalization,𝐒n=sn​𝐈4\\mathbf\{S\}\_\{n\}=s\_\{n\}\\mathbf\{I\}\_\{4\}; under CI normalization, its diagonal entries contain the channelwise scales\. After normalizing the observed context and target, we write

𝐗~n=\(𝐗n−𝟏L​𝝁n⊤\)​𝐒n−1,𝐙n=\(𝐘n−𝟏H​𝝁n⊤\)​𝐒n−1,\\widetilde\{\\mathbf\{X\}\}\_\{n\}=\\left\(\\mathbf\{X\}\_\{n\}\-\\mathbf\{1\}\_\{L\}\\bm\{\\mu\}\_\{n\}^\{\\top\}\\right\)\\mathbf\{S\}\_\{n\}^\{\-1\},\\qquad\\mathbf\{Z\}\_\{n\}=\\left\(\\mathbf\{Y\}\_\{n\}\-\\mathbf\{1\}\_\{H\}\\bm\{\\mu\}\_\{n\}^\{\\top\}\\right\)\\mathbf\{S\}\_\{n\}^\{\-1\},\(13\)where the statistics are computed exclusively from𝐗n\\mathbf\{X\}\_\{n\}\. The forecasting backbone produces a normalized prediction

𝐙^n=f𝜽​\(𝐗~n\),\\widehat\{\\mathbf\{Z\}\}\_\{n\}=f\_\{\\bm\{\\theta\}\}\\left\(\\widetilde\{\\mathbf\{X\}\}\_\{n\}\\right\),\(14\)and the corresponding physical\-scale prediction is recovered as

𝐘^n=𝟏H​𝝁n⊤\+𝐙^n​𝐒n\.\\widehat\{\\mathbf\{Y\}\}\_\{n\}=\\mathbf\{1\}\_\{H\}\\bm\{\\mu\}\_\{n\}^\{\\top\}\+\\widehat\{\\mathbf\{Z\}\}\_\{n\}\\mathbf\{S\}\_\{n\}\.\(15\)
Conventional RevIN\-based training commonly restores the prediction through Equation \([15](https://arxiv.org/html/2609.11206#Sx3.E15)\) before evaluating MSE\. We refer to this objective as original\-space RevIN training:

ℒR​\(𝜽\)=12​N​∑n=1N‖𝐘^n−𝐘n‖F2\.\\mathcal\{L\}\_\{\\mathrm\{R\}\}\\left\(\\bm\{\\theta\}\\right\)=\\frac\{1\}\{2N\}\\sum\_\{n=1\}^\{N\}\\left\\\|\\widehat\{\\mathbf\{Y\}\}\_\{n\}\-\\mathbf\{Y\}\_\{n\}\\right\\\|\_\{F\}^\{2\}\.\(16\)In contrast,*Two\-Phase RevIN*\(TP\-RevIN\) separates the normalized training phase from the physical\-scale inference phase\. During training, the objective is evaluated directly between the normalized prediction and normalized target:

ℒTP​\(𝜽\)=12​N​∑n=1N‖𝐙^n−𝐙n‖F2\.\\mathcal\{L\}\_\{\\mathrm\{TP\}\}\\left\(\\bm\{\\theta\}\\right\)=\\frac\{1\}\{2N\}\\sum\_\{n=1\}^\{N\}\\left\\\|\\widehat\{\\mathbf\{Z\}\}\_\{n\}\-\\mathbf\{Z\}\_\{n\}\\right\\\|\_\{F\}^\{2\}\.\(17\)Inverse normalization is therefore required for inference and reporting, but it does not participate in the training objective\.

##### Scale\-induced reweighting in original\-space RevIN\.

Letrvec⁡\(⋅\)\\operatorname\{rvec\}\(\\cdot\)stack the rows of anH×4H\\times 4matrix, and define

𝐃n=𝐈H⊗𝐒n∈ℝ4​H×4​H\.\\mathbf\{D\}\_\{n\}=\\mathbf\{I\}\_\{H\}\\otimes\\mathbf\{S\}\_\{n\}\\in\\mathbb\{R\}^\{4H\\times 4H\}\.\(18\)Define the vectorized normalized residual and its parameter Jacobian as

𝐞n​\(𝜽\)=rvec⁡\(𝐙^n−𝐙n\),𝐉n​\(𝜽\)=∂rvec⁡\(𝐙^n\)∂𝜽∈ℝ4​H×p\.\\mathbf\{e\}\_\{n\}\\left\(\\bm\{\\theta\}\\right\)=\\operatorname\{rvec\}\\left\(\\widehat\{\\mathbf\{Z\}\}\_\{n\}\-\\mathbf\{Z\}\_\{n\}\\right\),\\qquad\\mathbf\{J\}\_\{n\}\\left\(\\bm\{\\theta\}\\right\)=\\frac\{\\partial\\operatorname\{rvec\}\\left\(\\widehat\{\\mathbf\{Z\}\}\_\{n\}\\right\)\}\{\\partial\\bm\{\\theta\}\}\\in\\mathbb\{R\}^\{4H\\times p\}\.\(19\)
Proposition 1\.Assume that𝝁n\\bm\{\\mu\}\_\{n\}and𝐃n\\mathbf\{D\}\_\{n\}are computed from the observed context and are independent of𝜽\\bm\{\\theta\}\. Then original\-space RevIN MSE satisfies

ℒR​\(𝜽\)=12​N​∑n=1N𝐞n⊤​𝐃n2​𝐞n,\\mathcal\{L\}\_\{\\mathrm\{R\}\}\\left\(\\bm\{\\theta\}\\right\)=\\frac\{1\}\{2N\}\\sum\_\{n=1\}^\{N\}\\mathbf\{e\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{2\}\\mathbf\{e\}\_\{n\},\(20\)whereas TP\-RevIN satisfies Equation \([17](https://arxiv.org/html/2609.11206#Sx3.E17)\)\. Their parameter gradients are

∇𝜽ℒR=1N​∑n=1N𝐉n⊤​𝐃n2​𝐞n,\\nabla\_\{\\bm\{\\theta\}\}\\mathcal\{L\}\_\{\\mathrm\{R\}\}=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{2\}\\mathbf\{e\}\_\{n\},\(21\)and

∇𝜽ℒTP=1N​∑n=1N𝐉n⊤​𝐞n\.\\nabla\_\{\\bm\{\\theta\}\}\\mathcal\{L\}\_\{\\mathrm\{TP\}\}=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}\.\(22\)Consequently, original\-space RevIN introduces an explicit scale\-dependent weighting into both the empirical objective and the shared parameter update, while TP\-RevIN removes this weighting\.

##### Proof\.

From Equations \([13](https://arxiv.org/html/2609.11206#Sx3.E13)\) and \([15](https://arxiv.org/html/2609.11206#Sx3.E15)\),

rvec⁡\(𝐘^n−𝐘n\)\\displaystyle\\operatorname\{rvec\}\\left\(\\widehat\{\\mathbf\{Y\}\}\_\{n\}\-\\mathbf\{Y\}\_\{n\}\\right\)=𝐃n​rvec⁡\(𝐙^n−𝐙n\)\\displaystyle=\\mathbf\{D\}\_\{n\}\\operatorname\{rvec\}\\left\(\\widehat\{\\mathbf\{Z\}\}\_\{n\}\-\\mathbf\{Z\}\_\{n\}\\right\)\(23\)=𝐃n​𝐞n\.\\displaystyle=\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\}\.Therefore,

‖𝐘^n−𝐘n‖F2=𝐞n⊤​𝐃n⊤​𝐃n​𝐞n\.\\left\\\|\\widehat\{\\mathbf\{Y\}\}\_\{n\}\-\\mathbf\{Y\}\_\{n\}\\right\\\|\_\{F\}^\{2\}=\\mathbf\{e\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\}\.\(24\)Because𝐃n\\mathbf\{D\}\_\{n\}is diagonal and positive,𝐃n⊤​𝐃n=𝐃n2\\mathbf\{D\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}=\\mathbf\{D\}\_\{n\}^\{2\}, which proves Equation \([20](https://arxiv.org/html/2609.11206#Sx3.E20)\)\.

Differentiating the per\-sample term in Equation \([20](https://arxiv.org/html/2609.11206#Sx3.E20)\) and treating𝐃n\\mathbf\{D\}\_\{n\}as constant with respect to𝜽\\bm\{\\theta\}gives

∇𝜽\(12​𝐞n⊤​𝐃n2​𝐞n\)=𝐉n⊤​𝐃n2​𝐞n\.\\nabla\_\{\\bm\{\\theta\}\}\\left\(\\frac\{1\}\{2\}\\mathbf\{e\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{2\}\\mathbf\{e\}\_\{n\}\\right\)=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{2\}\\mathbf\{e\}\_\{n\}\.\(25\)Summing over all samples yields Equation \([21](https://arxiv.org/html/2609.11206#Sx3.E21)\)\. Similarly, differentiating the TP\-RevIN term12​𝐞n⊤​𝐞n\\frac\{1\}\{2\}\\mathbf\{e\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}gives𝐉n⊤​𝐞n\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}, establishing Equation \([22](https://arxiv.org/html/2609.11206#Sx3.E22)\)\.□\\square

For CD normalization, where𝐃n=sn​𝐈4​H\\mathbf\{D\}\_\{n\}=s\_\{n\}\\mathbf\{I\}\_\{4H\}, Proposition 1 reduces to

ℒR=12​N​∑n=1Nsn2​‖𝐞n‖22,\\mathcal\{L\}\_\{\\mathrm\{R\}\}=\\frac\{1\}\{2N\}\\sum\_\{n=1\}^\{N\}s\_\{n\}^\{2\}\\left\\\|\\mathbf\{e\}\_\{n\}\\right\\\|\_\{2\}^\{2\},\(26\)and

∇𝜽ℒR=1N​∑n=1Nsn2​𝐉n⊤​𝐞n\.\\nabla\_\{\\bm\{\\theta\}\}\\mathcal\{L\}\_\{\\mathrm\{R\}\}=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}s\_\{n\}^\{2\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}\.\(27\)Thus two samples with comparable normalized residuals and comparable model sensitivity contribute to the original\-space update in proportion to the squares of their context scales\. This is the precise sense in which scale dominance arises in RevIN\-based MSE training\. Large numerical scale alone does not determine the complete gradient, since the contribution also depends on𝐉n⊤​𝐞n\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}; however, the factorsn2s\_\{n\}^\{2\}is an unavoidable multiplicative component of the objective whenever the loss is evaluated after inverse normalization\.

The same result can be expressed at the asset level\. Letπa\\pi\_\{a\}denote the sampling probability of assetaa, and define its normalized forecasting risk as

Ra​\(𝜽\)=12​𝔼​\[‖𝐞⁡\(𝜽\)‖22\|a\]\.R\_\{a\}\\left\(\\bm\{\\theta\}\\right\)=\\frac\{1\}\{2\}\\mathbb\{E\}\\left\[\\left\\\|\\mathbf\{e\}\\left\(\\bm\{\\theta\}\\right\)\\right\\\|\_\{2\}^\{2\}\\,\\middle\|\\,a\\right\]\.\(28\)If assetaahas approximately constant context scalesas\_\{a\}, then original\-space RevIN minimizes

RR​\(𝜽\)=∑aπa​sa2​Ra​\(𝜽\),R\_\{\\mathrm\{R\}\}\\left\(\\bm\{\\theta\}\\right\)=\\sum\_\{a\}\\pi\_\{a\}s\_\{a\}^\{2\}R\_\{a\}\\left\(\\bm\{\\theta\}\\right\),\(29\)whereas TP\-RevIN minimizes

RTP​\(𝜽\)=∑aπa​Ra​\(𝜽\)\.R\_\{\\mathrm\{TP\}\}\\left\(\\bm\{\\theta\}\\right\)=\\sum\_\{a\}\\pi\_\{a\}R\_\{a\}\\left\(\\bm\{\\theta\}\\right\)\.\(30\)Up to a positive global constant, Equation \([29](https://arxiv.org/html/2609.11206#Sx3.E29)\) is equivalent to replacing the declared asset distributionπa\\pi\_\{a\}with

qa=πa​sa2∑bπb​sb2\.q\_\{a\}=\\frac\{\\pi\_\{a\}s\_\{a\}^\{2\}\}\{\\sum\_\{b\}\\pi\_\{b\}s\_\{b\}^\{2\}\}\.\(31\)Original\-space RevIN therefore changes the effective optimization priority toward assets with larger context scales\. TP\-RevIN preserves the declared sampling weights and removes this implicit scale\-dependent redistribution\.

This distinction is particularly important for cryptocurrency panels, where assets can span many orders of magnitude\. Under comparable normalized forecasting errors and model sensitivities, a high\-scale asset can dominate the aggregate parameter update even when low\-scale assets are sampled equally often\. TP\-RevIN does not make all assets equally difficult or guarantee equal gradient norms; differences in forecastability, residual structure, sampling frequency, and model sensitivity remain\. It specifically removes the multiplicative weighting introduced solely by the RevIN scale and thereby yields a training objective that is neutral to positive affine changes in the numerical units of each asset\. Further proofs based on Jacobian and Hessian analyses are provided in Appendix[B](https://arxiv.org/html/2609.11206#A2)\.

Table 3:Comparison of Fixed vs\. Dynamic Epsilon Normalization \(Values report Horizon 5 / 30 for MAPE\)\.Table 4:Analysis of training models using auxiliary physics loss on different methods

### Normalized\-Space OHLC Constraint Loss

Under TP\-RevIN, both the forecasting objective and the auxiliary OHLC constraint loss are evaluated in normalized coordinates\. Let𝐙^n\\widehat\{\\mathbf\{Z\}\}\_\{n\}denote the normalized prediction, with componentsO^n,τ\\widehat\{O\}\_\{n,\\tau\},H^n,τ\\widehat\{H\}\_\{n,\\tau\},L^n,τ\\widehat\{L\}\_\{n,\\tau\}, andC^n,τ\\widehat\{C\}\_\{n,\\tau\}at forecast stepτ\\tau\. Defining\[x\]\+=max⁡\(0,x\)\[x\]\_\{\+\}=\\max\(0,x\), the normalized\-space constraint loss is

ℒphy=1N​H​∑n=1N∑τ=1H\(CLOSE\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{phy\}\}=\\frac\{1\}\{NH\}\\sum\_\{n=1\}^\{N\}\\sum\_\{\\tau=1\}^\{H\}\\Bigl\(\[O^n,τ−H^n,τ\]\+\+\[C^n,τ−H^n,τ\]\+\\displaystyle\[\\widehat\{O\}\_\{n,\\tau\}\-\\widehat\{H\}\_\{n,\\tau\}\]\_\{\+\}\+\[\\widehat\{C\}\_\{n,\\tau\}\-\\widehat\{H\}\_\{n,\\tau\}\]\_\{\+\}\(32\)\+\[L^n,τ−O^n,τ\]\+\+\[L^n,τ−C^n,τ\]\+\\displaystyle\+\[\\widehat\{L\}\_\{n,\\tau\}\-\\widehat\{O\}\_\{n,\\tau\}\]\_\{\+\}\+\[\\widehat\{L\}\_\{n,\\tau\}\-\\widehat\{C\}\_\{n,\\tau\}\]\_\{\+\}OPEN\+\[L^n,τ−H^n,τ\]\+\)\.\\displaystyle\+\[\\widehat\{L\}\_\{n,\\tau\}\-\\widehat\{H\}\_\{n,\\tau\}\]\_\{\+\}\\Bigr\)\.The complete TP\-RevIN training objective is therefore

ℒtotal=λMSE​ℒTP\+λphy​ℒphy,\\mathcal\{L\}\_\{\\mathrm\{total\}\}=\\lambda\_\{\\mathrm\{MSE\}\}\\mathcal\{L\}\_\{\\mathrm\{TP\}\}\+\\lambda\_\{\\mathrm\{phy\}\}\\mathcal\{L\}\_\{\\mathrm\{phy\}\},\(33\)where both terms are computed before inverse normalization\. This design prevents the auxiliary loss from reintroducing the scale\-dependent weighting removed by TP\-RevIN\. Additional mathematical results concerning normalized\-space OHLC constraints are provided in Appendix[B](https://arxiv.org/html/2609.11206#A2)\.

### CryptoL Framework

CryptoL combines dynamic\-epsilon normalization, TP\-RevIN, and the normalized\-space OHLC constraint loss within a unified multivariate cryptocurrency forecasting framework\. The input context and target are first transformed using either CI or CD statistics together with the dynamic epsilon in Equation \([12](https://arxiv.org/html/2609.11206#Sx3.E12)\)\. The forecasting backbone predicts the future sequence in normalized coordinates, where both the MSE objective and the auxiliary constraint loss are evaluated according to Equation \([33](https://arxiv.org/html/2609.11206#Sx3.E33)\)\. Inverse normalization is applied only after optimization to recover predictions in their original price units\.

The selection between CI and CD remains configurable because the two schemes provide different practical trade\-offs\. CD preserves OHLC ordering under normalization, whereas CI can provide stronger channelwise forecasting accuracy\. Accordingly, both CI and CD variants of CryptoL achieve low physical\-consistency errors when combined with the constraint loss, allowing the normalization axis to be selected according to the dataset, backbone, and target evaluation criterion\.

## Experiments

### Dataset

We construct a large\-scale cryptocurrency forecasting dataset containing approximately15\.515\.5million historical OHLC observations from1616assets\. The data were collected through the Binance API at multiple temporal resolutions, enabling evaluation across different forecasting time frames and horizons\. The selected assets exhibit extreme scale heterogeneity, with quoted values ranging from approximately10−710^\{\-7\}for low\-valued assets such as PEPE to above10510^\{5\}for high\-valued assets such as BTC\. This broad dynamic range provides a suitable setting for evaluating normalization behavior, cross\-asset learning, and the scale\-balancing properties of TP\-RevIN\. Complete dataset details, including temporal coverage, row counts, and asset\-specific volatility distributions, are provided in Appendix[C](https://arxiv.org/html/2609.11206#A3)\.

### Training Configuration

We evaluate CryptoL using three decoder\-only time\-series forecasting backbones: Timer\([Liu et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib11)\), Timer\-XL\([Liu et al\. 2025](https://arxiv.org/html/2609.11206#bib.bib12)\), and Time\-MoE\([Shi et al\. 2025](https://arxiv.org/html/2609.11206#bib.bib13)\)\. All models are optimized using AdamW with a linearly scheduled learning rate initialized at10−510^\{\-5\}\. Each model is trained for one epoch using90%90\\%of the available data, while the remaining10%10\\%is reserved for evaluation\. All reported forecasting metrics are computed after restoring predictions to the original physical space, ensuring a consistent and fair comparison among normalization and training configurations\.

Each experiment is conducted separately for a specific time frame and forecast horizon\. Within each configuration, however, the model is trained jointly on all1616assets rather than independently for each cryptocurrency\. This protocol evaluates whether a shared forecasting backbone can learn transferable temporal and cross\-asset patterns despite substantial differences in asset scale, volatility, and market behavior\.

### Hardware and Software Environment

All experiments are conducted on eight TPU v5e accelerators, each providing16​GB16\\,\\mathrm\{GB\}of accelerator memory, for a total distributed memory capacity of128​GB128\\,\\mathrm\{GB\}\. The implementation is developed using JAX\([Bradbury et al\. 2018](https://arxiv.org/html/2609.11206#bib.bib19)\)and Flax\([Heek et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib20)\)\.111The experiments use JAX version 0\.9\.2 and Flax version 0\.12\.6\.Both model parameters and intermediate activations are represented in FP32 throughout training\.

To distribute the training workload, we employ fully sharded data\-parallel training across all eight TPUs\. The model parameters, optimizer states, and input data are sharded over the complete accelerator mesh, reducing the memory allocated to each device and enabling efficient training of the selected forecasting backbones\.

## Results and Analysis

We begin by evaluating the predictive performance of different normalization strategies, as summarized in Table[1](https://arxiv.org/html/2609.11206#Sx3.T1)and Table[2](https://arxiv.org/html/2609.11206#Sx3.T2)\. Without any RevIN\([Kim et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib8)\), the models struggle to converge effectively, resulting in extremely high MSE values due to the immense scale discrepancies among different cryptocurrency assets\. Introducing standard CI and CD RevIN significantly mitigates this problem\. However, TP\-RevIN\([Feng et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib14);[Berthelier et al\. 2026](https://arxiv.org/html/2609.11206#bib.bib15)\)formulations consistently yield further reductions in both MSE and MAE across various forecasting horizons and model backbones\. When comparing TP\-RevIN to alternative temporal normalization schemes such as FAN\([Ye et al\. 2024](https://arxiv.org/html/2609.11206#bib.bib10)\)and SAN\([Liu et al\. 2023](https://arxiv.org/html/2609.11206#bib.bib9)\)in Table[2](https://arxiv.org/html/2609.11206#Sx3.T2), we observe that both FAN and SAN yield substantially higher MAE and Mean Absolute Percentage Error \(MAPE\) values\. This indicates that while those methods are designed to handle non\-stationarity, they struggle to balance the extreme cross\-asset scale differences present in joint cryptocurrency panels\. In contrast, TP\-RevIN preserves representation stability across diverse scales and maintains lower physical violation \(PHY\) rates\.

To evaluate the impact of numerical stabilization across assets of varying price scales, we analyze the fixed versus dynamic epsilon formulations in Table[3](https://arxiv.org/html/2609.11206#Sx3.T3)\. For high\-value assets such as Bitcoin \(BTC\) and Ethereum \(ETH\), the choice between a fixed and dynamic stabilizer \(ϵ\\epsilon\) has negligible impact on the forecasting performance\. However, for low\-value assets like Shiba Inu \(SHIB\), which has an average price scale on the order of10−510^\{\-5\}, the fixed epsilon stabilizer \(ϵfix=10−5\\epsilon^\{\\text\{fix\}\}=10^\{\-5\}\) introduces severe numerical distortions during normalization\. This distortion results in highly elevated MAPE values, such as17362\.617362\.6for Time\-MoE under standard RevIN\. Utilizing the scale\-adaptive dynamic epsilon \(ϵdyn\\epsilon^\{\\text\{dyn\}\}\) resolves this numerical instability, reducing the SHIB MAPE to1\.871\.87/3\.513\.51under TP\-RevIN and stabilizing the optimization process without compromising the predictive accuracy of the high\-value assets\.

Finally, we analyze the effectiveness of the auxiliary physics\-informed constraint loss, as presented in Table[4](https://arxiv.org/html/2609.11206#Sx3.T4)\. The "Unconstrained Space" method\([Wang et al\. 2021](https://arxiv.org/html/2609.11206#bib.bib1)\)mathematically guarantees zero physical violations \(PHY=0\.0\\text\{PHY\}=0\.0\) through its structural design\. However, this rigid constraint comes at a severe cost to overall predictive accuracy, leading to MAE values that are orders of magnitude higher than those of alternative approaches \(for example, an MAE of3307\.533307\.53for Timer\-XL on a 30m timeframe at horizon 5, compared to34\.7734\.77for TP\-RevIN \(CI\)\)\. Conversely, the proposed TP\-RevIN models trained with the auxiliary physics loss find a more balanced trade\-off\. They reduce physical candlestick violations to near\-zero levels while keeping the MAE highly competitive\.

## Conclusion

In this work, we presented CryptoL, a framework for multivariate cryptocurrency forecasting that addresses cross\-asset scale heterogeneity, numerical instability, and physical candlestick constraint violations\. By combining Two\-Phase RevIN, dynamic epsilon stabilization, and a normalized\-space auxiliary physics loss, the framework balances optimization across highly diverse price scales while encouraging structural validity\. Empirical evaluations indicate that this integrated approach offers a practical compromise between forecasting accuracy and physical consistency, providing a stable foundation for training shared models on heterogeneous financial time series\.

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## Appendix ADiscussion

Cryptocurrency OHLC forecasting combines several difficulties that are usually studied separately in general\-purpose time\-series forecasting\. First, cryptocurrency assets exhibit extreme numerical heterogeneity: the quoted value of one asset may be below10−710^\{\-7\}, while another may exceed10510^\{5\}\. A shared forecasting model must therefore learn from series whose numerical scales differ by more than twelve orders of magnitude\. Second, OHLC observations are structurally constrained\. A valid candlestick must satisfy

H≥max⁡\(O,C\),L≤min⁡\(O,C\),H\\geq\\max\(O,C\),\\qquad L\\leq\\min\(O,C\),\(34\)and a model may produce an invalid financial object even when its aggregate pointwise forecasting error is small\.

Reversible instance normalization reduces temporal distribution shift by transforming each context into normalized coordinates before it is processed by the forecasting backbone\. However, normalization is not a single unambiguous operation in multivariate OHLC forecasting\. When each channel is normalized independently, the open, high, low, and close channels undergo different affine transformations\. Their original cross\-channel ordering is therefore not guaranteed to remain valid in normalized coordinates\. In contrast, when all four channels share one positive affine transformation, their pairwise order is preserved exactly\. This distinction motivates the configurable channel\-independent and channel\-dependent components of CryptoL\.

Figure 2:Difference between channel\-dependent and channel\-independent normalization on OHLC data and how they perform on order preservation\.A second issue concerns the coordinate system in which optimization is performed\. Under conventional RevIN\-style training, the model predicts in normalized coordinates, but the prediction may be inverse normalized before mean\-squared error is evaluated\. This operation reintroduces the context scale into the empirical objective\. For a sample with scalar context scalesns\_\{n\}, the raw\-space residual issns\_\{n\}times the normalized residual, and its squared loss is therefore weighted bysn2s\_\{n\}^\{2\}\. In a shared multi\-asset model, this weighting can alter the aggregate gradient direction and the local curvature of the optimization problem\. TP\-RevIN avoids this effect by evaluating the forecasting objective directly in normalized target coordinates\.

CryptoL additionally evaluates its OHLC consistency penalty in normalized coordinates\. Consequently, both the forecasting objective and the auxiliary structural objective are prevented from inheriting raw numerical scale\. Under CD normalization, normalized\-space OHLC constraints are exactly equivalent to the original\-space constraints because all channels share the same positive affine transformation\. Under CI normalization, this exact equivalence does not hold; the constraint term should instead be interpreted as a representation\-space regularizer, and financial validity must still be evaluated after inverse normalization\. Empirically, the auxiliary loss substantially reduces original\-space violations under both normalization schemes, allowing the normalization axis to remain a configurable design choice\.

The final component is the stabilizing term used in the normalization denominator\. A fixed epsilon has an absolute numerical scale and may dominate the empirical variance of extremely low\-valued assets while being negligible for high\-valued assets\. CryptoL therefore studies a dynamic epsilon proportional to the squared context mean\. This choice is approximately equivariant under positive changes of numerical units and makes the regularization strength depend primarily on relative rather than absolute scale\. The following sections formalize these claims and delimit their assumptions\.

## Appendix BMathematical Analysis

Let

𝒟=\{\(𝐗n,𝐘n\)\}n=1N,𝐗n∈ℝL×4,𝐘n∈ℝH×4,\\mathcal\{D\}=\\left\\\{\\left\(\\mathbf\{X\}\_\{n\},\\mathbf\{Y\}\_\{n\}\\right\)\\right\\\}\_\{n=1\}^\{N\},\\qquad\\mathbf\{X\}\_\{n\}\\in\\mathbb\{R\}^\{L\\times 4\},\\qquad\\mathbf\{Y\}\_\{n\}\\in\\mathbb\{R\}^\{H\\times 4\},\(35\)denote a set of OHLC forecasting samples\. The four channels are ordered as

𝐗n,t,:=\(On,t,Hn,t,Ln,t,Cn,t\)\.\\mathbf\{X\}\_\{n,t,:\}=\\left\(O\_\{n,t\},H\_\{n,t\},L\_\{n,t\},C\_\{n,t\}\\right\)\.\(36\)All normalization statistics are assumed to be measurable functions of the observed context𝐗n\\mathbf\{X\}\_\{n\}only\. They are treated as constants with respect to the forecasting parameters𝜽\\bm\{\\theta\}\.

For the optimization analysis, the forecast horizon and channel dimensions are vectorized into

d=4​H,𝐳n,𝐳^n∈ℝd\.d=4H,\\qquad\\mathbf\{z\}\_\{n\},\\widehat\{\\mathbf\{z\}\}\_\{n\}\\in\\mathbb\{R\}^\{d\}\.\(37\)
### Order Preservation Under CD and CI Normalization

#### Channel\-dependent normalization

Under CD normalization, one shared location and one shared positive scale are computed from the complete context:

μnCD=14​L​∑t=1L∑c=14Xn,t,c,\\mu\_\{n\}^\{\\mathrm\{CD\}\}=\\frac\{1\}\{4L\}\\sum\_\{t=1\}^\{L\}\\sum\_\{c=1\}^\{4\}X\_\{n,t,c\},\(38\)vnCD=14​L​∑t=1L∑c=14\(Xn,t,c−μnCD\)2,v\_\{n\}^\{\\mathrm\{CD\}\}=\\frac\{1\}\{4L\}\\sum\_\{t=1\}^\{L\}\\sum\_\{c=1\}^\{4\}\\left\(X\_\{n,t,c\}\-\\mu\_\{n\}^\{\\mathrm\{CD\}\}\\right\)^\{2\},\(39\)and

snCD=vnCD\+ϵn\>0\.s\_\{n\}^\{\\mathrm\{CD\}\}=\\sqrt\{v\_\{n\}^\{\\mathrm\{CD\}\}\+\\epsilon\_\{n\}\}\>0\.\(40\)The shared affine transformation is

TnCD​\(x\)=x−μnCDsnCD\.T\_\{n\}^\{\\mathrm\{CD\}\}\(x\)=\\frac\{x\-\\mu\_\{n\}^\{\\mathrm\{CD\}\}\}\{s\_\{n\}^\{\\mathrm\{CD\}\}\}\.\(41\)
###### Theorem B\.1\(Order preservation under CD normalization\)\.

Letx1,x2∈ℝx\_\{1\},x\_\{2\}\\in\\mathbb\{R\}, and letTnCDT\_\{n\}^\{\\mathrm\{CD\}\}be defined by[Equation41](https://arxiv.org/html/2609.11206#A2.E41)withsnCD\>0s\_\{n\}^\{\\mathrm\{CD\}\}\>0\. Then

x1≥x2⟺TnCD\(x1\)≥TnCD\(x2\)\.x\_\{1\}\\geq x\_\{2\}\\quad\\Longleftrightarrow\\quad T\_\{n\}^\{\\mathrm\{CD\}\}\(x\_\{1\}\)\\geq T\_\{n\}^\{\\mathrm\{CD\}\}\(x\_\{2\}\)\.\(42\)Consequently, every valid OHLC observation remains valid after CD normalization\.

###### Proof\.

Subtracting the two transformed values gives

TnCD​\(x1\)−TnCD​\(x2\)=x1−x2snCD\.T\_\{n\}^\{\\mathrm\{CD\}\}\(x\_\{1\}\)\-T\_\{n\}^\{\\mathrm\{CD\}\}\(x\_\{2\}\)=\\frac\{x\_\{1\}\-x\_\{2\}\}\{s\_\{n\}^\{\\mathrm\{CD\}\}\}\.\(43\)BecausesnCD\>0s\_\{n\}^\{\\mathrm\{CD\}\}\>0,

sign⁡\(TnCD​\(x1\)−TnCD​\(x2\)\)=sign⁡\(x1−x2\)\.\\operatorname\{sign\}\\left\(T\_\{n\}^\{\\mathrm\{CD\}\}\(x\_\{1\}\)\-T\_\{n\}^\{\\mathrm\{CD\}\}\(x\_\{2\}\)\\right\)=\\operatorname\{sign\}\(x\_\{1\}\-x\_\{2\}\)\.\(44\)Thereforex1−x2≥0x\_\{1\}\-x\_\{2\}\\geq 0if and only if the transformed difference is nonnegative, proving[Equation42](https://arxiv.org/html/2609.11206#A2.E42)\.

Applying the result to the OHLC inequalities gives

H≥O⟹H~CD≥O~CD,H\\geq O\\Longrightarrow\\widetilde\{H\}^\{\\mathrm\{CD\}\}\\geq\\widetilde\{O\}^\{\\mathrm\{CD\}\},\(45\)H≥C⟹H~CD≥C~CD,H\\geq C\\Longrightarrow\\widetilde\{H\}^\{\\mathrm\{CD\}\}\\geq\\widetilde\{C\}^\{\\mathrm\{CD\}\},\(46\)L≤O⟹L~CD≤O~CD,L\\leq O\\Longrightarrow\\widetilde\{L\}^\{\\mathrm\{CD\}\}\\leq\\widetilde\{O\}^\{\\mathrm\{CD\}\},\(47\)and

L≤C⟹L~CD≤C~CD\.L\\leq C\\Longrightarrow\\widetilde\{L\}^\{\\mathrm\{CD\}\}\\leq\\widetilde\{C\}^\{\\mathrm\{CD\}\}\.\(48\)Hence membership in the OHLC feasible set is preserved\. The figure[2](https://arxiv.org/html/2609.11206#A1.F2)shows how CD and CI perform different on order preservation for OHLC data\. ∎

###### Corollary B\.2\(Equivalence of CD normalized and raw constraints\)\.

For everysnCD\>0s\_\{n\}^\{\\mathrm\{CD\}\}\>0,

\[O~CD−H~CD\]\+=1snCD​\[O−H\]\+\.\\left\[\\widetilde\{O\}^\{\\mathrm\{CD\}\}\-\\widetilde\{H\}^\{\\mathrm\{CD\}\}\\right\]\_\{\+\}=\\frac\{1\}\{s\_\{n\}^\{\\mathrm\{CD\}\}\}\\left\[O\-H\\right\]\_\{\+\}\.\(49\)Analogous identities hold for every pairwise OHLC constraint\. Therefore, the zero set of the normalized\-space CD physics loss is identical to the zero set of the corresponding raw\-space OHLC constraint loss\.

###### Proof\.

Using the shared affine transformation,

O~CD−H~CD=O−HsnCD\.\\widetilde\{O\}^\{\\mathrm\{CD\}\}\-\\widetilde\{H\}^\{\\mathrm\{CD\}\}=\\frac\{O\-H\}\{s\_\{n\}^\{\\mathrm\{CD\}\}\}\.\(50\)Fora\>0a\>0, the positive\-part function satisfies

\[a​x\]\+=a​\[x\]\+\.\\left\[ax\\right\]\_\{\+\}=a\\left\[x\\right\]\_\{\+\}\.\(51\)Settinga=1/snCDa=1/s\_\{n\}^\{\\mathrm\{CD\}\}proves[Equation49](https://arxiv.org/html/2609.11206#A2.E49)\. ∎

#### Channel\-independent normalization

Under CI normalization, each channelc∈\{O,H,L,C\}c\\in\\\{O,H,L,C\\\}has its own statistics:

μn,cCI=1L​∑t=1LXn,t,c,\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}=\\frac\{1\}\{L\}\\sum\_\{t=1\}^\{L\}X\_\{n,t,c\},\(52\)vn,cCI=1L​∑t=1L\(Xn,t,c−μn,cCI\)2,v\_\{n,c\}^\{\\mathrm\{CI\}\}=\\frac\{1\}\{L\}\\sum\_\{t=1\}^\{L\}\\left\(X\_\{n,t,c\}\-\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}\\right\)^\{2\},\(53\)and

sn,cCI=vn,cCI\+ϵn,c\>0\.s\_\{n,c\}^\{\\mathrm\{CI\}\}=\\sqrt\{v\_\{n,c\}^\{\\mathrm\{CI\}\}\+\\epsilon\_\{n,c\}\}\>0\.\(54\)The channel\-specific transformation is

Tn,cCI​\(x\)=x−μn,cCIsn,cCI\.T\_\{n,c\}^\{\\mathrm\{CI\}\}\(x\)=\\frac\{x\-\\mu\_\{n,c\}^\{\\mathrm\{CI\}\}\}\{s\_\{n,c\}^\{\\mathrm\{CI\}\}\}\.\(55\)
###### Proposition B\.3\(CI normalization does not guarantee OHLC order\)\.

There exist valid OHLC values satisfyingH\>OH\>Ofor which

Tn,HCI​\(H\)<Tn,OCI​\(O\)\.T\_\{n,H\}^\{\\mathrm\{CI\}\}\(H\)<T\_\{n,O\}^\{\\mathrm\{CI\}\}\(O\)\.\(56\)Therefore, validity in original coordinates does not imply validity in CI\-normalized coordinates\.

###### Proof\.

Consider

O=10,H=11,O=10,\\qquad H=11,\(57\)so thatH\>OH\>O\. Let the channel statistics be

μO=9,sO=0\.5,μH=10\.8,sH=0\.2\.\\mu\_\{O\}=9,\\qquad s\_\{O\}=0\.5,\\qquad\\mu\_\{H\}=10\.8,\\qquad s\_\{H\}=0\.2\.\(58\)The normalized values are

O~CI=10−90\.5=2,\\widetilde\{O\}^\{\\mathrm\{CI\}\}=\\frac\{10\-9\}\{0\.5\}=2,\(59\)and

H~CI=11−10\.80\.2=1\.\\widetilde\{H\}^\{\\mathrm\{CI\}\}=\\frac\{11\-10\.8\}\{0\.2\}=1\.\(60\)Hence

H\>ObutH~CI<O~CI\.H\>O\\qquad\\text\{but\}\\qquad\\widetilde\{H\}^\{\\mathrm\{CI\}\}<\\widetilde\{O\}^\{\\mathrm\{CI\}\}\.\(61\)The original ordering is therefore not guaranteed\. ∎

### Mathematical Analysis of Dynamic Epsilon

Letx1,…,xLx\_\{1\},\\ldots,x\_\{L\}denote one context sequence with empirical mean and variance

μ=1L​∑t=1Lxt,\\mu=\\frac\{1\}\{L\}\\sum\_\{t=1\}^\{L\}x\_\{t\},\(63\)v=1L​∑t=1L\(xt−μ\)2\.v=\\frac\{1\}\{L\}\\sum\_\{t=1\}^\{L\}\(x\_\{t\}\-\\mu\)^\{2\}\.\(64\)
The fixed\-epsilon normalization is

x~tfix=xt−μv\+ϵ0,ϵ0=10−5\.\\widetilde\{x\}\_\{t\}^\{\\mathrm\{fix\}\}=\\frac\{x\_\{t\}\-\\mu\}\{\\sqrt\{v\+\\epsilon\_\{0\}\}\},\\qquad\\epsilon\_\{0\}=10^\{\-5\}\.\(65\)CryptoL uses

ϵdyn=λ⁡\(μ2\+δ\),λ=10−5,δ=10−12,\\epsilon^\{\\mathrm\{dyn\}\}=\\lambda\\left\(\\mu^\{2\}\+\\delta\\right\),\\qquad\\lambda=10^\{\-5\},\\qquad\\delta=10^\{\-12\},\(66\)and

x~tdyn=xt−μv\+λ⁡\(μ2\+δ\)\.\\widetilde\{x\}\_\{t\}^\{\\mathrm\{dyn\}\}=\\frac\{x\_\{t\}\-\\mu\}\{\\sqrt\{v\+\\lambda\(\\mu^\{2\}\+\\delta\)\}\}\.\(67\)
###### Proposition B\.6\(Non\-equivariance of fixed epsilon\)\.

Let

xt′=a​xt,a\>0\.x\_\{t\}^\{\\prime\}=ax\_\{t\},\\qquad a\>0\.\(68\)Then fixed\-epsilon normalization satisfies

x~t′fix=xt−μv\+ϵ0/a2\.\\widetilde\{x\}\_\{t\}^\{\\prime\\,\\mathrm\{fix\}\}=\\frac\{x\_\{t\}\-\\mu\}\{\\sqrt\{v\+\\epsilon\_\{0\}/a^\{2\}\}\}\.\(69\)Consequently,

x~t′fix≠x~tfix\\widetilde\{x\}\_\{t\}^\{\\prime\\,\\mathrm\{fix\}\}\\neq\\widetilde\{x\}\_\{t\}^\{\\mathrm\{fix\}\}\(70\)in general\.

###### Proof\.

Under[Equation68](https://arxiv.org/html/2609.11206#A2.E68),

μ′=a​μ,v′=a2​v\.\\mu^\{\\prime\}=a\\mu,\\qquad v^\{\\prime\}=a^\{2\}v\.\(71\)Therefore,

x~t′fix\\displaystyle\\widetilde\{x\}\_\{t\}^\{\\prime\\,\\mathrm\{fix\}\}=a​xt−a​μa2​v\+ϵ0\\displaystyle=\\frac\{ax\_\{t\}\-a\\mu\}\{\\sqrt\{a^\{2\}v\+\\epsilon\_\{0\}\}\}=a⁡\(xt−μ\)a​v\+ϵ0/a2\\displaystyle=\\frac\{a\(x\_\{t\}\-\\mu\)\}\{a\\sqrt\{v\+\\epsilon\_\{0\}/a^\{2\}\}\}=xt−μv\+ϵ0/a2,\\displaystyle=\\frac\{x\_\{t\}\-\\mu\}\{\\sqrt\{v\+\\epsilon\_\{0\}/a^\{2\}\}\},\(72\)wherea\>0a\>0was used\. The denominator depends onaa, proving the claim\. ∎

###### Theorem B\.7\(Approximate scale equivariance of dynamic epsilon\)\.

Under the rescaling in[Equation68](https://arxiv.org/html/2609.11206#A2.E68), dynamic\-epsilon normalization satisfies

x~t′dyn=xt−μv\+λ​μ2\+λ​δ/a2\.\\widetilde\{x\}\_\{t\}^\{\\prime\\,\\mathrm\{dyn\}\}=\\frac\{x\_\{t\}\-\\mu\}\{\\sqrt\{v\+\\lambda\\mu^\{2\}\+\\lambda\\delta/a^\{2\}\}\}\.\(73\)Ifδ=0\\delta=0, then

x~t′dyn=x~tdyn\\widetilde\{x\}\_\{t\}^\{\\prime\\,\\mathrm\{dyn\}\}=\\widetilde\{x\}\_\{t\}^\{\\mathrm\{dyn\}\}\(74\)for everya\>0a\>0\. Forδ\>0\\delta\>0, the transformation is approximately equivariant whenever

δ≪μ2andδa2≪v\+λ​μ2\.\\delta\\ll\\mu^\{2\}\\qquad\\text\{and\}\\qquad\\frac\{\\delta\}\{a^\{2\}\}\\ll v\+\\lambda\\mu^\{2\}\.\(75\)

###### Proof\.

Using[Equation71](https://arxiv.org/html/2609.11206#A2.E71),

x~t′dyn\\displaystyle\\widetilde\{x\}\_\{t\}^\{\\prime\\,\\mathrm\{dyn\}\}=a⁡\(xt−μ\)a2​v\+λ⁡\(a2​μ2\+δ\)\\displaystyle=\\frac\{a\(x\_\{t\}\-\\mu\)\}\{\\sqrt\{a^\{2\}v\+\\lambda\(a^\{2\}\\mu^\{2\}\+\\delta\)\}\}=a⁡\(xt−μ\)a​v\+λ​μ2\+λ​δ/a2\\displaystyle=\\frac\{a\(x\_\{t\}\-\\mu\)\}\{a\\sqrt\{v\+\\lambda\\mu^\{2\}\+\\lambda\\delta/a^\{2\}\}\}=xt−μv\+λ​μ2\+λ​δ/a2,\\displaystyle=\\frac\{x\_\{t\}\-\\mu\}\{\\sqrt\{v\+\\lambda\\mu^\{2\}\+\\lambda\\delta/a^\{2\}\}\},\(76\)which proves[Equation73](https://arxiv.org/html/2609.11206#A2.E73)\. Ifδ=0\\delta=0, the right\-hand side equals[Equation67](https://arxiv.org/html/2609.11206#A2.E67)\. Forδ\>0\\delta\>0, the only scale\-dependent difference is the termλ​δ/a2\\lambda\\delta/a^\{2\}, which is negligible under[Equation75](https://arxiv.org/html/2609.11206#A2.E75)\. ∎

###### Proposition B\.8\(Dependence on relative variability\)\.

Assumeμ≠0\\mu\\neq 0, and define the squared coefficient of variation as

CV2=vμ2\.\\operatorname\{CV\}^\{2\}=\\frac\{v\}\{\\mu^\{2\}\}\.\(77\)Ignoring the negligibleδ\\deltaterm, the ratio between the dynamic regularizer and the empirical variance is

ϵdynv=λCV2\.\\frac\{\\epsilon^\{\\mathrm\{dyn\}\}\}\{v\}=\\frac\{\\lambda\}\{\\operatorname\{CV\}^\{2\}\}\.\(78\)Thus, for assets with comparable relative volatility, the effect of dynamic epsilon is approximately independent of their absolute quoted price\.

###### Proof\.

From[Equation66](https://arxiv.org/html/2609.11206#A2.E66), withδ\\deltaomitted,

ϵdynv=λ​μ2v=λv/μ2=λCV2\.\\frac\{\\epsilon^\{\\mathrm\{dyn\}\}\}\{v\}=\\frac\{\\lambda\\mu^\{2\}\}\{v\}=\\frac\{\\lambda\}\{v/\\mu^\{2\}\}=\\frac\{\\lambda\}\{\\operatorname\{CV\}^\{2\}\}\.\(79\)∎

### Jacobian Analysis of Scale\-Dominance Mitigation

Let

𝐗~n=𝒩n​\(𝐗n\)\\widetilde\{\\mathbf\{X\}\}\_\{n\}=\\mathcal\{N\}\_\{n\}\(\\mathbf\{X\}\_\{n\}\)\(81\)denote the normalized context, and let the forecasting backbone produce

𝐳^n=f𝜽​\(𝐗~n\)\.\\widehat\{\\mathbf\{z\}\}\_\{n\}=f\_\{\\bm\{\\theta\}\}\\left\(\\widetilde\{\\mathbf\{X\}\}\_\{n\}\\right\)\.\(82\)The normalized target and normalized residual are

𝐳n=𝐃n−1​\(𝐲n−𝝁n\),𝐞n​\(𝜽\)=𝐳^n−𝐳n,\\mathbf\{z\}\_\{n\}=\\mathbf\{D\}\_\{n\}^\{\-1\}\\left\(\\mathbf\{y\}\_\{n\}\-\\bm\{\\mu\}\_\{n\}\\right\),\\qquad\\mathbf\{e\}\_\{n\}\(\\bm\{\\theta\}\)=\\widehat\{\\mathbf\{z\}\}\_\{n\}\-\\mathbf\{z\}\_\{n\},\(83\)where𝐃n≻0\\mathbf\{D\}\_\{n\}\\succ 0is diagonal\. The reconstructed prediction is

𝐲^n=𝝁n\+𝐃n​𝐳^n\.\\widehat\{\\mathbf\{y\}\}\_\{n\}=\\bm\{\\mu\}\_\{n\}\+\\mathbf\{D\}\_\{n\}\\widehat\{\\mathbf\{z\}\}\_\{n\}\.\(84\)
The normalized prediction Jacobian is

𝐉n=∂𝐳^n∂𝜽∈ℝd×p\.\\mathbf\{J\}\_\{n\}=\\frac\{\\partial\\widehat\{\\mathbf\{z\}\}\_\{n\}\}\{\\partial\\bm\{\\theta\}\}\\in\\mathbb\{R\}^\{d\\times p\}\.\(85\)
###### Theorem B\.10\(Residual\-Jacobian scaling under original\-space RevIN\)\.

Assume that𝛍n\\bm\{\\mu\}\_\{n\}and𝐃n\\mathbf\{D\}\_\{n\}are independent of𝛉\\bm\{\\theta\}\. Define the original\-space residual

𝐫nRaw=𝐲^n−𝐲n\.\\mathbf\{r\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\widehat\{\\mathbf\{y\}\}\_\{n\}\-\\mathbf\{y\}\_\{n\}\.\(86\)Then

𝐫nRaw=𝐃n​𝐞n,\\mathbf\{r\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\},\(87\)and the Jacobian of the original\-space residual is

𝐉nRaw=∂𝐫nRaw∂𝜽=𝐃n​𝐉n\.\\mathbf\{J\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\frac\{\\partial\\mathbf\{r\}\_\{n\}^\{\\mathrm\{Raw\}\}\}\{\\partial\\bm\{\\theta\}\}=\\mathbf\{D\}\_\{n\}\\mathbf\{J\}\_\{n\}\.\(88\)Consequently, the original\-space MSE gradient contribution is

𝐠nRaw=\(𝐉nRaw\)⊤​𝐫nRaw=𝐉n⊤​𝐃n2​𝐞n\.\\mathbf\{g\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\left\(\\mathbf\{J\}\_\{n\}^\{\\mathrm\{Raw\}\}\\right\)^\{\\top\}\\mathbf\{r\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{2\}\\mathbf\{e\}\_\{n\}\.\(89\)In contrast, the TP\-RevIN gradient contribution is

𝐠nTP=𝐉n⊤​𝐞n\.\\mathbf\{g\}\_\{n\}^\{\\mathrm\{TP\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}\.\(90\)

###### Proof\.

From[Equation84](https://arxiv.org/html/2609.11206#A2.E84)and the identity

𝐲n=𝝁n\+𝐃n​𝐳n,\\mathbf\{y\}\_\{n\}=\\bm\{\\mu\}\_\{n\}\+\\mathbf\{D\}\_\{n\}\\mathbf\{z\}\_\{n\},\(91\)we obtain

𝐫nRaw\\displaystyle\\mathbf\{r\}\_\{n\}^\{\\mathrm\{Raw\}\}=𝝁n\+𝐃n​𝐳^n−𝝁n−𝐃n​𝐳n\\displaystyle=\\bm\{\\mu\}\_\{n\}\+\\mathbf\{D\}\_\{n\}\\widehat\{\\mathbf\{z\}\}\_\{n\}\-\\bm\{\\mu\}\_\{n\}\-\\mathbf\{D\}\_\{n\}\\mathbf\{z\}\_\{n\}=𝐃n​\(𝐳^n−𝐳n\)\\displaystyle=\\mathbf\{D\}\_\{n\}\\left\(\\widehat\{\\mathbf\{z\}\}\_\{n\}\-\\mathbf\{z\}\_\{n\}\\right\)=𝐃n​𝐞n,\\displaystyle=\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\},\(92\)proving[Equation87](https://arxiv.org/html/2609.11206#A2.E87)\. Since𝐃n\\mathbf\{D\}\_\{n\}is independent of𝜽\\bm\{\\theta\},

𝐉nRaw=∂\(𝐃n​𝐞n\)∂𝜽=𝐃n​∂𝐞n∂𝜽=𝐃n​𝐉n,\\mathbf\{J\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\frac\{\\partial\(\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\}\)\}\{\\partial\\bm\{\\theta\}\}=\\mathbf\{D\}\_\{n\}\\frac\{\\partial\\mathbf\{e\}\_\{n\}\}\{\\partial\\bm\{\\theta\}\}=\\mathbf\{D\}\_\{n\}\\mathbf\{J\}\_\{n\},\(93\)because the target𝐳n\\mathbf\{z\}\_\{n\}is also independent of𝜽\\bm\{\\theta\}\.

For a least\-squares objective, the per\-sample gradient is the transpose of the residual Jacobian multiplied by the residual:

𝐠nRaw\\displaystyle\\mathbf\{g\}\_\{n\}^\{\\mathrm\{Raw\}\}=\(𝐃n​𝐉n\)⊤​\(𝐃n​𝐞n\)\\displaystyle=\\left\(\\mathbf\{D\}\_\{n\}\\mathbf\{J\}\_\{n\}\\right\)^\{\\top\}\\left\(\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\}\\right\)=𝐉n⊤​𝐃n⊤​𝐃n​𝐞n\\displaystyle=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}\\mathbf\{e\}\_\{n\}=𝐉n⊤​𝐃n2​𝐞n,\\displaystyle=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}^\{2\}\\mathbf\{e\}\_\{n\},\(94\)where the last equality follows because𝐃n\\mathbf\{D\}\_\{n\}is diagonal and positive\. The TP\-RevIN result follows directly from the normalized residual𝐞n\\mathbf\{e\}\_\{n\}and Jacobian𝐉n\\mathbf\{J\}\_\{n\}\. ∎

###### Corollary B\.11\(Scalar\-scale gradient weighting\)\.

Under CD normalization,

𝐃n=sn​𝐈d,\\mathbf\{D\}\_\{n\}=s\_\{n\}\\mathbf\{I\}\_\{d\},\(95\)and therefore

𝐠nRaw=sn2​𝐉n⊤​𝐞n=sn2​𝐠nTP\.\\mathbf\{g\}\_\{n\}^\{\\mathrm\{Raw\}\}=s\_\{n\}^\{2\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}=s\_\{n\}^\{2\}\\mathbf\{g\}\_\{n\}^\{\\mathrm\{TP\}\}\.\(96\)Thus, for equal normalized residual\-Jacobian products, original\-space RevIN weights the sample contributions in proportion tosn2s\_\{n\}^\{2\}\.

#### Stacked Jacobian geometry

Stack the normalized residuals and Jacobians as

𝐞=col⁡\(𝐞1,…,𝐞N\),𝐉=col⁡\(𝐉1,…,𝐉N\)\.\\mathbf\{e\}=\\operatorname\{col\}\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{N\}\),\\qquad\\mathbf\{J\}=\\operatorname\{col\}\(\\mathbf\{J\}\_\{1\},\\ldots,\\mathbf\{J\}\_\{N\}\)\.\(98\)Define

𝐒=diag⁡\(𝐃1,…,𝐃N\)\.\\mathbf\{S\}=\\operatorname\{diag\}\(\\mathbf\{D\}\_\{1\},\\ldots,\\mathbf\{D\}\_\{N\}\)\.\(99\)Then

𝐫Raw=𝐒𝐞,𝐉Raw=𝐒𝐉,\\mathbf\{r\}^\{\\mathrm\{Raw\}\}=\\mathbf\{S\}\\mathbf\{e\},\\qquad\\mathbf\{J\}^\{\\mathrm\{Raw\}\}=\\mathbf\{S\}\\mathbf\{J\},\(100\)whereas TP\-RevIN uses

𝐫TP=𝐞,𝐉TP=𝐉\.\\mathbf\{r\}^\{\\mathrm\{TP\}\}=\\mathbf\{e\},\\qquad\\mathbf\{J\}^\{\\mathrm\{TP\}\}=\\mathbf\{J\}\.\(101\)
###### Theorem B\.13\(Conditional Jacobian conditioning result\)\.

Assume CD normalization and an orthogonal decomposition

ℝp=V1⊕⋯⊕VA\.\\mathbb\{R\}^\{p\}=V\_\{1\}\\oplus\\cdots\\oplus V\_\{A\}\.\(102\)Suppose the normalized Jacobian of assetaasatisfies

𝐉a⊤​𝐉a=𝐏a,\\mathbf\{J\}\_\{a\}^\{\\top\}\\mathbf\{J\}\_\{a\}=\\mathbf\{P\}\_\{a\},\(103\)where𝐏a\\mathbf\{P\}\_\{a\}is the orthogonal projector ontoVaV\_\{a\}, and

∑a=1A𝐏a=𝐈p\.\\sum\_\{a=1\}^\{A\}\\mathbf\{P\}\_\{a\}=\\mathbf\{I\}\_\{p\}\.\(104\)If assetaahas scalar scalesa\>0s\_\{a\}\>0, then

κ2​\(𝐉TP\)=1,\\kappa\_\{2\}\\left\(\\mathbf\{J\}^\{\\mathrm\{TP\}\}\\right\)=1,\(105\)whereas

κ2​\(𝐉Raw\)=smaxsmin\.\\kappa\_\{2\}\\left\(\\mathbf\{J\}^\{\\mathrm\{Raw\}\}\\right\)=\\frac\{s\_\{\\max\}\}\{s\_\{\\min\}\}\.\(106\)

###### Proof\.

Under the stated assumptions,

\(𝐉TP\)⊤​𝐉TP=∑a=1A𝐏a=𝐈p\.\\left\(\\mathbf\{J\}^\{\\mathrm\{TP\}\}\\right\)^\{\\top\}\\mathbf\{J\}^\{\\mathrm\{TP\}\}=\\sum\_\{a=1\}^\{A\}\\mathbf\{P\}\_\{a\}=\\mathbf\{I\}\_\{p\}\.\(107\)All singular values of𝐉TP\\mathbf\{J\}^\{\\mathrm\{TP\}\}are therefore equal to one\.

For original\-space RevIN,

\(𝐉Raw\)⊤​𝐉Raw=∑a=1Asa2​𝐏a\.\\left\(\\mathbf\{J\}^\{\\mathrm\{Raw\}\}\\right\)^\{\\top\}\\mathbf\{J\}^\{\\mathrm\{Raw\}\}=\\sum\_\{a=1\}^\{A\}s\_\{a\}^\{2\}\\mathbf\{P\}\_\{a\}\.\(108\)For any𝐮∈Va\\mathbf\{u\}\\in V\_\{a\},

\(∑b=1Asb2​𝐏b\)​𝐮=sa2​𝐮\.\\left\(\\sum\_\{b=1\}^\{A\}s\_\{b\}^\{2\}\\mathbf\{P\}\_\{b\}\\right\)\\mathbf\{u\}=s\_\{a\}^\{2\}\\mathbf\{u\}\.\(109\)Thus the singular values of𝐉Raw\\mathbf\{J\}^\{\\mathrm\{Raw\}\}are the valuessas\_\{a\}, with multiplicities determined bydim\(Va\)\\dim\(V\_\{a\}\)\. Their ratio issmax/smins\_\{\\max\}/s\_\{\\min\}\. ∎

### Hessian Analysis of Scale\-Dominance Mitigation

For samplenn, define the normalized\-space MSE

ℓnTP=12​‖𝐞n‖22\.\\ell\_\{n\}^\{\\mathrm\{TP\}\}=\\frac\{1\}\{2\}\\left\\lVert\\mathbf\{e\}\_\{n\}\\right\\rVert\_\{2\}^\{2\}\.\(110\)The original\-space RevIN MSE is

ℓnRaw=12​𝐞n⊤​𝐖n​𝐞n,𝐖n=𝐃n⊤​𝐃n=𝐃n2\.\\ell\_\{n\}^\{\\mathrm\{Raw\}\}=\\frac\{1\}\{2\}\\mathbf\{e\}\_\{n\}^\{\\top\}\\mathbf\{W\}\_\{n\}\\mathbf\{e\}\_\{n\},\\qquad\\mathbf\{W\}\_\{n\}=\\mathbf\{D\}\_\{n\}^\{\\top\}\\mathbf\{D\}\_\{n\}=\\mathbf\{D\}\_\{n\}^\{2\}\.\(111\)
Let

z^n,k\\widehat\{z\}\_\{n,k\}\(112\)denote output componentkk, and define its parameter Hessian as

𝐇n,kf=∇𝜽2z^n,k\.\\mathbf\{H\}\_\{n,k\}^\{f\}=\\nabla\_\{\\bm\{\\theta\}\}^\{2\}\\widehat\{z\}\_\{n,k\}\.\(113\)
###### Theorem B\.15\(Exact TP\-RevIN and original\-space Hessians\)\.

Assume𝐖n\\mathbf\{W\}\_\{n\}is independent of𝛉\\bm\{\\theta\}\. Then the exact TP\-RevIN Hessian is

𝐇nTP=𝐉n⊤​𝐉n\+∑k=1den,k​𝐇n,kf\.\\mathbf\{H\}\_\{n\}^\{\\mathrm\{TP\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{J\}\_\{n\}\+\\sum\_\{k=1\}^\{d\}e\_\{n,k\}\\mathbf\{H\}\_\{n,k\}^\{f\}\.\(114\)The exact original\-space RevIN Hessian is

𝐇nRaw=𝐉n⊤​𝐖n​𝐉n\+∑k=1d\(𝐖n​𝐞n\)k​𝐇n,kf\.\\mathbf\{H\}\_\{n\}^\{\\mathrm\{Raw\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{W\}\_\{n\}\\mathbf\{J\}\_\{n\}\+\\sum\_\{k=1\}^\{d\}\\left\(\\mathbf\{W\}\_\{n\}\\mathbf\{e\}\_\{n\}\\right\)\_\{k\}\\mathbf\{H\}\_\{n,k\}^\{f\}\.\(115\)Under CD normalization, where

𝐖n=sn2​𝐈d,\\mathbf\{W\}\_\{n\}=s\_\{n\}^\{2\}\\mathbf\{I\}\_\{d\},\(116\)the complete per\-sample Hessians satisfy

𝐇nRaw=sn2​𝐇nTP\.\\mathbf\{H\}\_\{n\}^\{\\mathrm\{Raw\}\}=s\_\{n\}^\{2\}\\mathbf\{H\}\_\{n\}^\{\\mathrm\{TP\}\}\.\(117\)

###### Proof\.

The TP\-RevIN gradient is

∇𝜽ℓnTP=𝐉n⊤​𝐞n\.\\nabla\_\{\\bm\{\\theta\}\}\\ell\_\{n\}^\{\\mathrm\{TP\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{e\}\_\{n\}\.\(118\)Differentiating by the product rule gives

∇𝜽2ℓnTP=𝐉n⊤​𝐉n\+∑k=1den,k​∇𝜽2z^n,k,\\nabla\_\{\\bm\{\\theta\}\}^\{2\}\\ell\_\{n\}^\{\\mathrm\{TP\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{J\}\_\{n\}\+\\sum\_\{k=1\}^\{d\}e\_\{n,k\}\\nabla\_\{\\bm\{\\theta\}\}^\{2\}\\widehat\{z\}\_\{n,k\},\(119\)which proves[Equation114](https://arxiv.org/html/2609.11206#A2.E114)\.

For original\-space RevIN,

∇𝜽ℓnRaw=𝐉n⊤​𝐖n​𝐞n\.\\nabla\_\{\\bm\{\\theta\}\}\\ell\_\{n\}^\{\\mathrm\{Raw\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{W\}\_\{n\}\\mathbf\{e\}\_\{n\}\.\(120\)Differentiating again gives a term from the derivative of the residual and a term from the derivative of the Jacobian:

∇𝜽2ℓnRaw=𝐉n⊤​𝐖n​𝐉n\+∑k=1d\(𝐖n​𝐞n\)k​𝐇n,kf,\\nabla\_\{\\bm\{\\theta\}\}^\{2\}\\ell\_\{n\}^\{\\mathrm\{Raw\}\}=\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{W\}\_\{n\}\\mathbf\{J\}\_\{n\}\+\\sum\_\{k=1\}^\{d\}\\left\(\\mathbf\{W\}\_\{n\}\\mathbf\{e\}\_\{n\}\\right\)\_\{k\}\\mathbf\{H\}\_\{n,k\}^\{f\},\(121\)proving[Equation115](https://arxiv.org/html/2609.11206#A2.E115)\.

If𝐖n=sn2​𝐈d\\mathbf\{W\}\_\{n\}=s\_\{n\}^\{2\}\\mathbf\{I\}\_\{d\}, then

𝐇nRaw\\displaystyle\\mathbf\{H\}\_\{n\}^\{\\mathrm\{Raw\}\}=sn2​𝐉n⊤​𝐉n\+sn2​∑k=1den,k​𝐇n,kf\\displaystyle=s\_\{n\}^\{2\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{J\}\_\{n\}\+s\_\{n\}^\{2\}\\sum\_\{k=1\}^\{d\}e\_\{n,k\}\\mathbf\{H\}\_\{n,k\}^\{f\}=sn2​𝐇nTP,\\displaystyle=s\_\{n\}^\{2\}\\mathbf\{H\}\_\{n\}^\{\\mathrm\{TP\}\},\(122\)which proves[Equation117](https://arxiv.org/html/2609.11206#A2.E117)\. ∎

#### Gauss–Newton curvature

The Gauss–Newton matrices retain the positive\-semidefinite first term of the exact Hessians:

𝐆TP=1N​∑n=1N𝐉n⊤​𝐉n,\\mathbf\{G\}^\{\\mathrm\{TP\}\}=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{J\}\_\{n\},\(123\)and

𝐆Raw=1N​∑n=1N𝐉n⊤​𝐖n​𝐉n\.\\mathbf\{G\}^\{\\mathrm\{Raw\}\}=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{W\}\_\{n\}\\mathbf\{J\}\_\{n\}\.\(124\)
Under scalar CD scales,

𝐆Raw=1N​∑n=1Nsn2​𝐉n⊤​𝐉n\.\\mathbf\{G\}^\{\\mathrm\{Raw\}\}=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}s\_\{n\}^\{2\}\\mathbf\{J\}\_\{n\}^\{\\top\}\\mathbf\{J\}\_\{n\}\.\(125\)
###### Theorem B\.17\(Loewner and eigenvalue bounds\)\.

Assume scalar scales satisfying

0<smin≤sn≤smax<∞\.0<s\_\{\\min\}\\leq s\_\{n\}\\leq s\_\{\\max\}<\\infty\.\(126\)Then

smin2​𝐆TP⪯𝐆Raw⪯smax2​𝐆TP\.s\_\{\\min\}^\{2\}\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\preceq\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\preceq s\_\{\\max\}^\{2\}\\mathbf\{G\}^\{\\mathrm\{TP\}\}\.\(127\)On any common parameter subspace on which both matrices are positive definite,

λmin​\(𝐆Raw\)≥smin2​λmin​\(𝐆TP\),\\lambda\_\{\\min\}\\left\(\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\right\)\\geq s\_\{\\min\}^\{2\}\\lambda\_\{\\min\}\\left\(\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\right\),\(128\)λmax​\(𝐆Raw\)≤smax2​λmax​\(𝐆TP\),\\lambda\_\{\\max\}\\left\(\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\right\)\\leq s\_\{\\max\}^\{2\}\\lambda\_\{\\max\}\\left\(\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\right\),\(129\)and

κ⁡\(𝐆Raw\)≤\(smaxsmin\)2​κ​\(𝐆TP\)\.\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\right\)\\leq\\left\(\\frac\{s\_\{\\max\}\}\{s\_\{\\min\}\}\\right\)^\{2\}\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\right\)\.\(130\)

###### Proof\.

For any𝐮∈ℝp\\mathbf\{u\}\\in\\mathbb\{R\}^\{p\},

𝐮⊤​𝐆Raw​𝐮\\displaystyle\\mathbf\{u\}^\{\\top\}\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\mathbf\{u\}=1N​∑n=1Nsn2​‖𝐉n​𝐮‖22\.\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}s\_\{n\}^\{2\}\\left\\lVert\\mathbf\{J\}\_\{n\}\\mathbf\{u\}\\right\\rVert\_\{2\}^\{2\}\.\(131\)Applying[Equation126](https://arxiv.org/html/2609.11206#A2.E126)termwise gives

smin2​1N​∑n=1N‖𝐉n​𝐮‖22\\displaystyle s\_\{\\min\}^\{2\}\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\left\\lVert\\mathbf\{J\}\_\{n\}\\mathbf\{u\}\\right\\rVert\_\{2\}^\{2\}≤𝐮⊤​𝐆Raw​𝐮\\displaystyle\\leq\\mathbf\{u\}^\{\\top\}\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\mathbf\{u\}≤smax2​1N​∑n=1N‖𝐉n​𝐮‖22\.\\displaystyle\\leq s\_\{\\max\}^\{2\}\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\left\\lVert\\mathbf\{J\}\_\{n\}\\mathbf\{u\}\\right\\rVert\_\{2\}^\{2\}\.\(132\)The outer expressions are

smin2​𝐮⊤​𝐆TP​𝐮s\_\{\\min\}^\{2\}\\mathbf\{u\}^\{\\top\}\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\mathbf\{u\}\(133\)and

smax2​𝐮⊤​𝐆TP​𝐮,s\_\{\\max\}^\{2\}\\mathbf\{u\}^\{\\top\}\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\mathbf\{u\},\(134\)which proves the Loewner bounds\. The eigenvalue inequalities follow from the Rayleigh quotient, and their ratio gives[Equation130](https://arxiv.org/html/2609.11206#A2.E130)\. ∎

###### Theorem B\.18\(Exact scale\-induced conditioning under orthogonal asset subspaces\)\.

Assume the parameter space decomposes into mutually orthogonal subspacesV1,…,VAV\_\{1\},\\ldots,V\_\{A\}, and suppose the normalized Gauss–Newton operator for assetaais

𝐆a=λa​𝐏a,λa\>0,\\mathbf\{G\}\_\{a\}=\\lambda\_\{a\}\\mathbf\{P\}\_\{a\},\\qquad\\lambda\_\{a\}\>0,\(135\)where𝐏a\\mathbf\{P\}\_\{a\}projects ontoVaV\_\{a\}\. Let the asset sampling probabilities beπa\>0\\pi\_\{a\}\>0\. Then

𝐆TP=⨁a=1Aπa​λa​𝐈Va,\\mathbf\{G\}^\{\\mathrm\{TP\}\}=\\bigoplus\_\{a=1\}^\{A\}\\pi\_\{a\}\\lambda\_\{a\}\\mathbf\{I\}\_\{V\_\{a\}\},\(136\)and

𝐆Raw=⨁a=1Aπa​sa2​λa​𝐈Va\.\\mathbf\{G\}^\{\\mathrm\{Raw\}\}=\\bigoplus\_\{a=1\}^\{A\}\\pi\_\{a\}s\_\{a\}^\{2\}\\lambda\_\{a\}\\mathbf\{I\}\_\{V\_\{a\}\}\.\(137\)Consequently,

κ⁡\(𝐆TP\)=maxa⁡πa​λamina⁡πa​λa,\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\right\)=\\frac\{\\max\_\{a\}\\pi\_\{a\}\\lambda\_\{a\}\}\{\\min\_\{a\}\\pi\_\{a\}\\lambda\_\{a\}\},\(138\)whereas

κ⁡\(𝐆Raw\)=maxa⁡πa​sa2​λamina⁡πa​sa2​λa\.\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\right\)=\\frac\{\\max\_\{a\}\\pi\_\{a\}s\_\{a\}^\{2\}\\lambda\_\{a\}\}\{\\min\_\{a\}\\pi\_\{a\}s\_\{a\}^\{2\}\\lambda\_\{a\}\}\.\(139\)Ifπa\\pi\_\{a\}andλa\\lambda\_\{a\}are equal across assets, then

κ⁡\(𝐆Raw\)=\(smaxsmin\)2,κ⁡\(𝐆TP\)=1\.\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\right\)=\\left\(\\frac\{s\_\{\\max\}\}\{s\_\{\\min\}\}\\right\)^\{2\},\\qquad\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\right\)=1\.\(140\)

###### Proof\.

Because the subspaces are mutually orthogonal, the aggregate Gauss–Newton matrices are block diagonal\. The eigenvalue associated withVaV\_\{a\}isπa​λa\\pi\_\{a\}\\lambda\_\{a\}under TP\-RevIN andπa​sa2​λa\\pi\_\{a\}s\_\{a\}^\{2\}\\lambda\_\{a\}under original\-space RevIN\. The condition numbers are therefore the ratios of the largest and smallest block eigenvalues, proving[Equations138](https://arxiv.org/html/2609.11206#A2.E138)and[139](https://arxiv.org/html/2609.11206#A2.E139)\. Equalπa\\pi\_\{a\}andλa\\lambda\_\{a\}reduce these expressions to[Equation140](https://arxiv.org/html/2609.11206#A2.E140)\. ∎

###### Example B\.20\(Scale weighting may leave conditioning unchanged\)\.

Let

𝐆1=𝐆2=𝐈p\.\\mathbf\{G\}\_\{1\}=\\mathbf\{G\}\_\{2\}=\\mathbf\{I\}\_\{p\}\.\(142\)Then

𝐆TP=2​𝐈p,𝐆Raw=\(s12\+s22\)​𝐈p\.\\mathbf\{G\}^\{\\mathrm\{TP\}\}=2\\mathbf\{I\}\_\{p\},\\qquad\\mathbf\{G\}^\{\\mathrm\{Raw\}\}=\(s\_\{1\}^\{2\}\+s\_\{2\}^\{2\}\)\\mathbf\{I\}\_\{p\}\.\(143\)Both matrices have condition number one, even whens1≠s2s\_\{1\}\\neq s\_\{2\}\. Thus, scale heterogeneity does not necessarily worsen conditioning when all samples excite identical isotropic parameter directions\.

###### Example B\.21\(Scale weighting can compensate for intrinsic curvature\)\.

Let

𝐆1=\[1000\],𝐆2=\[000100\]\.\\mathbf\{G\}\_\{1\}=\\begin\{bmatrix\}1&0\\\\ 0&0\\end\{bmatrix\},\\qquad\\mathbf\{G\}\_\{2\}=\\begin\{bmatrix\}0&0\\\\ 0&100\\end\{bmatrix\}\.\(144\)Then

𝐆TP=\[100100\],κ⁡\(𝐆TP\)=100\.\\mathbf\{G\}^\{\\mathrm\{TP\}\}=\\begin\{bmatrix\}1&0\\\\ 0&100\\end\{bmatrix\},\\qquad\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{TP\}\}\\right\)=100\.\(145\)Choosing

s12=100,s22=1s\_\{1\}^\{2\}=100,\\qquad s\_\{2\}^\{2\}=1\(146\)gives

𝐆Raw=\[10000100\],κ⁡\(𝐆Raw\)=1\.\\mathbf\{G\}^\{\\mathrm\{Raw\}\}=\\begin\{bmatrix\}100&0\\\\ 0&100\\end\{bmatrix\},\\qquad\\kappa\\left\(\\mathbf\{G\}^\{\\mathrm\{Raw\}\}\\right\)=1\.\(147\)This example confirms that universal conditioning superiority cannot be claimed\. TP\-RevIN provides scale neutrality, not an unconditional minimum\-condition\-number guarantee\.

## Appendix CDataset Details and Volatility Analysis

This section provides a detailed breakdown of the large\-scale cryptocurrency forecasting dataset utilized in our empirical evaluations\. The dataset consists of historical Open\-High\-Low\-Close \(OHLC\) observations collected via the Binance API\. In total, the dataset contains15,490,30015,490,300data rows spanning1616heterogeneous assets across77distinct temporal resolutions \(5\-minute, 15\-minute, 30\-minute, 1\-hour, 2\-hour, 4\-hour, and 1\-day intervals\), yielding a total of112112distinct time\-series panels\.

### Dataset Overview and Coverage

The temporal coverage and active periods for each of the1616selected cryptocurrency assets are detailed in Table[5](https://arxiv.org/html/2609.11206#A3.T5)\. The assets cover highly mature coins such as Bitcoin \(BTC\) and Ethereum \(ETH\) with histories dating back to 2017, alongside more recently launched tokens such as SUI and TON, which provide evaluation paths for lower\-history, high\-volatility regimes\.

Table 5:Temporal coverage per cryptocurrency asset in the dataset\.The sample density varies across timeframes\. Table[6](https://arxiv.org/html/2609.11206#A3.T6)outlines the exact number of recorded data rows for each asset across the seven temporal resolutions\.

Table 6:Observed data row counts partitioned by coin and time frame\.
### Empirical Volatility Analysis and Optimization Difficulty

To understand the optimization challenges across different settings, we analyze the volatility profiles of the dataset\. Volatility is defined here as the absolute percentage change in the close price of a candle relative to the preceding candle:

vt=\|Ct−Ct−1\|Ct−1×100%v\_\{t\}=\\frac\{\|C\_\{t\}\-C\_\{t\-1\}\|\}\{C\_\{t\-1\}\}\\times 100\\%\(148\)We evaluate the proportion of candles exceeding specified threshold levels \(\>1%\>1\\%,\>2%\>2\\%,\>3%\>3\\%,\>4%\>4\\%, and\>5%\>5\\%\)\.

Table[7](https://arxiv.org/html/2609.11206#A3.T7)reports the aggregate volatility thresholds across all sixteen assets\. As the time frame increases, the likelihood of substantial price changes scales non\-linearly\. For instance, only0\.99%0\.99\\%of candles in the55\-minute resolution experience a price deviation greater than1%1\\%, and a mere0\.01%0\.01\\%exceed5%5\\%\. Conversely, in the daily \(11\-day\) timeframe,37\.46%37\.46\\%of the candles exceed a1%1\\%change, and11\.33%11\.33\\%experience fluctuations larger than5%5\\%\.

Table 7:Aggregate volatility distribution across all assets partitioned by timeframe\.This disparity directly aligns with the empirical observations presented in the main paper\. As demonstrated in our experimental results, forecasting backbones experience a reduction in predictive accuracy \(higher MAE and MAPE metrics\) when operating on larger timeframes\. The high density of extreme returns on larger temporal resolutions represents a highly volatile dynamical regime with increased variance in the target distributions, presenting a fundamentally more challenging forecasting objective\.

### Asset\-Specific Volatility Distributions

The following tables report the granular, asset\-specific volatility distributions across all experimental resolutions\. To accommodate the size of the dataset without compilation issues, the records are divided into two distinct parts: Table[8](https://arxiv.org/html/2609.11206#A3.T8)covers assets from ADAUSDT to LTCUSDT, and Table[9](https://arxiv.org/html/2609.11206#A3.T9)covers assets from PEPEUSDT to XRPUSDT\.

Table 8:Granular volatility thresholds for assets ADAUSDT through LTCUSDT across experimental resolutions\.Table 9:Granular volatility thresholds for assets PEPEUSDT through XRPUSDT across experimental resolutions\.

## Appendix DExperimental Details

To facilitate full reproducibility, this section details the concrete training configurations, hyperparameters, and optimization protocols employed across the four primary experimental tables of the manuscript\.

For the evaluation presented inTable 1\(comparing models without normalization, standard RevIN, and our proposed TP\-RevIN\), all backbones are trained strictly using the standard Mean Squared Error \(MSE\) forecasting objective without the auxiliary physics constraint loss \(λphy=0\\lambda\_\{\\text\{phy\}\}=0\)\. The input context window is fixed toL=480L=480steps\. A separate, independent experiment is conducted for each timeframe \(55m,3030m\) and target forecast horizon \(H∈\{5,15\}H\\in\\\{5,15\\\}\)\. All configurations utilize a fixed numerical stabilization constant ofϵ=10−5\\epsilon=10^\{\-5\}in their normalization denominators\. The Full results of Table 1 in the main paper are presented in Table[10](https://arxiv.org/html/2609.11206#A4.T10)\.

The comparison against alternative temporal normalization baselines inTable 2\(FAN and SAN\) is conducted using the11h timeframe with a lookback window ofL=480L=480\. All models in this evaluation are trained using the MSE objective only, without any auxiliary physics\-informed losses\. To ensure a fair and optimal implementation of each baseline, we adopt their recommended training schedules:

- •FAN:The forecasting backbone and the frequency adaptive projection module are optimized jointly for33epochs per experiment\.
- •SAN:A decoupled, two\-stage training strategy is adopted\. First, the non\-stationary slice projection module is optimized independently for33epochs until convergence\. This projection module is subsequently frozen, and the main forecasting backbone is trained for11epoch\.
- •TP\-RevIN:The model is trained for11epoch using the scale\-adaptive dynamic epsilon formulation \(ϵdyn=10−5​\(μ2\+10−12\)\\epsilon^\{\\text\{dyn\}\}=10^\{\-5\}\(\\mu^\{2\}\+10^\{\-12\}\)\) to handle cross\-asset heterogeneity\.

For the numerical stabilization analysis inTable 3\(comparing fixed versus dynamic epsilon\), all configurations are evaluated on the22h timeframe with an input context length ofL=480L=480\. The models are trained strictly under the MSE objective to isolate the structural impact of denominator regularization on low\-valued assets \(e\.g\., SHIB\) versus high\-valued assets \(e\.g\., BTC\)\.

For the physical consistency evaluations inTable 4, all models are trained with a lookback window ofL=480L=480across different historical sampling intervals \(3030m and11h\)\. Under the standard RevIN configuration, the auxiliary candlestick constraint loss is evaluated in the original physical coordinate space, which reintroduces absolute context scales\. All of the evaluations are done under original space for fair comparison\.

Table 10:Comparison of Models with different Normalization Techniques\. Lower numbers indicate lower errors and better performances\.
Best metrics in each category are colored withRedand Second best metrics are colored withBlue\.