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Temporal Predictive Multiplicity: Equally Accurate Time Series Models Yield Different Forecast Trajectories

Emanuele Albini, Francesca Toni, Saumitra Mishra, Francesco Leofante

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 187

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%

Last extracted: 10/11/2026, 3:38:56 AM

Summary

The paper introduces 'temporal predictive multiplicity,' a framework addressing how models with near-identical predictive performance in time-series forecasting can yield substantially different forecast trajectories. It establishes a hierarchy of multiplicity (pointwise, horizon-wise, temporal) and demonstrates that constraining aggregate performance or even horizon-wise disagreement does not eliminate trajectory-level variability. Experiments with 19 neural architectures on 11 datasets confirm that near-optimal models often produce divergent temporal patterns, exposing a gap in existing multiplicity studies.

Entities (8)

Temporal Predictive Multiplicity · concept · 98%Horizon-Wise Predictive Multiplicity · concept · 95%Pointwise Predictive Multiplicity · concept · 95%Rashomon Effect · concept · 92%Imperial College London · organization · 90%JP Morgan AI Research · organization · 90%Neural Forecasting Architectures · model-class · 90%Breiman · person · 85%

Relation Signals (8)

Temporal Predictive Multiplicity → includeslevel → Horizon-Wise Predictive Multiplicity

confidence 95% · our framework organizes multiplicity into a three-level hierarchy: pointwise, horizon-wise, and temporal

Temporal Predictive Multiplicity → includeslevel → Temporal Predictive Multiplicity

confidence 95% · our framework organizes multiplicity into a three-level hierarchy: pointwise, horizon-wise, and temporal

Temporal Predictive Multiplicity → includeslevel → Pointwise Predictive Multiplicity

confidence 95% · our framework organizes multiplicity into a three-level hierarchy: pointwise, horizon-wise, and temporal

Temporal Predictive Multiplicity → isa → Framework

confidence 95% · we introduce temporal predictive multiplicity, a framework that characterizes disagreement over complete forecast trajectories

Rashomon Effect → causes → Predictive multiplicity

confidence 90% · This phenomenon, known as the Rashomon Effect (Breiman, 2001), gives rise to predictive multiplicity

Temporal Predictive Multiplicity → extends → Predictive multiplicity

confidence 90% · In time-series forecasting, however, predictions across horizons jointly define a trajectory... To address this problem, we introduce temporal predictive multiplicity

Breiman → proposed → Rashomon Effect

confidence 90% · articulated by Breiman as the Rashomon Effect (Breiman, 2001)

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Abstract

Abstract:Models with near-identical predictive performance can yield substantially different predictions, a phenomenon known as predictive multiplicity. Prior work has mostly studied this at the level of individual scalar outputs. In time-series forecasting, however, predictions across horizons jointly define a trajectory, and horizon-wise comparisons can hide important differences in predictive behavior. To address this problem, we introduce temporal predictive multiplicity, a framework that characterizes disagreement over complete forecast trajectories among models with near-identical predictive performance. We show that constraining predictive performance alone can still admit a broad range of different trajectories. We further show that constraining multiplicity at individual horizons partially reduces, but does not eliminate, trajectory-level multiplicity. Experiments with 19 neural forecasting architectures on 11 datasets confirm that near-optimal models can exhibit substantial variability in the forecast trajectories they produce, and trajectory-level disagreement is largely unrelated to horizon-wise disagreement. Our framework, therefore, exposes a gap in existing multiplicity studies: models with indistinguishable predictive performance imply fundamentally different temporal trajectories, with consequential downstream effects.

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Temporal Predictive Multiplicity: Equally Accurate Time Series Models Yield Different Forecast Trajectories Emanuele Albini †thanks: Corresponding Author Affiliation: J.P. Morgan AI Research Affiliation: Imperial College London Affiliation: London, UK Email: emanuele.albini@jpmorgan.com Francesca Toni Affiliation: Imperial College London Affiliation: London, UK Email: ft@imperial.ac.uk Saumitra Mishra Affiliation: J.P. Morgan AI Research Affiliation: London, UK Email: saumitra.mishra@jpmorgan.com Francesco Leofante Affiliation: Imperial College London Affiliation: London, UK Email: f.leofante@imperial.ac.uk Abstract Models with near-identical predictive performance can yield substantially different predictions, a phenomenon known as predictive multiplicity. Prior work has mostly studied this at the level of individual scalar outputs. In time-series forecasting, however, predictions across horizons jointly define a trajectory, and horizon-wise comparisons can hide important differences in predictive behavior. To address this problem, we introduce temporal predictive multiplicity, a framework that characterizes disagreement over complete forecast trajectories among models with near-identical predictive performance. We show that constraining predictive performance alone can still admit a broad range of different trajectories. We further show that constraining multiplicity at individual horizons partially reduces, but does not eliminate, trajectory-level multiplicity. Experiments with 19 neural forecasting architectures on 11 datasets confirm that near-optimal models can exhibit substantial variability in the forecast trajectories they produce, and trajectory-level disagreement is largely unrelated to horizon-wise disagreement. Our framework, therefore, exposes a gap in existing multiplicity studies: models with indistinguishable predictive performance imply fundamentally different temporal trajectories, with consequential downstream effects. 1 Introduction In machine learning, multiple models can often achieve nearly indistinguishable predictive performance while producing different predictions. This phenomenon, known as the Rashomon Effect (Breiman, 2001), gives rise to predictive multiplicity: the fact that models with comparable performance can make different predictions for the same instance. A growing literature has studied the existence and consequences of competing models within the Rashomon set, the collection of models whose predictive performance is close to optimal (Fisher et al., 2019; Semenova et al., 2022), showing that model choice can affect individual predictions (Marx et al., 2020), feature attributions (Dong & Rudin, 2020), and downstream decisions (Du et al., 2025), even when aggregate performance remains essentially unchanged. Time-series forecasting is a particularly natural setting in which to study this phenomenon. Despite the proliferation of deep forecasting architectures (Lim & Zohren, 2021; Benidis et al., 2022), no model class consistently dominates: simple linear models remain competitive with more complex neural architectures (Zeng et al., 2023; Toner & Darlow, 2024), while model rankings shift considerably with hyper-parameter choices and evaluation protocols (Brigato et al., 2026). Forecasting practitioners therefore routinely train a heterogeneous pool of models with near-identical aggregate performance, yet must select a single model for deployment. In the absence of a principled framework that characterizes how these models differ temporally, this choice is effectively arbitrary. y yh11223344y yh11223344y yh11223344Model 1Model 2Origin 1Origin 2Origin 3 Figure 1: Examples of forecast trajectories that are indistinguishable under current multiplicity frameworks. At every horizon, the disagreement among forecasts is identical across the three panels. Yet, the panels exhibit markedly different patterns of temporal disagreement at the trajectory level. Current multiplicity frameworks offer little guidance for selecting among competing forecasting models. Existing work has largely focused on individual predictive multiplicity, in which models produce a single scalar prediction for each instance, and multiple outputs are treated as independent. Forecasts, however, are temporal objects that require a different treatment: they are sequences of predictions over multiple time points whose joint behavior defines a forecast trajectory, as illustrated in Figure 1. Ignoring this structure would render markedly different trajectories indistinguishable. Motivated by these observations, we introduce a new theoretical framework for predictive multiplicity in time-series forecasting. At its core, our framework organizes multiplicity into a three-level hierarchy: pointwise, horizon-wise, and temporal, making explicit which aspects of forecast structure each level can capture. We then use this hierarchy to reveal two fundamental limitations of conventional notions of multiplicity. First, near-optimal predictive performance provides weak control over forecast trajectories: constraining aggregate performance alone allows models to differ substantially over an entire trajectory, with this freedom increasing at longer horizons. Second, even when disagreement is tightly controlled at individual horizons, at pointwise and horizon-wise levels, substantial temporal differences can remain, including the direction of predicted changes, the timing of events, and the shape and frequency structure of forecast trajectories. We complement these theoretical results with experiments on 19 neural forecasting architectures across 11 public datasets. We find that near-optimal models frequently produce substantially different forecast trajectories, including predictions of changes in opposite directions. Moreover, trajectory-level disagreement is largely unrelated to disagreement measured at individual horizons, confirming that temporal multiplicity cannot be recovered from conventional pointwise or horizon-wise diagnostics. Finally, through a case study, we show that these differences can translate into materially different downstream decisions. Taken together, these results show that model choice is consequential; our framework makes the underlying temporal differences explicit and measurable, giving practitioners a principled basis for auditing near-optimal candidates along application-relevant dimensions and for justifying the model they ultimately deploy. 2 Related Work Model Multiplicity in Machine Learning. That many models may explain the same data approximately equally well was articulated by Breiman as the Rashomon Effect (Breiman, 2001), and has resurfaced in modern machine learning through studies of underspecification and training instability (Madhyastha & Jain, 2019; D’Amour et al., 2022; Teney et al., 2022). Models within a Rashomon set have been shown to differ in variable importance and feature attributions (Fisher et al., 2019; Dong & Rudin, 2020; Donnelly et al., 2023; Laberge et al., 2023) and, at the prediction level, to assign conflicting predictions to the same instance, a phenomenon formalized by Marx et al. (2020) as predictive multiplicity. A growing literature distinguishes predictive multiplicity from its procedural, explanatory, and dataset-induced forms (Black et al., 2022; Heljakka et al., 2023; Ganesh et al., 2025), investigates its root causes (Parikh, 2026), weighs its opportunities and risks (Rudin et al., 2024; Hsu et al., 2026) and reconcile its implications for downstream decisions (Du et al., 2025). All of these works treat each instance as an independent prediction problem: their notions of multiplicity live at the pointwise level of our hierarchy, and are provably blind to the complex structure of time series, as we will later show. Model Equivalence and Multiplicity in Forecasting. Forecasting research has long recognized that several models may be statistically indistinguishable in predictive ability: the Model Confidence Set of Hansen et al. (2011) retains the models whose differences cannot be statistically established, Fosten & Gutknecht (2021) let this set vary with the horizon through horizon confidence sets, while Quaedvlieg (2021) and Monschang et al. (2026) select models with one joint constraint on all horizons. Such statistical tests compare models through their losses, either globally, per horizon, or jointly, and thus determine which models are performance-equivalent; they do not, however, characterize how the forecasts of equivalent models differ, which is precisely the gap that this paper addresses. Recent work has begun to bring a Rashomon-set perspective to forecasting. This includes studies of predictive and procedural multiplicity in recovery-rate forecasting (Hibbeln et al., 2026), functional differences among volatility forecasters with tied loss (Cortesi et al., 2026), and variation in feature reliance among time-series forecasting models (Berry et al., 2024). Closest to our setting, Kale et al. (2026) introduce horizon-constrained Rashomon sets to track predictive multiplicity across forecast horizons in chaotic dynamical systems. All these works sit at the horizon-wise level of our hierarchy, which again cannot fully characterize trajectory-level multiplicity. 3 A Hierarchy of Predictive Multiplicity Notions This section develops our hierarchy of predictive multiplicity notions for forecasting. We begin by introducing the forecasting setup and notation. We then define three kinds of multiplicity: pointwise, horizon-wise, and temporal, which differ in whether they compare isolated predictions, horizon-specific collections of predictions, or complete forecast trajectories. Finally, we show that these levels are not merely alternative aggregations of the same disagreement: each retains a different amount of information about the forecasts. Setup and Notation. Consider a class of models ℱF. Given a model f∈ℱf∈F, let ℒ⁡(f)L(f) be the loss of f evaluated over a collection of origins T. For a tolerance ϵ≥0ε≥ 0, the corresponding Rashomon set is (Fisher et al., 2019): ℛϵ=f∈ℱ:ℒ⁡(f)≤ℒ⋆+ϵwhereℒ⋆=ℒ⁡(f⋆),f⋆∈argminf∈ℱ⁡ℒ(f).R_ε= \f∈F:L(f) +ε \ =L(f ), f ∈ _f∈FL(f). (1) Given a model f∈ℛϵf∈R_ε, let ^f∈ℝT×H Y^f ^T× H be its forecast matrix, ∈ℝT×H Y ^T× H the target matrix, and f=−^f E^f= Y- Y^f its error matrix, with rows indexed by origins t∈t∈T and columns by forecast horizons h∈1,…,Hh∈\1,…,H\. We consider three temporal trajectories extracted from these matrices: • the forecast trajectory ^tf=(y^t+1f,…,y^t+Hf)⊤ y^f_t=( y^f_t+1,…, y^f_t+H) , the t-th row of ^f Y^f: forecasts across horizons; • the revision trajectory ^tf r^f_t, the t-th anti-diagonal of ^f Y^f: successive forecasts of the same target; • the error trajectory tf=t−^tf e^f_t= y_t- y^f_t, the t-th row of f E^f: forecast errors across horizons. Forecast and revision trajectories also arise in the forecast-stability literature, where their fluctuations underlie the definitions of horizontal and vertical (in)stability, respectively (Van Belle et al., 2023; Godahewa et al., 2025). That literature regularizes such fluctuations within the forecasts of a single model; here, instead, we use the same geometry to characterize disagreement among the members of the Rashomon set: distinct models with equivalent loss. Model 1Model 2Origin 1Origin 2 y y (a) Loss one error bag per model y y (b) Pointwise Multiplicity per-cell connections among models y yh112233 (c) Horizon-Wise Multiplicity per-horizon bands y yh112233 (d) Temporal Multiplicity whole trajectories Figure 2: Pictorial representation of the structure that each of the loss and the three levels of the multiplicity hierarchy is aware of on a running example. A Hierarchy of Predictive Multiplicity in Forecasting. Having established the forecasting setup, we now introduce the three levels of our hierarchy. The pointwise level treats every prediction in isolation, while the horizon-wise level assesses disagreement separately at each forecast horizon. We then introduce temporal predictive multiplicity, a trajectory-level notion that compares how competing models behave jointly across horizons. The first two levels generalize existing multiplicity measures to forecasting and compare scalar predictions through their absolute difference (Watson-Daniels et al., 2023; Hibbeln et al., 2026, e.g.,); temporal multiplicity instead compares complete trajectories using distances designed to capture distinct aspects of their temporal behavior. Definition 3.1 (Pointwise Predictive Multiplicity). Let τ≥0τ≥ 0. The pointwise prediction range ν, pointwise prediction diameter ω, pointwise ambiguity ατ _τ, and pointwise discrepancy ςτ _τ of ℛϵR_ε are: ν=t∈1≤h≤H[maxf,g∈ℛϵ⁡|y^t+hf−y^t+hg|]ω=max⁡t∈1≤h≤Hf,g∈ℛϵ⁡[|y^t+hf−y^t+hg|]ατ=t∈1≤h≤H[maxf,g∈ℛϵ|y^ft+h−y^gt+h|>τ]ςτ=maxf,g∈ℛϵt∈1≤h≤H[|y^ft+h−y^gt+h|>τ] array[]@c@ c@ ν= E_ subarrayct∈T\\ 1≤ h≤ H subarray [ _f,g∈R_ε | y^f_t+h- y^g_t+h | ]& ω= _f,g∈R_ε\, E_ subarrayct∈T\\ 1≤ h≤ H subarray [ | y^f_t+h- y^g_t+h | ]\\[4.30554pt] _τ= E_ subarrayct∈T\\ 1≤ h≤ H subarray [ 1 \ _f,g∈R_ε | y^f_t+h- y^g_t+h |>τ \ ]& _τ= _f,g∈R_ε\, E_ subarrayct∈T\\ 1≤ h≤ H subarray [ 1 \ | y^f_t+h- y^g_t+h |>τ \ ] array The range and ambiguity average the per-instance maximal disagreement, while the diameter and discrepancy ask how far a single pair of models drifts apart on average, hence ω≤νω≤ν and ς≤α ≤α. Intuitively, pointwise ambiguity (ατ _τ) and discrepancy (ςτ _τ) generalize Marx et al. (2020) and Watson-Daniels et al. (2023) by maximizing over all pairs of competing models, not against f⋆f , to capture all the multiplicity in ℛϵR_ε, and by treating the (t,h)(t,h) cells as instances. The prediction range (ν) averages the width of the viable prediction range of Watson-Daniels et al. (2023) over the forecast matrix; the prediction diameter (ω) is a novel metric that we propose as its complement. Following a similar pattern, we formalize the per-horizon notion of multiplicity of Kale et al. (2026). Definition 3.2 (Horizon-Wise Predictive Multiplicity). Let τ≥0τ≥ 0. For each horizon h∈1,…,Hh∈\1,…,H\, the horizon-wise prediction range ν(h)ν^(h), prediction diameter ω(h)ω^(h), ambiguity ατ(h)α^(h)_τ, and discrepancy ςτ(h) ^(h)_τ of ℛϵR_ε, producing profiles indexed by h, are defined as follows: ν(h)=t∈[maxf,g∈ℛϵ⁡|y^t+hf−y^t+hg|]ω(h)=max⁡t∈f,g∈ℛϵ⁡[|y^t+hf−y^t+hg|]α(h)τ=t∈[maxf,g∈ℛϵ|y^ft+h−y^gt+h|>τ]ς(h)τ=maxf,g∈ℛϵt∈[|y^ft+h−y^gt+h|>τ] array[]@c@ c@ ν^(h)= E_t∈T [ _f,g∈R_ε | y^f_t+h- y^g_t+h | ]& ω^(h)= _f,g∈R_ε\, E_t∈T [ | y^f_t+h- y^g_t+h | ]\\[4.30554pt] α^(h)_τ= E_t∈T [ 1 \ _f,g∈R_ε | y^f_t+h- y^g_t+h |>τ \ ]& ^(h)_τ= _f,g∈R_ε\, E_t∈T [ 1 \ | y^f_t+h- y^g_t+h |>τ \ ] array We now introduce temporal predictive multiplicity. Unlike the pointwise and horizon-wise levels, it compares complete forecast trajectories and can therefore capture how competing models disagree across forecast horizons or across successive revisions. Definition 3.3 (Temporal Predictive Multiplicity). Let τ≥0τ≥ 0, let tf v^f_t denote any of the trajectories we defined earlier (forecast ^tf y^f_t, revision ^tf r^f_t, or error tf e^f_t) and let d:ℝH×ℝH→[0,∞)d:R^H×R^H→[0,∞) be a trajectory distance function. The temporal trajectory range νd _d, trajectory diameter ωd _d, ambiguity αd,τ _d,τ, and discrepancy ςd,τ _d,τ of ℛϵR_ε for such a trajectory are defined as: νd=t∈[maxf,g∈ℛϵ⁡d⁡(tf,tg)]ωd=max⁡t∈f,g∈ℛϵ⁡[d⁡(tf,tg)]αd,τ=t∈[maxf,g∈ℛϵd(ft,gt)>τ]ςd,τ=maxf,g∈ℛϵt∈[d(ft,gt)>τ] array[]@c@ c@ _d= E_t∈T [ _f,g∈R_εd ( v^f_t, v^g_t ) ]& _d= _f,g∈R_ε\, E_t∈T [d ( v^f_t, v^g_t ) ]\\[4.30554pt] _d,τ= E_t∈T [ 1 \ _f,g∈R_εd ( v^f_t, v^g_t )>τ \ ]& _d,τ= _f,g∈R_ε\, E_t∈T [ 1 \d ( v^f_t, v^g_t )>τ \ ] array The choice of dd determines which aspect of temporal disagreement is assessed. Models may differ in the magnitude, direction, volatility, shape, spectral structure, or timing of their trajectories. No single distance captures all of these forms of disagreement without conflating them. We therefore leave dd unspecified in the general definition and provide concrete instantiations in Section 5. Information Awareness Across Multiplicity Levels. The temporal statistics defined above summarize disagreement over complete trajectories. We next show that the three levels retain different information about the forecast matrix. We formalize this distinction through their sensitivity to rearrangements of forecast data, demonstrating that pointwise and horizon-wise statistics are blind to aspects of disagreement that depend on how predictions are coupled across horizons. Definition 3.4 (Multiplicity-, Horizon-, and Trajectory-Aware Statistics). A statistic is any function of the target matrix Y and the forecast matrices ^ff∈ℛϵ\ Y^f\_f∈R_ε. A statistic is: • multiplicity-aware if not invariant under per-model cell permutations f↦πf(f) E^f π^f( E^f), with πfπ^f a permutation of the cells ×1,…,HT×\1,…,H\ that may depend on f; • horizon-aware if not invariant under cell permutations, per-model cell permutations with a common πf≡π^f≡π; • trajectory-aware if not invariant under row permutations, cell permutations π⁡(t,h)=(πh(t),h)π(t,h)=( _h(t),h) that preserve horizons and re-index the origins within each horizon. These statistics are nested, since row permutations are cell permutations, and cell permutations are per-model cell permutations. As illustrated in Figure 2, the loss retains none of this structure; pointwise statistics retain model alignment but treat all THTH cells as an unordered collection; and horizon-wise statistics additionally distinguish horizons, but not the ordering of origins within them. Only temporal statistics retain full trajectory structure. For example, swapping origins at one horizon flattens the trajectories in the final panel of Figure 2 without changing any lower-level statistic, despite changing whether models agree on the direction of change. The proof is in Appendix C. 4 How Loosely Is Temporal Multiplicity Constrained? Knowing what each level captures, we now quantify how loosely information below the temporal level constrains temporal multiplicity. We consider losses of the following common form: ℒ⁡(f)=1TH∑t∈∑h=1Hℓ⁡(et,hf),L(f)= 1TH _t∈T _h=1^H (e^f_t,h ), (2) for some continuous ℓ:ℝ→[0,∞) :R→[0,∞) with ℓ⁡(0)=0 (0)=0 that is nondecreasing in |e||e| and unbounded (ℓ⁡(e)→∞ (e)→∞ as |e|→∞|e|→∞), with inverse ℓ−1(L)=max⁡e≥0:ℓ⁡(e)≤L ^-1(L)= \e≥ 0: (e)≤ L\. This covers most standard losses, including MSE (ℓ⁡(e)=e2 (e)=e^2), MAE (ℓ⁡(e)=|e| (e)=|e|), as well as MASE-type losses. We begin by quantifying how loosely the performance constraint defining the Rashomon set constrains temporal multiplicity. The bound below is tight for sufficiently rich model classes: we call ℱF shell-rich (at level ϵε) if for every ∈ℝT×H E ^T× H with ℒ⋆≤ℒ⁡()≤ℒ⋆+ϵL ( E) +ε there exists f∈ℱf∈F with f= E^f= E. This is a mild condition, far weaker than universal approximation (Hornik, 1991): ℱF need only realize error patterns with loss near ℒ⋆L . It is readily satisfied by sufficiently overparameterized networks, which can interpolate arbitrary values at finitely many inputs (Zhang et al., 2017). Theorem 4.1 (Multiplicity Bounds Under Rashomon Set Membership). For all f,g∈ℛϵf,g∈R_ε: |y^t+hf−y^t+hg|≤ 2ℓ−1(TH(ℒ⋆+ϵ))∀t∈,h∈1,…,H. | y^f_t+h- y^g_t+h |\;≤\;2\, ^-1\! (TH\,(L +ε) ) ∀ t∈T,\ h∈\1,…,H\. If ℱF is also shell-rich, the bound is attained, i.e., maxf,g∈ℛϵ⁡|y^t+hf−y^t+hg|=2ℓ−1(TH(ℒ⋆+ϵ)) _f,g∈R_ε | y^f_t+h- y^g_t+h |=2\, ^-1\! (TH\,(L +ε) ). Theorem 4.1 shows that Rashomon set membership imposes a single scalar budget on the THTH-dimensional error matrix, without constraining where errors fall within such a matrix. It thus uncovers a failure mode of performance-based model selection in temporal settings: beyond leaving open how errors distribute across instances (Marx et al., 2020), the loss is blind to how they distribute across horizons within each instance, i.e., to the shape of every forecast trajectory. Moreover, this freedom scales with the horizon: for ℓ⁡(e)=|e|p (e)=|e|^p, the theorem’s pointwise bound grows as H1/pH^1/p (H H under MSE and linearly under MAE). Longer horizons therefore weaken the control that Rashomon membership provides over individual forecasts, limiting what membership alone guarantees about the resulting trajectories. At the opposite extreme of our hierarchy lies horizon-wise multiplicity. We now quantify how constraining at this level affects temporal multiplicity by imposing the strongest requirement afforded by its definition, vanishing horizon-wise ambiguity: ατh(h)=0α^(h)_ _h=0 for every horizon h, at tolerances =(τ1,…,τH) τ=( _1,…, _H). This implies ν(h),ω(h)≤τhν^(h),ω^(h)≤ _h and ςτh(h)=0 ^(h)_ _h=0. When τh≡τ _h≡τ, this condition also recovers the pointwise case, ατ=0 _τ=0. Theorem 4.2 (Multiplicity Bound Under Horizon-Wise Control). Fix tolerances =(τ1,…,τH) τ=( _1,…, _H). Then, every pair f,g∈ℱf,g∈F with ατh(h)(f,g)=0α^(h)_ _h(\f,g\)=0 for all h satisfies: |y^t+hf−y^t+hg|≤τh∀t∈,h∈1,…,H. | y^f_t+h- y^g_t+h |\;≤\; _h ∀ t∈T,\ h∈\1,…,H\. If, moreover, ℱF is shell-rich, the bound is attained by pairs in ℛϵR_ε whenever it is loss-feasible, i.e., whenever the loss budget of Theorem 4.1 does not bind first: ∑hℓ⁡(τh/2)≤TH(ℒ⋆+ϵ) _h ( _h/2)≤ TH(L +ε). Indeed, horizon-wise multiplicity imposes a stronger constraint on predictions, bounding the absolute difference between models at each horizon by τh _h. Yet, except in the degenerate case where τh=0 _h=0 for every horizon and all models in the Rashomon set produce identical forecasts for every sample, this constraint leaves many temporal properties of the trajectories unrestricted. For example, it provides no control over differences in their direction, spectral behavior, or shape. 5 Instantiations of Temporal Multiplicity Definition 3.3 deliberately leaves the choice of trajectory distance function unspecified. We now instantiate this general definition with a family of interpretable distances, each designed to capture a distinct aspect of temporal disagreement. As there is no unique trajectory distance appropriate for all forecasting applications, we select distances that are complementary along two criteria: each has a clear interpretation in terms of a recognizable aspect of temporal behavior, and together they distinguish differences in level, local dynamics, and global trajectory structure. We begin with a magnitude distance, which aggregates the pointwise gaps between two trajectories ,∈ℝH u, v ^H and therefore retains the least temporal structure. It is consequently the temporal distance most closely related to pointwise and horizon-wise multiplicity. • Magnitude: dmagnitude(,)=1H‖−‖2d_magnitude( u, v)= 1 H\| u- v\|_2, the root-mean-square pointwise gap. The next three distances operate on trajectory increments and compare patterns of change independently of the trajectories’ absolute levels. • Increment: dincrement(,)=1H−1‖Δ−Δ‖2d_increment( u, v)= 1 H-1\| u- v\|_2, with Δ=(Δu1,…,ΔuH−1) u=( u_1,…, u_H-1). • Volatility: dvolatility(,)=1H−1|∑h|Δuh|−∑h|Δvh||d_volatility( u, v)= 1H-1 | _h| u_h|- _h| v_h| |, the gap between the mean absolute increments. • Direction: ddirection(,)=1H−1|h:sign⁡(Δuh)≠sign⁡(Δvh)|d_direction( u, v)= 1H-1 |\h: ( u_h)≠ ( v_h)\ |, the fraction of increments of opposite sign. These final three distances are scale-free, comparing the form of trajectories rather than their absolute values.11 1 Here, ρ denotes Pearson correlation, ϝ the normalized discrete Fourier transform, and ρτ _τ the lag-τ cross-correlation. See Section A.4 for precise numerical implementations. • Shape: dshape(,)=12(1−ρ⁡(,))∈[0,1]d_shape( u, v)= 12 (1-ρ( u, v) )∈[0,1] using the Pearson correlation. • Spectral: dspectral(,)=12‖|ϝ⁡()|−|ϝ⁡()|‖2∈[0,1]d_spectral( u, v)= 1 2 \|\,| ( u)|-| ( v)|\, \|_2∈[0,1], the magnitude spectrum gap. • Timing: dtiming(,)=|τ⋆|/(H−2)∈[0,1]d_timing( u, v)=|τ |/(H-2)∈[0,1] with τ⋆=argmax|τ|≤H−2⁡ρτ(,)τ = _|τ|≤ H-2 _τ( u, v), the best-alignment lag as a fraction of the largest admissible lag. Each distance can be applied to forecast, revision, or error trajectories. On forecast trajectories ^tf y^f_t, it measures disagreement about the future path of the target; on revision trajectories ^tf r^f_t, it measures differences in how models update forecasts of a common target as new observations become available; on error trajectories tf e^f_t, it measures differences in the temporal structure of model errors. The instantiations above also make the general results of Section 4 operational: substituting each distance into our theoretical framework yields metric-specific bounds on temporal disagreement. We defer these bounds, their proofs, and an empirical assessment of their tightness to Appendix D, and turn next to experiments that characterize temporal multiplicity among trained forecasting models. 6 Experiments We evaluate temporal predictive multiplicity in trained forecasting models guided by three questions: Q1: How much do near-optimal models differ across the temporal aspects captured by our trajectory distances? Q2: Can pointwise or horizon-wise disagreement serve as a proxy for such differences? Q3: Can temporal multiplicity alter the conclusions or decisions drawn from a forecast? Across the experiments, we find that near-optimal models frequently produce markedly different trajectories, that these differences are largely invisible to conventional multiplicity measures, and that they can have consequential downstream implications. Experimental Setup. We consider 11 datasets from diverse domains. For each dataset, we train models spanning 19 neural forecasting architectures, including transformer, MLP/mixer-based, and recurrent families, over 10 random seeds and several hyper-parameter configurations. The empirical Rashomon set ℛ^ϵ R_ε contains all models within ϵ=5%ε=5\% of the best validation MASE. We compute all multiplicity statistics from rolling-origin test forecasts of its members. Full details of the datasets, models, training procedure, and evaluation protocol appear in Appendix A. Q1. How large is temporal multiplicity? We first quantify the extent to which near-optimal models differ in the temporal aspects captured by our trajectory distances. For each distance, we compute the trajectory range and diameter of the empirical Rashomon set. Table 1 reports the results.22 2 Section B.1 reports the complete pointwise, horizon-wise, and temporal multiplicity statistics, including ambiguity and discrepancy, per target. Table 1: Pointwise, horizon-wise, and temporal predictive multiplicity in empirical Rashomon sets. Each cell reports the trajectory range / diameter (νd/ωd _d/ _d), maximized across forecast, revision, and error trajectories. For multi-target datasets, multiplicity statistics are maximized and MASE is averaged across targets. Magnitude, increment, and volatility are reported in seasonal-naive units; direction, timing, shape, and spectral distances are scale-free and reported as percentages. Dataset dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ν(1)ν^(1) / ω(1)ω^(1) ν(H)ν^(H) / ω(H)ω^(H) ν / ω MASE∗ ATM 1D 1.38 / 0.66 1.55 / 0.81 0.87 / 0.35 100% / 59% 96% / 54% 85% / 36% 100% / 54% 1.79 / 0.59 1.94 / 0.62 1.85 / 0.55 1.05 ATM 1W 1.46 / 0.67 1.38 / 0.65 0.94 / 0.49 100% / 55% 99% / 56% 85% / 32% 100% / 58% 1.75 / 0.71 1.67 / 0.61 1.68 / 0.59 0.93 Bitcoin 7.00 / 2.99 3.66 / 2.08 2.01 / 0.88 100% / 62% 99% / 61% 82% / 42% 100% / 69% 5.16 / 2.35 9.41 / 3.34 7.10 / 2.63 2.67 Electricity D. A 1.56 / 0.62 0.47 / 0.33 0.21 / 0.15 79% / 49% 83% / 39% 82% / 43% 98% / 33% 1.59 / 0.46 2.19 / 0.58 1.93 / 0.53 0.46 Electricity D. Z 0.80 / 0.30 0.20 / 0.15 0.07 / 0.05 69% / 50% 79% / 44% 77% / 49% 99% / 40% 0.62 / 0.22 1.17 / 0.32 0.97 / 0.24 0.24 Electricity Price 0.95 / 0.45 0.40 / 0.29 0.18 / 0.11 81% / 46% 72% / 35% 76% / 38% 96% / 30% 0.98 / 0.34 1.29 / 0.48 1.24 / 0.37 0.82 ETT 1H 1.42 / 0.56 0.50 / 0.33 0.23 / 0.14 83% / 47% 77% / 33% 79% / 37% 95% / 31% 1.13 / 0.44 1.97 / 0.60 1.66 / 0.47 0.48 Exchange Rate 5.31 / 1.78 5.25 / 2.31 3.05 / 1.32 100% / 100% 100% / 100% 97% / 87% 100% / 100% 4.09 / 1.59 6.26 / 2.02 5.35 / 1.36 2.36 FRED-MD 0.16 / 0.12 0.13 / 0.09 0.09 / 0.05 51% / 35% 49% / 36% 0% / 0% 61% / 45% 0.11 / 0.08 0.21 / 0.13 0.16 / 0.11 0.14 Gasoline 2.74 / 1.33 2.13 / 1.00 1.23 / 0.56 100% / 72% 100% / 66% 94% / 46% 100% / 79% 3.03 / 1.32 3.29 / 1.29 3.10 / 1.22 0.74 Weather 2.41 / 1.25 0.72 / 0.51 0.39 / 0.27 66% / 52% 88% / 50% 77% / 47% 99% / 48% 2.73 / 1.39 3.12 / 1.10 2.96 / 1.06 0.95 Mean 2.29 / 0.98 1.49 / 0.78 0.84 / 0.40 84% / 57% 86% / 52% 76% / 41% 95% / 53% 2.09 / 0.86 2.96 / 1.01 2.55 / 0.83 0.99 Temporal multiplicity is substantial across all temporal aspects. At a typical origin, some pair of near-optimal models moves in opposite directions on 84%84\% of trajectory increments, reaching 100%100\% on Gasoline; even a fixed pair does so on 57%57\% of increments.A shape range of 86%86\% corresponds to a Pearson correlation of approximately −0.7-0.7 between the most divergent trajectories, with one trajectory tending to rise at the time steps where the other falls. The timing range reaches 95%95\%, indicating that the most divergent trajectories often require a shift of about half the forecasting horizon to align, and the spectral range of 76%76\% lies well above the midpoint of its scale. Thus, models with indistinguishable validation performance can imply sharply different trajectories. The pointwise and horizon-wise results are also consistent with prior work: pointwise multiplicity increases with the forecast horizon (Marx et al., 2020; Watson-Daniels et al., 2023; Kale et al., 2026). However, the temporal results are not simply a rescaling of the pointwise ones. The two measures rank datasets differently; for example, Weather has the fourth-largest pointwise range but the second-smallest directional disagreement. We examine this distinction in the next experiment. Q2. Can conventional frameworks capture temporal multiplicity? The preceding theoretical results suggest that temporal multiplicity is not simply a rescaling of pointwise or horizon-wise multiplicity. We test this directly at the level of individual model pairs and forecast origins. For each pair of models in the empirical Rashomon set and each test forecast origin, we compute the Spearman correlation between their mean pointwise gap and their trajectory distance. We consider both a pointwise reading and a horizon-normalized reading, which scales gaps by the best validation MASE at each horizon: rd=ϱ⁡([δ¯tfg]f,g∈(ℛ^ϵ2)t∈,[d⁡(tf,tg)]f,g∈(ℛ^ϵ2)t∈),δ¯tfg=1H∑h=1H|y^t+hf♮−y^t+hg♮|ℒh⋆,r_d= ( [ δ^fg_t ]_ subarrayc\f,g\∈ R_ε2\\ t∈T subarray,\; [d( v^f_t, v^g_t) ]_ subarrayc\f,g\∈ R_ε2\\ t∈T subarray ), δ^fg_t= 1H _h=1^H | -0.5mu y^f_t+h- -0.5mu y^g_t+h |L _h, (3) where (ℛ^ϵ2) R_ε2 is the set of unordered pairs of distinct models in ℛ^ϵ R_ε, y^t+hf♮ -0.5mu y^f_t+h and y^t+hg♮ -0.5mu y^g_t+h are the forecasts in naive units, ℒh⋆=minf∈ℛ^ϵ⁡ℒh(f)L _h= _f∈ R_εL_h(f) is the best validation MASE at horizon h, and ϱ denotes Spearman rank correlation. The horizon-normalized reading places gaps from different horizons on a common scale; the unweighted pointwise reading is obtained by setting ℒh⋆≡1L _h≡ 1. Table 2 reports the weakest and strongest correlations across trajectories and, for multi-target datasets, targets. The results are aligned with our theory. Correlation is nearly perfect for magnitude, which directly aggregates pointwise gaps, and decreases for increment and volatility distances. In contrast, disagreement in direction, shape, spectral behavior, and timing has little or no association with pointwise or horizon-wise disagreement. On the least correlated trajectory, average correlations for these aspects range from 0.040.04 to 0.160.16, while even the strongest correlations do not exceed 0.470.47. Thus, knowing how far apart two near-optimal forecasts are at individual horizons provides little information about whether they agree on the direction, shape, spectrum, or timing of the trajectory. Table 2: Spearman rank correlation, over all (model pair, origin) samples of the empirical Rashomon set, between the pair’s mean pointwise gap δ¯tfg δ^fg_t and its trajectory distance d⁡(tf,tg)d( v^f_t, v^g_t) on the same forecast origin, read as in Equation 3 with ℒh⋆≡1L _h≡ 1 (Pointwise) or as is (Horizon-Wise). Each row reports the minimum / maximum over the three trajectories and, for multi-target datasets, its targets. Dataset Pointwise Horizon-Wise dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ATM 1D 0.98 / 0.98 0.58 / 0.75 0.35 / 0.37 0.15 / 0.28 0.24 / 0.61 0.05 / 0.42 0.07 / 0.24 0.98 / 0.98 0.58 / 0.75 0.35 / 0.37 0.15 / 0.28 0.24 / 0.61 0.05 / 0.42 0.07 / 0.24 ATM 1W 0.99 / 0.99 0.56 / 0.70 0.39 / 0.47 0.13 / 0.18 0.17 / 0.40 0.04 / 0.24 0.14 / 0.20 0.99 / 0.99 0.56 / 0.70 0.39 / 0.47 0.13 / 0.18 0.17 / 0.40 0.04 / 0.24 0.14 / 0.20 Bitcoin 0.99 / 1.00 0.61 / 0.69 0.43 / 0.49 -0.02 / 0.19 -0.01 / 0.32 -0.05 / 0.26 0.02 / 0.14 0.96 / 0.98 0.60 / 0.67 0.41 / 0.48 -0.03 / 0.17 -0.03 / 0.30 -0.05 / 0.25 -0.01 / 0.13 Electricity D. A 0.99 / 0.99 0.46 / 0.66 0.16 / 0.36 -0.03 / 0.26 -0.00 / 0.64 -0.08 / 0.49 0.05 / 0.36 0.96 / 0.97 0.47 / 0.65 0.16 / 0.36 -0.03 / 0.26 -0.03 / 0.63 -0.10 / 0.48 0.04 / 0.35 Electricity D. Z 0.99 / 0.99 0.23 / 0.52 0.09 / 0.28 -0.08 / 0.23 0.07 / 0.52 -0.05 / 0.38 0.10 / 0.37 0.98 / 0.99 0.24 / 0.52 0.09 / 0.28 -0.07 / 0.23 0.07 / 0.51 -0.06 / 0.38 0.10 / 0.37 Electricity Price 0.99 / 0.99 0.59 / 0.64 0.20 / 0.31 -0.02 / 0.06 0.17 / 0.35 0.08 / 0.23 0.10 / 0.14 0.98 / 0.98 0.59 / 0.64 0.20 / 0.31 -0.02 / 0.06 0.15 / 0.34 0.07 / 0.22 0.10 / 0.14 ETT 1H 0.99 / 0.99 0.56 / 0.68 0.30 / 0.39 0.32 / 0.36 0.43 / 0.51 0.31 / 0.39 0.17 / 0.29 0.98 / 0.98 0.58 / 0.69 0.31 / 0.40 0.33 / 0.38 0.41 / 0.50 0.30 / 0.39 0.17 / 0.29 Exchange Rate 0.99 / 0.99 0.50 / 0.83 0.29 / 0.51 0.09 / 0.22 0.14 / 0.47 0.01 / 0.28 0.06 / 0.22 0.96 / 0.98 0.49 / 0.82 0.29 / 0.51 0.09 / 0.21 0.14 / 0.45 0.01 / 0.27 0.06 / 0.20 FRED-MD 0.98 / 1.00 0.61 / 0.69 0.45 / 0.64 0.03 / 0.40 0.05 / 0.42 -0.11 / 0.04 0.02 / 0.40 0.97 / 0.99 0.59 / 0.67 0.44 / 0.62 0.03 / 0.39 0.04 / 0.42 -0.10 / 0.04 0.03 / 0.39 Gasoline 0.99 / 0.99 0.56 / 0.61 0.35 / 0.45 0.09 / 0.22 0.12 / 0.43 -0.01 / 0.29 0.10 / 0.23 0.99 / 0.99 0.56 / 0.61 0.35 / 0.45 0.09 / 0.22 0.12 / 0.43 -0.01 / 0.29 0.10 / 0.23 Weather 1.00 / 1.00 0.69 / 0.72 0.51 / 0.52 0.35 / 0.43 0.34 / 0.52 0.24 / 0.43 0.27 / 0.35 0.99 / 0.99 0.70 / 0.73 0.52 / 0.54 0.37 / 0.44 0.34 / 0.52 0.24 / 0.43 0.27 / 0.35 Mean 0.99 / 0.99 0.54 / 0.68 0.32 / 0.44 0.09 / 0.26 0.16 / 0.47 0.04 / 0.31 0.10 / 0.27 0.98 / 0.98 0.54 / 0.68 0.32 / 0.44 0.09 / 0.26 0.15 / 0.46 0.04 / 0.31 0.10 / 0.26 Q3. Does temporal multiplicity affect downstream decisions? We finally examine whether trajectory-level disagreement can lead to different downstream decisions. Such decisions often depend on properties of an entire forecast trajectory, rather than on individual forecast cells. As an illustrative case study, we consider disagreement in the value and timing of trajectory extrema and in the direction of trajectory increments. For a forecast trajectory, define its maximum vtf,Mv^f,M_t, the location of its maximum ℓtf,M ^f,M_t, and its increments Δt,hf ^f_t,h as: vtf,M=max1≤h≤H⁡y^t+hf♮,ℓtf,M=1Hargmax1≤h≤H⁡y^t+hf♮,Δt,hf=y^t+hf♮−y^t+h−1f♮.v^f,M_t= _1≤ h≤ H -0.5mu y^f_t+h, ^f,M_t= 1H _1≤ h≤ H -0.5mu y^f_t+h, ^f_t,h= -0.5mu y^f_t+h- -0.5mu y^f_t+h-1. (4) We define vtf,mv^f,m_t and ℓtf,m ^f,m_t analogously for the minimum and its location. For each origin, disagreement about an extremum is measured by its range across members of ℛ^ϵ R_ε, in both value and location: ranget()=maxf∈ℛ^ϵ⁡vtf−minf∈ℛ^ϵ⁡vtfℒ⁡(f⋆),ranget(ℓ)=maxf∈ℛ^ϵ⁡ℓtf−minf∈ℛ^ϵ⁡ℓtf.range_t( v)= _f∈ R_εv^f_t- _f∈ R_εv^f_tL(f ), _t( )= _f∈ R_ε ^f_t- _f∈ R_ε ^f_t. (5) The value range is expressed in multiples of the best validation MASE, while the location range is expressed as a fraction of the horizon. For directional disagreement, we consider the minority share of models moving against the majority direction, the share of contested cells on which at least one model predicts an increase and another a decrease, and the range of increment angles θt,hf=arctan⁡Δt,hfθ^f_t,h= ^f_t,h across models. Table 3 reports these quantities for each dataset. Table 3: Disagreement across empirical Rashomon-set members about the value and location of forecast minima and maxima, and about the direction of forecast increments. Value ranges are expressed in multiples of the best validation MASE; locations are fractions of the forecast horizon; and increment-angle ranges are in degrees (up to 180∘180 , opposite steep moves). Ranges are reported as mean (maximum) across origins, or across forecast cells for increment angles; directional shares are averaged across cells. We report the maximum for multi-target datasets. Dataset Extremes Value Extremes Location Increment Direction range⁡(M)range( v^M) range⁡(m)range( v^m) range⁡(ℓM)range( ^M) range⁡(ℓm)range( ^m) Contested Cells Minority Share range⁡()range( θ) ATM 1D 1.91 (6.22) 1.43 (5.40) 18.7% (85.7%) 48.3% (85.7%) 51.0% 9.2% 55.8 (159) ATM 1W 1.83 (6.82) 1.56 (8.17) 68.3% (75.0%) 70.7% (75.0%) 89.8% 21.8% 69.2 (153) Bitcoin 2.83 (7.35) 2.48 (9.21) 84.7% (85.7%) 85.2% (85.7%) 100% 28.7% 101 (164) Electricity D. A 5.44 (39.4) 3.13 (17.1) 43.9% (97.9%) 28.2% (97.9%) 60.9% 10.3% 34.6 (143) Electricity D. Z 4.18 (20.9) 2.19 (7.52) 30.3% (99.0%) 17.7% (99.0%) 70.1% 13.3% 17.2 (89.3) Electricity Price 1.45 (4.13) 1.53 (4.59) 45.3% (95.8%) 30.8% (95.8%) 66.0% 10.0% 34.4 (110) ETT 1H 3.33 (8.40) 3.24 (8.07) 76.6% (95.8%) 85.1% (95.8%) 100% 23.2% 46.2 (105) Exchange Rate 2.72 (482) 2.28 (64.5) 80.0% (80.0%) 80.0% (80.0%) 100% 41.3% 118 (177) FRED-MD 1.30 (3.12) 1.01 (4.82) 27.7% (66.7%) 27.4% (66.7%) 47.0% 8.2% 5.50 (19.0) Gasoline 3.91 (17.2) 4.26 (21.0) 74.4% (75.0%) 74.0% (75.0%) 92.7% 21.9% 83.6 (160) Weather 4.09 (15.0) 2.11 (12.0) 79.7% (99.3%) 69.6% (99.3%) 100% 33.4% 60.3 (157) Mean 3.00 (55.5) 2.29 (14.8) 57.2% (86.9%) 56.1% (86.9%) 79.8% 20.1% 56.8 (131) The consequences are substantial. On average, the most and least optimistic near-optimal models place the forecast peak 3.003.00 error budgets apart and disagree about its timing by 57%57\% of the horizon; at the worst origin, they locate it at opposite ends of the horizon on every dataset. Moreover, at least one model predicts an increase while another predicts a decrease on 80%80\% of forecast cells. At a typical cell, 20%20\% of models oppose the majority direction and the members’ increment angles span 57∘57 on average, i.e., steep rises against steep falls. Thus, performance-indistinguishable models can support different downstream decisions. Horizon-wise statistics reveal disagreement at individual steps, but not whether models agree on the direction of changes or on which peak is largest. Figure 3 illustrates these differences. For Bitcoin, 75 of 173 near-optimal models predict that the price will rise over the following week, while 98 predict a fall. For Spanish day-ahead electricity prices, all 208 models predict the same intraday shape, yet 98 forecast a higher price after 24 hours and 110 a lower one. The demand examples show analogous disagreement about peaks: in New South Wales, predicted peak increases range from 1.21.2 to 10.910.9 naive units; in Queensland, 165 of 332 models place the peak on the same evening and 167 on the following morning. Comparable predictive performance therefore leaves unresolved both the magnitude and timing of peak demand. Figure 3: Examples of temporal multiplicity within empirical Rashomon sets. Each panel shows member forecast trajectories at one origin, as deviations from the realized level at that origin, together with the realized trajectory (black). Left: Bitcoin and Spanish electricity-price forecasts, colored by net direction over the horizon (blue: up; orange: down). Right: Australian electricity-demand forecasts, colored by the timing of their maximum (dots); annotations report disagreement in peak value and timing. 7 Conclusions We introduced a framework for characterizing predictive multiplicity in time-series forecasting at the level of complete trajectories. By distinguishing pointwise, horizon-wise, and temporal disagreement, we revealed differences in direction, shape, timing, and other temporal features that conventional multiplicity metrics cannot capture. Using this framework, we showed that near-optimal performance, and even tight horizon-wise agreement, provide limited control over the trajectories competing models may produce. Experiments across forecasting architectures and datasets confirmed that these differences arise in practice and can affect downstream decisions. Our framework therefore provides practitioners with a means to assess temporal differences among models that standard accuracy measures deem interchangeable. Several directions follow. First, time-series attributions and counterfactuals are themselves temporal objects (Tonekaboni et al., 2020; Ates et al., 2021); applying our framework to them could reveal a temporal dimension of explanatory multiplicity (Dong & Rudin, 2020; Laberge et al., 2023). Second, extending the framework to quantile and probabilistic forecasts would enable the study of distributional disagreement. For time-series foundation models (Woo et al., 2024; Ansari et al., 2025), competing forecasts may also result from different contexts, prompts, or covariates, raising the question of whether Rashomon sets should encompass such conditioning choices, as a temporal counterpart to prompt multiplicity in language models (Sclar et al., 2023; Ganesh et al., 2026). Finally, combining deterministic Rashomon-set thresholds with model confidence sets (Hansen et al., 2011) could account for uncertainty in model performance when estimating temporal multiplicity. Acknowledgements Toni was partially funded by J.P. Morgan and by the Royal Academy of Engineering under the Research Chairs and Senior Research Fellowships scheme. Toni was partially funded by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 101020934). Disclaimer This paper was prepared for informational purposes by the Artificial Intelligence Research group of JPMorgan Chase & Co. and its affiliates (”JP Morgan”) and is not a product of the Research Department of JP Morgan. JP Morgan makes no representation and warranty whatsoever and disclaims all liability, for the completeness, accuracy or reliability of the information contained herein. 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Dataset Domain Source License Distribution Channel UK ATM Withdrawals - Daily finance Crone (2008); Godahewa et al. (2021) C BY 4.0 GluonTS (Alexandrov et al., 2020) UK ATM Withdrawals - Weekly finance Crone (2008); Godahewa et al. (2021) C BY 4.0 GluonTS (Alexandrov et al., 2020) Bitcoin Price finance Godahewa et al. (2021) C BY 4.0 GIFT-Eval (Aksu et al., 2024) Australian Electricity Demand energy AEMO (); Godahewa et al. (2021) C BY 4.0 GluonTS (Alexandrov et al., 2020) Zurich Electricity Consumption energy Stadt-Zürich () C0 Public Domain Darts (Herzen et al., 2022) Hourly Energy Pricing - Spain energy Jhana (2019) C0 Public Domain Darts (Herzen et al., 2022) Electricity Transformer Temperature 1 industry Zhou et al. (2021) C BY-ND 4.0 Darts (Herzen et al., 2022) Currency Exchange Rates (USD) finance Lai et al. (2018) – GluonTS (Alexandrov et al., 2020) FRED-MD (US Macroeconomic Data) economy McCracken & Ng (2016) C0 Public Domain GluonTS (Alexandrov et al., 2020) US Finished Motor Gasoline Supplied energy EIA () C0 Public Domain Darts (Herzen et al., 2022) Jena Weather nature Kolle (2025); Wu et al. (2021) C BY 4.0 Darts (Herzen et al., 2022) Table 5: Properties of the datasets used in our experiments. Dataset Frequency Variables Targets Timestamps (T) Input Size† Horizon Seasonality Poolable∗ UK ATM Withdrawals - Daily D 108 108 791 28 (4W) / 56 (8W) 7 (1W) 7 (1W) Yes UK ATM Withdrawals - Weekly W 108 108 113 16 (16W) / 26 (26W) 4 (4W) 52 (52W) Only Bitcoin Price D 18 1 2629 28 (4W) / 56 (8W) 7 (1W) – No Australian Electricity Demand 30min 5 5 230736 336 (1W) 48 (1D) 48 (1D), 336 (1W), 17,520 (52W) Yes Zurich Electricity Consumption 15min 10 2 268705 672 (1W) 96 (1D) 96 (1D), 672 (1W), 35,040 (52W) No Hourly Energy Pricing - Spain h 20 1 35064 168 (1W) 24 (1D) 24 (1D), 168 (1W), 8,760 (52W) No Electricity Transformer Temperature 1 h 7 1 17420 168 (1W) / 192 (1.1W) 24 (1D) 24 (1D), 168 (1W) No Currency Exchange Rates (USD) B 8 8 6221 20 (4W) / 40 (8W) 5 (1W) – Yes FRED-MD (US Macroeconomic Data) ME 107 3 728 12 (1Y) / 24 (2Y) 3 (3M) 12 (1Y) No US Finished Motor Gasoline Supplied W-FRI 1 1 1578 16 (16W) / 32 (32W) 4 (4W) 52 (52W) – Jena Weather 10min 21 1 52704 1,008 (1W) 144 (1D) 144 (1D) No ∗Which representations a dataset’s targets admit: No, the targets are heterogeneous, so pooling is not meaningful; Yes, they are homogeneous, so a pooled model is available alongside the per-target and joint ones; Only, pooling is the one feasible representation, as for mixed panels of many unrelated series. A dash (–) marks a genuinely univariate dataset: a single series, with nothing to pool. †Every input size the sweep spans, in sweep order, where a dataset is swept at more than one; the runs of one dataset differ in nothing else. We use 11 publicly available forecasting datasets spanning finance, economics, energy, industry, and nature domains. Table 4 reports their provenance; Table 5 reports their properties; Table 6 reports the train/validation/test splits. Table 6: Train/validation/test splits of the datasets used in our experiments. Dataset Length Samples Samples∗ Forecasts Train Val & Test Train Train† Val & Test Train Train† Val & Test Train Val & Test UK ATM Withdrawals - Daily 553 119 526 / 498 531 / 503 113 56808 / 53784 57348 / 54324 12204 56808 / 53784 12204 UK ATM Withdrawals - Weekly 79 17 – – – 6912 / 5832 7128 / 6048 1512 6912 / 5832 1512 Bitcoin Price 1841 394 1814 / 1786 1819 / 1791 388 – – – 1814 / 1786 388 Australian Electricity Demand 161516 34610 161181 161227 34563 805905 806135 172815 805905 172815 Zurich Electricity Consumption 188093 40306 187422 187516 40211 – – – 374844 80422 Hourly Energy Pricing - Spain 24544 5260 24377 24399 5237 – – – 24377 5237 Electricity Transformer Temperature 1 12194 2613 12027 / 12003 12049 / 12025 2590 – – – 12027 / 12003 2590 Currency Exchange Rates (USD) 4355 933 4336 / 4316 4339 / 4319 929 34688 / 34528 34712 / 34552 7432 34688 / 34528 7432 FRED-MD (US Macroeconomic Data) 510 109 499 / 487 500 / 488 107 – – – 1497 / 1461 321 US Finished Motor Gasoline Supplied 1104 237 1089 / 1073 1091 / 1075 234 – – – 1089 / 1073 234 Jena Weather 36892 7906 35885 36027 7763 – – – 35885 7763 ∗Samples of the pooled representation, windows × targets; the unmarked samples are those of the per-target and joint representations, one per rolling-origin window. A count is empty where its representation is infeasible: the pooled one where the dataset is not poolable, the unmarked one where pooling is its one feasible representation. †Training samples of the recurrent architectures, which unfold training windows of the input size plus 2 steps rather than the input size plus the horizon; every other count is shared by all architectures. A cell listing several counts holds one per input size the sweep spans, in the order of the properties table’s input sizes. Pre-processing. Placeholder codes (sentinel values, fabricated calendar days such as the 31st of a 30-day month) are converted to missing values before any other step. Three datasets whose series are strictly positive and span orders of magnitude (the currency exchange rates, the Bitcoin indicators, and the FRED-MD macroeconomic levels) are modeled in natural-log space, as is standard for those series in such domains; the nine FRED-MD series that take non-positive values (interest-rate spreads and non-borrowed reserves) are kept in raw levels. Missing Values Handling. Missing values are handled without future leakage: leading gaps are trimmed, interior gaps are forward-filled, and an availability mask records which positions were originally missing. The mask follows the data through the whole pipeline: imputed target positions do not contribute to the training loss, to the seasonal-naive scale of MASE and RMSSE, or to any evaluation metric. Standardization. Every architecture standardizes each input window by its own mean and standard deviation and inverts the transformation on its outputs, so all models see the same raw series. Refer to the model training section for more details on normalization. Exogenous Variables. Non-target variables are offered to models as historical exogenous inputs whenever the architecture allows it (see Section A.2). Seasonality. When a dataset declares one or more seasonal periods s, 4 Fourier terms of those periods are passed to the model as future-known exogenous features whenever the architecture allows it: sin⁡(2πts),cos⁡(2πts),sin⁡(4πts),cos⁡(4πts) \! ( 2π ts ), \! ( 2π ts ), \! ( 4π ts ), \! ( 4π ts ) Dataset Splits. Each dataset is split chronologically into a training, a validation, and a test portion, the validation portion immediately preceding the test tail and both holding out, by default, 15% of the calendar (see Table 6 for more details). Within each held-out portion the forecasts are rolling-origin forecasts at a dense stride of one timestamp, every origin producing one H-step trajectory, so that the multiplicity statistics of Definition 3.2 are computed over every origin of the test portion. A.2 Model Training Hardware Setup. To train and evaluate our models we used a Ray cluster of machines with one NVIDIA RTX PRO 6000 Blackwell Server Edition GPU, an 88-core CPU running at 3.23.2 GHz, and 128128 GB of RAM. To run our experiments on the trained models we instead used a CPU-only machine with 6464 cores, running at 3.73.7 GHz, and 256256 GB of RAM. Model Architectures. To generate the empirical Rashomon set we train models spanning the main families of neural forecasting architectures: Transformer-based, MLP/mixer-based, convolutional, recurrent, and near-linear models. Table 7 lists the 19 architectures used in our experiments together with the references of the original proposals. We use NeuralForecast (Olivares et al., 2022) to train all such model architectures. Table 7: Neural forecasting architectures used to generate the empirical Rashomon set, grouped by model family. Family Model Reference Transformer PatchTST Nie et al. (2022) iTransformer Liu et al. (2024) TimeXer Wang et al. (2024b) TFT Lim et al. (2021) MLP / Mixer NHITS Challu et al. (2023) NBEATSx Olivares et al. (2023) TiDE Das et al. (2023) TimeMixer Wang et al. (2024a) TSMixer Chen et al. (2023) TSMixerx Chen et al. (2023) SOFTS Han et al. (2024) Convolutional TimesNet Wu et al. (2022) BiTCN Sprangers et al. (2023) TCN Bai et al. (2018) Recurrent xLSTM Beck et al. (2024) GRU Cho et al. (2014) DilatedRNN Chang et al. (2017) Near-linear DLinear Zeng et al. (2023) xLinear Chen et al. (2026) Fixed Hyper-Parameters. The following hyper-parameters are fixed on all trained models: • Loss: MSE (mean squared error). • Optimizer: Adam with no decay schedule. • Regularization: every dropout rate set to zero. • Input scaling: per-window standard scaling of the inputs. • Precision: train-time operations in bfloat16 mixed precision whenever possible (parameters and optimizer state stay in fp32); every evaluation—for every model and every checkpoint—is run in fp32 so that no precision confound enters the comparison of models. Hyper-Parameters Swept to Generate the Full Set of Models. For every dataset the full set of trained models is the Cartesian product of the following hyper-parameters: • Model Architectures: the 19 architectures of Table 7 • Random Seeds: ten random seeds (00 to 99). • Learning Rate: 10−310^-3 and 3⋅10−43· 10^-4 • Batch Size: 3232 and 6464 • Input Size: four and eight times the forecast horizon, where the dataset admits both. Table 5 reports the corresponding input sizes, while Table 6 lists the training samples each input size yields. • Target: Univariate and Multivariate ∗† • Exogenous Variables: Disabled and Enabled ∗ • Pooled Targets: Unpooled and Pooled targets ∗ (as supported by each dataset, see Table 5) ∗ Note that the combinations an architecture cannot realize (e.g. a purely univariate architecture in the multivariate representation) are pruned per (architecture, dataset) pair. † Also note that when multi-target datasets are trained in univariate mode the training fans out into one model per target, which share one run’s compute budget as described below. Finally, note that dataset–architecture combinations that exceeded the RAM or VRAM capacity of the hardware described above were excluded. All other hyper-parameters that are not mentioned in these two paragraphs are left at their default value in NeuralForecast 3.1.9 (Olivares et al., 2022). Randomness Control. The random seed drives every source of randomness in a run: parameter initialization, the sampling of training origins, and any stochastic kernel. Deterministic kernels are requested wherever PyTorch offers them. Compute Budget. The step budget of every (architecture, dataset, representation) cell is determined by the following rules. • Wall-time budget: a short profiling run measures the isolated training throughput (steps per second) on a dedicated NVIDIA RTX PRO 6000 Blackwell Server Edition GPU, and the step budget is the number of steps that spends 900900 seconds (1515 minutes) at that throughput. • Epoch cap: the step budget is capped at 250 expected-coverage epochs, an epoch being the number of steps after which every training origin has been sampled once in expectation, i.e., ⌈Norigins/Nbatch⌉ N_origins/N_batch . • Per-target fan-out: the step budget is divided by the number of per-target models a run fans out into, so that a per-target univariate run spends the same total compute as a joint or pooled run on the same dataset. We estimate that training and evaluating our models required 3,0003,000 GPU-hours, with an additional 300300 hours of CPU-only compute used to run experiments. Checkpoints. Along every run we save periodic snapshots of the weights, and the end-of-training weights as a final checkpoint. The snapshot interval is 10001000 optimizer steps, shortened for slow models to at most 6060 seconds of isolated wall-time and to at most one expected-coverage epoch, whichever is smallest. Checkpoint Selection. Every checkpoint, including the final one, is scored with the above rolling-origin protocol. We keep the single checkpoint with the lowest validation MASE (Hyndman & Koehler, 2006), aggregated over the whole horizon and over every validation origin. Table 8: Validation and test performance of the full set of trained models of each dataset: minimum and mean across all models of the horizon-averaged MAE, RMSE, MASE, and RMSSE, on the validation portion (left) and on the test portion (right). Dataset Val Test MAE RMSE MASE RMSSE MAE RMSE MASE RMSSE Models Min Mean Min Mean Min Mean Min Mean Min Mean Min Mean Min Mean Min Mean ATM 1D 2080 4.23 4.68 6.88 7.56 1.05 1.16 1.09 1.20 3.16 3.45 4.95 5.23 0.73 0.80 0.71 0.76 ATM 1W 1520 15.4 18.0 22.3 25.9 0.93 1.11 1.05 1.26 12.9 13.9 18.2 19.5 0.75 0.81 0.80 0.86 Bitcoin 2800 0.06 0.07 0.08 0.10 2.67 3.26 2.58 3.03 0.05 0.06 0.07 0.09 2.35 3.01 2.16 2.76 Electricity D. A (NSW) 2160 171 206 274 323 0.38 0.45 0.41 0.48 171 206 274 320 0.38 0.46 0.41 0.48 Electricity D. A (QLD) 2160 100 118 153 176 0.41 0.48 0.40 0.46 114 141 174 205 0.47 0.58 0.46 0.54 Electricity D. A (SA) 2160 61.8 73.0 103 116 0.52 0.62 0.55 0.63 74.7 87.2 121 136 0.63 0.73 0.65 0.73 Electricity D. A (TAS) 2160 22.6 24.7 31.4 34.2 0.53 0.58 0.50 0.54 23.5 25.7 32.5 35.2 0.56 0.62 0.53 0.57 Electricity D. A (VIC) 2160 151 179 246 285 0.43 0.52 0.46 0.53 166 200 275 321 0.48 0.57 0.51 0.60 Electricity D. Z (Value_NE5) 1680 524 864 745 1195 0.21 0.35 0.18 0.28 543 888 819 1244 0.23 0.37 0.20 0.31 Electricity D. Z (Value_NE7) 1680 1019 1823 1485 2481 0.27 0.49 0.23 0.39 934 1734 1421 2400 0.26 0.48 0.23 0.39 Electricity Price 1400 4.88 5.65 6.88 7.90 0.82 0.95 0.81 0.93 2.65 3.12 3.63 4.17 0.43 0.51 0.42 0.48 ETT 1H 2800 1.14 1.27 1.50 1.64 0.48 0.53 0.46 0.50 1.27 1.46 1.75 1.96 0.57 0.65 0.57 0.64 Exchange Rate (AUD) 4320 0.01 0.01 0.02 0.02 2.78 3.35 2.86 3.40 0.01 0.01 0.01 0.01 1.66 2.03 1.50 1.84 Exchange Rate (CAD) 4320 0.01 0.01 0.01 0.02 3.61 4.35 2.89 3.47 0.01 0.01 0.01 0.01 1.70 2.04 1.39 1.74 Exchange Rate (CHF) 4320 0.01 0.01 0.01 0.01 1.56 1.90 1.51 1.82 0.01 0.01 0.01 0.01 1.51 1.89 1.55 1.91 Exchange Rate (CNY) 4320 0.00 0.00 0.00 0.00 1.83 3.77 0.21 0.43 0.00 0.00 0.00 0.00 2.52 4.75 0.28 0.60 Exchange Rate (GBP) 4320 0.01 0.01 0.01 0.02 2.21 2.65 2.16 2.56 0.01 0.01 0.01 0.01 1.31 1.59 1.13 1.40 Exchange Rate (JPY) 4320 0.01 0.01 0.01 0.02 1.90 2.28 1.78 2.08 0.01 0.01 0.01 0.01 1.22 1.51 1.11 1.36 Exchange Rate (NZD) 4320 0.01 0.02 0.02 0.02 2.98 3.59 2.90 3.43 0.01 0.01 0.01 0.01 1.75 2.14 1.55 1.89 Exchange Rate (SGD) 4320 0.00 0.01 0.01 0.01 1.99 2.43 1.62 1.98 0.00 0.00 0.01 0.01 1.77 2.13 1.34 1.65 FRED-MD (CPIAUCSL) 3240 0.00 0.01 0.01 0.01 0.09 0.13 0.12 0.16 0.00 0.00 0.00 0.00 0.06 0.08 0.06 0.09 FRED-MD (INDPRO) 3240 0.01 0.01 0.01 0.02 0.14 0.21 0.17 0.27 0.00 0.01 0.01 0.01 0.10 0.15 0.11 0.16 FRED-MD (UNRATE) 3240 0.02 0.03 0.03 0.04 0.19 0.28 0.18 0.28 0.02 0.03 0.03 0.04 0.17 0.22 0.15 0.22 Gasoline 1520 207 224 262 286 0.74 0.80 0.73 0.80 330 400 538 700 1.15 1.39 1.48 1.93 Weather 1240 8.22 9.14 12.0 13.3 0.95 1.06 0.88 0.98 8.93 10.4 12.3 14.3 1.02 1.19 0.89 1.03 Model Performance. Table 8 reports, for each dataset, the minimum and the mean of the validation and of the test performance across the whole set of trained models, under the four error metrics MAE, RMSE, MASE, and RMSSE (Hyndman & Koehler, 2006; Makridakis et al., 2022), each averaged over the whole horizon and over every origin of the portion. A.3 Rashomon Set Selection Rule. The empirical Rashomon set ℛ^ϵ R_ε of a dataset is selected once again using the validation MASE as our ℒ⁡(f)L(f). Table 9: Empirical Rashomon sets: forecast horizon H, the size of the full set of trained models |ℱ^|| F|, Rashomon set size |ℛ^ϵ|| R_ε|, the Rashomon ratio, and the best and worst validation score of the set’s members on the selection metric (MASE, in bold) and on the other reported metrics. Dataset MASE MAE RMSE RMSSE H |ℱ^|| F| |ℛϵ||R_ε| Ratio Best Worst Best Worst Best Worst Best Worst ATM 1D 7 2080 88 0.04 1.05 1.10 4.23 4.44 6.88 7.51 1.09 1.19 ATM 1W 4 1520 81 0.05 0.93 0.98 15.4 16.2 22.3 24.5 1.05 1.16 Bitcoin 7 2800 173 0.06 2.67 2.81 0.06 0.06 0.08 0.09 2.58 2.81 Electricity D. A (NSW) 48 2160 135 0.06 0.38 0.40 171 179 274 298 0.41 0.44 Electricity D. A (QLD) 48 2160 332 0.15 0.41 0.43 100 105 153 168 0.40 0.44 Electricity D. A (SA) 48 2160 164 0.08 0.52 0.55 61.8 64.8 103 110 0.55 0.59 Electricity D. A (TAS) 48 2160 876 0.41 0.53 0.56 22.6 23.7 31.4 33.4 0.50 0.53 Electricity D. A (VIC) 48 2160 259 0.12 0.43 0.46 151 158 246 270 0.46 0.50 Electricity D. Z (Value_NE5) 96 1680 107 0.06 0.21 0.22 524 550 745 827 0.18 0.20 Electricity D. Z (Value_NE7) 96 1680 109 0.06 0.27 0.29 1019 1070 1485 1603 0.23 0.25 Electricity Price 24 1400 208 0.15 0.82 0.86 4.88 5.12 6.88 7.27 0.81 0.86 ETT 1H 24 2800 748 0.27 0.48 0.51 1.14 1.20 1.50 1.59 0.46 0.49 Exchange Rate (AUD) 5 4320 1435 0.33 2.78 2.92 0.01 0.01 0.02 0.02 2.86 3.10 Exchange Rate (CAD) 5 4320 1339 0.31 3.61 3.79 0.01 0.01 0.01 0.02 2.89 3.10 Exchange Rate (CHF) 5 4320 1326 0.31 1.56 1.64 0.01 0.01 0.01 0.01 1.51 1.60 Exchange Rate (CNY) 5 4320 100 0.02 1.83 1.92 0.00 0.00 0.00 0.00 0.21 0.23 Exchange Rate (GBP) 5 4320 1201 0.28 2.21 2.33 0.01 0.01 0.01 0.01 2.16 2.32 Exchange Rate (JPY) 5 4320 1334 0.31 1.90 2.00 0.01 0.01 0.01 0.01 1.78 1.90 Exchange Rate (NZD) 5 4320 1238 0.29 2.98 3.13 0.01 0.01 0.02 0.02 2.90 3.10 Exchange Rate (SGD) 5 4320 1208 0.28 1.99 2.09 0.00 0.00 0.01 0.01 1.62 1.83 FRED-MD (CPIAUCSL) 3 3240 128 0.04 0.09 0.09 0.00 0.00 0.01 0.01 0.12 0.13 FRED-MD (INDPRO) 3 3240 13 0.00 0.14 0.15 0.01 0.01 0.01 0.01 0.17 0.19 FRED-MD (UNRATE) 3 3240 1 0.00 0.19 0.19 0.02 0.02 0.03 0.03 0.18 0.18 Gasoline 4 1520 436 0.29 0.74 0.77 207 218 262 284 0.73 0.79 Weather 144 1240 153 0.12 0.95 1.00 8.22 8.63 12.0 13.0 0.88 0.96 Mean – – – 0.14 0.99 1.03 – – – – 0.95 1.03 Since the best achievable MASE differs from dataset to dataset, a common absolute tolerance would admit sets of very different quality across datasets; we therefore choose ϵε as a fixed fraction of the best validation score, so that the results remain comparable across datasets: ϵ=5%⋅ℒ⁡(f⋆)ε=5\%·L(f ), i.e., every model within 5% of the best validation MASE is a member. For a dataset with several targets that admit per-target modeling, one set is selected and measured per target. Every multiplicity statistic is then computed on the members’ test forecasts, so that the selection and the reported disagreement never share data. Table 9 reports, for every dataset, several metrics on its Rashomon set including the size |ℛ^ϵ|| R_ε|, its Rashomon ratio, and the best and the worst validation score among the members of the set, on four metrics: MASE, MAE, RMSE, and RMSSE. The best MASE among the models of the Rashomon set is by construction the best of the full set of trained models, ℒ⁡(f⋆)L(f ). A.4 Temporal Multiplicity Trajectory Scaling. Since the series differ from dataset to dataset (and, within a dataset, from series to series) in unit and in scale, a distance between two forecasts, or a threshold τ on such a distance, expressed in raw units would not be comparable across them; we therefore read every magnitude-like quantity in scale-free units, so that the results remain comparable across series and datasets. Unless stated otherwise, before any multiplicity statistic is computed, each prediction and each trajectory is divided by its series’ in-sample seasonal-naive mean absolute error—the MASE denominator (Hyndman & Koehler, 2006). The thresholds τ are also expressed in the same unit. We note that (ddirectiond_direction, dshaped_shape, dspectrald_spectral, dtimingd_timing) are invariant to positive per-series rescaling and are unaffected. For the other trajectory distance functions we note that scaling is a fixed positive constant per series, applied identically to every model and to the realized values: as such, every result of the paper holds verbatim in the rescaled space, while the statistics gain a unit-free, cross-dataset interpretation. Numerical Implementation: Shape Distance. ρ⁡(,)ρ( u, v) is the Pearson correlation between the entries of u and v, so that dshape=12(1−ρ)d_shape= 12(1-ρ) ranges from 00 (proportional trajectories) to 11 (anti-proportional ones). The Pearson correlation is undefined at constant trajectories and unstable near them. Calling u flat if sd⁡()≤ζsd( u)≤ζ for a small threshold ζ≥0ζ≥ 0, with sdsd the population standard deviation of its entries, we adopt the convention ρ⁡(,)=1ρ( u, v)=1 if both trajectories are flat and ρ⁡(,)=0ρ( u, v)=0 if exactly one is; the same threshold governs the direction, spectral, and timing trajectory distance functions, as detailed below, and the increments. We use ζ=10−8ζ=10^-8 (in naive units) in our experiments, so that a trajectory is flat only when it is constant up to floating-point noise. Numerical Implementation: Direction Distance. The sign of an increment is read with the flatness threshold ζ above: an increment is flat, sign⁡(Δuh)=0 ( u_h)=0, when it does not clear ζ in absolute value, |Δuh|≤ζ| u_h|≤ζ, and it goes up or down, sign⁡(Δuh)=±1 ( u_h)=± 1, otherwise. A flat increment thus counts as a third direction: two flat increments agree, whereas a flat increment against an upward or a downward one counts as a disagreement, exactly as an upward move against a downward one does. Two flat trajectories are therefore at distance 00 and a flat trajectory against a strictly monotone one at distance 11, mirroring the convention adopted for ρ above; the direction shares read the increments with the same rule. Numerical Implementation: Spectral Distance. The trajectories are first standardized, ˇ=(−u¯)/‖−u¯‖2 u=( u- u 1)/\| u- u 1\|_2 with u¯=1H∑huh u= 1H _hu_h, and ϝ is applied to ˇ u and ˇ v; ϝ denotes the normalized discrete Fourier transform, (ϝ)k=1H∑h=1Huhe−2πik(h−1)/H( u)_k= 1 H _h=1^Hu_h\,e^-2π ik(h-1)/H, so that ‖ϝ‖2=‖2\| u\|_2=\| u\|_2. The standardization is undefined at flat trajectories; with the flatness threshold ζ above, we set ˇ= u= 0 whenever sd⁡()≤ζsd( u)≤ζ. Two unit-norm magnitude spectra are at most 2 2 apart, so dspectral∈[0,1]d_spectral∈[0,1], and the maximum 11 is attained exactly when the two spectra have disjoint supports. Numerical Implementation: Timing Distance. For a lag τ≥0τ≥ 0, the overlapping segments of u and v are (u1+τ,…,uH)(u_1+τ,…,u_H) and (v1,…,vH−τ)(v_1,…,v_H-τ), of length H−τH-τ. Their means are denoted by u¯τ u_τ and v¯τ v_τ, while u¯ u and v¯ v denote the means of the full trajectories. Negative lags are handled by swapping the roles of u and v. The lag-τ Pearson correlation ρτ(,) _τ( u, v) is computed on the overlapping segments. Following the shape-distance convention, a segment is flat if its population standard deviation is at most ζ, computed over that segment’s own length. We set ρτ=1 _τ=1 if both segments are flat and ρτ=0 _τ=0 if exactly one is flat. Pearson correlation measures agreement within the overlap alone, so a short overlap can achieve a high correlation even when the full trajectories align poorly. We limit this effect in two ways. First, we require the overlap to contain at least two points and cover at least half of the horizon, restricting admissible lags to |τ|≤L|τ|≤ L, where L=H−max⁡2,⌈H/2⌉.L=H- \2, H/2 \. Second, we normalize the centered cross-product over the overlap by the norms of the two full centered trajectories. For τ≥0τ≥ 0, the resulting score is ρ~τ(,)=∑k=1H−τ(uk+τ−u¯τ)(vk−v¯τ)∑i=1H(ui−u¯)2∑i=1H(vi−v¯)2. ρ_τ( u, v)= _k=1^H-τ(u_k+τ- u_τ)(v_k- v_τ) _i=1^H(u_i- u)^2 _i=1^H(v_i- v)^2. The denominator is fixed across lags, so an overlap containing little of the full trajectories’ variation receives a smaller score in magnitude, even if its Pearson correlation is high. At zero lag, ρ~0=ρ0 ρ_0= _0. If either full trajectory is flat, we use ρτ _τ instead. The best-alignment lag is τ⋆=argmax|τ|≤L⁡ρ~τ(,)τ = _|τ|≤ L ρ_τ( u, v). We define the timing distance as dtiming(,)=|τ⋆|/Ld_timing( u, v)=|τ |/L, setting it to zero when L=0L=0. The distance therefore lies in [0,1][0,1], and the thresholds τ0=0.25 _0=0.25 and τ1=0.5 _1=0.5 correspond to one quarter and one half of the largest admissible lag, respectively. This implementation differs from the definition in Section 5 by capping the lag at L rather than H−2H-2, maximizing ρ~τ ρ_τ rather than ρτ _τ, and normalizing the selected lag by L. The theoretical statements in Appendix D apply to the definition in Section 5; the modifications above concern the numerical implementation used in the experiments. Error Trajectories. Consistent with the evaluation protocol of Section A.1, a forecast cell whose ground truth was imputed at load has no observed error, so the error trajectories drop every forecast origin carrying at least one such cell, whereas the forecast and revision trajectories keep it (a prediction is a prediction whatever it is scored against). Extremes. The location of the maximum and of the minimum of a forecast trajectory, ℓtf,M ^f,M_t and ℓtf,m ^f,m_t in Equation 4, is the first step at which the extreme is reached whenever several steps attain it. Increments. The first increment is Δt,1f=y^t+1f♮−yt♮ ^f_t,1= -0.5mu y^f_t+1- -0.5muy_t, i.e., its first term is the realized level at the origin, that is, y^tf♮=yt♮ -0.5mu y^f_t= -0.5muy_t in Equation 4. Appendix B Additional Experimental Results This appendix presents additional experimental results omitted from the main text due to space constraints. Note that while the main text presents results aggregated at the dataset level, this section reports the results from Section 6 at a finer grain: separately for each target in multi-target datasets. Section B.1 reports all pointwise and horizon-wise multiplicity metrics separately for each target. Similarly, Section B.2 reports all temporal multiplicity statistics separately for each target. Section B.3 examines the relationship between pointwise or horizon-wise disagreement and temporal disagreement. It reports the Spearman rank correlations separately for each target, together with their minimum and maximum across the three trajectory types, showing how the relationships summarized in the main text vary across targets and temporal aspects. B.1 Pointwise and Horizon-Wise Predictive Multiplicity In this subsection we provide more details on the pointwise and horizon-wise multiplicity results, reported per target. In particular, Table 10 reports range, diameter, ambiguity, and discrepancy, with horizon-wise statistics shown at the first and last horizons. For ambiguity and discrepancy, we report results at two thresholds relative to the best validation MASE: τ0=0.25⋅ℒ⁡(f⋆) _0=0.25·L(f ) and τ1=0.5⋅ℒ⁡(f⋆) _1=0.5·L(f ). Table 10: Pointwise and horizon-wise predictive multiplicity statistics. Ambiguity α and discrepancy ς are measured at thresholds τ0=0.25⋅ℒ⁡(f⋆) _0=0.25·L(f ) and τ1=0.5⋅ℒ⁡(f⋆) _1=0.5·L(f ). Horizon-wise statistics (range, diameter, ambiguity, and discrepancy) are reported at the first and the last horizon only. Dataset Pointwise Horizon-Wise ν ω α⁡(τ0)α( _0) ς⁡(τ0) ( _0) α⁡(τ1)α( _1) ς⁡(τ1) ( _1) ν(1)ν^(1) ν(H)ν^(H) ω(1)ω^(1) ω(H)ω^(H) α(1)(τ0)α^(1)( _0) α(H)(τ0)α^(H)( _0) ς(1)(τ0) ^(1)( _0) ς(H)(τ0) ^(H)( _0) α(1)(τ1)α^(1)( _1) α(H)(τ1)α^(H)( _1) ς(1)(τ1) ^(1)( _1) ς(H)(τ1) ^(H)( _1) ATM 1D 1.85 0.55 1.00 0.67 1.00 0.41 1.79 1.94 0.59 0.62 1.00 1.00 0.70 0.76 1.00 1.00 0.44 0.49 ATM 1W 1.68 0.59 0.99 0.72 0.98 0.48 1.75 1.67 0.71 0.61 0.99 0.99 0.76 0.75 0.98 0.98 0.55 0.51 Bitcoin 7.10 2.63 1.00 0.79 0.98 0.62 5.16 9.41 2.35 3.34 1.00 1.00 0.77 0.85 0.93 1.00 0.59 0.69 Electricity D. A (NSW) 1.38 0.35 1.00 0.78 1.00 0.58 0.76 1.64 0.24 0.42 1.00 1.00 0.76 0.83 1.00 1.00 0.53 0.67 Electricity D. A (QLD) 1.93 0.48 1.00 0.85 1.00 0.70 1.37 2.19 0.45 0.56 1.00 1.00 0.85 0.87 1.00 1.00 0.70 0.75 Electricity D. A (SA) 1.92 0.53 1.00 0.82 1.00 0.65 1.26 2.17 0.41 0.58 1.00 1.00 0.81 0.84 1.00 1.00 0.61 0.67 Electricity D. A (TAS) 1.90 0.44 1.00 0.79 1.00 0.60 1.59 2.04 0.46 0.48 1.00 1.00 0.83 0.83 1.00 1.00 0.65 0.66 Electricity D. A (VIC) 1.71 0.46 1.00 0.81 1.00 0.63 1.01 2.00 0.34 0.52 1.00 1.00 0.78 0.83 1.00 1.00 0.58 0.67 Electricity D. Z (Value_NE5) 0.64 0.17 1.00 0.76 1.00 0.55 0.47 0.78 0.14 0.20 1.00 1.00 0.78 0.81 1.00 1.00 0.56 0.63 Electricity D. Z (Value_NE7) 0.97 0.24 1.00 0.80 1.00 0.61 0.62 1.17 0.22 0.32 1.00 1.00 0.82 0.85 1.00 1.00 0.64 0.71 Electricity Price 1.24 0.37 1.00 0.63 1.00 0.35 0.98 1.29 0.34 0.48 1.00 1.00 0.63 0.72 1.00 1.00 0.32 0.47 ETT 1H 1.66 0.47 1.00 0.81 1.00 0.64 1.13 1.97 0.44 0.60 1.00 1.00 0.83 0.87 1.00 1.00 0.64 0.73 Exchange Rate (AUD) 5.35 1.19 1.00 0.57 0.99 0.29 4.09 6.26 1.23 1.74 1.00 1.00 0.68 0.86 0.99 1.00 0.35 0.54 Exchange Rate (CAD) 5.08 1.18 1.00 0.60 0.99 0.22 4.07 5.90 1.20 1.86 1.00 1.00 0.57 0.88 0.96 1.00 0.23 0.50 Exchange Rate (CHF) 3.67 0.94 1.00 0.79 1.00 0.48 3.07 4.37 1.05 1.34 1.00 1.00 0.92 0.98 1.00 1.00 0.55 0.71 Exchange Rate (CNY) 4.47 1.29 0.99 0.77 0.97 0.52 3.78 5.00 1.59 1.97 0.99 1.00 0.96 0.95 0.97 0.97 0.74 0.76 Exchange Rate (GBP) 3.58 1.02 1.00 0.72 1.00 0.37 3.09 4.18 1.06 1.43 1.00 1.00 0.78 0.91 1.00 1.00 0.41 0.54 Exchange Rate (JPY) 3.19 0.81 1.00 0.64 1.00 0.29 2.57 3.89 0.90 1.14 1.00 1.00 0.77 0.86 1.00 1.00 0.37 0.51 Exchange Rate (NZD) 4.85 1.36 1.00 0.73 1.00 0.34 4.03 5.84 1.43 2.02 1.00 1.00 0.79 0.97 1.00 1.00 0.42 0.65 Exchange Rate (SGD) 4.22 1.12 1.00 0.67 0.99 0.42 3.38 5.08 1.03 1.50 1.00 1.00 0.78 0.94 0.98 1.00 0.43 0.66 FRED-MD (CPIAUCSL) 0.09 0.04 1.00 0.82 0.93 0.33 0.06 0.12 0.04 0.05 1.00 1.00 0.73 1.00 0.82 1.00 0.36 0.69 FRED-MD (INDPRO) 0.16 0.11 0.99 0.80 0.88 0.58 0.11 0.21 0.08 0.13 0.98 1.00 0.84 0.91 0.79 0.94 0.55 0.68 FRED-MD (UNRATE) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Gasoline 3.10 1.22 1.00 0.89 1.00 0.74 3.03 3.29 1.32 1.29 1.00 1.00 0.91 0.94 1.00 1.00 0.75 0.78 Weather 2.96 1.06 1.00 0.81 1.00 0.64 2.73 3.12 1.39 1.10 1.00 1.00 0.88 0.85 1.00 1.00 0.76 0.68 Mean 2.41 0.79 0.97 0.74 0.96 0.52 1.99 2.80 0.81 0.95 0.97 0.97 0.76 0.82 0.95 0.97 0.55 0.62 Two observations are worth noting: 1. Pointwise multiplicity is pervasive: on most of the datasets the pointwise ambiguity is almost one at both thresholds, i.e., at almost every (t,h)(t,h) cell some pair of near-optimal models disagrees by more than half of the best MASE, and a single pair does so on close to half of the cells on average; the exceptions are datasets whose small and accurate sets fail to contain enough models to register large multiplicity. 2. The horizon-wise statistics grow with the horizon on most datasets, in line with the findings of Kale et al. (2026). However, there are a few exceptions, such as ATM 1W. These exceptions may reflect differences in the underlying dynamics: Kale et al. (2026) study chaotic systems, whereas not all forecasting tasks considered here necessarily exhibit chaotic behavior. B.2 Temporal Predictive Multiplicity In this appendix we provide more details on the temporal multiplicity results. We report the temporal multiplicity statistics per target. In particular, Table 11 provides the equivalent per-target results for range and diameter in Table 1. We also report the same results for ambiguity and discrepancy in Table 12 using a threshold τ1=0.5⋅ℒ⁡(f⋆) _1=0.5·L(f ). Table 11: Temporal multiplicity of the empirical Rashomon Sets. Each cell reports the range and diameter (νd/ωd _d/ _d), maximized over the three trajectories (forecast, revision, and error). dmagnituded_magnitude, dincrementd_increment and dvolatilityd_volatility are in seasonal-naive units while ddirectiond_direction, dtimingd_timing, dshaped_shape, and dspectrald_spectral are scale-free on [0,1][0,1] and reported as percentages to allow comparisons across different datasets. Dataset dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ATM 1D 1.38 / 0.66 1.55 / 0.81 0.87 / 0.35 99.7% / 58.9% 95.9% / 54.1% 85.1% / 36.0% 99.9% / 54.1% ATM 1W 1.46 / 0.67 1.38 / 0.65 0.94 / 0.49 99.9% / 55.3% 99.4% / 56.0% 85.4% / 31.6% 100% / 57.6% Bitcoin 7.00 / 2.99 3.66 / 2.08 2.01 / 0.88 99.9% / 61.7% 99.4% / 61.1% 81.5% / 41.6% 100% / 68.9% Electricity D. A (NSW) 1.17 / 0.42 0.32 / 0.23 0.14 / 0.09 75.4% / 48.5% 81.1% / 39.2% 78.9% / 42.9% 97.9% / 32.7% Electricity D. A (QLD) 1.55 / 0.57 0.41 / 0.28 0.19 / 0.13 76.6% / 48.9% 83.3% / 38.6% 81.7% / 43.3% 98.3% / 33.1% Electricity D. A (SA) 1.56 / 0.62 0.47 / 0.33 0.21 / 0.15 73.4% / 47.4% 75.1% / 33.9% 76.8% / 41.8% 94.9% / 29.0% Electricity D. A (TAS) 1.41 / 0.52 0.42 / 0.27 0.19 / 0.11 78.6% / 49.4% 79.8% / 33.8% 81.6% / 40.4% 97.9% / 28.5% Electricity D. A (VIC) 1.45 / 0.56 0.40 / 0.26 0.17 / 0.10 76.6% / 48.8% 80.5% / 37.1% 80.1% / 42.4% 97.8% / 32.4% Electricity D. Z (Value_NE5) 0.54 / 0.20 0.12 / 0.08 0.04 / 0.02 65.9% / 48.1% 76.7% / 36.8% 75.9% / 41.8% 97.8% / 31.6% Electricity D. Z (Value_NE7) 0.80 / 0.30 0.20 / 0.15 0.07 / 0.05 68.8% / 50.0% 79.4% / 44.3% 77.3% / 49.1% 99.0% / 40.0% Electricity Price 0.95 / 0.45 0.40 / 0.29 0.18 / 0.11 80.6% / 45.8% 72.3% / 35.0% 76.5% / 38.2% 95.5% / 30.2% ETT 1H 1.42 / 0.56 0.50 / 0.33 0.23 / 0.14 83.4% / 46.5% 76.6% / 32.8% 78.7% / 36.6% 95.5% / 31.2% Exchange Rate (AUD) 5.31 / 1.78 5.25 / 2.25 3.05 / 1.32 100% / 100% 100% / 99.6% 97.2% / 70.9% 100% / 100% Exchange Rate (CAD) 4.79 / 1.41 4.46 / 1.75 2.61 / 1.22 100% / 100% 100% / 98.5% 97.0% / 79.5% 100% / 100% Exchange Rate (CHF) 3.32 / 1.04 2.84 / 1.20 1.60 / 0.76 100% / 100% 100% / 97.2% 97.0% / 76.0% 100% / 100% Exchange Rate (CNY) 3.85 / 1.70 4.84 / 2.31 2.83 / 1.05 100% / 100% 99.7% / 97.5% 89.9% / 75.3% 100% / 100% Exchange Rate (GBP) 3.26 / 1.14 3.10 / 1.43 1.79 / 0.67 100% / 100% 100% / 97.2% 96.7% / 71.9% 100% / 100% Exchange Rate (JPY) 2.93 / 0.95 2.38 / 1.00 1.36 / 0.61 100% / 100% 100% / 99.6% 96.9% / 76.7% 100% / 100% Exchange Rate (NZD) 4.50 / 1.51 3.40 / 1.55 2.02 / 0.87 100% / 100% 100% / 98.5% 96.8% / 86.7% 100% / 100% Exchange Rate (SGD) 3.90 / 1.23 3.18 / 1.20 1.83 / 0.56 100% / 100% 100% / 99.6% 96.3% / 58.4% 100% / 100% FRED-MD (CPIAUCSL) 0.09 / 0.04 0.06 / 0.03 0.05 / 0.03 51.4% / 27.1% 49.4% / 24.4% 0.0% / 0.0% 61.0% / 23.8% FRED-MD (INDPRO) 0.16 / 0.12 0.13 / 0.09 0.09 / 0.05 46.7% / 34.6% 45.7% / 35.6% 0.0% / 0.0% 57.9% / 44.9% FRED-MD (UNRATE) 0.00 / 0.00 0.00 / 0.00 0.00 / 0.00 0.0% / 0.0% 0.0% / 0.0% 0.0% / 0.0% 0.0% / 0.0% Gasoline 2.74 / 1.33 2.13 / 1.00 1.23 / 0.56 100% / 71.6% 100% / 65.6% 94.3% / 45.8% 100% / 78.6% Weather 2.41 / 1.25 0.72 / 0.51 0.39 / 0.27 66.0% / 51.9% 88.0% / 49.5% 76.6% / 46.9% 99.4% / 48.3% Mean 2.14 / 0.92 1.33 / 0.70 0.75 / 0.35 82.3% / 55.4% 83.7% / 50.0% 75.5% / 39.9% 93.2% / 50.8% Table 12: Temporal multiplicity of the empirical Rashomon Sets. Each cell reports the ambiguity and discrepancy (αd/ςd _d/ _d) with threshold τ1=0.5⋅ℒ⁡(f⋆) _1=0.5·L(f ), maximized over the three trajectories (forecast, revision, and error). Dataset dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ATM 1D 100% / 62% 100% / 76% 90% / 16% 100% / 53% 100% / 58% 100% / 23% 100% / 56% ATM 1W 98% / 63% 98% / 61% 90% / 44% 100% / 58% 100% / 58% 100% / 23% 100% / 41% Bitcoin 99% / 75% 96% / 70% 73% / 24% 100% / 62% 100% / 65% 100% / 34% 100% / 79% Electricity D. A (NSW) 100% / 95% 98% / 67% 13% / 2% 100% / 42% 95% / 27% 99% / 30% 98% / 29% Electricity D. A (QLD) 100% / 99% 100% / 77% 31% / 10% 100% / 44% 96% / 29% 99% / 32% 98% / 30% Electricity D. A (SA) 100% / 94% 99% / 71% 20% / 7% 100% / 37% 88% / 22% 97% / 28% 95% / 25% Electricity D. A (TAS) 100% / 96% 92% / 48% 10% / 1% 100% / 47% 92% / 21% 99% / 25% 98% / 25% Electricity D. A (VIC) 100% / 97% 98% / 51% 19% / 6% 100% / 43% 94% / 27% 99% / 30% 98% / 28% Electricity D. Z (Value_NE5) 100% / 90% 41% / 12% 3% / 1% 100% / 36% 93% / 24% 99% / 25% 98% / 29% Electricity D. Z (Value_NE7) 100% / 99% 95% / 75% 4% / 1% 100% / 50% 97% / 39% 99% / 48% 99% / 39% Electricity Price 100% / 50% 40% / 13% 0% / 0% 100% / 34% 88% / 21% 98% / 18% 96% / 27% ETT 1H 100% / 98% 99% / 83% 37% / 8% 100% / 38% 89% / 22% 96% / 18% 96% / 28% Exchange Rate (AUD) 99% / 34% 96% / 45% 75% / 8% 100% / 100% 100% / 100% 100% / 100% 100% / 100% Exchange Rate (CAD) 98% / 27% 95% / 29% 55% / 8% 100% / 100% 100% / 100% 100% / 100% 100% / 100% Exchange Rate (CHF) 100% / 60% 100% / 75% 95% / 35% 100% / 100% 100% / 100% 100% / 100% 100% / 100% Exchange Rate (CNY) 97% / 76% 98% / 88% 92% / 47% 100% / 100% 100% / 100% 100% / 100% 100% / 100% Exchange Rate (GBP) 100% / 40% 99% / 46% 74% / 12% 100% / 100% 100% / 100% 100% / 100% 100% / 100% Exchange Rate (JPY) 100% / 34% 100% / 46% 78% / 11% 100% / 100% 100% / 100% 100% / 99% 100% / 100% Exchange Rate (NZD) 100% / 38% 100% / 46% 69% / 10% 100% / 100% 100% / 100% 100% / 100% 100% / 100% Exchange Rate (SGD) 98% / 51% 98% / 50% 84% / 12% 100% / 100% 100% / 100% 100% / 78% 100% / 100% FRED-MD (CPIAUCSL) 99% / 42% 85% / 21% 61% / 19% 28% / 10% 46% / 21% 0% / 0% 61% / 24% FRED-MD (INDPRO) 92% / 66% 92% / 54% 55% / 23% 33% / 18% 37% / 29% 0% / 0% 58% / 45% FRED-MD (UNRATE) 0% / 0% 0% / 0% 0% / 0% 0% / 0% 0% / 0% 0% / 0% 0% / 0% Gasoline 100% / 92% 100% / 90% 100% / 61% 100% / 79% 100% / 74% 100% / 49% 100% / 70% Weather 100% / 89% 70% / 50% 27% / 9% 100% / 66% 98% / 51% 99% / 41% 100% / 49% Mean 96% / 73% 84% / 57% 51% / 18% 93% / 53% 90% / 47% 90% / 34% 93% / 49% B.3 Temporal Multiplicity Is Uncorrelated with Other Notions of Multiplicity In this appendix we report the correlation results of Table 2 per target. Table 13 reports the same Spearman rank correlations of Equation 3, for both the pointwise and the horizon-wise reading, but with one row per target rather than aggregated over the targets of each multi-target dataset. The picture is the same as in the main text: the correlation is close to one for magnitude, decreases for increment and volatility, and is weak or absent for direction, shape, spectral behavior, and timing on every target. Table 13: Spearman rank correlation, over all (pair, origin) samples of the empirical Rashomon set, between the pair’s mean pointwise gap and its trajectory distance dd on the same forecast origin, read as in Equation 3 with ℒh⋆≡1L _h≡ 1 (Pointwise) or as is (Horizon-Wise). Each row reports the minimum / maximum over the three trajectories. Dataset Pointwise Horizon-Wise dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ATM 1D 0.98 / 0.98 0.58 / 0.75 0.35 / 0.37 0.15 / 0.28 0.24 / 0.61 0.05 / 0.42 0.07 / 0.24 0.98 / 0.98 0.58 / 0.75 0.35 / 0.37 0.15 / 0.28 0.24 / 0.61 0.05 / 0.42 0.07 / 0.24 ATM 1W 0.99 / 0.99 0.56 / 0.70 0.39 / 0.47 0.13 / 0.18 0.17 / 0.40 0.04 / 0.24 0.14 / 0.20 0.99 / 0.99 0.56 / 0.70 0.39 / 0.47 0.13 / 0.18 0.17 / 0.40 0.04 / 0.24 0.14 / 0.20 Bitcoin 0.99 / 1.00 0.61 / 0.69 0.43 / 0.49 -0.02 / 0.19 -0.01 / 0.32 -0.05 / 0.26 0.02 / 0.14 0.96 / 0.98 0.60 / 0.67 0.41 / 0.48 -0.03 / 0.17 -0.03 / 0.30 -0.05 / 0.25 -0.01 / 0.13 Electricity D. A (NSW) 0.99 / 0.99 0.51 / 0.66 0.16 / 0.36 -0.01 / 0.16 0.04 / 0.63 -0.08 / 0.46 0.08 / 0.33 0.97 / 0.97 0.52 / 0.65 0.16 / 0.36 -0.01 / 0.16 0.03 / 0.61 -0.09 / 0.45 0.07 / 0.32 Electricity D. A (QLD) 0.99 / 0.99 0.53 / 0.60 0.19 / 0.33 -0.01 / 0.23 0.04 / 0.63 -0.05 / 0.44 0.08 / 0.36 0.97 / 0.97 0.54 / 0.59 0.19 / 0.33 -0.01 / 0.23 0.03 / 0.63 -0.06 / 0.44 0.07 / 0.35 Electricity D. A (SA) 0.99 / 0.99 0.46 / 0.61 0.16 / 0.29 -0.03 / 0.26 -0.00 / 0.61 -0.08 / 0.45 0.05 / 0.34 0.97 / 0.97 0.47 / 0.58 0.16 / 0.30 -0.03 / 0.26 -0.03 / 0.59 -0.10 / 0.43 0.04 / 0.33 Electricity D. A (TAS) 0.99 / 0.99 0.55 / 0.55 0.23 / 0.30 0.02 / 0.21 0.11 / 0.43 0.04 / 0.35 0.11 / 0.24 0.96 / 0.97 0.54 / 0.55 0.23 / 0.29 0.02 / 0.21 0.09 / 0.42 0.02 / 0.34 0.10 / 0.24 Electricity D. A (VIC) 0.99 / 0.99 0.53 / 0.66 0.21 / 0.31 -0.02 / 0.19 0.04 / 0.64 -0.07 / 0.49 0.08 / 0.33 0.96 / 0.97 0.55 / 0.64 0.21 / 0.31 -0.01 / 0.18 0.02 / 0.62 -0.08 / 0.48 0.07 / 0.32 Electricity D. Z (Value_NE5) 0.99 / 0.99 0.33 / 0.51 0.19 / 0.28 -0.08 / 0.23 0.08 / 0.40 -0.05 / 0.26 0.12 / 0.28 0.98 / 0.99 0.33 / 0.51 0.19 / 0.28 -0.07 / 0.23 0.08 / 0.39 -0.06 / 0.26 0.11 / 0.28 Electricity D. Z (Value_NE7) 0.99 / 0.99 0.23 / 0.52 0.09 / 0.24 -0.01 / 0.18 0.07 / 0.52 -0.03 / 0.38 0.10 / 0.37 0.98 / 0.98 0.24 / 0.52 0.09 / 0.24 -0.00 / 0.18 0.07 / 0.51 -0.03 / 0.38 0.10 / 0.37 Electricity Price 0.99 / 0.99 0.59 / 0.64 0.20 / 0.31 -0.02 / 0.06 0.17 / 0.35 0.08 / 0.23 0.10 / 0.14 0.98 / 0.98 0.59 / 0.64 0.20 / 0.31 -0.02 / 0.06 0.15 / 0.34 0.07 / 0.22 0.10 / 0.14 ETT 1H 0.99 / 0.99 0.56 / 0.68 0.30 / 0.39 0.32 / 0.36 0.43 / 0.51 0.31 / 0.39 0.17 / 0.29 0.98 / 0.98 0.58 / 0.69 0.31 / 0.40 0.33 / 0.38 0.41 / 0.50 0.30 / 0.39 0.17 / 0.29 Exchange Rate (AUD) 0.99 / 0.99 0.57 / 0.74 0.36 / 0.45 0.09 / 0.21 0.14 / 0.39 0.02 / 0.24 0.09 / 0.15 0.97 / 0.98 0.57 / 0.73 0.36 / 0.45 0.09 / 0.21 0.14 / 0.37 0.02 / 0.23 0.09 / 0.14 Exchange Rate (CAD) 0.99 / 0.99 0.56 / 0.71 0.34 / 0.43 0.10 / 0.17 0.16 / 0.38 0.01 / 0.24 0.07 / 0.12 0.97 / 0.98 0.55 / 0.70 0.34 / 0.43 0.09 / 0.17 0.14 / 0.36 0.01 / 0.22 0.07 / 0.11 Exchange Rate (CHF) 0.99 / 0.99 0.55 / 0.70 0.34 / 0.43 0.10 / 0.18 0.17 / 0.39 0.02 / 0.24 0.10 / 0.15 0.97 / 0.98 0.55 / 0.69 0.34 / 0.43 0.10 / 0.17 0.16 / 0.37 0.02 / 0.23 0.10 / 0.15 Exchange Rate (CNY) 0.99 / 0.99 0.72 / 0.83 0.45 / 0.51 0.10 / 0.20 0.15 / 0.39 0.01 / 0.26 0.11 / 0.22 0.97 / 0.98 0.71 / 0.82 0.44 / 0.51 0.10 / 0.19 0.14 / 0.38 0.01 / 0.25 0.10 / 0.20 Exchange Rate (GBP) 0.99 / 0.99 0.52 / 0.69 0.32 / 0.41 0.13 / 0.21 0.22 / 0.47 0.02 / 0.28 0.11 / 0.18 0.96 / 0.98 0.51 / 0.68 0.32 / 0.40 0.13 / 0.21 0.20 / 0.45 0.02 / 0.27 0.11 / 0.16 Exchange Rate (JPY) 0.99 / 0.99 0.50 / 0.67 0.29 / 0.38 0.11 / 0.22 0.17 / 0.38 0.01 / 0.23 0.12 / 0.17 0.96 / 0.98 0.49 / 0.66 0.29 / 0.38 0.10 / 0.21 0.16 / 0.35 0.02 / 0.22 0.12 / 0.16 Exchange Rate (NZD) 0.99 / 0.99 0.53 / 0.68 0.32 / 0.40 0.11 / 0.19 0.18 / 0.38 0.01 / 0.23 0.09 / 0.15 0.97 / 0.98 0.52 / 0.67 0.32 / 0.40 0.11 / 0.18 0.17 / 0.36 0.01 / 0.22 0.09 / 0.14 Exchange Rate (SGD) 0.99 / 0.99 0.58 / 0.74 0.38 / 0.47 0.10 / 0.13 0.19 / 0.32 0.04 / 0.19 0.06 / 0.16 0.97 / 0.98 0.58 / 0.73 0.37 / 0.47 0.10 / 0.12 0.18 / 0.30 0.04 / 0.19 0.06 / 0.15 FRED-MD (CPIAUCSL) 0.99 / 0.99 0.63 / 0.63 0.47 / 0.64 0.04 / 0.19 0.05 / 0.28 -0.03 / 0.04 0.03 / 0.16 0.97 / 0.97 0.59 / 0.60 0.44 / 0.60 0.03 / 0.18 0.04 / 0.26 -0.04 / 0.04 0.03 / 0.15 FRED-MD (INDPRO) 0.99 / 1.00 0.69 / 0.69 0.45 / 0.49 0.20 / 0.40 0.35 / 0.42 0.00 / 0.04 0.13 / 0.40 0.98 / 0.99 0.66 / 0.67 0.44 / 0.48 0.19 / 0.39 0.34 / 0.42 0.00 / 0.04 0.12 / 0.39 FRED-MD (UNRATE) 0.98 / 0.99 0.61 / 0.67 0.51 / 0.64 0.03 / 0.23 0.22 / 0.38 -0.11 / -0.01 0.02 / 0.16 0.97 / 0.99 0.61 / 0.67 0.51 / 0.62 0.04 / 0.24 0.23 / 0.38 -0.10 / 0.00 0.03 / 0.17 Gasoline 0.99 / 0.99 0.56 / 0.61 0.35 / 0.45 0.09 / 0.22 0.12 / 0.43 -0.01 / 0.29 0.10 / 0.23 0.99 / 0.99 0.56 / 0.61 0.35 / 0.45 0.09 / 0.22 0.12 / 0.43 -0.01 / 0.29 0.10 / 0.23 Weather 1.00 / 1.00 0.69 / 0.72 0.51 / 0.52 0.35 / 0.43 0.34 / 0.52 0.24 / 0.43 0.27 / 0.35 0.99 / 0.99 0.70 / 0.73 0.52 / 0.54 0.37 / 0.44 0.34 / 0.52 0.24 / 0.43 0.27 / 0.35 Mean 0.99 / 0.99 0.54 / 0.68 0.32 / 0.44 0.09 / 0.26 0.16 / 0.47 0.04 / 0.31 0.10 / 0.27 0.98 / 0.98 0.54 / 0.68 0.32 / 0.44 0.09 / 0.26 0.15 / 0.46 0.04 / 0.31 0.10 / 0.26 Appendix C Predictive Multiplicity Hierarchy Characterization Table 14: Classification of the loss and of the multiplicity statistics of the hierarchy. Statistic Multiplicity-Aware not invariant to per-model cell permutations Horizon-Aware not invariant to cell permutations Trajectory-Aware not invariant to row permutations Loss ℒL × × × Pointwise Multiplicity ✓ × × Horizon-wise Multiplicity ✓ ✓ × Temporal Multiplicity ✓ ✓ ✓ This appendix provides the deferred proofs and develops the theoretical results underlying our hierarchy of predictive multiplicity. In particular, it establishes the distinction between the information retained by its three levels: pointwise and horizon-wise statistics are invariant to rearrangements that preserve comparisons within each horizon, whereas temporal statistics can detect the resulting changes in trajectory structure. Table 14 provides a summary of the theoretical results. We begin by establishing what the loss and the pointwise and horizon-wise multiplicity statistics cannot detect. The key observation is that predictions can be rearranged across origins within each horizon, changing the trajectories they form while preserving identical pointwise and horizon-wise multiplicity statistics. The following proposition formalizes this limitation through the invariance notions of Definition 3.4, showing why agreement at these levels does not fully characterize temporal disagreement. Proposition C.1 (Classification of Loss, and Pointwise and Horizon-Wise Multiplicity Statistics). The loss ℒL is not multiplicity-aware; for every tolerance τ, the horizon-wise statistics of Definition 3.2 are not trajectory-aware, and the pointwise statistics of Definition 3.1 are neither horizon-aware nor trajectory-aware. Proof. (i) The loss (Equation 2) of each model is the entrywise sum over its own error matrix, which is invariant to any permutation of that matrix’s cells; hence ℒL is invariant to per-model cell permutations, i.e., it is not multiplicity-aware, nor, a fortiori, horizon- or trajectory-aware. (i) Each pointwise statistic (Definition 3.1) is built from THTH per-cell quantities (|y^t+hf−y^t+hg|)f,g∈ℛϵ (| y^f_t+h- y^g_t+h| )_f,g∈R_ε, each of which depends only on the forecasts at the single pair (t,h)(t,h): the prediction range and ambiguity are symmetric functions of these quantities, while the prediction diameter and discrepancy are maxima over pairs (f,g)(f,g) of symmetric functions of them. A cell permutation moves each such pair as a unit and therefore merely permutes these THTH quantities, under which a symmetric function—and hence a maximum of symmetric functions—is invariant; hence no pointwise statistic is horizon-aware, nor, a fortiori, trajectory-aware, since every row permutation is a cell permutation. (i) The horizon-wise statistics (Definition 3.2) are, for each h, symmetric functions of the same per-cell quantities restricted to horizon h; a row permutation permutes these quantities within each horizon, so no horizon-wise statistic is trajectory-aware. ∎ Before proving that the temporal statistics proposed in the main text are trajectory-aware in the sense of Definition 3.4, we establish a useful reduction that the proof will later use: for a pair of models f and g, the four statistics at each level of multiplicity reduce to two, as range coincides with diameter and ambiguity with discrepancy. Remark C.2 (Multiplicity Statistics of a Pair of Models). Let dd be symmetric and vanish on the diagonal, i.e., d⁡(,)=d⁡(,)d( u, v)=d( v, u) and d⁡(,)=0d( u, u)=0 for all ,∈ℝH u, v ^H. Then, for every pair f,g∈ℱf,g∈F, every τ≥0τ≥ 0, every horizon h∈1,…,Hh∈\1,…,H\, and every trajectory t v_t, we have: ν⁡(f,g)=ω⁡(f,g)=t∈1≤h≤H[|y^t+hf−y^t+hg|]ν(\f,g\)=ω(\f,g\)= E_ subarrayct∈T\\ 1≤ h≤ H subarray [ | y^f_t+h- y^g_t+h | ] ατ(f,g)=ςτ(f,g)=t∈1≤h≤H[|y^t+hf−y^t+hg|>τ] _τ(\f,g\)= _τ(\f,g\)= E_ subarrayct∈T\\ 1≤ h≤ H subarray [ 1 \ | y^f_t+h- y^g_t+h |>τ \ ] ν(h)(f,g)=ω(h)(f,g)=t∈[|y^t+hf−y^t+hg|]ν^(h)(\f,g\)=ω^(h)(\f,g\)= E_t∈T [ | y^f_t+h- y^g_t+h | ] ατ(h)(f,g)=ςτ(h)(f,g)=t∈[|y^t+hf−y^t+hg|>τ]α^(h)_τ(\f,g\)= ^(h)_τ(\f,g\)= E_t∈T [ 1 \ | y^f_t+h- y^g_t+h |>τ \ ] νd(f,g)=ωd(f,g)=t∈[d⁡(tf,tg)] _d(\f,g\)= _d(\f,g\)= E_t∈T [d ( v^f_t, v^g_t ) ] αd,τ(f,g)=ςd,τ(f,g)=t∈[d(tf,tg)>τ]. _d,τ(\f,g\)= _d,τ(\f,g\)= E_t∈T [ 1 \d ( v^f_t, v^g_t )>τ \ ]. Proof. On a two-element set, every maximum over f′,g′∈f,gf ,g ∈\f,g\ ranges over the two zero distances between a model and itself and the two copies of the pair’s distance. The same conclusion therefore holds whether the maximum is taken inside the expectation (range, ambiguity) or outside it (diameter, discrepancy). The hypotheses of the remark hold for every trajectory distance function dd of Section 5, each of which is symmetric and vanishes at identical trajectories. They also hold trivially for the absolute difference used at the pointwise and horizon-wise levels. ∎ For a pair of models, multiplicity is therefore summarized by the expected distance between their predictions and the rate at which that distance exceeds τ. Consequently, the inequalities ω≤νω≤ν and ς≤α ≤α noted after Definition 3.1 can be strict only for sets of three or more models. With that in mind, we now prove that the statistics induced by the trajectory distances of Section 5 are trajectory-aware and hence, by the nesting of Definition 3.4, also horizon- and multiplicity-aware. Proposition C.3 (Temporal Multiplicity Statistics Are Trajectory-Aware). The seven distances in Section 5 give trajectory-aware range and diameter for forecast, revision, and error trajectories. Their ambiguity and discrepancy are also trajectory-aware, with the following restrictions on the tolerance: • for magnitude, increment, and volatility, any τ≥0τ≥ 0; • for direction, shape, spectral, and timing, any 0≤τ<10≤τ<1. For the latter four distances, ambiguity and discrepancy are always zero when τ≥1τ≥ 1. Specifically, for each statistic at the stated tolerances, we can choose a horizon length and two ensembles that differ only by a row permutation, yet have different temporal multiplicity. Their pointwise and horizon-wise multiplicity remain unchanged. Proof. A row permutation exchanges origins within a horizon, using the same exchange for every model. It preserves the comparisons between models at that horizon, but can change which values belong to the same trajectory. We will show that this can change each of the temporal statistics. It is enough to use two models, f and g, and two trajectories per model. By Remark C.2, their range and diameter both equal the average distance between the models’ trajectories. Their ambiguity and discrepancy both equal the fraction of trajectories for which this distance exceeds τ. We therefore only need examples where a row permutation changes this average or this fraction. We begin with error trajectories, so each example consists of two error matrices with two rows. Whenever we exchange entries within a column, we do so in both matrices. We then explain how to use the same examples for forecasts and revisions, and how to include both models in a Rashomon set. 1. Magnitude, increment, and volatility. Consider a model whose errors are constant along each trajectory, and compare it with a model whose errors are all zero: f=(1100),g=(0000). E^f= pmatrix1&1\\ 0&0 pmatrix, E^g= pmatrix0&0\\ 0&0 pmatrix. Exchange the entries in the second column. The first matrix becomes (1001), pmatrix1&0\\ 0&1 pmatrix, while the second stays zero. Before the exchange, the two magnitude distances are 11 and 00; afterward, both are 1/21/ 2. Their average therefore changes from 1/21/2 to 1/21/ 2, changing both range and diameter. The same exchange also changes increment and volatility distances. Initially, all trajectories are constant, so both distances are zero at both origins. Afterward, each trajectory of f changes by 11 in absolute value, while those of g remain constant. Both distances are now 11 at both origins, so their range and diameter change from 00 to 11. To handle ambiguity and discrepancy at any given τ≥0τ≥ 0, replace every 11 in this example by a number a>2τa> 2τ. The magnitude distances are then (a,0)(a,0) before the exchange and (a/2,a/2)(a/ 2,a/ 2) afterward: the fraction exceeding τ changes from 1/21/2 to 11. For increment and volatility, this fraction changes from 00 to 11. 2. Direction and shape. This time, start with two models whose trajectories are constant but at opposite levels: f=(11−1−1),g=(−1−111). E^f= pmatrix1&1\\ -1&-1 pmatrix, E^g= pmatrix-1&-1\\ 1&1 pmatrix. At each origin, both models have zero increments, so their direction distance is zero. Their shape distance is also zero by the convention for two constant trajectories in Section A.4. Exchange the entries in the second column. At each origin, the two models now have opposite trajectories, (1,−1)(1,-1) and (−1,1)(-1,1). One decreases while the other increases, and their correlation is −1-1. Thus both direction and shape distances change from 00 to 11 at both origins. Their range and diameter change from 00 to 11, as do their ambiguity and discrepancy for every 0≤τ<10≤τ<1. 3. Spectral. We need four horizons for this example: f=(1−11−11−11−1),g=(10−10−1010). E^f= pmatrix1&-1&1&-1\\ 1&-1&1&-1 pmatrix, E^g= pmatrix1&0&-1&0\\ -1&0&1&0 pmatrix. The two models initially vary at different frequencies. More explicitly, after standardizing each trajectory, their Fourier magnitude spectra are (0,0,1,0)for f,(0,1/2,0,1/2)for g.(0,0,1,0) f, (0,1/ 2,0,1/ 2) g. These spectra have no nonzero entries in common, so their spectral distance is 11 at both origins. Now exchange the two entries within column 33 and within column 44. The trajectories of f stay unchanged, while those of g become (1,0,1,0)(1,0,1,0) and (−1,0,−1,0)(-1,0,-1,0). Subtracting their means gives 12(1,−1,1,−1) 12(1,-1,1,-1) and its negative. After standardization, both therefore have the same magnitude spectrum as the trajectories of f. The spectral distance changes from 11 to 00 at both origins. Hence range and diameter change from 11 to 00, as do ambiguity and discrepancy for every 0≤τ<10≤τ<1. 4. Timing. The timing distance in Section 5 measures the shift that gives the largest Pearson correlation. When several shifts give the same correlation, we choose the one with the smallest absolute value. Take three horizons and let f=(10−110−1),g=(10−1−101). E^f= pmatrix1&0&-1\\ 1&0&-1 pmatrix, E^g= pmatrix1&0&-1\\ -1&0&1 pmatrix. The possible shifts are −1-1, 00, and 11. At the first origin, both trajectories decrease linearly, so their correlation is 11 at every shift. At the second origin, one decreases linearly and the other increases linearly, so their correlation is −1-1 at every shift. In both cases, shifting gives no improvement over zero, and the tie-breaking rule selects zero. Both timing distances are therefore zero. Exchange the entries in the third column. The trajectories of f stay unchanged, while those of g become (1,0,1)(1,0,1) and (−1,0,−1)(-1,0,-1). Without a shift, each has correlation 00 with the corresponding trajectory of f. A one-step shift, however, can match two decreasing segments or two increasing segments, giving correlation 11. Thus the best shift now has absolute value 11 at both origins. Since H−2=1H-2=1, both timing distances are 11. Their range and diameter change from 00 to 11, as do their ambiguity and discrepancy for every 0≤τ<10≤τ<1. These examples also work for forecast trajectories. Choose all targets to be zero, so every forecast is the negative of its error. Negating both trajectories leaves each of the seven distances unchanged. The changes in the statistics are therefore exactly the same. For revisions, the only difference is where we place the trajectories in the matrix: along anti-diagonals instead of rows. Take T=H+1T=H+1 origins, which gives exactly two complete revision trajectories. Place the first row of each example on the first complete anti-diagonal and the second row on the next, setting the remaining cells to zero. The k-th entries of these two trajectories occupy cells (k,H+1−k)and(k+1,H+1−k).(k,H+1-k) (k+1,H+1-k). They lie in the same column. We can therefore exchange them by swapping origins k and k+1k+1 within that column, for both models. This reproduces every exchange used above through a row permutation. With zero targets, the resulting revision trajectories are again the negatives of the error sequences, so all the distance calculations still apply. Finally, the examples satisfy the Rashomon-set requirement. For each construction, including any choice of a in the first example, take a model class containing just f and g, and choose ϵ≥|ℒ⁡(f)−ℒ⁡(g)|.ε≥ |L(f)-L(g) |. The Rashomon set then contains both models. Each exchange preserves each model’s loss, so the same tolerance includes both models after the exchange as well. By Proposition C.1, it also preserves all pointwise and horizon-wise multiplicity statistics. The temporal statistics change as shown above, proving the claim for forecast, revision, and error trajectories. ∎ Appendix D Temporal Multiplicity Bounds Characterization Table 15: Summary of the bounds proved in Sections D.1 and D.2: how Rashomon set membership and additional pointwise or horizon-wise multiplicity constraints bound disagreement between forecast, revision, and error trajectories at each origin. The first column is reported for the MAE. † For forecast trajectories, the bounds are attained under conditions that rule out degenerate cases. § For revision trajectories, the bounds hold when the trajectories are traversed in reverse order. Temporal Statistic ϵ R_ε only + + pointwise = _τ=0 + + horizon-wise ()= α^(h)_ _h=0 Pointwise |y^f−y^g|| y^f- y^g| 2TH(ℒ⋆+ϵ)2TH(L +ε) τ τh _h Magnitude dmagnituded_magnitude 2HT(ℒ⋆+ϵ)2 H\,T(L +ε) τ 1H∑hτh2 1H _h _h^2 Increment dincrement§d_increment @sectionsign 22H−1TH(ℒ⋆+ϵ) 2 2 H-1\,TH(L +ε) 2τ2τ 1H−1∑h<H(τh+τh+1)2 1H-1 _h<H( _h+ _h+1)^2 Volatility dvolatility†§d_volatility @sectionsign 2H−1TH(ℒ⋆+ϵ) 2H-1\,TH(L +ε) 2τ2τ 1H−1∑h<H(τh+τh+1) 1H-1 _h<H( _h+ _h+1) Direction ddirection†d_direction free free free Shape dshape†d_shape free free free Spectral dspectral†d_spectral free free free Timing dtiming†d_timing free free free This appendix provides the deferred proofs for the temporal multiplicity bounds, characterizes how these bounds constrain forecast trajectories, and tests how closely trained models approach them. It is organized in three parts. Section D.1 studies what Rashomon set membership alone implies for temporal multiplicity. It first proves Theorem 4.1, which bounds the gap between any two models’ forecasts at a single cell. It then translates this pointwise bound into bounds on the trajectory distances of Section 5: magnitude, increment, and volatility disagreement are bounded, whereas direction, shape, spectral, and timing disagreement can remain maximal. Section D.2 studies what changes when, in addition, pointwise or horizon-wise ambiguity is required to vanish. It first proves Theorem 4.2, which tightens the per-cell gap to the chosen tolerances. It then derives the corresponding trajectory-distance bounds, showing that the stronger control bounds magnitude, increment, and volatility disagreement more tightly, yet still leaves direction, shape, spectral, and timing disagreement unconstrained under the stated assumptions. Section D.3 evaluates these bounds empirically. For each trajectory distance, it measures the largest disagreement among trained models in the empirical Rashomon set and reports what fraction of the corresponding theoretical bound it attains. Table 15 summarizes the theoretical results of the first two parts, comparing the bounds under Rashomon membership alone, specialized to MAE, with those under additional pointwise or horizon-wise control. D.1 The Performance Constraint of the Rashomon Set Membership: Exact Characterization and Bounds for Each Temporal Aspect We first restate and prove Theorem 4.1, which bounds pointwise disagreement under the performance constraint defining the Rashomon set. We then derive bounds and attaining constructions for the seven trajectory distances introduced in Section 5, identifying how much disagreement can remain between models that are members of the Rashomon set. We conclude by explaining how these results extend to error and revision trajectories. The MAE results are summarized in the Rashomon-membership column of Table 15. See 4.1 Proof. Fix an origin t∈t∈T and a horizon h∈1,…,Hh∈\1,…,H\, and write B:=TH(ℒ⋆+ϵ),u:=ℓ−1(B).B:=TH(L +ε), u:= ^-1(B). Here, B is each model’s total loss budget, and u is the largest error magnitude that this budget allows at a single entry. For any f∈ℛϵf∈R_ε, nonnegativity of the pointwise losses gives ℓ⁡(et,hf)≤∑t′∈∑h′=1Hℓ⁡(et′,h′f)=THℒ(f)≤B. (e^f_t,h)≤ _t ∈T _h =1^H (e^f_t ,h )=TH\,L(f)≤ B. Since ℓ depends only on the error magnitude, the definition of u implies |et,hf|≤u=ℓ−1(TH(ℒ⋆+ϵ)).|e^f_t,h|≤ u= ^-1 (TH(L +ε) ). Every Rashomon forecast therefore lies within distance u of the same target. Since this target cancels when subtracting two forecasts, y^t+hf−y^t+hg=et,hg−et,hf, y^f_t+h- y^g_t+h=e^g_t,h-e^f_t,h, and hence |y^t+hf−y^t+hg|≤|et,hg|+|et,hf|≤2u. | y^f_t+h- y^g_t+h |≤|e^g_t,h|+|e^f_t,h|≤ 2u. To show attainment under shell-richness, we concentrate each model’s entire budget at the chosen entry, with opposite error signs. First, ℓ⁡(u)=B: (u)=B: otherwise, continuity would allow a slightly larger error with loss still below B, contradicting the definition of u. Define [±]t′,h′=±u,(t′,h′)=(t,h),0,otherwise.[ E^±]_t ,h = cases± u,&(t ,h )=(t,h),\\ 0,&otherwise. cases Because ℓ⁡(0)=0 (0)=0 and ℓ⁡(−u)=ℓ⁡(u) (-u)= (u), both matrices have loss ℒ⁡(±)=ℓ⁡(u)TH=ℒ⋆+ϵ.L( E^±)= (u)TH=L +ε. They therefore lie in the admissible loss shell ℬϵ:=∈ℝT×H:ℒ⋆≤ℒ⁡()≤ℒ⋆+ϵ.B_ε:= \ E ^T× H:L ( E) +ε \. Shell-richness guarantees models f+,f−∈ℛϵf^+,f^-∈R_ε realizing these matrices. Their forecasts at the chosen entry are yt+h−uy_t+h-u and yt+h+uy_t+h+u, hence maxf,g∈ℛϵ⁡|y^t+hf−y^t+hg|=2u=2ℓ−1(TH(ℒ⋆+ϵ)). _f,g∈R_ε | y^f_t+h- y^g_t+h |=2u=2\, ^-1 (TH(L +ε) ). Thus, two models with identical loss attain the bound by assigning opposite errors to a single entry while matching the targets everywhere else. ∎ Theorem 4.1 tells us how far apart two models in the Rashomon set can be at a single forecast horizon. Under shell-richness, its upper bound can be reached. We now ask how much these models can disagree over an entire trajectory, using the distances in Section 5. The following corollaries quantify how much forecast trajectories can differ despite identical aggregate loss. Each construction gives a lower bound on the largest disagreement within the Rashomon set; when a matching upper bound is available, it identifies the exact maximum. Fix an origin t∈t∈T, and write B:=TH(ℒ⋆+ϵ),u:=ℓ−1(B),ϱ:=ℓ−1(B/H).B:=TH(L +ε), u:= ^-1(B), := ^-1(B/H). Here, B is each model’s total loss budget. An error of magnitude u uses this budget at a single entry, whereas H errors of magnitude ϱ spread it equally across one trajectory: ℓ⁡(u)=Hℓ(ϱ)=B. (u)=H ( )=B. These equalities follow from continuity and the definition of ℓ−1 ^-1, as in the preceding proof. Every Rashomon model satisfies the row budget ∑h=1Hℓ⁡(et,hf)≤B. _h=1^H (e^f_t,h)≤ B. In all constructions below, errors outside origin t are zero and the row loss equals B. The resulting matrices therefore have average loss ℒ⋆+ϵL +ε, so shell-richness realizes them as models in ℛϵR_ε. This establishes both membership and equality of aggregate loss without repeating the argument for each construction. Let (j)∈ℝH b^(j) ^H denote the vector with a 11 at entry j and zeros elsewhere. For any pair of models, their forecast gap is :=^tf−^tg=tg−tf. w:= y^f_t- y^g_t= e^g_t- e^f_t. Thus, distances that depend only on this gap, including magnitude and increment, do not depend on the target trajectory. We begin with magnitude, which measures the root-mean-square gap between forecasts. Its maximum depends on how a model can distribute its loss budget to obtain the largest Euclidean error norm. Two allocations are relevant: concentrating the budget at one horizon, or spreading it equally across all horizons. Which allocation attains the maximum depends on ϕ⁡(s):=ℓ⁡(s)φ(s):= ( s), the loss associated with a squared error of size s: • when ϕφ is concave, concentrating the budget at one horizon attains the maximum; • when ϕφ is convex, spreading it equally attains the maximum. The following corollary establishes this distinction. For power losses ℓ⁡(e)=|e|p (e)=|e|^p, these cases correspond to 0<p≤20<p≤ 2 and p≥2p≥ 2, respectively; at p=2p=2, both allocations attain the same maximum. Corollary D.1 (Magnitude Multiplicity Bound Under Rashomon Set Membership). Let ϕ⁡(s):=ℓ⁡(s)φ(s):= ( s) for s≥0s≥ 0. If ℱF is shell-rich, then maxf,g∈ℛϵ⁡dmagnitude(^tf,^tg)=2u/H,if ϕ is concave,2ϱ,if ϕ is convex. _f,g∈R_εd_magnitude ( y^f_t, y^g_t )= cases2u/ H,&if $φ$ is concave,\\[4.30554pt] 2 ,&if $φ$ is convex. cases For MAE, the maximum is 2HT(ℒ⋆+ϵ)2 H\,T(L +ε); for MSE, it is 2T⁡(ℒ⋆+ϵ)2 T(L +ε). Both maxima are attained by models with opposite error rows and identical pointwise absolute errors. Proof. The triangle inequality gives dmagnitude(^tf,^tg)=‖tg−tf‖2H≤‖tf‖2+‖tg‖2H.d_magnitude ( y^f_t, y^g_t )= \| e^g_t- e^f_t\|_2 H≤ \| e^f_t\|_2+\| e^g_t\|_2 H. We therefore need the largest Euclidean norm that one error row can have within the budget. Write sh=eh2s_h=e_h^2, so that ∑hϕ⁡(sh)≤B _hφ(s_h)≤ B. If ϕφ is concave, ϕ⁡(0)=0φ(0)=0 implies subadditivity, and hence ℓ⁡(‖2)=ϕ⁡(∑hsh)≤∑hϕ⁡(sh)≤B. (\| e\|_2)=φ ( _hs_h )≤ _hφ(s_h)≤ B. Thus ‖2≤u\| e\|_2≤ u, giving the upper bound 2u/H2u/ H. It is attained by concentrating the budget at one entry: tf=u(j),tg=−u(j). e^f_t=u b^(j), e^g_t=-u b^(j). Their gap has Euclidean norm 2u2u. If ϕφ is convex, Jensen’s inequality instead gives ℓ⁡(‖2H)=ϕ⁡(1H∑hsh)≤1H∑hϕ⁡(sh)≤B/H. ( \| e\|_2 H )=φ ( 1H _hs_h )≤ 1H _hφ(s_h)≤ B/H. Thus ‖2≤Hϱ\| e\|_2≤ H\, , giving the upper bound 2ϱ2 . It is attained by spreading the budget equally: tf=ϱ,tg=−ϱ. e^f_t= 1, e^g_t=- 1. Their gap has magnitude 2ϱ2 at every entry. Finally, MAE has ϕ⁡(s)=sφ(s)= s and u=Bu=B, whereas MSE has ϕ⁡(s)=sφ(s)=s and ϱ=B/H = B/H, giving the stated expressions. ∎ We next consider increment distance, which measures disagreement in how forecasts change from one horizon to the next. Here, the arrangement of the errors matters as well as their size: an error concentrated at a horizon other than the first or last creates two abrupt changes, while alternating errors create changes throughout the trajectory. The following corollary uses these patterns to obtain lower bounds for general losses and identifies the exact maxima for MAE and MSE. Corollary D.2 (Increment Multiplicity Bound Under Rashomon Set Membership). Let ϕ⁡(s):=ℓ⁡(s)φ(s):= ( s). If ℱF is shell-rich and H≥3H≥ 3, then maxf,g∈ℛϵ⁡dincrement(^tf,^tg)≥22u/H−1,if ϕ is concave,4ϱ,if ϕ is convex. _f,g∈R_εd_increment ( y^f_t, y^g_t )≥ cases2 2\,u/ H-1,&if $φ$ is concave,\\[4.30554pt] 4 ,&if $φ$ is convex. cases For MAE, the exact maximum is 22TH(ℒ⋆+ϵ)H−1. 2 2\,TH(L +ε) H-1. For MSE, it is 4cos⁡(π2H)TH(ℒ⋆+ϵ)H−1.4 ( π2H ) TH(L +ε)H-1. The displayed lower bounds and exact maxima are attained by pairs with opposite error rows and identical pointwise absolute errors. Proof. Increment distance measures how quickly the forecast gap changes: dincrement(^tf,^tg)=‖Δ⁡(tg−tf)‖2H−1.d_increment ( y^f_t, y^g_t )= \| ( e^g_t- e^f_t)\|_2 H-1. For the first lower bound, take opposite errors concentrated at an interior entry: tf=u(j)=−tg,1<j<H. e^f_t=u b^(j)=- e^g_t, 1<j<H. The gap changes by 2u2u when entering this entry and by 2u2u when leaving it. Its two nonzero increments therefore give dincrement=(2u)2+(2u)2H−1=22uH−1.d_increment= (2u)^2+(2u)^2 H-1= 2 2\,u H-1. For the second lower bound, take opposite alternating errors, et,hf=(−1)hϱ=−et,hg.e^f_t,h=(-1)^h =-e^g_t,h. Every increment of their gap has magnitude 4ϱ4 , so dincrement=4ϱd_increment=4 . Both constructions use the full budget. To establish the exact MAE maximum, observe that ‖1≤B=u\| e\|_1≤ B=u and ‖Δ‖2=‖∑hehΔ(h)‖2≤∑h|eh|‖Δ(h)‖2≤2‖1.\| e\|_2= \| _he_h b^(h) \|_2≤ _h|e_h|\,\| b^(h)\|_2≤ 2\,\| e\|_1. The last inequality holds because a single entry affects at most two increments. Applying this bound to both models gives 22u/H−12 2\,u/ H-1, attained by the interior-entry construction. For MSE, the budget is ‖2≤B\| e\|_2≤ B. The largest singular value of the increment operator is σmax(Δ)=2cos⁡(π2H), _ ( )=2 ( π2H ), as follows from the eigenvalues of the path Laplacian Δ⊤Δ (Brouwer & Haemers, 2012, Section 1.4.4). Consequently, dincrement≤2σmax(Δ)BH−1.d_increment≤ 2 _ ( ) B H-1. Equality is obtained by placing opposite error rows along a unit right singular vector for σmax(Δ) _ ( ): tf=B=−tg,vh=2H(−1)h−1sin⁡((2h−1)π2H). e^f_t= B\, v=- e^g_t, v_h= 2H\,(-1)^h-1 ( (2h-1)π2H ). Both rows have squared norm B, and their gap is amplified by Δ by exactly σmax(Δ) _ ( ). This proves the stated MSE maximum. ∎ Magnitude and increment distances depend only on the difference between the two forecast trajectories, from which the common target cancels. The remaining distances compare properties of the trajectories that are not determined by this difference alone. Their values can therefore depend on the target trajectory as well as on the errors. For volatility and direction, this dependence is visible in the forecast increments: Δy^hf=Δyh−Δet,hf,Δyh:=yt+h+1−yt+h. y^f_h= y_h- e^f_t,h, y_h:=y_t+h+1-y_t+h. The target’s own changes can reinforce or offset the changes in the errors, affecting both the magnitude and the sign of the predicted increments. To isolate the disagreement produced by the errors, the volatility, shape, spectral, and timing results below assume a constant target t=c y_t=c 1. Under this assumption, each forecast trajectory is simply its error trajectory reflected and shifted by c. The direction result allows a nonconstant target, provided its increments are small enough for the constructed errors to reverse the predicted directions. Volatility distance compares how much the two forecasts fluctuate, measured by their mean absolute increments. To produce a large disagreement, we pair a flat forecast with one that fluctuates strongly. The following corollary bounds this disagreement and constructs pairs that attain the stated bounds. Corollary D.3 (Volatility Multiplicity Bound Under Rashomon Set Membership). If ℱF is shell-rich, t=c y_t=c 1, and H≥3H≥ 3, then maxf,g∈ℛϵ⁡dvolatility(^tf,^tg)=2u/(H−1),if ℓ is concave on [0,∞),≥2ϱ,if ℓ is convex on [0,∞). _f,g∈R_εd_volatility ( y^f_t, y^g_t ) cases=2u/(H-1),&if $ $ is concave on $[0,∞)$,\\[4.30554pt] ≥ 2 ,&if $ $ is convex on $[0,∞)$. cases For MAE, the maximum is 2TH(ℒ⋆+ϵ)/(H−1)2TH(L +ε)/(H-1). For MSE, it is H⁡(4H−6)H−1T⁡(ℒ⋆+ϵ). H(4H-6)H-1 T(L +ε). Each displayed bound is attained by a pair with identical aggregate loss, one of which forecasts a flat trajectory. Proof. For a constant target, forecast increments equal error increments up to sign. Thus dvolatility(^tf,^tg)=|‖Δtf‖1−‖Δtg‖1|H−1.d_volatility ( y^f_t, y^g_t )= |\| e^f_t\|_1-\| e^g_t\|_1 |H-1. Because both norms are nonnegative, their difference is at most the larger one. We therefore seek the largest possible total variation ‖Δ‖1\| e\|_1 of a single error row. If ℓ is concave, subadditivity and the row budget give ℓ⁡(‖1)≤∑hℓ⁡(|eh|)≤B,‖1≤u. (\| e\|_1)≤ _h (|e_h|)≤ B, \| e\|_1≤ u. Since each error contributes to at most two increments, ‖Δ‖1≤2‖1≤2u.\| e\|_1≤ 2\| e\|_1≤ 2u. This gives the upper bound 2u/(H−1)2u/(H-1). To attain it, take the flat error row tf=ϱ e^f_t= 1 and the concentrated row tg=u(j) e^g_t=u b^(j), with 1<j<H1<j<H. Both use the full budget. The first forecast is flat, whereas the second has two increments of magnitude u, giving distance 2u/(H−1)2u/(H-1). This also proves the MAE formula. For the convex-case lower bound, keep tf=ϱ e^f_t= 1 and take et,hg=(−1)hϱe^g_t,h=(-1)^h . Every increment of the second forecast has magnitude 2ϱ2 , giving distance 2ϱ2 . For the exact MSE maximum, retain the endpoint contributions in the variation bound: ‖Δ‖1≤|e1|+2∑h=2H−1|eh|+|eH|≤4H−6‖2≤(4H−6)B.\| e\|_1≤|e_1|+2 _h=2^H-1|e_h|+|e_H|≤ 4H-6\,\| e\|_2≤ (4H-6)B. The middle step is Cauchy–Schwarz with weights =(1,2,…,2,1) a=(1,2,…,2,1). All inequalities are attained by an alternating row whose magnitudes follow these weights: et,hg=(−1)hahB4H−6.e^g_t,h=(-1)^ha_h B4H-6. Indeed, ‖tg‖22=B\| e^g_t\|_2^2=B, and opposite signs at consecutive entries make every increment equal the sum of its endpoint magnitudes. Pairing this row with tf=ϱ e^f_t= 1 yields dvolatility=(4H−6)BH−1,d_volatility= (4H-6)BH-1, which is the stated MSE maximum. ∎ Direction distance asks whether the two forecasts rise or fall together at each step, regardless of the size of those changes. The following corollary shows that models with identical pointwise absolute errors can predict opposite directions at every step, provided the target’s own changes are small enough for the errors to reverse the predicted directions. Corollary D.4 (Direction Multiplicity Bound Under Rashomon Set Membership). If ℱF is shell-rich, H≥2H≥ 2, and |Δyh|<2ϱfor all h=1,…,H−1,| y_h|<2 all h=1,…,H-1, then maxf,g∈ℛϵ⁡ddirection(^tf,^tg)=1. _f,g∈R_εd_direction ( y^f_t, y^g_t )=1. The condition holds for a constant target whenever ϱ>0 >0. The threshold 2ϱ2 is 2T(ℒ⋆+ϵ)2T(L +ε) for MAE and 2T⁡(ℒ⋆+ϵ)2 T(L +ε) for MSE. The maximum is attained by models with opposite error rows and identical pointwise absolute errors. Proof. Take opposite alternating errors, et,hf=(−1)hϱ=−et,hg.e^f_t,h=(-1)^h =-e^g_t,h. Their forecast increments are Δy^hf=Δyh+2(−1)hϱ,Δy^hg=Δyh−2(−1)hϱ. y^f_h= y_h+2(-1)^h , y^g_h= y_h-2(-1)^h . The error contribution dominates the target increment because |Δyh|<2ϱ| y_h|<2 . The two predicted increments therefore have opposite signs at every step. Their direction distance is 11, the largest possible value. ∎ Shape distance compares the overall patterns of the trajectories after removing their levels and scales. For a constant target, opposite error trajectories produce forecasts with exactly opposite shapes. The following corollary shows that this yields the largest possible shape disagreement while preserving absolute errors at every horizon. Corollary D.5 (Shape Multiplicity Bound Under Rashomon Set Membership). If ℱF is shell-rich, t=c y_t=c 1, H≥2H≥ 2, and B>0B>0, then maxf,g∈ℛϵ⁡dshape(^tf,^tg)=1. _f,g∈R_εd_shape ( y^f_t, y^g_t )=1. Every admissible pair with opposite, nonconstant error rows attains this maximum. Proof. With a constant target, centering a forecast removes the target and leaves the negative centered error. Thus, opposite nonconstant error rows produce opposite nonzero centered forecasts, whose Pearson correlation is −1-1. Such a pair exists: take tf=u(j)=−tg. e^f_t=u b^(j)=- e^g_t. Since B>0B>0, we have u>0u>0, and these rows are nonconstant for H≥2H≥ 2. Both use the full budget, and dshape=1−ρ2=1−(−1)2=1.d_shape= 1-ρ2= 1-(-1)2=1. This attains the largest possible shape distance. ∎ Spectral distance compares how the variation in each trajectory is distributed across frequencies. The following construction assigns different frequencies to the two forecasts and adjusts their amplitudes to give both models the same loss. For a constant target, their spectral disagreement reaches its largest possible value. Corollary D.6 (Spectral Multiplicity Bound Under Rashomon Set Membership). If ℱF is shell-rich, t=c y_t=c 1, H≥4H≥ 4, and B>0B>0, then maxf,g∈ℛϵ⁡dspectral(^tf,^tg)=1. _f,g∈R_εd_spectral ( y^f_t, y^g_t )=1. The maximum is attained by models with identical aggregate loss whose error rows contain different single frequencies. Proof. For two nonconstant trajectories, their standardized magnitude spectra a and b are nonnegative unit vectors. Hence dspectral2=12‖−‖22=1−⟨,⟩≤1,d_spectral^2= 12\| a- b\|_2^2=1- a, b ≤ 1, with equality when the spectra have disjoint supports. The flat-trajectory convention also preserves this upper bound. To attain it, choose two distinct frequencies from 1,…,⌊H/2⌋\1,…, H/2 \, which is possible because H≥4H≥ 4. For each chosen frequency k, define ch(k)=cos⁡(2πkh/H),tfk=ak(k),c^(k)_h= (2π kh/H), e^f_k_t=a_k c^(k), where ak>0a_k>0 is chosen so that ∑h=1Hℓ⁡(akch(k))=B. _h=1^H (a_kc^(k)_h)=B. Such an amplitude exists because this sum is continuous, starts at zero, and grows without bound. Each cosine row has mean zero. Since the target is constant, centering and normalizing its forecast gives −(k)/∥(k)∥2- c^(k)/\| c^(k)\|_2. Its Fourier transform is supported on k,H−k\k,H-k\. Distinct frequencies in the chosen range have disjoint supports, so their magnitude spectra are orthogonal and their spectral distance is 11. ∎ Timing distance measures the shift needed to best align the two forecast trajectories. The following corollary constructs equally accurate forecasts from shifted portions of the same cosine wave. Their best alignment requires the largest admissible lag, showing that equal aggregate loss can coexist with maximal timing disagreement. Corollary D.7 (Timing Multiplicity Bound Under Rashomon Set Membership). If ℱF is shell-rich, t=c y_t=c 1, H≥3H≥ 3, and B>0B>0, then maxf,g∈ℛϵ⁡dtiming(^tf,^tg)=1. _f,g∈R_εd_timing ( y^f_t, y^g_t )=1. The maximum is attained by models with identical aggregate loss whose forecast trajectories are samples of the same cosine wave shifted by H−2H-2 steps. Proof. The timing distance is at most 11 because admissible lags satisfy |τ|≤H−2|τ|≤ H-2. We construct a pair whose best alignment occurs at τ=H−2τ=H-2. Let θ:=π/Hθ:=π/H and define uh=cos⁡(θh),vh=cos⁡(θ⁡(h+H−2))=−cos⁡(θ⁡(h−2)).u_h= (θ h), v_h= (θ(h+H-2) )=- (θ(h-2) ). These rows have the same absolute entries up to permutation: v1=uH−1,v2=uH,vh=−uh−2(h≥3).v_1=u_H-1, v_2=u_H, v_h=-u_h-2 (h≥ 3). Choose a>0a>0 so that the error row −a-a u uses budget B; continuity and unboundedness guarantee its existence. Evenness of ℓ implies that −a-a v has the same loss. Shell-richness therefore provides forecasts ^tf=c+a,^tg=c+a. y^f_t=c 1+a u, y^g_t=c 1+a v. Their lagged correlations equal those of u and v, since Pearson correlation is unchanged by an offset and positive scale. At lag H−2H-2, the overlapping segments are (uH−1,uH)=(v1,v2).(u_H-1,u_H)=(v_1,v_2). They are nonconstant, so their correlation is 11. It remains to exclude a better or equally good alignment at any other lag. The row u is strictly decreasing, whereas v1=v3>v2,v_1=v_3>v_2, and v is strictly increasing from its second entry onward. For 0≤τ≤H−30≤τ≤ H-3, the overlap includes the first three entries of v. Their decrease followed by an increase prevents them from being a positive affine transformation of the strictly decreasing segment of u. Their correlation is therefore strictly less than 11. For τ<0τ<0, the overlap pairs a strictly decreasing segment of u with a strictly increasing segment of v, so its correlation is negative. Thus the unique maximizing lag is τ⋆=H−2τ =H-2, and dtiming=|τ⋆|H−2=1.d_timing= |τ |H-2=1. ∎ Together, Corollaries D.1–D.7 show that models with identical aggregate loss can disagree about forecast magnitude, increments, volatility, direction, shape, frequency content, and timing. We close with three remarks explaining how these results relate to revision and error trajectories. Remark D.8 (Error Multiplicity). Forecast and error trajectories have the same cross-model gaps up to sign: tf−tg=−(^tf−^tg). e^f_t- e^g_t=- ( y^f_t- y^g_t ). Consequently, symmetric distances that depend only on these gaps, including magnitude and increment, give identical multiplicity values for forecasts and errors. Pointwise and horizon-wise multiplicity statistics also coincide. For the other distances, the two trajectories can differ: forecast trajectories describe disagreement about the future, whereas error trajectories describe disagreement in the patterns of the models’ mistakes. Remark D.9 (Error Multiplicity for Each Temporal Aspect). The bounds and constructions above also apply to error trajectories, without requiring a constant target or small target increments. To see why, recall that for a constant target, ^tf=c−tf. y^f_t=c 1- e^f_t. Thus, forecasts are obtained by reflecting and shifting the error trajectories. Applying this transformation to both trajectories leaves every distance considered here unchanged. The constructions therefore already produce the stated disagreement between error trajectories. Under shell-richness, the same error patterns can be realized for any target whenever they lie in the admissible loss shell. Only the restrictions on the target disappear; the assumptions on the loss, model class, horizon length, and budget are retained. Remark D.10 (Revision Multiplicity). The bounds and constructions above also apply to any complete revision trajectory. A revision trajectory collects H forecasts of the same target, issued at successive origins. We can therefore place the constructed errors along its anti-diagonal instead of a row without changing their total loss; shell-richness guarantees models realizing these patterns. Because every entry forecasts the same target, the constant-target assumption holds automatically. The direction condition also holds whenever ϱ>0 >0. All remaining assumptions on the loss, model class, horizon length, and budget remain in force. D.2 The Horizon-Wise Multiplicity Constraint: Exact Characterization and Bounds for Each Temporal Aspect We first restate and prove Theorem 4.2. We then show how its pointwise bounds control magnitude, increment, and volatility distances. In contrast, direction, shape, spectral behavior, and timing disagreement can remain maximal even under arbitrarily small positive tolerances, subject to the stated conditions on the target trajectory. These results are summarized in Table 15 at the start of this appendix. We now ask what controlling disagreement at every horizon guarantees about entire trajectories. Vanishing horizon-wise ambiguity means that ατh(h)(f,g)=0⟺|y^t+hf−y^t+hg|≤τhfor every t∈.α^(h)_ _h(\f,g\)=0 | y^f_t+h- y^g_t+h|≤ _h every t∈T. Thus, τh _h limits the forecast gap at horizon h. Taking τh≡τ _h≡τ recovers the pointwise constraint. The resulting trajectory bounds are summarized in Table 15. Write B:=TH(ℒ⋆+ϵ)B:=TH(L +ε) for the total loss budget. For the attainment results below, assume T≥2T≥ 2, so that errors can also be assigned outside the origin being studied. A small distinction matters when constructing models: shell-richness guarantees realization only when their average losses lie in [ℒ⋆,ℒ⋆+ϵ][L ,L +ε]. A construction below the upper limit may still fall below ℒ⋆L . When two constructed error rows have the same total loss S≤BS≤ B, we can resolve this by assigning an identical additional error to both models at another origin, contributing max⁡THℒ⋆−S,0 \THL -S,0\ to each total loss. Such an error exists by continuity and unboundedness of ℓ . Both models then lie in the admissible loss shell, while their forecast gaps remain unchanged. We first restate and prove the multiplicity bound under horizon-wise control. See 4.2 Proof. The upper bound follows directly from the definition of vanishing horizon-wise ambiguity. To attain all the tolerances at a chosen origin t, take eft,h=−τh/2,egt,h=τh/2.e^f_t,h=- _h/2, e^g_t,h= _h/2. Their forecast gap is exactly τh _h at every horizon, and each row has total loss S=∑h=1Hℓ⁡(τh/2)≤B.S= _h=1^H ( _h/2)≤ B. Adding identical errors at another origin if necessary, as described above, places both matrices in the admissible shell without introducing further disagreement. Shell-richness then provides the required models. ∎ Magnitude, increment, and volatility distances admit upper bounds directly in terms of the tolerances. Magnitude depends on the sizes of the forecast gaps; increment also depends on how those gaps change between successive horizons. Volatility compares the amount of fluctuation in the two forecasts, whose difference is likewise controlled by changes in their gap. Throughout the following results, maxima run over pairs f,g∈ℛϵf,g∈R_ε satisfying |y^t′+hf−y^t′+hg|≤τh| y^f_t +h- y^g_t +h|≤ _h at every origin t′t and horizon h. Corollary D.11 (Magnitude, Increment, and Volatility Multiplicity Bounds Under Horizon-Wise Control). Fix an origin t, let H≥2H≥ 2, and assume τh≥0 _h≥ 0. If ℱF is shell-rich and ∑h=1Hℓ⁡(τh/2)≤B, _h=1^H ( _h/2)≤ B, then max⁡dmagnitude(^tf,^tg) \,d_magnitude ( y^f_t, y^g_t ) =1H∑h=1Hτh2, = 1H _h=1^H _h^2, max⁡dincrement(^tf,^tg) \,d_increment ( y^f_t, y^g_t ) =1H−1∑h=1H−1(τh+τh+1)2. = 1H-1 _h=1^H-1( _h+ _h+1)^2. If additionally t=c y_t=c 1 and ∑hℓ⁡(τh)≤B _h ( _h)≤ B, then max⁡dvolatility(^tf,^tg)=1H−1∑h=1H−1(τh+τh+1). \,d_volatility ( y^f_t, y^g_t )= 1H-1 _h=1^H-1( _h+ _h+1). The magnitude and increment maxima are attained by models with identical pointwise absolute errors. The volatility maximum is attained by models with identical aggregate loss, one of which forecasts a flat trajectory at origin t. Proof. Write =^tf−^tg w= y^f_t- y^g_t for the forecast gap. Since |wh|≤τh|w_h|≤ _h, we have dmagnitude=‖2H≤1H∑hτh2.d_magnitude= \| w\|_2 H≤ 1H _h _h^2. Each change in the gap is bounded by the two adjacent tolerances: |Δwh|≤|wh+1|+|wh|≤τh+τh+1.| w_h|≤|w_h+1|+|w_h|≤ _h+ _h+1. This gives the increment bound, dincrement=‖Δ‖2H−1≤1H−1∑h=1H−1(τh+τh+1)2.d_increment= \| w\|_2 H-1≤ 1H-1 _h=1^H-1( _h+ _h+1)^2. For volatility, the reverse triangle inequality gives dvolatility=|‖Δ^tf‖1−‖Δ^tg‖1|H−1≤‖Δ‖1H−1≤1H−1∑h=1H−1(τh+τh+1).d_volatility= |\| y^f_t\|_1-\| y^g_t\|_1 |H-1≤ \| w\|_1H-1≤ 1H-1 _h=1^H-1( _h+ _h+1). To attain the magnitude bound, use the opposite errors eft,h=−τh/2=−egt,he^f_t,h=- _h/2=-e^g_t,h from the preceding proof. Their gap has magnitude τh _h at every horizon. To attain the increment bound, alternate the signs: et,hf=(−1)hτh/2=−et,hg.e^f_t,h=(-1)^h _h/2=-e^g_t,h. Consecutive gaps now have opposite signs, so |Δwh|=τh+τh+1| w_h|= _h+ _h+1. Both constructions have equal row losses within the budget and can be placed in the admissible shell by adding identical errors at another origin. For volatility, we need one flat forecast and one alternating forecast. Let zh=(−1)hτhz_h=(-1)^h _h, choose another origin s≠ts≠ t, and assign tf e^f_t =a+, =a 1+ z, tg e^g_t =a, =a 1, sf e^f_s =a, =a 1, sg e^g_s =a+, =a 1+ z, with errors zero elsewhere. Swapping the rows at origin s gives the models identical total loss: C⁡(a)=∑h=1Hℓ⁡(a+zh)+Hℓ(a).C(a)= _h=1^H (a+z_h)+H (a). At a=0a=0, this equals ∑hℓ⁡(τh)≤B _h ( _h)≤ B. If it is below THℒ⋆THL , continuity and unboundedness allow us to choose a>0a>0 so that C⁡(a)=THℒ⋆C(a)=THL . Otherwise, take a=0a=0. Both matrices then lie in the admissible shell. The gaps at origins t and s have magnitude τh _h, so all horizon-wise constraints hold. At origin t, the forecast of g is flat, while every increment of the forecast of f has magnitude τh+τh+1 _h+ _h+1. Their volatility distance therefore attains the upper bound. ∎ The preceding bounds shrink as the tolerances decrease. For a constant target, direction, shape, spectral, and timing disagreement can instead remain maximal even when all tolerances are arbitrarily small but positive. The reason is that their constructions can be reduced in amplitude while preserving the signs or normalized patterns that these distances compare. The direction result also allows a nonconstant target, provided its changes are sufficiently small. Corollary D.12 (Direction, Shape, Spectral and Timing Multiplicity Structural Freedom Under Horizon-Wise Control). Fix an origin t, let H≥2H≥ 2, and assume ℱF is shell-rich, B>0B>0, and τh>0 _h>0 for every horizon. Define ϱ:=ℓ−1(B/H),κh:=min⁡τh,2ϱ,κ:=minh⁡κh. := ^-1(B/H), _h:= \ _h,2 \, κ:= _h _h. Here, κh _h limits the constructed forecast gap using both the horizon-wise tolerance and the available loss budget. Then max⁡ddirection(^tf,^tg) \,d_direction ( y^f_t, y^g_t ) =1 =1 if |Δyh|<κh+κh+12 for every h, | y_h|< _h+ _h+12 for every h, max⁡dshape(^tf,^tg) \,d_shape ( y^f_t, y^g_t ) =1 =1 if t=c, y_t=c 1, max⁡dspectral(^tf,^tg) \,d_spectral ( y^f_t, y^g_t ) =1 =1 if t=c and H≥4, y_t=c 1 and H≥ 4, max⁡dtiming(^tf,^tg) \,d_timing ( y^f_t, y^g_t ) =1 =1 if t=c and H≥3. y_t=c 1 and H≥ 3. All maxima are attained by models with identical aggregate loss. The direction and shape constructions also preserve pointwise absolute errors. Proof. All four distances are at most 11, so it suffices to construct admissible pairs attaining this value. In each construction, the error magnitude at horizon h is at most κh/2 _h/2 for both models. Their gap is therefore at most κh≤τh _h≤ _h, and each row satisfies the loss budget: ∑h=1Hℓ⁡(et,h)≤Hℓ(ϱ)=B. _h=1^H (e_t,h)≤ H ( )=B. We choose the two rows to have equal loss, allowing identical errors at another origin to place both models in the admissible shell if necessary. Direction. Take opposite alternating errors, et,hf=(−1)hκh/2=−et,hg.e^f_t,h=(-1)^h _h/2=-e^g_t,h. Their forecast increments are Δy^hf=Δyh+(−1)hκh+κh+12,Δy^hg=Δyh−(−1)hκh+κh+12. y^f_h= y_h+(-1)^h _h+ _h+12, y^g_h= y_h-(-1)^h _h+ _h+12. Under the stated condition, the error contribution exceeds the magnitude of the target increment. The forecasts therefore move in opposite directions at every step, giving distance 11. Shape. For a constant target, the same construction gives opposite, nonconstant centered forecast trajectories. Their correlation is −1-1, so their shape distance is 11. Spectral. Choose two distinct frequencies j,k∈1,…,⌊H/2⌋j,k∈\1,…, H/2 \ and let ch(r)=cos⁡(2πrh/H),tf=aj(j),tg=ak(k).c^(r)_h= (2π rh/H), e^f_t=a_j c^(j), e^g_t=a_k c^(k). Choose positive amplitudes aj,ak≤κ/2a_j,a_k≤κ/2 giving both rows total loss ℓ⁡(κ/2) (κ/2). This is possible by continuity because each cosine row contains an entry equal to 11. If ℓ⁡(κ/2)=0 (κ/2)=0, simply take both amplitudes equal to κ/2κ/2. As in Corollary D.6, centering removes the constant target and normalization removes the amplitudes. The resulting magnitude spectra have disjoint supports, giving spectral distance 11. Timing. Use the shifted cosine patterns from Corollary D.7: uh=cos⁡(πh/H),vh=cos⁡(π⁡(h+H−2)/H),u_h= (π h/H), v_h= (π(h+H-2)/H ), and take tf=−(κ/2),tg=−(κ/2). e^f_t=-(κ/2) u, e^g_t=-(κ/2) v. Their entries have magnitude at most κ/2κ/2, and their absolute entries are permutations of one another, so the rows have equal loss. For a constant target, the resulting forecasts have the same lagged correlations as u and v. Their best alignment occurs at lag H−2H-2, as established in Corollary D.7, giving timing distance 11. ∎ Thus, horizon-wise agreement controls the sizes of forecast gaps and the resulting magnitude, increment, and volatility disagreement. Under the stated target conditions, it can nevertheless leave direction, shape, spectral, and timing disagreement maximal. Remark D.13 (Revision and Error Trajectories). The results also apply to complete revision trajectories. As forecasts of a fixed target are updated, their horizons decrease from H to 11, so the corresponding tolerances are (τH,…,τ1)( _H,…, _1). Reversing this order leaves the bounds unchanged, and the constructions transfer to the relevant anti-diagonal without changing their loss. Since every entry forecasts the same target, the constant-target condition holds automatically. For error trajectories, the horizon-wise constraints are unchanged because |et,hf−et,hg|=|y^t+hf−y^t+hg|.|e^f_t,h-e^g_t,h|=| y^f_t+h- y^g_t+h|. The same error patterns therefore attain the stated bounds directly, without assumptions on the target trajectory. The constant-target and small-target-increment conditions are needed only to obtain the corresponding disagreement between forecasts. In both cases, the remaining assumptions on shell-richness, horizon length, tolerances, and the loss budget still apply. D.3 How Close Does the Empirical Rashomon Set Come to the Theoretical Bounds? We test whether trained models in the empirical Rashomon set ℛ^ϵ R_ε attain the theoretical temporal disagreement allowed by Rashomon membership alone and by vanishing pointwise or horizon-wise ambiguity (Table 15). Since these bounds concern individual origins, we compute the largest disagreement at any origin, rather than the origin-averaged multiplicity statistics we measured in Section 6. The question is whether the extreme disagreement permitted by the theory occurs among near-optimal trained models, and which distances remain far from their theoretical maxima. Rashomon Membership Alone. For MASE, the loss budget is TH(ℒ⋆+ϵ)TH(L +ε), where T is the number of validation origins, H the horizon, ℒ⋆=ℒ⁡(f⋆)L =L(f ) is the best validation MASE, and ϵ=5%⋅ℒ⋆ε=5\%·L (Section A.3). For each trajectory distance dd, we define the worst-origin range and its utilization of the membership bound UdU_d as: ν¯d=maxt∈⁡maxf,g∈ℛ^ϵ⁡d⁡(tf,tg),ν¯dUd. ν_d= _t∈T\, _f,g∈ R_εd ( v^f_t, v^g_t ), ν_dU_d. (6) The corresponding bounds (Theorems 4.1 and D.1) are: Ud=2TH(ℒ⋆+ϵ),d=dmagnitude,22TH−1(ℒ⋆+ϵ),d=dincrement,2TH−1(ℒ⋆+ϵ),d=dvolatility,1,d∈ddirection,dshape,dspectral,dtiming.U_d= cases2T H\,(L +ε),&d=d_magnitude,\\[6.0pt] 2 2\,TH H-1\,(L +ε),&d=d_increment,\\[6.0pt] 2THH-1\,(L +ε),&d=d_volatility,\\[6.0pt] 1,&d∈\d_direction,d_shape,d_spectral,d_timing\. cases For the four scale-free distances, the bound 11 is reported as 100%100\%. The worst-origin range selects the most separated pair at the most adverse origin and is at least the origin-averaged trajectory range νd _d of Table 1. Utilization expresses this observed distance as a fraction of the theoretical bound. These maxima measure whether large disagreement occurs, rather than how typical it is across pairs or origins. Table 16 shows that the scale-free bounds are readily attained: direction, shape, and timing reach 100%100\% on every dataset, while spectral distance reaches 99.7%99.7\%. Scale-free distances compare signs or normalized patterns, so attaining their range does not require large errors: reducing the amplitude of a disagreement need not reduce its distance. Indeed, this confirms empirically what the theoretical findings of Corollaries D.4, D.5, D.7 and D.6 predicted. Magnitude, increment, and volatility use at most 6.8%6.8\% of their bounds. This is expected: the bounds for these distance functions grow with T; however, the attaining constructions would concentrate an entire model’s error budget in a single forecast cell, an extreme pattern unlikely to result from training using gradient descent. As such, a finite set of trained models need not contain such constructions, even when the model class can represent them. Although the observed distances reach only a small fraction of their theoretical bounds, they remain large in absolute terms: between 16.916.9 and 21.421.4 multiples of the naive error scale on average, with most datasets attaining values well above 33 multiples of the naive error scale. Rashomon membership alone therefore leaves extremely large temporal multiplicity at some individual forecast origins, even without the extreme error patterns needed to attain the bounds. Table 16: Worst-origin range and utilization of the Rashomon-membership bound (Equation 6). Magnitude, increment, and volatility are in naive units, with utilization in parentheses; scale-free distances are percentages of their bound 11. Values are maxima over trajectories, averaged over targets where applicable; the last row averages over datasets. Dataset ℛϵR_ε only dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ATM 1D 6.54 (0.0%) 6.20 (0.0%) 3.52 (0.0%) 100% 100% 97.6% 100% ATM 1W 6.66 (0.1%) 5.47 (0.1%) 3.62 (0.1%) 100% 100% 99.4% 100% Bitcoin 19.7 (0.3%) 11.1 (0.1%) 7.39 (0.3%) 100% 100% 95.4% 100% Electricity D. A 7.86 (0.0%) 1.66 (0.0%) 0.75 (0.0%) 90.2% 98.8% 95.0% 100% Electricity D. Z 3.73 (0.0%) 0.72 (0.0%) 0.31 (0.0%) 83.2% 98.4% 93.2% 100% Electricity Price 2.74 (0.0%) 0.85 (0.0%) 0.50 (0.0%) 95.7% 96.5% 92.4% 100% ETT 1H 3.29 (0.0%) 1.21 (0.0%) 0.79 (0.0%) 100% 99.1% 96.5% 100% Exchange Rate 162 (1.5%) 236 (1.3%) 163 (2.5%) 100% 100% 99.5% 100% FRED-MD 0.36 (0.8%) 0.22 (0.3%) 0.17 (0.4%) 100% 100% 0.0% 100% Gasoline 12.2 (1.7%) 7.88 (0.7%) 4.90 (1.0%) 100% 100% 99.7% 100% Weather 10.6 (0.0%) 2.26 (0.0%) 1.31 (0.0%) 96.5% 99.8% 94.4% 100% Mean 21.4 (0.4%) 24.8 (0.2%) 16.9 (0.4%) 96.9% 99.3% 87.6% 100% Vanishing Pointwise and Horizon-Wise Ambiguity. We evaluate each pair f,g∈ℛ^ϵf,g∈ R_ε at the smallest tolerances for which its ambiguity vanishes: the largest absolute forecast gap over all test cells (pointwise), or separately at each horizon (horizon-wise): τfg=maxt∈⁡max1≤h≤H|y^t+hf♮−y^t+hg♮|,τhfg=maxt∈⁡|y^t+hf♮−y^t+hg♮|.τ^fg= _t∈T _1≤ h≤ H | -0.5mu y^f_t+h- -0.5mu y^g_t+h |, τ^fg_h= _t∈T | -0.5mu y^f_t+h- -0.5mu y^g_t+h |. (7) By construction, ατfg(f,g)=0 _τ^fg(\f,g\)=0 and ατhfg(h)(f,g)=0α^(h)_τ^fg_h(\f,g\)=0 for every h. Writing fg=(τ1fg,…,τHfg) τ^fg=(τ^fg_1,…,τ^fg_H), let Ud(τ)U_d(τ) and Ud()U_d( τ) denote the pointwise and horizon-wise bounds in Table 15 (Theorems 4.2, D.11 and D.2), namely: Ud(τ)=τ,d=dmagnitude,2τ,d=dincrement,2τ,d=dvolatility,1,d∈ddirection,dshape,dspectral,dtiming,U_d(τ)= casesτ,&d=d_magnitude,\\[6.0pt] 2τ,&d=d_increment,\\[6.0pt] 2τ,&d=d_volatility,\\[6.0pt] 1,&d∈\d_direction,d_shape,d_spectral,d_timing\, cases Ud()=1H∑h=1Hτh2,d=dmagnitude,1H−1∑h=1H−1(τh+τh+1)2,d=dincrement,1H−1∑h=1H−1(τh+τh+1),d=dvolatility,1,d∈ddirection,dshape,dspectral,dtiming.U_d( τ)= cases 1H _h=1^H _h^2,&d=d_magnitude,\\[6.0pt] 1H-1 _h=1^H-1( _h+ _h+1)^2,&d=d_increment,\\[6.0pt] 1H-1 _h=1^H-1( _h+ _h+1),&d=d_volatility,\\[6.0pt] 1,&d∈\d_direction,d_shape,d_spectral,d_timing\. cases This evaluates the tightness of the bounds at the tolerances each pair actually satisfies; it does not establish attainment at a single, prespecified tolerance shared by all pairs. The pair’s utilization is its largest distance at any origin divided by the corresponding bound: udfg=maxt∈⁡d⁡(tf,tg)Ud(τfg),ud,fg=maxt∈⁡d⁡(tf,tg)Ud(fg).u^fg_d= _t∈Td ( v^f_t, v^g_t )U_d(τ^fg), u^fg_d, τ= _t∈Td ( v^f_t, v^g_t )U_d( τ^fg). (8) For the scale-free distances, whose bound is 11, we restrict the maximum to origins with |Δyh♮|<(τhfg+τh+1fg)/2| -0.5muy_h|<(τ^fg_h+τ^fg_h+1)/2 at every step: the target-flatness criterion for direction in Corollary D.12. For shape, spectral, and timing distances, the corollary proves attainment for constant targets; the filter alone does not guarantee attainment. For revision trajectories, the tolerance vector follows the reversed horizon order. Table 17 reports the maximum utilization over pairs. Figure 4 plots all pairs and each dataset’s running maximum against normalized tolerance. Three observations are worth noting. Most Distances Attain or Approach Their Bounds. Magnitude reaches 99.8%99.8\% under pointwise control and 100%100\% under horizon-wise control; direction, shape, and timing reach 100%100\%, and spectral distance reaches 99.7%99.7\%. Increment and volatility remain further below: their dataset-averaged maxima are 65.4%65.4\% and 53.5%53.5\% pointwise, rising to 82.5%82.5\% and 67.9%67.9\% horizon-wise. The gap is smallest for Exchange Rate, where increment and volatility reach 99.8%99.8\% and 95.5%95.5\% of their horizon-wise bounds, showing that trained models can approach the theoretical maxima. A possible explanation is the limited predictability of exchange rates: if their dynamics are approximately those of a random walk, the loss may provide less guidance toward a common forecast structure, leaving more varied models with similar performance. More generally, as with the Rashomon-membership bounds, the remaining gaps may reflect the finite collection of models explored and the implicit restrictions imposed by training, rather than limitations on the disagreement that the full model class can express. Substantial Gaps Remain for Increment and Volatility on Most Datasets. The difference follows from the configurations needed for equality. Magnitude only requires the absolute gap to reach its tolerance at every horizon of one origin; a persistent offset can therefore approach the bound. Increment requires the gap to alternate between +τh+ _h and −τh+1- _h+1 at every consecutive horizon: the forecasts must cross at every step with maximal amplitude. Volatility additionally requires these changes to produce the largest possible difference in total variation; one attaining construction pairs a flat forecast with a full-amplitude zigzag. These extreme configurations are unlikely for trained forecasters. Smooth, persistent gaps can be large in magnitude while changing little across horizons, leaving increment and volatility well below their bounds. Horizon-Wise Control Gives Higher Utilization. Horizon-wise control gives higher utilization because each horizon receives its own tolerance, avoiding the slack introduced by applying the largest gap anywhere to every horizon. Table 17: Maximum pointwise / horizon-wise utilization over pairs (Equation 8), each evaluated at its smallest admissible tolerances (Equation 7). Scale-free distances use only origins satisfying the target-flatness filter. Aggregation over trajectories, targets, and datasets follows Table 16. Dataset dmagnituded_magnitude dincrementd_increment dvolatilityd_volatility ddirectiond_direction dshaped_shape dspectrald_spectral dtimingd_timing ATM 1D 90.4% / 100% 49.8% / 69.2% 41.5% / 47.7% 100% / 100% 100% / 100% 97.6% / 97.6% 100% / 100% ATM 1W 99.6% / 100% 62.7% / 74.8% 54.4% / 65.1% 100% / 100% 100% / 100% 99.4% / 99.4% 100% / 100% Bitcoin 99.3% / 100% 53.4% / 71.0% 32.8% / 44.8% 100% / 100% 100% / 100% 95.4% / 95.4% 100% / 100% Electricity D. A 85.5% / 99.1% 18.1% / 23.5% 7.7% / 9.9% 90.2% / 90.2% 98.8% / 98.8% 95.0% / 95.0% 100% / 100% Electricity D. Z 81.5% / 95.4% 11.5% / 14.0% 4.8% / 6.5% 83.2% / 83.2% 98.4% / 98.4% 93.2% / 93.2% 100% / 100% Electricity Price 84.7% / 99.1% 26.3% / 31.9% 11.8% / 14.5% 95.7% / 95.7% 96.5% / 96.5% 92.4% / 92.4% 100% / 100% ETT 1H 94.1% / 100% 33.7% / 53.6% 23.1% / 31.7% 100% / 100% 99.1% / 99.1% 96.5% / 96.5% 100% / 100% Exchange Rate 99.7% / 100% 76.7% / 97.7% 67.7% / 92.4% 100% / 100% 100% / 100% 99.3% / 99.3% 100% / 100% FRED-MD 95.0% / 100% 67.0% / 80.5% 49.6% / 62.4% 100% / 100% 100% / 100% 0.0% / 0.0% 100% / 100% Gasoline 99.8% / 100% 83.3% / 97.7% 69.3% / 86.3% 100% / 100% 100% / 100% 99.7% / 99.7% 100% / 100% Weather 95.4% / 99.4% 23.4% / 32.4% 9.4% / 14.5% 96.5% / 96.5% 99.8% / 99.8% 94.4% / 94.4% 100% / 100% Mean 93.2% / 99.4% 46.0% / 58.8% 33.8% / 43.3% 96.9% / 96.9% 99.3% / 99.3% 87.5% / 87.5% 100% / 100% Figure 4: Pairwise bound utilization against normalized tolerance: τfg/ℒ⋆τ^fg/L for pointwise control (top), and maxh⁡(τhfg/ℒh⋆) _h(τ^fg_h/L _h) for horizon-wise control (bottom). Each marker represents one pair; lines show each dataset’s running maximum.

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Temporal Predictive Multiplicity: Equally Accurate Time Series Models Yield Different Forecast Trajectories

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