HARN：用于事件驱动多时间框架预测的分层关联共振网络

# HARN: Hierarchical Associative Resonance Network for Event-Driven Multi-Timeframe Forecasting
Source: [https://arxiv.org/html/2609.26822](https://arxiv.org/html/2609.26822)
Nabeel Ahmad Saidd

###### Abstract

Financial time series evolve across multiple temporal resolutions, challenging forecasting systems to incorporate newly available information without repeatedly recomputing unchanged representations\. We introduce HARN, a Hierarchical Associative Resonance Network for event\-driven multi\-timeframe forecasting\. HARN maintains persistent representations across temporal levels and updates each level only when its corresponding completed bar becomes available\. The architecture combines causal multi\-scale temporal encoding, gated associative memory, cross\-level resonance, and hierarchical evidence aggregation, with forecasting performed in basis\-point space and reconstructed to the original price scale\. We evaluate HARN on four assets spanning equity, foreign exchange, and commodity markets using multiple random seeds and component ablations\. HARN achieves competitive reconstructed\-price forecasting errors against single\-timeframe PatchTST and TimeXer baselines, while ablations reveal the effects of removing individual components across assets and timeframes\. A code\-level audit further examines consistency between the implementation and the defined event\-driven causal protocol\. The results position HARN as a persistent multi\-timeframe forecasting framework rather than evidence of universal predictive superiority\.

## 1Introduction

A forecaster operating on live financial data faces a temporal structure that standard sequence\-modeling benchmarks often abstract away: observations at different resolutions do not become available at the same time\. A five\-minute bar may close while the corresponding hourly or four\-hour bar is still forming\. Thus, the*event time*, when a forecast is requested, and the*representation time*, the timestamp of the latest completed bar at each timeframe, need not coincide above the finest resolution\. A forecasting system must therefore distinguish newly available information from representations that remain unchanged\. Reading a still\-forming higher\-timeframe bar or refreshing a coarse representation when no new bar has closed changes the information set available to the model and can introduce look\-ahead leakage\[[7](https://arxiv.org/html/2609.26822#bib.bib7),[8](https://arxiv.org/html/2609.26822#bib.bib8),[9](https://arxiv.org/html/2609.26822#bib.bib9)\]\.

This problem is particularly relevant in financial time series, where volatility clustering, changing conditional dependence, and heavy\-tailed return distributions make the statistical environment non\-stationary\[[2](https://arxiv.org/html/2609.26822#bib.bib2),[5](https://arxiv.org/html/2609.26822#bib.bib5),[6](https://arxiv.org/html/2609.26822#bib.bib6)\]\. Regime\-switching and state\-space models provide established approaches for representing evolving market dynamics\[[4](https://arxiv.org/html/2609.26822#bib.bib4),[3](https://arxiv.org/html/2609.26822#bib.bib3)\], while modern neural forecasting methods learn temporal representations directly from historical observations\. However, regardless of the modeling paradigm, a live multi\-timeframe system must first specify which information is available at each event and when each temporal representation is allowed to change\.

Most neural forecasting architectures operate on fixed windows and map those windows to future values\. Recurrent models maintain sequential state\[[21](https://arxiv.org/html/2609.26822#bib.bib21),[22](https://arxiv.org/html/2609.26822#bib.bib22)\], while state\-space models provide alternative mechanisms for persistent temporal representations\[[23](https://arxiv.org/html/2609.26822#bib.bib23),[24](https://arxiv.org/html/2609.26822#bib.bib24),[25](https://arxiv.org/html/2609.26822#bib.bib25)\]\. These approaches, however, generally advance their state whenever the input sequence advances and do not directly encode a hierarchy in which an hourly representation remains unchanged across several intervening five\-minute events\. Transformers provide flexible interactions across historical windows\[[14](https://arxiv.org/html/2609.26822#bib.bib14)\], but conventional attention over resampled or regularly indexed contexts does not by itself specify when asynchronous representations should be refreshed\. Similarly, multi\-timeframe feature aggregation through resampling, concatenation, or forward filling can represent coarse information on a common grid without explicitly distinguishing a newly completed bar from a carried\-forward value\. Independent models for each timeframe avoid this issue but do not provide an integrated mechanism for cross\-scale interaction\.

These observations motivate an event\-driven formulation in which each timeframe maintains its own representation and updates only when new information at that resolution becomes available\. Such a formulation requires three properties: causal alignment of observations, persistent state between updates, and an explicit rule governing communication between temporal levels\. Existing work provides relevant building blocks\. Financial econometric models describe evolving conditional dependence and volatility\[[4](https://arxiv.org/html/2609.26822#bib.bib4),[3](https://arxiv.org/html/2609.26822#bib.bib3)\]; neural forecasting architectures exploit long temporal contexts through attention, patching, decomposition, and frequency\-aware representations\[[15](https://arxiv.org/html/2609.26822#bib.bib15),[16](https://arxiv.org/html/2609.26822#bib.bib16),[17](https://arxiv.org/html/2609.26822#bib.bib17),[18](https://arxiv.org/html/2609.26822#bib.bib18),[19](https://arxiv.org/html/2609.26822#bib.bib19),[20](https://arxiv.org/html/2609.26822#bib.bib20)\]; and associative\-memory approaches provide mechanisms for learned key–value storage and retrieval\[[27](https://arxiv.org/html/2609.26822#bib.bib27),[28](https://arxiv.org/html/2609.26822#bib.bib28),[26](https://arxiv.org/html/2609.26822#bib.bib26),[29](https://arxiv.org/html/2609.26822#bib.bib29)\]\. These components motivate the design considered here, without implying that the present study establishes an exhaustive distinction from all prior multi\-timeframe or event\-driven systems\.

We introduce HARN, a Hierarchical Associative Resonance Network for event\-driven multi\-timeframe forecasting\. HARN treats the fastest configured timeframe as an anchor event stream and maintains a persistent state for every configured temporal level\. Each level uses a causal multi\-scale encoder, gated associative memory, and forecasting representation\. At an anchor event, the model receives the most recent completed window available at each level\. A level’s state and memory are updated only when its completed\-bar indicator fires; otherwise, its previous representation is carried forward\. States available at the same event can interact through cross\-level resonance, while a directional bottom\-up readout provides information from lower to higher temporal levels\. Forecasting is performed in basis\-point change space and reconstructed to the original price scale for evaluation\. This makes HARN an event\-driven state\-transition system whose neural components provide one parameterization of the underlying update protocol\.

The central contribution of this work is therefore the explicit formulation and implementation of persistent, asynchronously updated multi\-timeframe representations rather than a claim of universal predictive superiority\. We evaluate HARN on four assets, AAPL, EURUSD, USDCHF, and XAUUSD, using the supplied preprocessing, training, and evaluation pipeline\. The empirical analysis includes reconstructed\-price forecasting results, comparisons with single\-timeframe PatchTST and TimeXer baselines, and component ablations\. Because the baselines are single\-timeframe models trained on the anchor timeframe, their comparison with HARN is descriptive rather than an information\-matched architectural comparison; differences in input information, target formulation, and training procedure are therefore relevant to interpretation\. A code\-level audit is additionally used to examine consistency between the implementation and the defined event\-level causal protocol\. The audit is a static code review and is not an empirical test for leakage\.

The contributions of this paper are as follows:

1. 1\.We formalize an event\-driven multi\-timeframe forecasting protocol based on completed\-bar alignment, anchor events, persistent per\-level state, and asynchronous update indicators, including its batched training formulation\.
2. 2\.We develop HARN as a neural parameterization of this protocol, combining causal multi\-scale encoding, gated associative memory, cross\-level resonance, and directional bottom\-up evidence aggregation\.
3. 3\.We report the supplied forecasting and ablation results with explicit experimental scope and provenance, distinguishing descriptive baseline comparisons from component\-level observations\.
4. 4\.We provide a code\-level causal\-consistency audit and document the available preprocessing, configuration, checkpoint, and per\-seed result artifacts to support reproducibility\.

The remainder of the paper first reviews related work, then formalizes the forecasting protocol and HARN architecture\. The experimental protocol, forecasting results, and ablation analysis follow, after which the implementation audit and discussion of limitations are presented\. The paper concludes with directions for further empirical validation\.

## 2Related Work

Financial forecasting begins from a setting in which the statistical properties of the observed process are neither fixed nor necessarily stable over time\. Prices, returns, volatility, and trading activity exhibit different forms of dependence, and the efficient\-markets literature provides a demanding benchmark under which persistent predictability must be supported empirically rather than assumed\[[1](https://arxiv.org/html/2609.26822#bib.bib1)\]\. Empirical financial returns exhibit heavy tails, volatility clustering, and changing dependence structures\[[2](https://arxiv.org/html/2609.26822#bib.bib2)\]\. ARCH and GARCH models formalize time\-varying conditional variance\[[5](https://arxiv.org/html/2609.26822#bib.bib5),[6](https://arxiv.org/html/2609.26822#bib.bib6)\], while state\-space and regime\-switching formulations explicitly allow the underlying data\-generating process to evolve over time\[[4](https://arxiv.org/html/2609.26822#bib.bib4),[3](https://arxiv.org/html/2609.26822#bib.bib3)\]\. These properties make evaluation protocol as important as model capacity: a forecasting system can exploit scale artifacts, normalization statistics, temporal dependence, or a favorable subperiod without learning a relationship that remains available at deployment time\. Consequently, random or otherwise inappropriate validation schemes can provide misleading estimates of forecasting performance in non\-stationary settings\[[9](https://arxiv.org/html/2609.26822#bib.bib9)\]\.

This concern extends beyond the choice of train–validation–test split\. Temporal leakage may arise through feature construction, normalization, resampling, target alignment, hyperparameter selection, or inadvertent reuse of future information\[[7](https://arxiv.org/html/2609.26822#bib.bib7)\]\. Forecast evaluation must therefore respect both the temporal dependence between forecast errors and the information set available at each forecast origin\[[8](https://arxiv.org/html/2609.26822#bib.bib8),[9](https://arxiv.org/html/2609.26822#bib.bib9)\]\. No single scale\-free error measure is uniformly appropriate across forecasting problems\[[10](https://arxiv.org/html/2609.26822#bib.bib10)\]; comparative forecast evaluation can additionally require tests based on loss differentials across dependent forecast origins, such as the Diebold–Mariano framework\[[11](https://arxiv.org/html/2609.26822#bib.bib11)\], while conditional predictive\-ability tests consider comparisons whose relative performance may depend on the available information set\[[12](https://arxiv.org/html/2609.26822#bib.bib12)\]\. When probabilistic forecasts are reported, proper scoring rules provide the corresponding formal basis for evaluating predictive distributions\[[13](https://arxiv.org/html/2609.26822#bib.bib13)\]\. The present work adopts the implementation\-level implications of this literature through chronological data separation, training\-only normalization, completed\-bar alignment, target isolation, causal feature construction, state\-reset boundaries, and validation\-only checkpoint selection\. These conditions are examined explicitly in the code\-level audit in[Section9](https://arxiv.org/html/2609.26822#S9)\. The forecasting target is expressed as a basis\-point change rather than an absolute price level, following the broader practice of modeling returns and related stationary transformations in financial time\-series analysis\[[3](https://arxiv.org/html/2609.26822#bib.bib3),[2](https://arxiv.org/html/2609.26822#bib.bib2)\]\. This choice is a design rationale rather than an experimentally established advantage here, since the reported experiments do not isolate target representation as an independent factor\.

Within this broader forecasting setting, neural sequence models have substantially expanded the class of temporal dependencies that can be represented\. DeepAR learns shared recurrent probabilistic models across related series\[[32](https://arxiv.org/html/2609.26822#bib.bib32)\], N\-BEATS constructs forecasts through backcast and forecast transformations\[[31](https://arxiv.org/html/2609.26822#bib.bib31)\], and Temporal Fusion Transformers combine recurrent processing, attention, and variable selection for multi\-horizon forecasting\[[33](https://arxiv.org/html/2609.26822#bib.bib33)\]\. The Transformer introduced global self\-attention as an alternative to recurrence\[[14](https://arxiv.org/html/2609.26822#bib.bib14)\], after which time\-series architectures explored sparse attention, decomposition, frequency\-domain representations, patch\-based tokenization, temporal reshaping, and cross\-attention mechanisms\. Informer reduces attention cost through sparse attention and distillation\[[15](https://arxiv.org/html/2609.26822#bib.bib15)\]; Autoformer and FEDformer introduce decomposition\-based approaches with autocorrelation or frequency\-enhanced attention\[[16](https://arxiv.org/html/2609.26822#bib.bib16),[17](https://arxiv.org/html/2609.26822#bib.bib17)\]; PatchTST represents time\-series segments as patches\[[18](https://arxiv.org/html/2609.26822#bib.bib18)\]; TimesNet maps temporal variation into a two\-dimensional representation\[[19](https://arxiv.org/html/2609.26822#bib.bib19)\]; and TimeXer uses cross\-attention to incorporate exogenous variables\[[20](https://arxiv.org/html/2609.26822#bib.bib20)\]\.

These architectures establish strong alternatives for temporal representation learning, but their usual forecasting protocol assumes a preconstructed sequence or covariate panel whose indices provide the sequence progression\. In a multi\-timeframe financial stream, however, different representations become newly informative at different event times\. A coarse bar may remain unchanged across several fine\-scale events, while a newly completed fine\-scale bar may require an immediate update\. Treating the resulting observations as a regular sequence therefore requires an explicit convention for when each representation is updated and how an inactive representation is carried forward\. HARN makes this convention part of the model specification rather than an external preprocessing assumption\. Each timeframe has a persistent state and an associated update indicator; when the indicator is inactive, the corresponding state and associative memory are preserved rather than overwritten \([Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)\)\. Cross\-level interaction is then performed through resonance \([Eq\.22](https://arxiv.org/html/2609.26822#S4.E22)\) and a defined lower\-to\-higher evidence pathway \([Eq\.23](https://arxiv.org/html/2609.26822#S4.E23)\)\. This distinction concerns the forecasting protocol and state\-transition semantics, not a rejection of attention\-based modeling: HARN itself uses attention\-like interactions across levels and within its evidence readout\.

Persistent state is also central to recurrent and state\-space approaches\. LSTM introduced gated memory mechanisms that substantially improved the practical representation of long\-range dependencies\[[21](https://arxiv.org/html/2609.26822#bib.bib21)\]\. Legendre Memory Units encode long histories through structured continuous\-time dynamics\[[22](https://arxiv.org/html/2609.26822#bib.bib22)\], while structured state\-space models provide explicit state\-transition formulations with efficient long\-sequence computation\[[23](https://arxiv.org/html/2609.26822#bib.bib23),[24](https://arxiv.org/html/2609.26822#bib.bib24)\]\. More recent selective state\-space models, including Mamba, condition state transitions on the input itself\[[25](https://arxiv.org/html/2609.26822#bib.bib25)\]\. HARN is related to this family through its use of compact persistent states, but differs in making the update event itself an explicit component of the transition rule\. Conventional recurrent formulations ordinarily advance a state at each sequence index; HARN instead applies a per\-timeframe mask so that a level can remain unchanged while other levels receive new observations \([Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)\)\. Its state is consequently a coupled collection of level\-specific recurrences operating under an event\-conditioned update schedule\. State\-space and recurrent models therefore provide relevant architectural alternatives, but they are not tested controls for the specific event\-conditioned state\-transition mechanism studied here\.

A second relevant line of work concerns associative memory\. Hopfield networks established content\-addressable memory through an energy\-based attractor formulation\[[26](https://arxiv.org/html/2609.26822#bib.bib26)\]\. Fast\-weight methods subsequently introduced temporary key–value structures that can be updated to encode recent context\[[27](https://arxiv.org/html/2609.26822#bib.bib27)\], while linear Transformers admit an interpretation in terms of fast\-weight programming and associative updates\[[28](https://arxiv.org/html/2609.26822#bib.bib28)\]\. Modern Hopfield networks further connect associative retrieval with attention\-like mechanisms and substantially extend the representational capacity of the original formulation\[[29](https://arxiv.org/html/2609.26822#bib.bib29)\]\. HARN uses a narrower mechanism designed for its event\-driven setting\. Its encoder and cross\-scale context produce a normalized key, value, and query; the difference between the current value and the retrieved memory content forms a retrieval residual, referred to as “surprise” in[Section4\.2](https://arxiv.org/html/2609.26822#S4.SS2), which controls a bounded outer\-product write \([Eq\.17](https://arxiv.org/html/2609.26822#S4.E17)\)\. The mechanism therefore separates retrieval from writing rather than treating memory solely as another recurrent hidden\-state transformation\. The A1 ablation replaces this associative cell with a gated recurrent\-style update, but the comparison is not parameter\-count matched and therefore does not isolate memory capacity as a single causal factor \([Section8](https://arxiv.org/html/2609.26822#S8)\)\.

The temporal hierarchy considered by HARN is also related to hierarchical forecasting, although the objectives differ\. Hierarchical forecasting traditionally concerns forecasts defined at multiple aggregation levels and, in many formulations, their reconciliation into a coherent collection of forecasts\. Optimal\-combination methods construct forecasts while respecting aggregation relationships\[[30](https://arxiv.org/html/2609.26822#bib.bib30)\], while neural forecasting architectures such as N\-BEATS and Temporal Fusion Transformers learn multi\-scale or multi\-horizon representations\[[31](https://arxiv.org/html/2609.26822#bib.bib31),[33](https://arxiv.org/html/2609.26822#bib.bib33)\]\. Decomposition\-based Transformer models similarly separate temporal variation at different scales\[[16](https://arxiv.org/html/2609.26822#bib.bib16),[17](https://arxiv.org/html/2609.26822#bib.bib17)\]\. HARN uses the term “hierarchical” in a related but distinct sense\. Its hierarchy consists of timeframes whose bars complete at different rates, and the hierarchy determines when a persistent representation is eligible for a state transition rather than imposing an output\-side aggregation constraint\. A coarse representation is not merely a lower\-frequency forecasting head: it retains its own state while receiving contemporaneous cross\-level context and lower\-level evidence\. Conversely, simply repeating the most recent coarse value does not record whether a new coarse bar has actually completed\. HARN therefore treats completion status as explicit information through the update indicators\. The reported experiments do not compare this mechanism with reconciled hierarchical forecasting methods or with explicitly matched multi\-scale neural baselines\.

Irregular and event\-driven sequence models provide another point of comparison\. Latent ODEs model hidden trajectories at arbitrary observation times by integrating a learned vector field\[[34](https://arxiv.org/html/2609.26822#bib.bib34)\], while Neural Controlled Differential Equations treat an observed path as a control signal driving continuous hidden dynamics\[[35](https://arxiv.org/html/2609.26822#bib.bib35)\]\. These approaches provide principled mechanisms for handling observations that are not regularly spaced\. The multi\-timeframe financial setting considered here is more structured: each timeframe has a known bar\-completion rule, observations have ordered timestamps, each input window contains completed bars, and the update status of a level can be determined from timestamp equality\. HARN consequently uses discrete event ordering and state retention rather than inferring a continuous hidden trajectory between observations\. This makes the causal contract directly inspectable, while also making the implementation dependent on the specified timestamp and bar\-construction conventions\. These assumptions are examined in[Section9](https://arxiv.org/html/2609.26822#S9)\.

Taken together, these literatures establish the components from which HARN is constructed: financial econometrics motivates careful treatment of non\-stationarity and information availability\[[4](https://arxiv.org/html/2609.26822#bib.bib4),[2](https://arxiv.org/html/2609.26822#bib.bib2)\]; recurrent and state\-space models provide mechanisms for compact persistent state\[[21](https://arxiv.org/html/2609.26822#bib.bib21),[25](https://arxiv.org/html/2609.26822#bib.bib25)\]; Transformers provide flexible cross\-temporal interaction\[[14](https://arxiv.org/html/2609.26822#bib.bib14),[18](https://arxiv.org/html/2609.26822#bib.bib18)\]; associative\-memory methods provide content\-addressable retrieval and update mechanisms\[[26](https://arxiv.org/html/2609.26822#bib.bib26),[28](https://arxiv.org/html/2609.26822#bib.bib28)\]; hierarchical forecasting addresses relationships across temporal aggregation levels\[[30](https://arxiv.org/html/2609.26822#bib.bib30)\]; irregular\-time models provide alternatives for asynchronous observations\[[34](https://arxiv.org/html/2609.26822#bib.bib34),[35](https://arxiv.org/html/2609.26822#bib.bib35)\]; and leakage and forecast\-evaluation research establishes the need to align computation with the information set available at each forecast origin\[[7](https://arxiv.org/html/2609.26822#bib.bib7),[9](https://arxiv.org/html/2609.26822#bib.bib9),[11](https://arxiv.org/html/2609.26822#bib.bib11)\]\. The more specific gap addressed by HARN lies in how these ideas are combined for a stream of completed financial bars observed at multiple resolutions\.

In particular, the literature does not provide a single established protocol that simultaneously specifies per\-timeframe completion events, persistence of inactive higher\-level states, within\-event cross\-level communication, and compact associative state updates under an explicit causal ordering\. This should be understood as a protocol and state\-transition gap rather than a claim that no prior method contains any individual component\. HARN addresses the combination by defining an event\-conditioned state transition, a bounded residual\-driven associative write \([Eq\.17](https://arxiv.org/html/2609.26822#S4.E17)\), cross\-level resonance around selective updates \([Eq\.22](https://arxiv.org/html/2609.26822#S4.E22)\), and a lower\-to\-higher evidence pathway with an explicitly defined history order \([Eq\.23](https://arxiv.org/html/2609.26822#S4.E23)\)\. The resulting formulation provides a testable specification for event\-driven multi\-timeframe state rather than introducing each component as an isolated architectural novelty\. The empirical evidence in this study is consequently interpreted within the scope of that specification and its single implementation, rather than as evidence of blanket superiority over the broader forecasting literature \([Sections6](https://arxiv.org/html/2609.26822#S6),[7](https://arxiv.org/html/2609.26822#S7)and[10](https://arxiv.org/html/2609.26822#S10)\)\.

## 3Problem Formulation: Event\-Driven Multi\-Timeframe Forecasting

This section formalizes the protocol that[Section2](https://arxiv.org/html/2609.26822#S2)identified as under\-specified\. It fixes the notation used throughout and states what a causally valid multi\-timeframe forecast may and may not depend on, which is the standard against which the audit of[Section9](https://arxiv.org/html/2609.26822#S9)checks the implementation\.

Let the levels be ordered from finest to coarsest asT1,…,TLT\_\{1\},\\ldots,T\_\{L\}\. A completed bar at levelkkisxtk=\(otk,htk,ℓtk,ctk,vtk\)∈ℝ5x\_\{t\}^\{k\}=\(o\_\{t\}^\{k\},h\_\{t\}^\{k\},\\ell\_\{t\}^\{k\},c\_\{t\}^\{k\},v\_\{t\}^\{k\}\)\\in\\mathbb\{R\}^\{5\}, the usual open–high–low–close–volume representation of a fixed time interval\.*Event time*τ\\tauis the sequence of anchor timestamps at which the system is queried for a forecast, and*representation time*is the timestamp of the most recently completed bar available at each level\. The two clocks coincide at the finest level by construction, since the anchor stream is defined by that level’s bar completions, but they need not coincide at coarser levels\. We writerτk,Mτkr\_\{\\tau\}^\{k\},M\_\{\\tau\}^\{k\}for the persistent state immediately before processing eventτ\\tau, andrτ\+1k,Mτ\+1kr\_\{\\tau\+1\}^\{k\},M\_\{\\tau\+1\}^\{k\}for the state stored immediately after that event and used by the forecast emitted atτ\\tau\. The subscriptτ\+1\\tau\+1thus denotes the post\-event state; it does not imply that the forecast waits for the next event\.

#### Update indicator\.

Events are indexed by the finest level\. For anchor timeτ\\tau, preprocessing selects the latest level\-kkbar whose timestamp is no later thanτ\\tauand sets

uτk=𝟏​\{selected timestamp equals​τ\}\.u\_\{\\tau\}^\{k\}=\\mathbf\{1\}\\\{\\text\{selected timestamp equals \}\\tau\\\}\.\(1\)The no\-later\-than rule prevents a future bar from ever entering an input window, while the equality test separately determines whether the persistent level state may be overwritten\. Thusuτk=0u\_\{\\tau\}^\{k\}=0means that the selected higher\-level window is retained for context but that no new completed bar closed exactly atτ\\tau, so the level contributes its most recent valid representation without its persistent state being overwritten\. Exact timestamp equality is an implementation assumption, not a claim that every market feed exposes perfectly aligned timestamps;[Section9](https://arxiv.org/html/2609.26822#S9)returns to this point\.

#### Persistent state and masked transition\.

For each level, HARN maintains a staterτk∈ℝdsr\_\{\\tau\}^\{k\}\\in\\mathbb\{R\}^\{d\_\{s\}\}and a matrix memoryMτk∈ℝds×dmM\_\{\\tau\}^\{k\}\\in\\mathbb\{R\}^\{d\_\{s\}\\times d\_\{m\}\}\. The state is a compact vector summary analogous to a recurrent hidden state, and the memory is a matrix\-valued associative store supporting content\-based retrieval, in the sense developed for fast\-weight and Hopfield\-style memories\[[26](https://arxiv.org/html/2609.26822#bib.bib26),[27](https://arxiv.org/html/2609.26822#bib.bib27),[29](https://arxiv.org/html/2609.26822#bib.bib29)\]\. Letr~τ\+1k\\tilde\{r\}\_\{\\tau\+1\}^\{k\}andM~τ\+1k\\tilde\{M\}\_\{\\tau\+1\}^\{k\}denote candidate updates computed from the current payload, that is, the state and memory that the level*would*adopt if it were active at this event\. For a single stream andk\>1k\>1, the persistent transition is

rτ\+1k\\displaystyle r\_\{\\tau\+1\}^\{k\}=\(1−uτk\)​rτk\+uτk​r~τ\+1k,\\displaystyle=\(1\-u\_\{\\tau\}^\{k\}\)\\,r\_\{\\tau\}^\{k\}\+u\_\{\\tau\}^\{k\}\\,\\tilde\{r\}\_\{\\tau\+1\}^\{k\},Mτ\+1k\\displaystyle M\_\{\\tau\+1\}^\{k\}=\(1−uτk\)​Mτk\+uτk​M~τ\+1k,\\displaystyle=\(1\-u\_\{\\tau\}^\{k\}\)\\,M\_\{\\tau\}^\{k\}\+u\_\{\\tau\}^\{k\}\\,\\tilde\{M\}\_\{\\tau\+1\}^\{k\},\(2\)whereuτk∈\{0,1\}u\_\{\\tau\}^\{k\}\\in\\\{0,1\\\}is a scalar that multiplies the whole state or memory\. The base level, whose bar closes at every anchor event, hasuτ1≡1u\_\{\\tau\}^\{1\}\\equiv 1and is updated without masking\. Because the indicator is binary rather than a learned scalar, this convex combination expresses “no new information at this level, so do not change its state” as an exact identity transition rather than an approximate one\.

#### Batched form used in training\.

Training and evaluation processBBcontiguous streams in parallel \([Section5](https://arxiv.org/html/2609.26822#S5)\)\. Letb∈\{1,…,B\}b\\in\\\{1,\\ldots,B\\\}index streams, and letub,τku\_\{b,\\tau\}^\{k\}be the indicator of[Eq\.1](https://arxiv.org/html/2609.26822#S3.E1)for streambbat its own current event; we writeτ\\taufor the step index and suppress each stream’s separate timestamp\. The scalar indicators of one step are stored as the mask vector

mτk=\(u1,τk,…,uB,τk\)⊤∈\{0,1\}B\.m\_\{\\tau\}^\{k\}=\\bigl\(u\_\{1,\\tau\}^\{k\},\\ldots,u\_\{B,\\tau\}^\{k\}\\bigr\)^\{\\\!\\top\}\\in\\\{0,1\\\}^\{B\}\.\(3\)With stacked statesRτk∈ℝB×dsR\_\{\\tau\}^\{k\}\\in\\mathbb\{R\}^\{B\\times d\_\{s\}\}and memoriesℳτk∈ℝB×ds×dm\\mathcal\{M\}\_\{\\tau\}^\{k\}\\in\\mathbb\{R\}^\{B\\times d\_\{s\}\\times d\_\{m\}\}, the implemented update is

Rτ\+1k=\(1−mτk\)⊙BRτk\+mτk⊙BR~τ\+1k,ℳτ\+1k=\(1−mτk\)⊙Bℳτk\+mτk⊙Bℳ~τ\+1k,R\_\{\\tau\+1\}^\{k\}=\(1\-m\_\{\\tau\}^\{k\}\)\\odot\_\{B\}R\_\{\\tau\}^\{k\}\+m\_\{\\tau\}^\{k\}\\odot\_\{B\}\\tilde\{R\}\_\{\\tau\+1\}^\{k\},\\qquad\\mathcal\{M\}\_\{\\tau\+1\}^\{k\}=\(1\-m\_\{\\tau\}^\{k\}\)\\odot\_\{B\}\\mathcal\{M\}\_\{\\tau\}^\{k\}\+m\_\{\\tau\}^\{k\}\\odot\_\{B\}\\tilde\{\\mathcal\{M\}\}\_\{\\tau\+1\}^\{k\},\(4\)where⊙B\\odot\_\{B\}multiplies thebb\-th slice of a tensor by thebb\-th entry of the mask, broadcasting over the trailing dimensions\. Slicebbof[Eq\.4](https://arxiv.org/html/2609.26822#S3.E4)is exactly[Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)for streambb, so the scalar definition and the batched implementation coincide\. The candidatesR~\\tilde\{R\}andℳ~\\tilde\{\\mathcal\{M\}\}are computed for the whole batch regardless of the mask, and masking selects between the two already\-computed branches\.

#### Causal requirement\.

Letℐτ=\{xsj:s≤τ,1≤j≤L\}\\mathcal\{I\}\_\{\\tau\}=\\\{x\_\{s\}^\{j\}:\\ s\\leq\\tau,\\ 1\\leq j\\leq L\\\}be the completed bars, at all levels, with timestamps no later thanτ\\tau\. A causally valid forecaster must satisfy

y^τ\+1k=fk\(ℐτ;r01:L,M01:L\),\\hat\{y\}\_\{\\tau\+1\}^\{k\}=f\_\{k\}\\bigl\(\\mathcal\{I\}\_\{\\tau\};\\,r\_\{0\}^\{1:L\},M\_\{0\}^\{1:L\}\\bigr\),\(5\)where\(r01:L,M01:L\)\(r\_\{0\}^\{1:L\},M\_\{0\}^\{1:L\}\)is the reset initial state and the dependence onℐτ\\mathcal\{I\}\_\{\\tau\}is mediated by the persistent states obtained by applying the transition to completed bars in order\. The requirement is stated at the level of*events*: no bar completed afterτ\\tau, and no state derived from one, may influencey^τ\+1k\\hat\{y\}\_\{\\tau\+1\}^\{k\}\. It does not require that the level\-kkupdate at eventτ\\taube independent of what other levels computed at the same event\. HARN’s bottom\-up evidence path lets a higher\-level update atτ\\tauread the lower\-level state that was already updated atτ\\tau, and its forecasting head consumes post\-update statesrτ\+1kr\_\{\\tau\+1\}^\{k\}\([Section4](https://arxiv.org/html/2609.26822#S4)\)\. HARN therefore satisfies[Eq\.5](https://arxiv.org/html/2609.26822#S3.E5)at event granularity, but it is not “past\-event\-only within an event”, because a higher\-level computation atτ\\taucan depend on a same\-event, already\-updated lower\-level read\. This qualification recurs as entry 9 of the audit \([Table6](https://arxiv.org/html/2609.26822#S9.T6)\)\. HARN enforces a stronger local property for its convolutions and shifted features \([Section4](https://arxiv.org/html/2609.26822#S4)\)\.

#### Implementation notes and data handling\.

The implementation computes candidate encodings for every supplied level on each forward call, including inactive ones; masking governs persistent overwrites, not computation skipping\. “Inactive” therefore means that the level’s persistent state is protected, not that no computation touched its window\. Within an event, lower\-level updates complete before their current reads are appended to the higher\-level history buffer\. Data are cleaned by timestamp parsing, invalid\-row removal, last\-duplicate retention, and ascending sort\. Splits are chronological 70%/15%/15%\[[8](https://arxiv.org/html/2609.26822#bib.bib8),[9](https://arxiv.org/html/2609.26822#bib.bib9)\], and feature and target statistics are computed on training samples only and reused for validation and test\[[7](https://arxiv.org/html/2609.26822#bib.bib7)\]\. The resulting formulation is an event\-driven state\-transition problem, not a regular\-grid forecasting problem with extra covariates\.

#### Stationary target\.

LetPτkP\_\{\\tau\}^\{k\}be the latest observed close andPτ\+1kP\_\{\\tau\+1\}^\{k\}the next completed close at levelkk\. The supervised target is the basis\-point change

yτ\+1k=10,000​\(Pτ\+1kPτk−1\)​bps,y\_\{\\tau\+1\}^\{k\}=10\{,\}000\\left\(\\frac\{P\_\{\\tau\+1\}^\{k\}\}\{P\_\{\\tau\}^\{k\}\}\-1\\right\)\\;\\mathrm\{bps\},\(6\)and the prediction is reconstructed asP^τ\+1k=Pτk​\(1\+y^τ\+1k/10,000\)\\hat\{P\}\_\{\\tau\+1\}^\{k\}=P\_\{\\tau\}^\{k\}\(1\+\\hat\{y\}\_\{\\tau\+1\}^\{k\}/10\{,\}000\)\. Modeling a relative change rather than a level is standard practice in financial time\-series analysis, since levels differ widely across assets and are highly persistent while relative changes are closer to stationary\[[3](https://arxiv.org/html/2609.26822#bib.bib3),[2](https://arxiv.org/html/2609.26822#bib.bib2)\]\. The BPS target is dimensionless with respect to price level and is expressed in the units conventionally used for small relative movements\. Two consequences matter for interpreting the reported results\. First, the reconstruction error is

P^τ\+1k−Pτ\+1k=Pτk​\(y^τ\+1k−yτ\+1k\)10,000,\\hat\{P\}\_\{\\tau\+1\}^\{k\}\-P\_\{\\tau\+1\}^\{k\}=\\frac\{P\_\{\\tau\}^\{k\}\\,\(\\hat\{y\}\_\{\\tau\+1\}^\{k\}\-y\_\{\\tau\+1\}^\{k\}\)\}\{10\{,\}000\},\(7\)so HARN’s reported price errors are price\-scaled BPS errors anchored at the last observed close, and a zero\-BPS forecast reproduces the last\-value \(persistence\) forecast\. Second, all reported evaluation metrics are computed on reconstructed absolute price, which reintroduces asset\-scale dependence; the BPS target is used for training and reconstruction and is not itself reported as an evaluation metric\.

## 4The HARN Architecture

[Figure1](https://arxiv.org/html/2609.26822#S4.F1)gives a conceptual overview\. The computational path at each event is: completed OHLCV windows→\\rightarrowper\-level causal encoders→\\rightarrowprior\-state resonance→\\rightarrowgated memory candidates→\\rightarrowordered bottom\-up evidence for higher levels→\\rightarrowposterior resonance→\\rightarrowper\-level forecasting heads\. Separate encoders, memories, histories, and heads are instantiated for every configured level, so the hierarchy is simultaneously a representation hierarchy and an update hierarchy: level identity determines both what a state represents and when it may change\. This section describes the components in execution order, and[Section5](https://arxiv.org/html/2609.26822#S5)assembles them into the per\-event protocol\. Everything in this section describes the implemented computation, a code observation, and not an experimental finding\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/harn_architecture_overview.png)Figure 1:Overview of HARN’s event\-driven computational path at one anchor event: per\-level causal encoding, prior\-state resonance, gated memory candidates, ordered bottom\-up evidence, posterior resonance, and per\-level forecasting\. Conceptual schematic, not an empirical result\.### 4\.1Causal multi\-scale encoder

The encoder summarizes a completed OHLCV window into a fixed\-size representation using only information within that window, at multiple temporal scales, without mixing in a future position \([Figure2](https://arxiv.org/html/2609.26822#S4.F2)\)\. ForXk∈ℝB×T×5X^\{k\}\\in\\mathbb\{R\}^\{B\\times T\\times 5\}, it projects the raw OHLCV features together with six derived features that make short\-horizon price structure explicit to the downstream convolutions,

dtk\\displaystyle d\_\{t\}^\{k\}=\(ct−ct−1,ct−ot,ht−ℓt,ht−max⁡\(ot,ct\),min⁡\(ot,ct\)−ℓt,vt\),\\displaystyle=\(c\_\{t\}\-c\_\{t\-1\},\\,c\_\{t\}\-o\_\{t\},\\,h\_\{t\}\-\\ell\_\{t\},\\,h\_\{t\}\-\\max\(o\_\{t\},c\_\{t\}\),\\,\\min\(o\_\{t\},c\_\{t\}\)\-\\ell\_\{t\},\\,v\_\{t\}\),\(8\)ztk\\displaystyle z\_\{t\}^\{k\}=Wraw​xtk\+Wderived​dtk,\\displaystyle=W\_\{\\rm raw\}x\_\{t\}^\{k\}\+W\_\{\\rm derived\}d\_\{t\}^\{k\},\(9\)where the previous close is shifted by one position and the first previous close is copied from the first close, avoiding an artificial pre\-window observation\. The six derived channels are close change, signed candle body, full candle range, upper wick, lower wick, and the already normalized volume channel, so volume is reused as a derived input rather than separately transformed\. The implementation receives normalized OHLCV windows and computes these derived quantities from that representation\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/harn_causal_multiscale_encoder.png)Figure 2:Causal multi\-scale encoder: left\-padded dilated branches, a learned branch mixture, and decayed pooling\. Conceptual schematic, not an empirical result\.Four depthwise\-separable gated convolution branches with kernel size three and dilationsD=\{1,2,4,8\}D=\\\{1,2,4,8\\\}use left\-only padding, so no branch output at positionttdepends on a later position\. This follows the causal dilated design used for autoregressive modeling of long sequences, in which left\-padding with exponentially increasing dilation gives a large receptive field at a cost that grows only logarithmically with length\[[36](https://arxiv.org/html/2609.26822#bib.bib36)\], while the depthwise\-separable factorization reduces parameters relative to a full convolution\[[37](https://arxiv.org/html/2609.26822#bib.bib37)\]:

Cd=Vd⊙σ⁡\(Gd\),C=∑d∈Dsoftmax⁡\(α\)d​Cd\.C\_\{d\}=V\_\{d\}\\odot\\sigma\(G\_\{d\}\),\\qquad C=\\sum\_\{d\\in D\}\\operatorname\{softmax\}\(\\alpha\)\_\{d\}C\_\{d\}\.\(10\)The gateσ⁡\(Gd\)\\sigma\(G\_\{d\}\)lets each branch modulate which channels and positions contribute, and the learned softmax mixture over dilations lets the model weight temporal scales rather than averaging them uniformly\. The mixture passes through a pointwise convolution, a residual connection, and LayerNorm\[[40](https://arxiv.org/html/2609.26822#bib.bib40)\]\. A learnable per\-channel decayλc=σ⁡\(ϕc\)\\lambda\_\{c\}=\\sigma\(\\phi\_\{c\}\)then pools the sequence with more weight on recent positions while retaining a causal contribution from the full window,

pc=∑t=1T\(1−λc\)​λcT−t​Ct,c\.p\_\{c\}=\\sum\_\{t=1\}^\{T\}\(1\-\\lambda\_\{c\}\)\\lambda\_\{c\}^\{T\-t\}C\_\{t,c\}\.\(11\)The encoder output combines this decayed pool with a tail projection of the last position and a short trailing mean, so that the most recent bar and the immediate local trend are represented directly,

ek=LN\(pk\+Wtail\[CTk;mean\(CT−min⁡\(4,T\)\+1:Tk\)\]\)∈ℝ72e^\{k\}=\\operatorname\{LN\}\\left\(p^\{k\}\+W\_\{\\rm tail\}\\bigl\[C\_\{T\}^\{k\};\\operatorname\{mean\}\(C\_\{T\-\\min\(4,T\)\+1:T\}^\{k\}\)\\bigr\]\\right\)\\in\\mathbb\{R\}^\{72\}\(12\)in the default configuration\. This local representation is the only signal drawn from the raw window; persistence across events enters through the memory cell described next\.

### 4\.2Gated associative memory with residual\-driven writes

Where the encoder summarizes the current window, the memory cell maintains a persistent, content\-addressable representation across events \([Figure3](https://arxiv.org/html/2609.26822#S4.F3)\)\. Its input is the encoder output concatenated with prior resonance context and, for levels above the base, bottom\-up evidence, giving an input dimension of 144 at the base level and 216 above it\. For normalized inputxtx\_\{t\}, statertr\_\{t\}, and memoryMtM\_\{t\}, the cell computes a key, value, and query in the manner of a standard attention or fast\-weight mechanism\[[28](https://arxiv.org/html/2609.26822#bib.bib28)\],

kt\\displaystyle k\_\{t\}=normalize⁡\(Wk​xt\),\\displaystyle=\\operatorname\{normalize\}\(W\_\{k\}x\_\{t\}\),vt\\displaystyle v\_\{t\}=tanh⁡\(Wv​xt\),\\displaystyle=\\tanh\(W\_\{v\}x\_\{t\}\),qt\\displaystyle q\_\{t\}=normalize⁡\(Wq​xt\),\\displaystyle=\\operatorname\{normalize\}\(W\_\{q\}x\_\{t\}\),\(13\)v¯t\\displaystyle\\bar\{v\}\_\{t\}=Mt​kt,\\displaystyle=M\_\{t\}k\_\{t\},st\\displaystyle s\_\{t\}=vt−v¯t\.\\displaystyle=v\_\{t\}\-\\bar\{v\}\_\{t\}\.\(14\)The residualsts\_\{t\}is the discrepancy between the value implied by the current input and what the memory retrieves for the corresponding key\. We call it “surprise” as a label only: it is a retrieval residual internal to the memory, it is not the forecast error against the downstream target, and no reported experiment tests whether it tracks prediction error, novelty, or market events\. Withgt=\[xt;rt;v¯t\]g\_\{t\}=\[x\_\{t\};r\_\{t\};\\bar\{v\}\_\{t\}\], learned gates control the write and blend rates,

ηt=σ\(Wηgt\),ρt=σ\(Wρgt\),ηt,ρt∈ℝB×ds\.\\eta\_\{t\}=\\sigma\(W\_\{\\eta\}g\_\{t\}\),\\qquad\\rho\_\{t\}=\\sigma\(W\_\{\\rho\}g\_\{t\}\),\\qquad\\eta\_\{t\},\\rho\_\{t\}\\in\\mathbb\{R\}^\{B\\times d\_\{s\}\}\.\(15\)The residual enters the*content*of the write, not the gate:sts\_\{t\}is not an explicit input toηt\\eta\_\{t\}orρt\\rho\_\{t\}\. The write is a rank\-one,tanh\\tanh\-bounded outer product of the residual and the key, scaled by the square root of the memory\-key dimension as in scaled dot\-product attention\[[14](https://arxiv.org/html/2609.26822#bib.bib14)\]:

Ωt\\displaystyle\\Omega\_\{t\}=tanh⁡\(st​kt⊤dm\),\\displaystyle=\\tanh\\left\(\\frac\{s\_\{t\}k\_\{t\}^\{\\top\}\}\{\\sqrt\{d\_\{m\}\}\}\\right\),\(16\)Mt\+1\\displaystyle M\_\{t\+1\}=\(1−ηt\)⊙BMt\+ηt⊙BΩt\.\\displaystyle=\(1\-\\eta\_\{t\}\)\\odot\_\{B\}M\_\{t\}\+\\eta\_\{t\}\\odot\_\{B\}\\Omega\_\{t\}\.\(17\)HereMt,Ωt∈ℝB×ds×dmM\_\{t\},\\Omega\_\{t\}\\in\\mathbb\{R\}^\{B\\times d\_\{s\}\\times d\_\{m\}\}, while each gate has shape\(B,ds\)\(B,d\_\{s\}\)and is unsqueezed to\(B,ds,1\)\(B,d\_\{s\},1\)before multiplication\. The notation⊙B\\odot\_\{B\}therefore means row\-wise broadcasting over thedmd\_\{m\}memory\-key columns; it is not a single scalar gate for the whole matrix\. The updated memory is then queried with the query vector rather than the key just written, so that retrieval reflects the memory’s content as a whole, and the state update combines a memory\-derived and a direct pathway,

μt\\displaystyle\\mu\_\{t\}=Mt\+1​qt,\\displaystyle=M\_\{t\+1\}q\_\{t\},\(18\)rt\+1\\displaystyle r\_\{t\+1\}=LN⁡\(rt\+ρt⊙Wr​μt\+\(1−ρt\)⊙tanh⁡\(Wc​gt\)\),\\displaystyle=\\operatorname\{LN\}\\left\(r\_\{t\}\+\\rho\_\{t\}\\odot W\_\{r\}\\mu\_\{t\}\+\(1\-\\rho\_\{t\}\)\\odot\\tanh\(W\_\{c\}g\_\{t\}\)\\right\),\(19\)followed by dropout\[[42](https://arxiv.org/html/2609.26822#bib.bib42)\]\. The default memory shape is\(B,72,18\)\(B,72,18\)\. Unlike a standard recurrent cell, which compresses temporal information into a single vector transition, this cell maintains a matrix of key–value associations and exposes the retrieval residual explicitly, in a manner structurally related to Hopfield\-style retrieval\[[26](https://arxiv.org/html/2609.26822#bib.bib26),[29](https://arxiv.org/html/2609.26822#bib.bib29)\]\. Relative to generic fast\-weight formulations\[[27](https://arxiv.org/html/2609.26822#bib.bib27),[28](https://arxiv.org/html/2609.26822#bib.bib28)\], the write is bounded bytanh\\tanh, scaled by the memory\-key dimension, and modulated by a state\-sized gate\. These choices are*intended*to keep the persistent memory numerically stable over long streams, but no stability analysis or capacity claim is made or tested\. The A1 ablation \([Section8](https://arxiv.org/html/2609.26822#S8)\) replaces this cell with a simpler gated recurrent update\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/harn_associative_memory.png)Figure 3:Gated associative memory cell: key, value, and query projections, the retrieval residualsts\_\{t\}, and the bounded gated write of[Eq\.17](https://arxiv.org/html/2609.26822#S4.E17)\. Conceptual schematic, not an empirical result\.
### 4\.3Cross\-scale communication and forecasting head

Resonance and evidence are the two communication paths across levels in HARN, and they differ in topology and timing \([Figure4](https://arxiv.org/html/2609.26822#S4.F4)\)\. States are stacked asR∈ℝB×L×dsR\\in\\mathbb\{R\}^\{B\\times L\\times d\_\{s\}\}, and shared projections of rankdr=36d\_\{r\}=36produce an all\-to\-all attention update over the level axis, following multi\-head self\-attention\[[14](https://arxiv.org/html/2609.26822#bib.bib14)\]but applied across levels rather than across time, with a learned pairwise level\-strength biasSS:

Q\\displaystyle Q=Wq​R,\\displaystyle=W\_\{q\}R,K\\displaystyle K=Wk​R,\\displaystyle=W\_\{k\}R,V\\displaystyle V=Wv​R,\\displaystyle=W\_\{v\}R,\(20\)A\\displaystyle A=softmax⁡\(Q​K⊤dr\+S\),\\displaystyle=\\operatorname\{softmax\}\\left\(\\frac\{QK^\{\\top\}\}\{\\sqrt\{d\_\{r\}\}\}\+S\\right\),\(21\)R′\\displaystyle R^\{\\prime\}=LN⁡\(R\+Dropout⁡\(Wo​A​V\)\)\.\\displaystyle=\\operatorname\{LN\}\\left\(R\+\\operatorname\{Dropout\}\(W\_\{o\}AV\)\\right\)\.\(22\)Resonance is applied twice per event: once to form prior context before the memory update, so that each level’s write is informed by the other levels’ states as they stood at the start of the event, and once to the updated states before forecasting\. It is all\-to\-all and contemporaneous: every configured level can attend to every other level’s state at the same event, with no directional constraint\.

Bottom\-up evidence, by contrast, is adjacent\-level and directional: a coarser level’s update is informed by what has happened at the level immediately below it\. For each adjacent pair, a length\-four history buffer stores recent lower\-level reads\. A learned query attends to projected keys and values with a learnable recency bias, and the readout combines the attention\-weighted summary with an explicit emphasis on the most recent read and the buffer mean,

Ei=LN⁡\(Attention⁡\(Hi\)\+0\.35​Hi,latest\+0\.15​mean⁡\(Hi\)\)\.E\_\{i\}=\\operatorname\{LN\}\\left\(\\operatorname\{Attention\}\(H\_\{i\}\)\+0\.35\\,H\_\{i,\\mathrm\{latest\}\}\+0\.15\\operatorname\{mean\}\(H\_\{i\}\)\\right\)\.\(23\)The coefficients0\.350\.35and0\.150\.15are hand\-set implementation constants, not learned parameters\. The supplied materials contain no derivation, tuning record, or sensitivity analysis for them, and it is not documented whether they were tuned or inherited from earlier code\. We therefore report them as manually chosen rather than empirically validated; the A3 variant zeros the entire readout and does not probe them individually\.

The current lower\-level read is appended toHiH\_\{i\}before the corresponding higher\-level evidence is computed, which establishes an ordered same\-event path\. This is the qualification stated in[Section3](https://arxiv.org/html/2609.26822#S3): the path is causal with respect to future events but not past\-state\-only within the event, because the lower level’s already\-completed update is what gets appended\. Neither path is computation\-free for inactive levels, since their encoders and history reads are evaluated and only the persistent state and memory overwrite is masked \([Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)\)\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/harn_resonance_and_evidence.png)Figure 4:The two cross\-level communication paths of HARN: all\-to\-all, contemporaneous resonance \([Eq\.22](https://arxiv.org/html/2609.26822#S4.E22)\) and directional, ordered bottom\-up evidence \([Eq\.23](https://arxiv.org/html/2609.26822#S4.E23)\)\. Conceptual schematic, not an empirical result\.#### Forecasting head\.

The forecasting head maps the cross\-scale\-informed state onto a one\-step BPS forecast\. It receives the current state, posterior resonance context, and six anchor features: one\-step and two\-step close changes, trailing close deviation, trailing range, trailing body, and mean normalized volume\. This gives a concatenated input of dimension72\+72\+6=15072\+72\+6=150in the default configuration\. A two\-hidden\-layer GELU MLP of width 80\[[41](https://arxiv.org/html/2609.26822#bib.bib41)\]with dropout 0\.1\[[42](https://arxiv.org/html/2609.26822#bib.bib42)\]is combined with a residual linear path from the current state and anchors, so that the head can fall back on a linear function of the state:

y^t\+1k=MLP⁡\(\[rt\+1k;Rt\+1′k;atk\]\)\+Wres​\[rt\+1k;atk\]\.\\hat\{y\}\_\{t\+1\}^\{k\}=\\operatorname\{MLP\}\\bigl\(\[r\_\{t\+1\}^\{k\};R\_\{t\+1\}^\{\\prime k\};a\_\{t\}^\{k\}\]\\bigr\)\+W\_\{\\rm res\}\[r\_\{t\+1\}^\{k\};a\_\{t\}^\{k\}\]\.\(24\)Linear layers use Xavier uniform initialization\[[38](https://arxiv.org/html/2609.26822#bib.bib38)\]and convolution layers Kaiming uniform initialization\[[39](https://arxiv.org/html/2609.26822#bib.bib39)\], with zero biases throughout; both set the initial weight variance from the layer’s fan\-in and fan\-out, and the choice follows the activation used downstream of each layer type\. The head outputs a one\-step BPS forecast for each configured level, and the absolute\-price forecast used for evaluation is obtained only afterwards by inserting the last observed close into the reconstruction of[Eq\.6](https://arxiv.org/html/2609.26822#S3.E6)\.

## 5Streaming Semantics and Training

At each anchor event the implementation selects completed\-bar windows, encodes every supplied window, resonates the previous states, constructs anchors, always updates the base level, sequentially computes lower\-level evidence and conditionally updates the higher levels, resonates the updated states, and forecasts at every level\. The ordered procedure is restated in[SectionA\.3](https://arxiv.org/html/2609.26822#A1.SS3)\. This ordering makes the masked transition operational: candidate transitions are computed for every level at every event, but per sample only those whose mask entry in[Eq\.3](https://arxiv.org/html/2609.26822#S3.E3)equals one overwrite persistent state \([Eq\.4](https://arxiv.org/html/2609.26822#S3.E4)\)\. Because masking follows a batched forward pass, candidate computation occurs for the whole batch even when only some streams are active\. Missing activity flags default to lock\-step updates \(mτk≡𝟏m\_\{\\tau\}^\{k\}\\equiv\\mathbf\{1\}for every level\), the degenerate case in which the mechanism reduces to ordinary recurrence\.[Figure5](https://arxiv.org/html/2609.26822#S5.F5)summarizes the event ordering\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/harn_event_timeline.png)Figure 5:Event\-driven multi\-timeframe timeline\. The anchor stream generates forecast events; a coarser representation is updated only when its completed\-bar timestamp equals the anchor timestamp and is otherwise carried forward\. Conceptual schematic of the protocol, not an empirical result\.#### State handling and loss\.

States and zero\-filled histories are reset at epoch and evaluation boundaries and whenever the batch size changes, which prevents state from one contiguous stream from leaking into an unrelated one, and the production loop detaches state, memory, and history tensors after every optimizer step\. With one optimizer step per streaming batch, the effective truncated\-backpropagation span is one batch transition\[[46](https://arxiv.org/html/2609.26822#bib.bib46)\]\. A longer gradient accumulation appears in the model demonstration script but is not the production training procedure\. The production loss sums per\-level BPS mean\-squared error without level normalization, so hierarchies with different numbers of levels do not share a loss scale, and a level with larger BPS variance can dominate the gradient:

ℒ=∑k=1L1B​∑b=1B\(y^b,t\+1k−yb,t\+1k\)2\.\\mathcal\{L\}=\\sum\_\{k=1\}^\{L\}\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\(\\hat\{y\}\_\{b,t\+1\}^\{k\}\-y\_\{b,t\+1\}^\{k\}\)^\{2\}\.\(25\)StreamingBatcherdivides each chronological split into non\-overlapping contiguous streams, discarding any remainder that cannot fill the configured partition, and preserves chronological continuity at each batch position\. Consecutive batches for a given stream position are therefore consecutive events in time, which is the property that the masking and detachment behavior assumes\.

#### Training configuration\.

The production parser’s defaults are batch size 256, eight evaluation streams, 50 maximum epochs, AdamW with learning rate10−310^\{\-3\}and weight decay10−510^\{\-5\}\[[43](https://arxiv.org/html/2609.26822#bib.bib43)\], gradient clipping at 1\.0\[[45](https://arxiv.org/html/2609.26822#bib.bib45)\], patience 15, seed 42, cosine annealing withTmax=epochsT\_\{\\max\}=\\text\{epochs\}\[[44](https://arxiv.org/html/2609.26822#bib.bib44)\], and CUDA automatic mixed precision when available\[[47](https://arxiv.org/html/2609.26822#bib.bib47)\]\. Its source sets NumPy and PyTorch seeds but does not provide the ablation trainer’s full Python\-random and deterministic\-cuDNN controls\. The configuration table accompanying the reported results specifies the same batch size and epoch budget and adds the seeds 42, 151, and 359, whereas the separate ablation trainer defaults to batch size 256 and five epochs\. These artifact\-level configurations are inventoried in[Table8](https://arxiv.org/html/2609.26822#A1.T8), and[Section6\.2](https://arxiv.org/html/2609.26822#S6.SS2)states which is associated with each reported result\. Validation inside the production trainer is stateful over its own contiguous streams, whereas the standalone evaluator uses a single stream and does not reproduce that default exactly\.

## 6Experimental Setup and Provenance

This section describes the data and the comparators and, because several configurations coexist in the supplied artifacts, states which configuration is associated with each reported result\. It also states the governing caveat for the baseline comparison, to which later sections refer\.

Per\-seed aggregate metric files, the dataset configuration, the complete preprocessing pipeline, the completed\-bar alignment and cleaning code, and seed\-specific checkpoints are available in the project repository \([Section12](https://arxiv.org/html/2609.26822#S12)\)\. Raw vendor market data cannot be redistributed, but the preprocessing pipeline is vendor\-neutral and can regenerate the dataset from a compatible feed\. Per\-event predictions and persistence outputs are not included as precomputed files in the current release, although they can be generated from the released checkpoints and preprocessing pipeline\. The zero\-BPS persistence baseline remains a required rerun, and no values for it are reported here\.

### 6\.1Data, splits, and comparators

The supplied dataset table covers four assets with different anchor frequencies and hierarchy depths: AAPL \(M15/H1\), EURUSD \(M15/H1/H4\), USDCHF \(M15/M30/H1\), and XAUUSD \(M5/M15\)\. The selection spans equities, foreign\-exchange pairs, and gold, but it does not separate asset\-specific effects from hierarchy\-depth or period effects\. All windows default to 32 bars unless explicitly overridden\. Each sample is aligned to an anchor event and carries the latest completed window for every configured level, together with update indicators and next\-bar targets, as formalized in[Section3](https://arxiv.org/html/2609.26822#S3)\.[Table1](https://arxiv.org/html/2609.26822#S6.T1)reports sample counts, periods, and mean update rates\.

Table 1:Dataset and split statistics as supplied\. Each row gives, for one asset and chronological split, the hierarchy, anchor timeframe, number of anchor\-event observations, period, span in days, and mean update rate \(the event\-update rate averaged across levels\)\. Per\-level update rates are listed beneath the table\.Per\-level update rates \(finest to coarsest\): AAPL M15/H1 = 1\.00/0\.232; EURUSD M15/H1/H4 = 1\.00/0\.250/0\.063; USDCHF M15/M30/H1 = 1\.00/0\.500/0\.250; XAUUSD M5/M15 = 1\.00/0\.333\.

The per\-level update rates in[Table1](https://arxiv.org/html/2609.26822#S6.T1)show the expected step pattern: exactly 1\.0 at the anchor level and decreasing at each coarser level, roughly in line with the ratio of bar durations\. For example, USDCHF’s H1 level closes once per four completed M15 bars, giving a rate of 0\.250\. This is a sanity check on the completed\-bar alignment of[Section3](https://arxiv.org/html/2609.26822#S3), consistent with the update indicator being computed correctly upstream of the masked transition; it is not an independent empirical claim\. The counts, dates, and update rates are transcriptions from the supplied dataset configuration\. The expected input format and cleaning procedure are described in[Section12](https://arxiv.org/html/2609.26822#S12)\.

PatchTST usesdmodel=128d\_\{\\rm model\}=128,df​f=256d\_\{ff\}=256, two layers, eight heads, patch length 16, stride 8, disabled RevIN and decomposition, and horizon one\[[18](https://arxiv.org/html/2609.26822#bib.bib18)\]\. TimeXer shares these principal dimensions, patches the close price, and treats the remaining OHLCV channels as exogenous variables\[[20](https://arxiv.org/html/2609.26822#bib.bib20)\]\. Both are single\-timeframe architectures and are trained on the anchor timeframe because they do not natively support multi\-timeframe, event\-driven state; anchor\-only training is their native configuration\. HARN is a multi\-timeframe, event\-driven model and is evaluated with all configured payload levels\.

The comparison in[Section7](https://arxiv.org/html/2609.26822#S7)is therefore a*descriptive, scope\-specific comparison of complete pipelines*\. As[Table2](https://arxiv.org/html/2609.26822#S6.T2)shows, HARN receives every configured payload level and trains with stationary BPS targets, stateful contiguous streams, cosine scheduling, patience 15, and CUDA AMP where available, whereas the baselines train on normalized absolute\-close targets with constant learning rates, patience 5, stateless shuffled batches, and no AMP\. The target difference is not cosmetic: by[Eq\.7](https://arxiv.org/html/2609.26822#S3.E7), HARN’s price error is a price\-scaled BPS error anchored at the last observed close, whereas the baselines regress a normalized absolute close\. The reported gap therefore does not isolate hierarchy, memory, resonance, target representation, training protocol, or any other single factor\. This is the governing caveat for the baseline comparison\.

Table 2:Scope and training differences between HARN and the single\-timeframe baseline trainers\. PatchTST and TimeXer are trained on the anchor timeframe, their native configuration, whereas HARN is evaluated as a multi\-timeframe model\. These differences make the comparison in[Table4](https://arxiv.org/html/2609.26822#S7.T4)descriptive: it does not isolate architecture, target definition, or training recipe\.
### 6\.2Which configuration produced which reported result

Because the supplied artifacts contain several configurations \([Table8](https://arxiv.org/html/2609.26822#A1.T8)\),[Table3](https://arxiv.org/html/2609.26822#S6.T3)records, for each reported item, the configuration that the manuscript’s own evidence associates with it and what cannot be determined from the current materials\. Where the materials are insufficient, the ambiguity is stated rather than resolved by inference\.

Table 3:Provenance of each reported item, based on statements in this manuscript and the repository artifacts\. For each item, the table gives the associated configuration and what is and is not determined from the current materials\.

## 7Metrics and Reported Results

For price errorsei=P^i−Pie\_\{i\}=\\hat\{P\}\_\{i\}\-P\_\{i\}, the supplied evaluators report four point\-forecast accuracy metrics that give both scale\-dependent and scale\-normalized views, consistent with the recommendation that no single measure suffices across series of different scales\[[10](https://arxiv.org/html/2609.26822#bib.bib10)\]:

MAE\\displaystyle\\mathrm\{MAE\}=n−1​∑i\|ei\|,\\displaystyle=n^\{\-1\}\\sum\_\{i\}\|e\_\{i\}\|,RMSE\\displaystyle\\mathrm\{RMSE\}=n−1​∑iei2,\\displaystyle=\\sqrt\{n^\{\-1\}\\sum\_\{i\}e\_\{i\}^\{2\}\},\(26\)sMAPE\\displaystyle\\mathrm\{sMAPE\}=100​n−1​∑i2​\|ei\|\|Pi\|\+\|P^i\|,\\displaystyle=100\\,n^\{\-1\}\\sum\_\{i\}\\frac\{2\|e\_\{i\}\|\}\{\|P\_\{i\}\|\+\|\\hat\{P\}\_\{i\}\|\},MASE\\displaystyle\\mathrm\{MASE\}=MAEmeanj⁡\|yj\+1train−yjtrain\|\.\\displaystyle=\\frac\{\\mathrm\{MAE\}\}\{\\operatorname\{mean\}\_\{j\}\|y^\{\\rm train\}\_\{j\+1\}\-y^\{\\rm train\}\_\{j\}\|\}\.\(27\)MAE and RMSE are in the asset’s native price units, and sMAPE is a symmetric percentage error\. The evaluator’s MASE divides reconstructed\-price MAE by the mean absolute first difference of the stored training target\-event sequence\. Because higher\-level target values can repeat while a coarser bar is carried forward, this denominator is an implementation\-defined scale rather than necessarily the chronological close\-series denominator of the standard MASE definition\[[10](https://arxiv.org/html/2609.26822#bib.bib10)\]\. The supplied evaluators do not emit a target\-space metric, so all reported values are computed on reconstructed absolute price\. The natural persistence comparator predictsy^i=0\\hat\{y\}\_\{i\}=0and thereforeP^i=Pilast\\hat\{P\}\_\{i\}=P\_\{i\}^\{\\rm last\}at each forecast origin\. No persistence output is included as a precomputed file in the current release, and this comparator is a required rerun, not a missing value that can be inferred from the aggregate price metrics\.

[Table4](https://arxiv.org/html/2609.26822#S7.T4)reports the results\. All values are supplied values transcribed as computed\. The±\\pmfigures are aggregate variability across three seeds, whose per\-seed outputs are given in[SectionA\.5](https://arxiv.org/html/2609.26822#A1.SS5), and[Section6\.2](https://arxiv.org/html/2609.26822#S6.SS2)records the configuration associated with each panel\.

Table 4:Forecasting results on the test split, reported as mean±\\pmstandard deviation across three seeds\. Lower is better for every metric, and all metrics are computed on reconstructed absolute price\. Panel A is a descriptive, scope\-specific comparison between HARN \(multi\-timeframe, event\-driven, BPS target, stateful training\) and single\-timeframe PatchTST and TimeXer trained on the anchor timeframe; it does not isolate architecture, target definition, or training recipe\. Panel B reports HARN at further timeframes, for which no baseline was run; its XAUUSD M5 row repeats the Panel A entry\. Values are reproduced as supplied\.### 7\.1Baseline comparison and further timeframes

In Panel A of[Table4](https://arxiv.org/html/2609.26822#S7.T4), the supplied HARN price errors are numerically lower than those of both baselines for all four assets at the anchor timeframe, which is M15 for AAPL, EURUSD, and USDCHF and M5 for XAUUSD\. The XAUUSD HARN entries equal the means of the per\-seed M5 values in[Table9](https://arxiv.org/html/2609.26822#A1.T9)\. As[Section6](https://arxiv.org/html/2609.26822#S6)explains, this is a statement about complete pipelines as configured \([Table2](https://arxiv.org/html/2609.26822#S6.T2)\), not an architectural result\. No reported run varies the available timeframes, the target representation, or the training protocol one at a time, so the results do not indicate whether any numerical difference arises from additional timeframes, the BPS target, stateful streaming, tuning effort, or architecture\. Two further points bear on interpretation\.

#### Missing persistence baseline\.

The reported results contain no persistence \(last\-value, that is, zero\-BPS\) or other random\-walk baseline\. Because HARN’s price error is a price\-scaled BPS error anchored at the last close \([Eq\.7](https://arxiv.org/html/2609.26822#S3.E7)\), such a reference is the natural yardstick, and its absence is a missing interpretability baseline\. The MASE column does not fill this role\. Its denominator is an implementation\-defined training\-sample scale, and HARN’s own MASE values range from below one \(EURUSD and USDCHF at the anchor\) to above one \(the AAPL and XAUUSD anchor rows and all higher levels\), so they do not settle whether HARN improves on persistence over the test period\.

#### Omitted percentage reductions\.

The supplied results also expressed HARN’s advantage over PatchTST as percentage reductions, which are omitted here\. They restate Panel A as an effect\-size\-like quantity for a scope\-specific comparison and so invite an attribution to architecture that the design cannot support\. They are ratios of aggregate means without uncertainty, the aggregation method is undocumented, and a few entries differ by about 0\.1 percentage points from the reduction implied by the rounded tabulated means\. Nothing has been recomputed or replaced\.

#### Further timeframes\.

Panel B reports HARN’s forecasts at the further levels of each hierarchy, which have different update rates \([Table1](https://arxiv.org/html/2609.26822#S6.T1)\)\. It documents forecasts at several resolutions and does not show superiority, cross\-level forecast coherence, or benefit from the hierarchy\. Its XAUUSD M5 row repeats the anchor entry of Panel A, and per\-seed results for the non\-anchor XAUUSD M15 level appear in[Table9](https://arxiv.org/html/2609.26822#A1.T9)\. MASE denominators at higher levels use repeated targets, so MASE values are not comparable across timeframes\.

### 7\.2Cross\-seed variance and statistical status

The reported HARN standard deviations across seeds are very small next to those of the baselines\. For AAPL MAE, HARN’s standard deviation is about 0\.1% of its mean, versus about 28% for both PatchTST and TimeXer; for EURUSD MAE the corresponding figures are under 1% and about 36% for PatchTST\. Several HARN entries, for example USDCHF MAE and RMSE in Panel A and RMSE at M30 and H1 in Panel B, round to±0\.000000\\pm 0\.000000at the reported precision, and the pattern holds across metrics and panels\.

We regard this as an unresolved variance observation\. The supplied evidence does not establish its cause, and we do not offer one\. Seed\-specific checkpoints and per\-seed aggregate metrics are available \([SectionA\.5](https://arxiv.org/html/2609.26822#A1.SS5)\), but per\-seed training trajectories and per\-event predictions are not included as precomputed files in the current release\. Seed independence therefore cannot be verified from the supplied CSVs and checkpoints alone, and demonstrating that the three seeds \(42, 151, 359\) produced independent training trajectories requires inspecting the seed\-specific checkpoints and their configuration metadata\. The small HARN spread should not be read as evidence of robustness, and the baselines’ larger spread should not be read as a property of those architectures, given the differing protocols\. Interpreting either requires the per\-event outputs and the verification identified here as necessary follow\-up work\.

[Figures6](https://arxiv.org/html/2609.26822#S7.F6)and[7](https://arxiv.org/html/2609.26822#S7.F7)are empirical visualizations taken from the supplied evaluation outputs, in contrast with the conceptual architecture and timeline figures\. Their captions do not identify the asset, timeframe, seed, or split shown, and the underlying predictions are not included as precomputed files in the current release, so the figures provide qualitative context only and no statistical comparison\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/representative_forecasts.png)Figure 6:Representative forecasts against actual trajectories, taken from the supplied evaluation outputs\. The asset, timeframe, seed, and split are not identified in the supplied materials, so the figure gives qualitative context only\.![Refer to caption](https://arxiv.org/html/2609.26822v1/forecast_error_distributions.png)Figure 7:Representative forecast\-error distributions across models and assets, taken from the supplied evaluation outputs\. The provenance limits are the same as for[Figure6](https://arxiv.org/html/2609.26822#S7.F6), and the figure gives qualitative context only\.A standard deviation over three aggregate seed scores is not a confidence interval for the temporal forecast\-error process, does not capture dependence across forecast origins, and cannot recover the paired model differences that a formal predictive\-accuracy test requires\[[11](https://arxiv.org/html/2609.26822#bib.bib11),[12](https://arxiv.org/html/2609.26822#bib.bib12)\]\. No confidence intervals, paired tests, Diebold–Mariano tests, Wilcoxon tests, effect sizes, or multiplicity corrections were computed, and none are introduced here\.[Section10](https://arxiv.org/html/2609.26822#S10)lists the corresponding analyses as proposed work\.

## 8Ablation Variants: Single\-Run Observations

The ablation study asks a narrower question than the baseline comparison: how do HARN’s own reported values change when one named internal mechanism is replaced or bypassed, with the surrounding streaming pipeline and tensor interfaces held fixed? The variants share HARN’s interfaces and pipeline but not its parameterization or parameter count, so they are component\-replacement interventions rather than matched counterfactuals\. A0 is full HARN\. A1 replaces the associative cell of[Section4\.2](https://arxiv.org/html/2609.26822#S4.SS2)with a gated recurrent\-style update while carrying the memory tensor unchanged\. A2 replaces resonance \([Eq\.22](https://arxiv.org/html/2609.26822#S4.E22)\) with the identity map while retaining bottom\-up evidence, so it removes only the all\-to\-all path and not all cross\-level communication\. A3 replaces the history readout \([Eq\.23](https://arxiv.org/html/2609.26822#S4.E23)\) with zero evidence while preserving the interface and history maintenance\. Each intervention can alter capacity and optimization dynamics in addition to the named mechanism: A1 changes the cell’s parameterization, A2 removes learnable resonance parameters, and A3 removes learned evidence content\.

[Table5](https://arxiv.org/html/2609.26822#S8.T5)covers only AAPL and EURUSD and is rounded to six decimals\. Each cell is a single aggregate value with no uncertainty estimate, and the supplied evidence does not document repeated runs, so the entries are treated as single\-run observations and not as statistically established sensitivity effects\. For metricmm, the supplied relative\-degradation definition is

Δi=m⁡\(Ai\)−m⁡\(A0\)\|m⁡\(A0\)\|,\\Delta\_\{i\}=\\frac\{m\(A\_\{i\}\)\-m\(A\_\{0\}\)\}\{\|m\(A\_\{0\}\)\|\},\(28\)where a positive value indicates higher error than the full model\.

Table 5:Ablation variants as supplied, for AAPL and EURUSD only\. Each cell is a single aggregate value without per\-seed uncertainty, and lower is better\. A0 is full HARN; A1 replaces the associative memory with a gated recurrent update; A2 replaces resonance with the identity map; A3 zeros the bottom\-up evidence readout\. Configuration provenance is given in[Table3](https://arxiv.org/html/2609.26822#S6.T3)\.Two features of[Table5](https://arxiv.org/html/2609.26822#S8.T5)limit what it can show\. First, the differences are small and non\-monotonic\. A0 is not uniformly best: A1 and A3 have lower AAPL H1 RMSE than A0, and A1–A3 all have lower EURUSD H1 MASE\. A variant scoring below A0 on one metric does not by itself indicate that the removed mechanism is harmful, given the capacity and optimization differences described above\. Second, A0 does not coincide with the corresponding HARN means in[Table4](https://arxiv.org/html/2609.26822#S7.T4)\. For AAPL M15 the A0 MAE is 0\.506070 against a Panel A mean of 0\.507696, and for AAPL H1 it is 1\.055134 against a Panel B mean of 1\.059066\. These gaps are of similar size to the gaps between A0 and A1–A3, and the manuscript does not establish whether A0 is a rerun, a different epoch budget, or a different evaluation path\. Variant differences of this size therefore cannot currently be separated from unexplained run\-to\-run or protocol differences\. A measured reading is that no single component dominates these single\-run observations, and that firmer statements would require the multi\-seed, capacity\-matched experiments identified as proposed work\.[Figure8](https://arxiv.org/html/2609.26822#S8.F8)plotsΔi\\Delta\_\{i\}and is descriptive only\.

![Refer to caption](https://arxiv.org/html/2609.26822v1/relative_component_degradation.png)Figure 8:Relative component degradationΔi\\Delta\_\{i\}\([Eq\.28](https://arxiv.org/html/2609.26822#S8.E28)\) of ablation variants, by asset, timeframe, and variant, derived from[Table5](https://arxiv.org/html/2609.26822#S8.T5)\. The plot summarizes single aggregate values; it is not a statistical test and carries no uncertainty\.
## 9Code\-Level Causality Audit

This section records a static review of the source code against the event\-level causal contract defined in[Section3](https://arxiv.org/html/2609.26822#S3)\. It is a code\-level causal\-consistency audit and*not*an empirical test: no experiment was conducted in which bars afterτ\\tauwere perturbed, shuffled, or withheld to verify that forecasts remain unchanged\. Code review also cannot detect upstream problems in the data itself, such as source\-level bar construction, timestamp conventions, clock synchronization, or price adjustment\. An empirical check of this kind remains proposed work\.[Table6](https://arxiv.org/html/2609.26822#S9.T6)summarizes the audit of chronological splitting, completed\-bar alignment, next\-bar targets, training\-only statistics, left padding, shifted derived features, state masking, reset and detachment behavior, contiguous streams, cleaning, and validation\-based checkpoint selection, following the implementation\-level considerations relevant to leakage and validation in the forecasting literature\[[7](https://arxiv.org/html/2609.26822#bib.bib7),[8](https://arxiv.org/html/2609.26822#bib.bib8),[9](https://arxiv.org/html/2609.26822#bib.bib9)\]\. The findings indicate consistency with the specified future\-event information\-ordering contract under the implementation’s event ordering\. Reproducibility is supported by the artifacts listed in[Section12](https://arxiv.org/html/2609.26822#S12), subject to the raw\-data redistribution constraint\.

Three qualifications apply\. First, the lower\-level state is appended before the higher\-level evidence is read within the same event\. The evidence path is thus ordered with respect to future events but is not strictly past\-event\-only within an event; this is the same\-event evidence\-ordering qualification of[Section3](https://arxiv.org/html/2609.26822#S3), recorded in row 9 of[Table6](https://arxiv.org/html/2609.26822#S9.T6)\. Second, inactive levels are state\-preserving but not computation\-free: their windows and anchors are processed at every event, and only the persistent overwrite is masked \([Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)\)\. Third, the exact\-timestamp\-equality rule underlyinguτku\_\{\\tau\}^\{k\}is a data\-interface assumption\. Feeds with different timestamp conventions, clock skew, or asynchronous bar\-close reporting would require an explicit alignment policy before the same audit could be applied unchanged\.

Table 6:Condensed code\-level causal\-consistency audit from a static review of the source code; it is not an empirical causality test\. “Consistent \(qualified\)” denotes consistency with the event\-level information\-ordering contract, subject to the same\-event evidence\-ordering qualification of[Section3](https://arxiv.org/html/2609.26822#S3)\.\#DimensionMechanism observed in the codeFinding1SplittingChronological 70/15/15 train–validation–test slicesConsistent2AlignmentLatest completed timestamp no later than the anchor; exact timestamp equality determines an updateConsistent3Windows/
targetsWindow ends at the selected completed bar; target is the next completed barConsistent4StatisticsTraining\-only feature and target statistics are reused during evaluationConsistent5Local causalityLeft\-only padding and shifted previous\-close features avoid dependence on future observationsConsistent6ResonanceOnly states available at the same event are mixedConsistent7MemoryCurrent input, previous state, and bounded residual\-driven write determine memory updatesConsistent8Masked updatesInactive higher\-level persistent states and memory are retained rather than overwrittenConsistent9HistoryCurrent lower\-level read is appended before higher\-level readoutConsistent \(qualified\)10LifecycleStates are reset at boundaries and states and histories are detached after stepsConsistent11StreamingNon\-overlapping chronological streams preserve batch\-position continuityConsistent12CleaningInvalid rows are removed, duplicates are resolved, and timestamps are sorted chronologicallyConsistent13SelectionValidation loss selects the checkpoint before test evaluationConsistent14Test isolationTest observations are not used for training or validationConsistent
## 10Discussion and Limitations

#### What the evidence supports\.

HARN is an event\-driven multi\-timeframe architecture that maintains a compact state for each configured resolution and updates it when new information completes at that resolution\. Its principal mechanisms are the residual\-driven associative memory, cross\-scale resonance, and ordered history readout, which together retain, exchange, and read information across asynchronously completing timeframes\. The experiments describe the behavior of this design under the specified completed\-bar protocol, and the code\-level audit finds the implementation consistent with the event\-driven alignment rules used in them\. Under the reported configuration, HARN achieves lower reconstructed\-price errors than the accompanying single\-timeframe, anchor\-only baselines on the evaluated datasets, but this comparison remains descriptive because the models differ in information access, target formulation, and training procedure\. The ablations indicate that replacing individual components changes the reported values only slightly, and the single\-run design does not establish the independent contribution of any component\. The results therefore show that the design is viable in the evaluated multi\-timeframe setting\. They do not show that it is superior to alternative forecasting architectures, nor do they support attributing the observed differences to any single design choice\.

#### Limitations\.

The main limitations concern comparison scope, reference baselines, ablation strength, generalization, computational evaluation, and design assumptions\.

- •*Comparison scope\.*The baseline comparison does not isolate architectural effects from differences in information access, target formulation, or training procedure, and it is therefore not a controlled architectural comparison\.
- •*Reference baselines and statistics\.*No persistence baseline, confidence intervals, or paired predictive\-accuracy tests are reported, and the unusually small cross\-seed spread of HARN remains unexplained\.
- •*Ablation strength\.*The ablations are single\-run experiments on a subset of assets and are not parameter\- or compute\-matched\. They provide component\-level observations but not a statistically robust attribution of performance to individual mechanisms\.
- •*Generalization\.*The study covers four assets and specific evaluation periods\. The available evidence does not establish how performance changes across a broader range of assets, market conditions, hierarchy depths, or time periods\.
- •*Computational evaluation\.*The study does not report systematic measurements of inference latency, peak memory, throughput, or energy consumption, so the practical cost of maintaining and updating multiple resolution\-specific states remains incompletely characterized\.
- •*Design assumptions\.*The evidence\-readout coefficients are hand\-set and unvalidated, and the update indicator relies on exact timestamp equality under the chosen bar\-construction protocol\.

#### Proposed experiments\.

The following studies have not been performed and are proposed work: \(i\) evaluation of the zero\-BPS persistence baseline; \(ii\) generation of per\-event predictions from the available checkpoints, enabling confidence intervals and paired tests such as Diebold–Mariano and conditional predictive\-ability tests; \(iii\) verification of seed independence from checkpoint metadata and training trajectories; \(iv\) controlled comparisons that vary the available timeframes, the target representation, and the training protocol one at a time, together with matched multi\-timeframe or event\-driven baselines; \(v\) multi\-seed, capacity\-matched ablations, including sensitivity to the hand\-set evidence coefficients; \(vi\) an empirical perturbation test in which bars after the anchor time are altered or withheld; \(vii\) systematic measurement of latency, memory, and throughput; and \(viii\) archiving of generated outputs with command lines, source hashes, data checksums, and software and device versions\.

## 11Conclusion

This paper specifies HARN as an event\-driven, stateful multi\-timeframe forecaster: causal local encoding, gated matrix memory, all\-to\-all cross\-scale resonance, directional bottom\-up evidence, masked asynchronous updates, and BPS reconstruction\. Its central contribution is the completed\-bar update protocol, formalized in scalar and batched form and reviewed at the code level for event\-level causal consistency\. The reported results show numerically lower absolute\-price error for HARN than for the accompanying PatchTST and TimeXer runs\. Those baselines are single\-timeframe models trained on the anchor timeframe, their native configuration, while HARN is evaluated as a multi\-timeframe, event\-driven model, so the comparison is descriptive and scope\-specific and does not isolate architecture, target definition, or training recipe\. The single\-run ablations show no dominant component\. Whether the reported differences reflect the multi\-timeframe scope, the target representation, or the training recipe remains open\. Open work comprises a broader comparison with other multi\-timeframe or event\-driven models, the persistence evaluation, per\-event predictions, and the paired statistical analysis described in[Section10](https://arxiv.org/html/2609.26822#S10)\.

## 12Data and Code Availability

#### Repository artifacts\.

The project repository contains the source code examined for this study, including the HARN implementation, the data\-cleaning and completed\-bar alignment code, the complete preprocessing pipeline, the dataset configuration, the production training script, the standalone evaluator, the ablation trainer, and the PatchTST and TimeXer trainers\. It also contains model checkpoints for each seed and per\-seed result CSVs\. The candidate public URL[https://github\.com/NabeelAhmad9/HARN](https://github.com/NabeelAhmad9/HARN)returned HTTP 404 when checked during revision and is therefore not claimed as a public release\. The reviewed repository or workspace is the source of the available artifacts listed here\.

#### Data sharing and vendor\-agnostic acquisition\.

The raw market data used in the reported runs cannot be redistributed because the author does not have permission to share the vendor\-provided data\. The preprocessing pipeline is vendor\-neutral and can be applied to compatible data obtained from any market\-data vendor\. The expected input is one row per completed bar and timeframe with columns for timestamp, open, high, low, close, and volume\. Timestamps must be parseable and use a consistent timezone and session convention, and bar construction follows the configured per\-level timeframes \(for example, M5, M15, M30, H1, and H4\)\. The cleaning procedure parses timestamps, removes invalid rows, resolves duplicate timestamps by retaining the last occurrence, and sorts rows in ascending timestamp order\. Completed\-bar alignment selects the latest level\-kkbar whose timestamp is no later than the anchor timestamp, and the update indicator is set by exact timestamp equality\. Splits are chronological 70%/15%/15%, and feature and target statistics are computed on training samples only\. A compatible vendor feed with the same schema and bar\-construction rules can therefore regenerate the dataset with the provided pipeline\.

#### Checkpoints, evaluator, and generated outputs\.

The production trainer writes seed\-specific checkpoints\. The standalone evaluator reconstructs missing configuration fields from defaults and tensor shapes and evaluates on a single stream rather than the trainer’s eight\-stream default\. Ablation checkpoints retain configuration metadata but are not parameter\-count matched across variants\. Per\-event predictions and persistence outputs are not included as precomputed files in the current per\-seed CSV release, but they can be generated from the available checkpoints and preprocessing pipeline with a compatible vendor feed\. Archiving those generated outputs alongside the exact command line, source hash, data checksums, Python, PyTorch, and CUDA versions, and device details remains proposed work\.

#### Command templates\.

The source\-level command templates are as follows: production HARN training withpython \-m src\.models\.train\_harnand\-\-asset ASSET; standalone evaluation withpython \-m src\.models\.evaluate\_harn,\-\-asset ASSET, and\-\-seed SEED; and ablation training withpython \-m src\.ablation\.train\_harn\_ablations,\-\-asset ASSET, and\-\-variant A0throughA3\. These templates are documented commands, not recovered historical commands for the supplied tables\.

## Conflicts of Interest

The author declares that there are no conflicts of interest associated with this work\. The research was conducted independently, with no financial, commercial, institutional, or personal interests that could reasonably be considered to have influenced the design, implementation, evaluation, or interpretation of the study\.

No external organization provided funding, directed the experimental methodology, or exercised control over the reported results\. The work is presented solely for research purposes and does not constitute investment advice, a recommendation to trade any financial instrument, or an endorsement of any particular financial product or service\.

## Appendix ATechnical and Reproducibility Appendix

This appendix collects material that supports reproduction and technical scrutiny but would interrupt the main argument: the event\-transition derivation underlying[Eqs\.2](https://arxiv.org/html/2609.26822#S3.E2)and[4](https://arxiv.org/html/2609.26822#S3.E4), tensor shapes, the ordered per\-event algorithm, the configuration inventory across the three supplied training scripts, and the per\-seed metrics \([SectionsA\.1](https://arxiv.org/html/2609.26822#A1.SS1),[A\.2](https://arxiv.org/html/2609.26822#A1.SS2),[A\.3](https://arxiv.org/html/2609.26822#A1.SS3),[A\.4](https://arxiv.org/html/2609.26822#A1.SS4)and[A\.5](https://arxiv.org/html/2609.26822#A1.SS5)\)\. All values are configuration values or values already reported in the main text, and no new experiments are introduced\.

### A\.1Event\-transition derivation

[Section3](https://arxiv.org/html/2609.26822#S3)states the masked transition in terms of the scalar update indicatoruτku\_\{\\tau\}^\{k\}\([Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)\) and in the batched form actually implemented \([Eq\.4](https://arxiv.org/html/2609.26822#S3.E4)\)\. The equivalent case\-based form is closer to the code and makes explicit that masking selects between two already\-computed branches rather than modifying either\. Lett¯τk=max\{t:t≤τ,tis a completed level\-ktimestamp\}\\bar\{t\}\_\{\\tau\}^\{k\}=\\max\\\{t:t\\leq\\tau,\\ t\\text\{ is a completed level\-\}k\\text\{ timestamp\}\\\}be the no\-later\-than selection rule anduτk=𝟏\{t¯τk=τ\}u\_\{\\tau\}^\{k\}=\\mathbf\{1\}\\\{\\bar\{t\}\_\{\\tau\}^\{k\}=\\tau\\\}the resulting indicator of[Eq\.1](https://arxiv.org/html/2609.26822#S3.E1)\. For a candidate transitionΦk\\Phi\_\{k\},

\(rτ\+1k,Mτ\+1k\)=\{Φk​\(rτk,Mτk,Xτk\),uτk=1,\(rτk,Mτk\),uτk=0,\(r\_\{\\tau\+1\}^\{k\},M\_\{\\tau\+1\}^\{k\}\)=\\begin\{cases\}\\Phi\_\{k\}\(r\_\{\\tau\}^\{k\},M\_\{\\tau\}^\{k\},X\_\{\\tau\}^\{k\}\),&u\_\{\\tau\}^\{k\}=1,\\\\ \(r\_\{\\tau\}^\{k\},M\_\{\\tau\}^\{k\}\),&u\_\{\\tau\}^\{k\}=0,\\end\{cases\}\(29\)which coincides with[Eq\.2](https://arxiv.org/html/2609.26822#S3.E2)becauseuτk∈\{0,1\}u\_\{\\tau\}^\{k\}\\in\\\{0,1\\\}, so the convex combination always selects exactly one branch\. In the batched implementation the same selection is applied per sample: entrybbof the mask vectormτkm\_\{\\tau\}^\{k\}\([Eq\.3](https://arxiv.org/html/2609.26822#S3.E3)\) is the scalar indicator of streambb, and[Eq\.4](https://arxiv.org/html/2609.26822#S3.E4)broadcasts it over the state and memory dimensions\. The base level hasuτ1≡1u\_\{\\tau\}^\{1\}\\equiv 1, since the anchor stream is defined by level\-1 bar completions\. This establishes preservation of persistent state, not computational skipping: candidate encodings are formed for all supplied windows regardless of the mask \([Section5](https://arxiv.org/html/2609.26822#S5)\)\.

The causal contract of[Eq\.5](https://arxiv.org/html/2609.26822#S3.E5)is event\-level\. The lower\-level read is appended to its history before the higher\-level evidence readout \([Eq\.23](https://arxiv.org/html/2609.26822#S4.E23)\), so no future*event*is used, but a higher\-level update can consume a lower\-level representation produced at the same event\. This is the reason for the qualified entry 9 of[table6](https://arxiv.org/html/2609.26822#S9.T6), and it is the one place in the specification where “causal” must be read at event granularity rather than at the granularity of individual computation steps within an event\.

### A\.2Tensor shapes across the pipeline

[Table7](https://arxiv.org/html/2609.26822#A1.T7)traces the tensor shape at each stage of[Section4](https://arxiv.org/html/2609.26822#S4)for the default configuration \(ds=72d\_\{s\}=72,dm=18d\_\{m\}=18, window lengthT=32T=32, rankdr=36d\_\{r\}=36\)\.

Table 7:Tensor shapes at each pipeline stage in the default configuration \(ds=72d\_\{s\}=72,dm=18d\_\{m\}=18,T=32T=32,dr=36d\_\{r\}=36\)\.BBis the batch size andLLthe number of configured levels for the asset in question\.At the base level, the memory input concatenates a 72\-dimensional encoder output with a 72\-dimensional prior resonance context\. Higher levels additionally receive 72\-dimensional bottom\-up evidence, which accounts for the 144\- versus 216\-dimensional memory input of[Section4\.2](https://arxiv.org/html/2609.26822#S4.SS2)\. The readv¯=M​k\\bar\{v\}=Mkand residuals=v−v¯s=v\-\\bar\{v\}enter[Eqs\.17](https://arxiv.org/html/2609.26822#S4.E17)and[19](https://arxiv.org/html/2609.26822#S4.E19)as described there\. The shapes make explicit that the association store is a72×1872\\times 18matrix per sample, not a 72\-dimensional vector state, which is the structural difference from a standard recurrent cell noted in[Section2](https://arxiv.org/html/2609.26822#S2)\.

### A\.3Ordered per\-event algorithm

The exact ordering of the per\-event computation described in[Section5](https://arxiv.org/html/2609.26822#S5)is what makes the causal qualification of[Section3](https://arxiv.org/html/2609.26822#S3)\(entry 9 of[table6](https://arxiv.org/html/2609.26822#S9.T6)\) precise, and it is restated here as an ordered procedure:

1. 1\.select the latest completed window and update indicator for each level \([Section3](https://arxiv.org/html/2609.26822#S3)\);
2. 2\.encode all supplied windows and construct anchor features \([Section4](https://arxiv.org/html/2609.26822#S4)\);
3. 3\.apply prior\-state resonance \([Eq\.22](https://arxiv.org/html/2609.26822#S4.E22)\);
4. 4\.update the base level, which is always active;
5. 5\.append each current lower\-level read and compute the adjacent higher\-level evidence \([Eq\.23](https://arxiv.org/html/2609.26822#S4.E23)\);
6. 6\.compute candidate higher\-level transitions and apply their masks \([Eq\.4](https://arxiv.org/html/2609.26822#S3.E4)\);
7. 7\.apply posterior resonance to the resulting states \([Eq\.22](https://arxiv.org/html/2609.26822#S4.E22)\); and
8. 8\.emit one BPS forecast per configured level \([Eq\.24](https://arxiv.org/html/2609.26822#S4.E24)\)\.

Steps 3 and 7 are the two resonance applications of[Section4](https://arxiv.org/html/2609.26822#S4), and step 5 is the ordered evidence computation responsible for the qualified entry 9\. Price reconstruction occurs only after the BPS forecast is emitted in step 8, consistent with the network being optimized purely in BPS space\.

### A\.4Configuration inventory across the supplied scripts

The production trainer, the reported configuration table, and the ablation trainer do not share identical defaults \([Sections5](https://arxiv.org/html/2609.26822#S5)and[6\.2](https://arxiv.org/html/2609.26822#S6.SS2)\), and these artifact\-level configurations must not be silently merged\.[Table8](https://arxiv.org/html/2609.26822#A1.T8)consolidates the values referenced individually in the main text\.

Table 8:Key training settings of the production trainer, the reported main\-results recipe, and the ablation trainer\. The configurations differ only in the seed lists and in the ablation epoch budget; shared values are shown explicitly rather than inferred\.One discrepancy should be resolved from the available configuration and checkpoint metadata: the reported recipe lists eight\-stream validation and test, whereas the standalone evaluator uses one stream and reconstructs missing checkpoint configuration fields from defaults and tensor shapes\. Because the production trainer writes seed\-specific checkpoints and the dataset configuration is available in the repository, the reported three\-seed aggregates can be regenerated from the repository together with a compatible vendor feed\. Per\-seed aggregate metrics and seed\-specific checkpoints are available, and per\-event outputs can be generated from those checkpoints\. No target\-space metric is emitted by the supplied evaluators, and no persistence baseline, confidence interval, or paired test is provided \([Section7](https://arxiv.org/html/2609.26822#S7)\)\. The ablation variants A1–A3 are interface\-compatible interventions, not capacity\-controlled counterfactuals \([Section8](https://arxiv.org/html/2609.26822#S8)\)\.

### A\.5Per\-seed metrics

The per\-seed aggregate metrics reported here are supplied values for seeds 42, 151, and 359\. They are aggregate metrics, not per\-event predictions\.[Table9](https://arxiv.org/html/2609.26822#A1.T9)lists HARN’s per\-seed metrics for each asset and timeframe, and[Table10](https://arxiv.org/html/2609.26822#A1.T10)lists those of PatchTST and TimeXer\. The baseline per\-seed CSV does not contain a timeframe column, so the timeframe labels in[Table10](https://arxiv.org/html/2609.26822#A1.T10)are inferred from[Tables4](https://arxiv.org/html/2609.26822#S7.T4)and[2](https://arxiv.org/html/2609.26822#S6.T2)\. The mean±\\pmstandard deviation entries in[Table4](https://arxiv.org/html/2609.26822#S7.T4)are computed from these per\-seed values, and the reported standard deviations correspond to the population convention, in which the sum of squared deviations is divided by the number of seeds\. Recomputing the means and standard deviations from[Tables9](https://arxiv.org/html/2609.26822#A1.T9)and[10](https://arxiv.org/html/2609.26822#A1.T10)reproduces the corresponding entries in[Table4](https://arxiv.org/html/2609.26822#S7.T4), and no discrepancies were found\. Per\-seed training trajectories and per\-event predictions are not included as precomputed files and can be generated from the checkpoints and preprocessing pipeline\.

Table 9:HARN per\-seed aggregate test metrics for seeds 42, 151, and 359, grouped by asset and timeframe\. The means and standard deviations in[Table4](https://arxiv.org/html/2609.26822#S7.T4)are computed from these values\.AssetTFSeedMAERMSEsMAPEMASEAAPLH1421\.058 7521\.058\\,7521\.801 0581\.801\\,0580\.425 9640\.425\\,9644\.815 8504\.815\\,8501511\.060 3291\.060\\,3291\.800 2361\.800\\,2360\.426 4670\.426\\,4674\.823 0214\.823\\,0213591\.058 1171\.058\\,1171\.799 3791\.799\\,3790\.425 7600\.425\\,7604\.812 9614\.812\\,961M15420\.506 9800\.506\\,9800\.933 0280\.933\\,0280\.204 6530\.204\\,6531\.325 7131\.325\\,7131510\.507 9870\.507\\,9870\.933 3960\.933\\,3960\.205 0640\.205\\,0641\.328 3451\.328\\,3453590\.508 1200\.508\\,1200\.934 0440\.934\\,0440\.205 1230\.205\\,1231\.328 6941\.328\\,694EURUSDH1420\.000 6170\.000\\,6170\.000 9070\.000\\,9070\.053 0210\.053\\,0213\.468 8553\.468\\,8551510\.000 6180\.000\\,6180\.000 9080\.000\\,9080\.053 1030\.053\\,1033\.474 2283\.474\\,2283590\.000 6130\.000\\,6130\.000 9040\.000\\,9040\.052 6700\.052\\,6703\.445 9653\.445\\,965H4420\.001 2270\.001\\,2270\.001 7460\.001\\,7460\.105 4500\.105\\,45013\.545 65413\.545\\,6541510\.001 2250\.001\\,2250\.001 7430\.001\\,7430\.105 2670\.105\\,26713\.522 09013\.522\\,0903590\.001 2290\.001\\,2290\.001 7460\.001\\,7460\.105 5760\.105\\,57613\.562 51713\.562\\,517M15420\.000 3050\.000\\,3050\.000 4650\.000\\,4650\.026 1730\.026\\,1730\.861 1500\.861\\,1501510\.000 3090\.000\\,3090\.000 4690\.000\\,4690\.026 5760\.026\\,5760\.874 4690\.874\\,4693590\.000 3050\.000\\,3050\.000 4660\.000\\,4660\.026 1820\.026\\,1820\.861 4440\.861\\,444USDCHFH1420\.000 5270\.000\\,5270\.000 7620\.000\\,7620\.066 7300\.066\\,7303\.237 4413\.237\\,4411510\.000 5250\.000\\,5250\.000 7630\.000\\,7630\.066 4590\.066\\,4593\.224 3943\.224\\,3943590\.000 5270\.000\\,5270\.000 7630\.000\\,7630\.066 7060\.066\\,7063\.236 3703\.236\\,370M15420\.000 2630\.000\\,2630\.000 3910\.000\\,3910\.033 2880\.033\\,2880\.800 2270\.800\\,2271510\.000 2630\.000\\,2630\.000 3910\.000\\,3910\.033 2650\.033\\,2650\.799 6680\.799\\,6683590\.000 2630\.000\\,2630\.000 3910\.000\\,3910\.033 2990\.033\\,2990\.800 4980\.800\\,498M30420\.000 3740\.000\\,3740\.000 5440\.000\\,5440\.047 2870\.047\\,2871\.610 3271\.610\\,3271510\.000 3720\.000\\,3720\.000 5430\.000\\,5430\.047 1280\.047\\,1281\.604 8941\.604\\,8943590\.000 3730\.000\\,3730\.000 5430\.000\\,5430\.047 2090\.047\\,2091\.607 6931\.607\\,693XAUUSDM15424\.532 4214\.532\\,4216\.820 1306\.820\\,1300\.108 4620\.108\\,4623\.485 1433\.485\\,1431514\.523 9204\.523\\,9206\.815 8186\.815\\,8180\.108 2630\.108\\,2633\.478 6063\.478\\,6063594\.530 0764\.530\\,0766\.826 7676\.826\\,7670\.108 4040\.108\\,4043\.483 3403\.483\\,340M5422\.550 0122\.550\\,0123\.844 5273\.844\\,5270\.061 0480\.061\\,0481\.135 0771\.135\\,0771512\.553 4652\.553\\,4653\.845 9443\.845\\,9440\.061 1310\.061\\,1311\.136 6141\.136\\,6143592\.552 8572\.552\\,8573\.842 4313\.842\\,4310\.061 1170\.061\\,1171\.136 3441\.136\\,344Table 10:Per\-seed aggregate test metrics for PatchTST and TimeXer, grouped by asset, model, and timeframe\. Timeframe labels are inferred from[Tables4](https://arxiv.org/html/2609.26822#S7.T4)and[2](https://arxiv.org/html/2609.26822#S6.T2)because the source CSV has no timeframe column\.AssetModelTFSeedMAERMSEsMAPEMASEAAPLPatchTSTM15421\.358 9671\.358\\,9671\.949 9871\.949\\,9870\.527 7550\.527\\,7553\.553 5913\.553\\,5911512\.095 4962\.095\\,4962\.865 2262\.865\\,2260\.791 9030\.791\\,9035\.479 5535\.479\\,5533592\.768 7402\.768\\,7403\.291 2603\.291\\,2601\.064 2831\.064\\,2837\.240 0327\.240\\,032TimeXerM15423\.855 0253\.855\\,0254\.600 7364\.600\\,7361\.480 7431\.480\\,74310\.080 58110\.080\\,5811511\.868 1871\.868\\,1872\.529 7582\.529\\,7580\.764 7240\.764\\,7244\.885 1584\.885\\,1583592\.922 4462\.922\\,4463\.709 8663\.709\\,8661\.100 4291\.100\\,4297\.641 9627\.641\\,962EURUSDPatchTSTM15420\.000 6580\.000\\,6580\.000 8370\.000\\,8370\.056 4280\.056\\,4281\.859 3081\.859\\,3081510\.001 1350\.001\\,1350\.001 3880\.001\\,3880\.097 0500\.097\\,0503\.206 5703\.206\\,5703590\.000 4840\.000\\,4840\.000 6760\.000\\,6760\.041 5540\.041\\,5541\.366 7521\.366\\,752TimeXerM15420\.011 2270\.011\\,2270\.013 7100\.013\\,7100\.963 5140\.963\\,51431\.726 97931\.726\\,9791510\.010 3850\.010\\,3850\.012 4350\.012\\,4350\.891 0520\.891\\,05229\.348 18629\.348\\,1863590\.010 9010\.010\\,9010\.013 0060\.013\\,0060\.935 5280\.935\\,52830\.805 51530\.805\\,515USDCHFPatchTSTM15420\.000 7800\.000\\,7800\.001 0120\.001\\,0120\.099 1580\.099\\,1582\.371 8722\.371\\,8721510\.000 5710\.000\\,5710\.000 7730\.000\\,7730\.072 2180\.072\\,2181\.738 6371\.738\\,6373590\.000 5630\.000\\,5630\.000 8050\.000\\,8050\.071 4150\.071\\,4151\.712 9441\.712\\,944TimeXerM15420\.014 5170\.014\\,5170\.015 9620\.015\\,9621\.828 8441\.828\\,84444\.169 50644\.169\\,5061510\.013 4300\.013\\,4300\.014 7610\.014\\,7611\.693 3731\.693\\,37340\.862 84040\.862\\,8403590\.014 4860\.014\\,4860\.016 2310\.016\\,2311\.825 6911\.825\\,69144\.073 77844\.073\\,778XAUUSDPatchTSTM5428\.634 8858\.634\\,88511\.674 33711\.674\\,3370\.205 6270\.205\\,6273\.843 6143\.843\\,61415111\.015 64211\.015\\,64213\.945 93013\.945\\,9300\.264 0410\.264\\,0414\.903 3514\.903\\,3513599\.039 8669\.039\\,86612\.415 75212\.415\\,7520\.215 3990\.215\\,3994\.023 8824\.023\\,882TimeXerM54216\.241 03116\.241\\,03121\.248 00821\.248\\,0080\.380 1070\.380\\,1077\.229 3097\.229\\,30915116\.801 72916\.801\\,72921\.078 02221\.078\\,0220\.395 5850\.395\\,5857\.478 8917\.478\\,89135912\.849 67012\.849\\,67014\.958 77314\.958\\,7730\.308 1000\.308\\,1005\.719 7265\.719\\,726

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