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Timer-M1: A Multivariate Time Series Foundation Model via Learning Primitives
Haoran Zhang, Haixuan Liu, Xingjian Su, Yong Liu, Zhi Chen, Yuxuan Wang, Jianmin Wang, Mingsheng Long
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
The paper introduces Timer-M1, a multivariate time series foundation model designed for zero-shot forecasting. It utilizes a primitive-based data synthesis pipeline that combines real-world data with synthetic series containing temporal (trend, periodicity) and relational (cross-variate correlation, delayed responses) primitives. The model employs gated two-dimensional Transformer blocks to dynamically allocate cross-variate attention. Timer-M1 achieves state-of-the-art results, ranking first on FEV and TIME benchmarks and second on GIFT-Eval.
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Timer-M1 → developedby → ByteDance
confidence 95% · 2 ByteDance, Beijing, China
Timer-M1 → developedby → Tsinghua University
confidence 95% · 1 School of Software, BNRist, Tsinghua University
Timer-M1 → ranksfirston → TIME
confidence 95% · Timer-M1 ranks first on both FEV and TIME
Timer-M1 → ranksfirston → FEV
confidence 95% · Timer-M1 ranks first on both FEV and TIME
Timer-M1 → rankssecondon → Gift-Eval
confidence 95% · second on GIFT-Eval among most recent time series foundation models
Timer-M1 → usesarchitecture → Gated Two-Dimensional Transformer
confidence 95% · Technically, Timer-M1 further adapts gated two-dimensional Transformer blocks
Relational Primitives → includes → Cross-variate correlation
confidence 90% · relational primitives describe cross-variate correlation, delayed responses, cointegration, and event effects
Temporal Primitives → includes → Trend
confidence 90% · temporal primitives describe trends, periodicity, stochastic dependence, and regime changes
Timer-M1 → →
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Abstract
Abstract:We introduce Timer-M1, a pretrained multivariate time series foundation model that learns with primitives for zero-shot forecasting. Across domains, time series share elementary temporal and relational patterns, termed primitives, yet differ in how these primitives manifest and evolve across different contexts. Despite progress in zero-shot and task-general forecasting, existing foundation models may still struggle to generalize to complex real-world scenarios. To this end, we develop a primitive-based data synthesis and pretraining pipeline. The synthesis pipeline generates series with temporal primitives shared across domains and then assembles real and generated series into multivariate samples using relational primitives. Afterwards, samples are organized into episodes by assigning distinct channel roles as target variates, past-only covariates, and known-future covariates, ensuring that the model is optimized on predictable variates using available exogenous information. Technically, Timer-M1 further adapts gated two-dimensional Transformer blocks that dynamically allocate cross-variate attention across layers. Across three large-scale forecasting benchmarks, Timer-M1 ranks first on both FEV and TIME and second on GIFT-Eval among most recent time series foundation models. These results support effective primitive-based pretraining as a route to robust general forecasting technique across domains and task settings.
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- Source: https://arxiv.org/abs/2610.11734v1
- Canonical: https://arxiv.org/abs/2610.11734v1
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TIMER-M1: A MULTIVARIATETIMESERIES FOUNDATIONMODEL VIALEARNINGPRIMITIVES Haoran Zhang *,1 , Haixuan Liu *,1 , Xingjian Su *,1 , Yong Liu *,2 , Zhi Chen 1 , Yuxuan Wang 1 , Jianmin Wang 1 , Mingsheng Long ⊠,1 1 School of Software, BNRist, Tsinghua University 2 ByteDance, Beijing, China zhang-hr24,liuhaixu21,suxj22@mails.tsinghua.edu.cn ⊠ mingsheng@tsinghua.edu.cn ABSTRACT We introduce Timer-M1, a pretrained multivariate time series foundation model that learns with primitives for zero-shot forecasting. Across domains, time series share elementary temporal and relational patterns, termedprimitives, yet differ in how these primitives manifest and evolve across different contexts. Despite progress in zero-shot and task-general forecasting, existing foundation models may still struggle to generalize to complex real-world scenarios. To this end, we develop a primitive-based data synthesis and pretraining pipeline. The synthe- sis pipeline generates series with temporal primitives shared across domains and then assembles real and generated series into multivariate samples using relational primitives. Afterwards, samples are organized intoepisodesby assigning distinct channel roles as target variates, past-only covariates, and known-future covariates, ensuring that the model is optimized on predictable variates using available exoge- nous information. Technically, Timer-M1 further adapts gated two-dimensional Transformer blocks that dynamically allocate cross-variate attention across lay- ers. Across three large-scale forecasting benchmarks, Timer-M1 ranks first on both FEV and TIME and second on GIFT-Eval among most recent time series foundation models. These results support effective primitive-based pretraining as a route to robust general forecasting technique across domains and task settings. 1 INTRODUCTION Time series forecasting spans domains and task configurations in real-world applications. Time series foundation models have advanced a shift from task-specific training toward cross-domain pretraining and zero-shot forecasting (Das et al.,2024;Ansari et al.,2024;Woo et al.,2024;Khwaja et al.,2026;Liu et al.,2024b), demonstrating competitive accuracy against statistical and supervised baselines on multi-domain and multi-task benchmarks (Aksu et al.,2024;Shchur et al.,2025;Qiao et al.,2026). Meanwhile, multivariate foundation models (Ansari et al.,2025;Khwaja et al.,2026; Liu et al.,2026a) have empowered native capability to jointly predict multiple targets with historical or known-future covariates under observation settings (Shchur et al.,2025;Cohen et al.,2026). Time series foundation models are targeted to generalize on cross-domain data and deal with het- erogeneous input context. Previous foundation models have continuously improved toward this goal by task-agnostic training on real-world and synthetic datasets ( Das et al.,2023;Dooley et al.,2023). However, real-world time series record complex and noisy dynamics but offer limited control over which patterns are observed; synthetic datasets provide such control but may omit the irregularities of real systems. This complementarity motivates a data construction process that combines observed dynamics with controllable relations, rather than relying on either source alone. Recurring temporal and relational patterns provide a common basis for learning across diverse do- mains and forecasting tasks. Classical time series analysis separates trend, seasonal, and remainder components to understand and forecast a series ( Hyndman & Athanasopoulos,2018). We refer to such elementary patterns shared across series and domains asprimitives: temporal primitives de- * Equal Contribution 1 arXiv:2610.11734v1 [cs.LG] 8 Oct 2026 (c)Episode Sampling ASample Target1 Target2 Noise Event Time Periodicity Tr e n d Bool A C Pre-training Timer-M1 New Forecasting Task Time HistoryForecast Ta r g e t Bool Period Median Interval B V T1 V T2 V N V E BSample CSample Ta r g e t Union Ta r g e t State Time 2 1 0 Time V T3 V U V T4 V S SelectedSample UnrelatedSample V T1 V N V T3 V U ... ... V T1 V N V E V E V T4 V T3 Sampling Actual Relation Masked PrimitiveDecomposition Tr e n d Periodicity Ta r g e t Bool ×+ Time Time ObservedTrend (a)RealWo r l dObservations(b)Primitive Compositions Schedules Figure 1:Learning with primitives. Temporal and relational patterns observed in real series are composed into diverse samples; episodes vary what is observed and predicted while introducing noise and interference to improve robustness, linking to generalization to new forecasting scenarios. scribe trends, periodicity, stochastic dependence, and regime changes, while relational primitives de- scribe cross-variate correlation, delayed responses, cointegration, and event effects. These primitives characterize individual regularities rather than specific forecasting scenarios. Just as learned con- cepts and motifs can be generalized in new combinations in language and vision ( Zhao et al.,2024; Liu et al.,2022), recurring primitives motivate learning transferable forecasting capabilities that remain useful when their surrounding signals and available observations change across contexts. Learning with primitives offers a systematic route for training generalizable time series foundation models. As shown in Figure1, we develop a compact set of operators to describe temporal patterns and cross-variate relations, and to combine synthetic and real-world time series. In addition to con- structing a bottom-up data pipeline, we extend the pretraining process of foundation models into a forecasting task pipeline. We build forecasting episodes that complement metadata of real-world tasks by specifying which variates are targets and which observations are available, coupling rela- tional structure with the information needed for forecasting. Repeated exposure under different roles and masks encourages general forecasting capabilities rather than reliance on a task configuration. Technically, we build Timer-M1 with native zero-shot and multivariate capabilities via a primitive- based data pipeline and episode-based pretraining. Synthesis operators provide temporal and rela- tional primitives, compose real and generated series into multivariate samples, and apply data aug- mentations. Episodes assign target, past-only, and known-future roles, via task-dependent supervi- sion masks to guarantee predictability. In terms of the architecture, we adapt gated two-dimensional Transformer blocks to learning from episodes along temporal and variate dimensions, using adap- tive gates learned to allocate cross-variate attention across layers. Timer-M1 ranks first on FEV and TIME and second on GIFT-Eval among the state-of-the-art foundation models; controlled primitive analyses further examine the forecasting behaviors underlying these results. Our contributions are: •We develop a unified data synthesis pipeline that generates elementary temporal and rela- tional primitives to enrich pretraining data and improve cross-domain generalization. •We present Timer-M1, a time series foundation model pretrained on carefully constructed forecasting episodes that unify univariate, covariate-aware, and multivariate forecasting. •Timer-M1 achieves strong zero-shot results across tasks, ranking first on FEV and TIME and second on GIFT-Eval, highlighting its potential for broader real-world applications. 2 RELATEDWORK 2.1 TIMESERIESPRETRAININGDATA Pretraining data shape the temporal patterns and cross-variate relations a model encounters. Fore- castPFN (Dooley et al.,2023) learns from synthetic tasks, Chronos (Ansari et al.,2024) combines real series with Gaussian-process compositions, and Chronos-2 ( Ansari et al.,2025) constructs multivariate dependencies synthetically. Real-data curation remains complementary: TimeBench ( Liu et al.,2025) supplies heterogeneous dynamics, while Timer-S1 (Liu et al.,2026b) combines corpus scaling with transformations to reduce predictive bias. Controlled studies further highlight 2 the importance of data composition and training protocol (Wen et al.,2026), motivating coordinated data construction and task design for generalizable forecasting. Learning from generated tasks offers a complementary approach to pretraining data construction. In the tabular domain, TabPFN (Hollmann et al.,2023;2025) uses structural-causal-model priors, TabICL (Qu et al.,2025) scales such pretraining to larger contexts, and TabFM (Kong & Das,2026) combines hybrid attention with causal-model generation. TabDPT (Ma et al.,2025) instead exploits signals in real tables that synthetic priors may miss. These approaches motivate combining con- trolled generation with empirical diversity, which we pursue by applying shared relation operators to both real and generated series, composing primitives while retaining observed dynamics. 2.2 TIMESERIESFOUNDATIONMODELS Time series foundation models explore different pretraining objectives and representations for forecasting across domains. Chronos (Ansari et al.,2024) models discretized values, whereas Sun- dial (Liu et al.,2025) develops continuous probabilistic generation. Autoregressive approaches learn transferable temporal patterns through next-token prediction (Liu et al.,2024b;2026b), while TiRex (Auer et al.,2025) adopts contiguous patch masking. Beyond individual series, Moirai (Woo et al., 2024) supports variable-dimensional inputs through any-variate attention, and Timer-XL (Liu et al., 2024a) extends attention to autoregressive forecasting under multivariate and covariate settings. Recent advances strengthen cross-variate modeling in time series foundation models. Chronos- 2 (Ansari et al.,2025) combines group attention with synthetic multivariate tasks, while TiRex-2 (Podest et al.,2026) extends masked pretraining to multivariate forecasting. Toto 2.0 (Khwaja et al., 2026) combines temporal–variate attention with large-scale pretraining; TimesFM-3 (Google Re- search,2026) adopts alternating attention and parallel forecasting, and Falcon-X (Liu et al.,2026a) mediates variate interaction through latent prototypes. As these architectures accommodate diverse inputs, pretraining must expose the dependencies needed to use them. This motivates us to organize primitive compositions into forecasting episodes with different variate roles and observation masks, aligning constructed relations with observation availability across task settings. 3 APPROACH Timer-M1 aligns primitive-based data construction, model architecture, and pretraining tasks around learning temporal and relational primitives. Specifically, it composes primitives into multivariate data, organizes forecasting episodes, and learns from their permitted observations with gated two- dimensional Transformer blocks. We denote history length byL, forecast horizon byH, and the total number of target and covariate variates byC. The model representation width isD. 3.1 DATA Our primitive-based data pipeline brings real and synthetic series into a shared data construction process. Synthesis operators provide controlled periodic, trend, autoregressive, and other stochastic patterns. Real sources, including TimeBench ( Liu et al.,2025;2026b), retain irregular dynamics, noise, and incomplete observations that are difficult to specify exhaustively. Relation operators accept either source origin or their combination, constructing signed responses, delayed effects, shared drivers, and other cross-variate dependencies. Thus, real observations serve not only as standalone training examples but also as inputs from which additional cross-variate re- lations can be generated. Transform operators derive varied responses from source series through sign inversion, resampling, clipping, local perturbations, and discrete-valued mappings. Applied during multivariate construction and subsequent augmentation, they diversify both cross-variate de- pendencies and the observed forms of primitives, with temporal transformations coordinated wher- ever alignment is required. Together, synthesis, relations, and transforms turn jointly sampled real and synthetic sources into diverse multivariate samples within a unified data construction pipeline. Original multivariate observations can also bypass synthesis to retain their existing dependencies. Formally, letξ∈R κ×τ containκsource series aligned overτconsecutive time steps. Each rowξ i is obtained from a real-data window or a synthesis operatorS γ i (ε i ), whereγ i specifies the generator parameters andε i its random inputs. A relation operatorR φ producesνdependent response series 3 (a)Data Primitives Relational Primitives f(t) t 1 (f(t)) RealDataPool Te m p o r a lPrimitives Sine Noise Tr e n d ...... t 2 (f(t)) t 3 (f(t)) f(t)g(t) r(f(t),g(t)) Relations Tr a n s f o r m s Synthetic DataPool Unified Sample ...... MultivariateSynthesis DataAugmentation negative clamp upsample Pre-LN TimeAttention Pre-LN VariateAttention Pre-LN FeedForward + + × + RoPE Masked Padding Known ⋯ HistoryFuture Episode 1 Episode 2 ...... Batch Adaptive Gate PredMask PaddingMask Patches Episode 1 ... Episode 2 PackedEpisodes Variates Future History TimeAttn.Map VariateAttn.Map Observed Episode 1 Episode 2 V T1 V N V E V T3 V T1 V N V E V T3 ⋯ (b)Pre-Tr a i n i n g (c)Architecture ×N Source1Source2 Source MultivariateSeriesSample Figure 2:Timer-M1 overview. (a) Temporal and relational primitives combine real and synthetic sources into multivariate samples through synthesis and augmentation. (b) Forecasting episodes specify variate roles and observation availability, with masks controlling forecasting and within- episode attention. (c) Gated two-dimensional Transformer blocks integrate temporal and cross- variate information, with learned layer-wise gates regulating cross-variate updates. η∈R ν×τ . For branches that retain the original source channels, the resulting construction is η=R φ (ξ), Ξ=A ψ ([ξ;η]). (1) Here,φspecifies the relation structure and its parameters, including mixing coefficients or temporal lags;A ψ denotes the selected transforms with parametersψ; and the semicolon denotes channel- wise concatenation. The outputΞis an aligned multivariate record from which forecasting episodes are subsequently sampled. Response-only branches omit the original source channels from the con- catenation. Generation-level dimensionsκ,ν,τare distinct from the episode dimensionsC,L,H, which are determined after task-specific channel selection and temporal sampling. 214.3M Records 45.1% 13.9% 41.1% Persisted Rows 3.02T Positions 32.8% 28.9% 38.3% Persisted Points Real-World CorpusSynthetic-Only Real-Synthetic Joint Generation Figure 3:Persisted records and scalar po- sitions by generation route, before sampling. Real–synthetic joint generation includes corpus- derived and mixed-source generation routes. For example, linear relations mix source chan- nels with signed coefficients, whereas nonlinear relations apply a sampled mapping to the source vector. Reusing a source across multiple re- sponses creates shared drivers, and lagged rela- tions introduce delayed dependencies. The same construction applies whether the source dynam- ics are observed or synthesized. Representative constructions and the generation procedure are described in Appendix A.2. The stable pretraining configuration references 227 dataset entries containing 214.3 million per- sisted records and 3.02 trillion scalar positions (Figure 3). These unweighted storage counts precede holdouts and sampling; they are not unique observations or training exposure. 3.2 ARCHITECTURE Timer-M1 uses gated two-dimensional Transformer blocks to model temporal and relational primitives. Following normalization with valid-history statistics, each variate is divided into length- Ppatches. A residual patch embedder maps the concatenated normalized values e x c,n ∈R P and observation maskm c,n ∈0,1 P to aD-dimensional token: h 0 c,n = PatchEmbed([ e x c,n ;m c,n ]),H 0 ∈R C×N×D ,(2) 4 wherecindexes variates,nindexes patches, andNcounts padded history and forecast patches; hidden target values and their observation masks are zeroed before embedding. Forecast tokens remain active despite their hidden values, while attention prevents interactions across episodes. Each layer applies temporal attention, a feed-forward network, and gated variate attention in that order. With separate pre-layer normalizationsLN, the residual updates are U ℓ =H ℓ + TimeAttention ℓ LN t,ℓ (H ℓ ) , Z ℓ =U ℓ + FFN ℓ LN f,ℓ (U ℓ ) , H ℓ+1 =Z ℓ +α ℓ VariateAttention ℓ LN v,ℓ (Z ℓ ) , α ℓ =σ(a ℓ ). (3) Here,α ℓ =σ(a ℓ )applies sigmoid to a learned layer-wise scalar. TimeAttention attends across patches within variates; VariateAttention attends across variates per patch. Both use multi-head scaled dot-product attention with RMS-normalized queries and keys; temporal attention uses rotary positional embeddings. The FFN comprises two linear projections with GELU. A shared quantile head maps final tokens to patch-level forecasts, restored to the original scale. Gates adapt during training but remain fixed at inference, whereas attention weights depend on the input. Thus, the gate regulates each layer’s cross-variate interaction strength rather than selecting in- put features. This retains standard temporal modeling while controlling the contribution of relational primitives across successive Transformer layers. 3.3 PRE-TRAINING Anepisodeturns a generated or original multivariate record into a forecasting task by specifying history and horizon, target and covariate roles, observation availability, and future supervision. A mode describes one target and its permitted inputs; aligned modes form an episode with a shared forecast origin and attention group. Sampling different channels, windows, and roles from a record therefore exposes its underlying temporal and relational primitives under different task settings. WriteE= (x,y,c;μ), with target historyx∈R C y ×L , target futurey∈R C y ×H , and covariates c∈R C c ×(L+H) . Here,C y andC c are the numbers of target and covariate variates, withC= C y +C c . Metadataμ= (L,H,r,M obs ,M loss ,g)specify lengths, variate rolesr, observation and supervision masks, and group identityg. They control packing and masking rather than supplying generating information. From permitted observations, the model predicts quantiles: b y (q) =f (q) θ x,c;L,H,r,M obs ,g , q∈Q.(4) Hereθdenotes model parameters andQthe set of quantile levels. Target futures are hidden. Past- only covariates are masked after the forecast origin, whereas known-future covariates remain visible where permitted by the task setting. The same variate can take different roles across episodes. Gen- erating annotations guide the selection of derived targets and source or auxiliary variates, exposing direct, shared, and partially observed relations without revealing operator identities to the model. Additional channels may be redundant or unrelated to the prediction target. Original multivariate observations use the same interface without requiring explicit generating annotations. Temporal sampling varies both the observed span of a primitive and the extent of its extrapolation. Episodes are packed with separate attention groups and optimized in three steps. First, define a binary supervision maskM loss ih , equal to one only at valid target-future entries, and apply horizon- decaying weights over the padded forecast lengthH pad ≥H: w h = h −1 H −1 pad P H pad j=1 j −1 , ω ih =M loss ih w h .(5) Second, average the pointwise pinball loss over quantiles, using targets and predictions normalized by observed-history statistics: ℓ ih = 2 |Q| X q∈Q ρ q ey ih − e by (q) ih , ρ q (v) =v q−1[v <0] .(6) For each target variate, tildes denote normalization of both targets and predictions using the same statistics computed from its observed history. Finally, aggregate over target variatesiand forecast 5 positionsh: L(θ) = P i,h ω ih ℓ ih P i,h ω ih +ε .(7) Hereε= 10 −8 stabilizes the denominator. This objective excludes unobserved supervision and em- phasizes nearer forecast positions without changing the observation mask. Large-scale multivariate pretraining is followed by long-context post-training. Together, the shared objective and varied episode metadata train forecasting across different primitive compositions and observation avail- ability. An optional GroupMix extension is described in Appendix B.3. 4 EXPERIMENTS Te m p o r a l Extrapolation Periodicity Trend Context ⋯ Relational Forecasting Signed Delayed Shared ⋯ Input Reliability Irrelevant Missing Order ⋯ PrimitiveCapability Univariate Multivariate Rank1 •MASE(FEV) Rank1 •SQL(FEV) Rank1 •FEVTa s kRank Rank1 •MASE(TIME) Rank1 •CRPS(TIME) Rank2 •MASE(GIFT) •CRPS(GIFT) Public Benchmarks Figure 4:Evaluation overview. Public bench- marks assess forecasting across settings, while primitive evaluations examine temporal extrap- olation, relational forecasting, and input relia- bility. Rankings are among compared methods. We evaluate Timer-M1 through public bench- marks, primitive evaluation, and ablation stud- ies. These examine forecasting accuracy, tem- poral extrapolation, relational forecasting, and input reliability, alongside the contributions of adaptive gates and data construction. Together, they connect performance with learning reusable primitives through coordinated data construction, episode-based pretraining, and model design. 4.1 PUBLICBENCHMARKS We evaluate 100 FEV ( Shchur et al.,2025), 97 GIFT-Eval ( Aksu et al.,2024), and 98 TIME (Qiao et al.,2026) configurations, reporting point and probabilistic scores over matched sets. Public baseline submissions are compared with a shared Timer-M1 checkpoint. FEV distinguishes five task types, while TIME and predominantly univariate GIFT-Eval complement coverage across temporal settings. Details are provided in AppendixB.4. FEV.Timer-M1 achieves MASE skill 0.3771 and SQL skill 0.4887 over the 100 matched config- urations, ranking first on both aggregate metrics among the compared methods. Its relative errors are lower than those of Chronos-2 by 3.43% and 3.02%, and lower than those of TimesFM-3 by 0.47% and 0.42%, respectively. Figure5further separates five mutually exclusive task types: uni- variate, multivariate, past-only covariates, known-future covariates, and their combination. Across these groups, Timer-M1 achieves the best unweighted average rank for both MASE (1.80) and SQL (1.60), placing among the top two models in every group under both metrics. This consistency complements its aggregate accuracy, demonstrating broad task coverage within a compact 121M- parameter backbone. A shared frozen checkpoint and inference recipe accommodate different vari- ate roles and observation conditions without task-specific parameter updates, connecting temporal and relational primitives with flexible episode construction. TIME.Timer-M1 obtains normalized MASE 0.6372 and CRPS 0.5363 over 98 configurations (Figure 6), reducing errors relative to Chronos-2 by 3.75% and 3.60%, respectively. These im- provements extend across both point and probabilistic aggregate scores, suggesting benefits beyond point estimation alone, complementing the results on FEV across diverse forecasting task settings. For comparison with TimesFM-3, a repeat under the same evaluation conditions yields 0.63977 and 0.53635, corresponding to reductions of 0.40% and less than 0.01%. We interpret this narrow margin as competitive performance with TimesFM-3 rather than statistically established superiority, while observing larger improvements over Chronos-2 on both reported metrics. GIFT-Eval.Across 97 configurations, Timer-M1 obtains normalized MASE 0.6806 and CRPS 0.4688 (Figure7), reducing errors relative to Chronos-2 by 2.47% and 3.43%, respectively. Timer- M1 also improves both metrics over TiRex-2 and Toto 2.0 (2.5B), ranking second among the com- pared methods, while TimesFM-3 retains the lowest aggregate errors. The benchmark spans diverse 6 0.00.20.40.60.81.01.2 Agg. Relative MASE Timer-M1 TimesFM-3 TimesFM-2.5 Chronos-2 TiRex-2 Toto 2.0 TiRex Moirai 2.0 Chronos-Bolt Sundial AutoTheta Seasonal Naive 0.623 0.626 0.644 0.645 0.663 0.675 0.699 0.720 0.735 0.748 0.890 1.000 0.00.20.40.60.81.01.2 Agg. Relative SQL Timer-M1 TimesFM-3 Chronos-2 TimesFM-2.5 TiRex-2 Toto 2.0 TiRex Moirai 2.0 Chronos-Bolt Sundial AutoTheta Seasonal Naive 0.511 0.513 0.527 0.533 0.545 0.556 0.573 0.598 0.611 0.661 0.945 1.000 Univariate (n=32)Multivariate (n=26)Past-only (n=12)Known-only (n=18)Known+past (n=12) 0.35 0.50 0.65 0.80 Relative SQL 0.617 0.615 0.630 0.646 0.644 0.405 0.405 0.421 0.433 0.397 0.533 0.562 0.576 0.561 0.605 0.464 0.454 0.474 0.503 0.566 0.570 0.583 0.575 0.623 0.698 StatisticalUnivariate pre-trainedCovariate-capable pre-trainedJoint-target pre-trained Timer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Figure 5:Forecasting results on FEV across 100 matched configurations. Top: aggregate MASE and SQL for 12 methods, normalized by Seasonal Naive. Bottom: SQL across five task types with different variate roles and observation availability. Timer-M1 achieves the lowest aggregate errors. Colors denote forecasting interfaces above and individual models below. 0.00.20.40.60.81.01.2 Agg. Relative MASE Timer-M1 TimesFM-3 Toto 2.0 Chronos-2 TimesFM-2.5 TiRex Timer-S1 Moirai 2.0 Chronos-Bolt Sundial Moirai Seasonal Naive 0.637 0.640 0.642 0.662 0.669 0.683 0.702 0.703 0.733 0.758 0.826 1.000 0.00.20.40.60.81.01.2 Agg. Relative CRPS Timer-M1 TimesFM-3 Toto 2.0 Chronos-2 TimesFM-2.5 TiRex Moirai 2.0 Timer-S1 Chronos-Bolt Sundial Moirai Seasonal Naive 0.536 0.536 0.539 0.556 0.567 0.573 0.588 0.589 0.620 0.663 0.666 1.000 StatisticalUnivariate pre-trainedCovariate-capable pre-trainedJoint-target pre-trained Figure 6:Comparison on TIME across 98 matched configurations. Timer-M1 achieves state-of-the- art aggregate performance, attaining the lowest normalized MASE and CRPS among the compared methods across diverse forecasting task settings. Colors distinguish forecasting interfaces. sampling frequencies and forecast lengths, testing whether a shared pretrained model remains ef- fective across heterogeneous temporal resolutions and task settings. These results on predominantly univariate GIFT-Eval complement FEV and TIME, showing that Timer-M1’s improvements extend beyond multivariate tasks to both point and probabilistic forecasting of individual series. 4.2 PRIMITIVEEVALUATION Primitives encompass recurring temporal and relational patterns that support forecasting across task settings. To examine how Timer-M1 learns and applies these primitives, we conductPrimitive Evaluation, assessing temporal extrapolation, relational forecasting, and input reliability through controlled tasks. Figure 8presents aggregate results over complete task groups alongside illustra- tive cases, connecting forecasting performance with responses to systematic changes in temporal patterns, variate relations, and observation availability, thereby examining primitive reuse under changing conditions. Details are provided in Appendices B.5,C.3, andD. Temporal extrapolation.Timer-M1 shows strong periodic extrapolation, achieving NMAE 0.02432across 13 periodic cases. Long-period continuation and frequency drift illustrate its ability to follow recurring patterns beyond the observed context. These results are consistent with learn- 7 TimesFM-3 Timer-M1 Toto 2.0 TiRex-2 Timer-S1 Chronos-2 TiRex TimesFM-2.5 Moirai 2.0 TimesFM-2.0 Sundial Chronos-Bolt 0.450 0.475 0.500 0.525 0.550 0.575 CRPS (↓) .456 .469 .476 .478 .485 .485 .488 .490 .516 .550 .559 .574 CRPS TimesFM-3 Timer-M1 Timer-S1 Toto 2.0 TiRex-2 Chronos-2 TimesFM-2.5 TiRex Moirai 2.0 Sundial TimesFM-2.0 Chronos-Bolt 0.70 0.75 0.80 MASE (↓) .667 .681 .693 .696 .697 .698 .705 .716 .728 .750 .758 .808 MASE Univariate pre-trainedCovariate-capable pre-trainedJoint-target pre-trainedTimer-M1 (ours) Figure 7:Point and probabilistic forecasting performance on GIFT-Eval across 97 matched configu- rations. Timer-M1 ranks second among 12 compared pretrained models on both MASE and CRPS, demonstrating competitive generalization across diverse domains and forecast horizons. ShiftPeriodicTrend 10 −2 10 −1 10 0 10 1 MSE (log scale) (a) Temporal extrapolation Sign Coint. Promo Lag Phase Scale Shift 10 −5 10 −3 10 −1 10 1 MSE (log scale) × (b) Variate relations 10 −2 10 −1 10 0 10 1 MSE (log scale) Old history Extra targets† Event inputs Corruption Missingness Order† (c) Reliability −1280128 −1 0 1 (d) Long-period continuation 3.8e-05 8.0e-05 −1280128 −3 0 3 (g) Frequency drift 0.330 0.660 −1280128 2 4 6 (e) Promotion-driven targets 0.275 0.338 −1280128 −4 0 4 (h) Shared regime change 0.162 0.309 −1280128 −2 0 2 (f) Random missing history 0.005 0.006 −1280128 4 8 (i) False-positive events 2.217 2.500 Timer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Timer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0TruthHistory Figure 8:Primitive evaluation. Top: predefined temporal groups, seven relational mechanisms, and six reliability categories, measured by MSE (lower is better). Unsupported interfaces are omit- ted; detailed results for the displayed groups appear in Appendix C.3. Bottom: six strength cases spanning distinct mechanisms, not an estimate of their frequency. Shading denotes M1’s marginal 10–90% quantiles.†marks unavailable squared-error archives for some baseline. ing reusable temporal primitives across diverse compositions, linking accurate periodic continuation with the ability to track evolving temporal patterns rather than repeat observed sequences. Relational forecasting.Timer-M1 achieves NMAE0.02160across six joint-target tasks, demon- strating strong cross-variate forecasting. Promotion-driven targets and shared regime changes il- lustrate how relational primitives connect observations to target dynamics. These findings sup- port learning reusable relational primitives through data construction and episode-based pretraining, which exposes dependencies under different variate roles and observation availability. Input reliability.Timer-M1 reduces target-missingness NMAE by32.5%relative to Chronos- 2, with selected cases illustrating resilience to incomplete histories and misleading inputs. These findings support composing primitives under varied observation conditions to broaden coverage of challenging forecasting scenarios. Episodes with observation masks and redundant or unrelated variates complement this construction by exposing primitives through imperfect inputs. 8 4.3 ABLATIONSTUDY We conduct ablation studies to examine how adaptive cross-variate interaction and primitive-based data construction contribute to forecasting. Gate ablations compare learned gates with fixed alterna- tives, testing the benefit of allocating interaction strength across layers. Data ablations compare four recipes to assess the contribution of joint real–synthetic generation beyond source diversity. All vari- ants use the same evaluation inputs and inference recipe. Figure9reports overall and task-specific FEV results, while Table1summarizes data ablations across benchmarks. OverallUni.Multi.PastKnownBoth 0.35 0.45 0.55 0.65 SQL Skill ( ↑ ) 0.489 0.383 0.595 0.467 0.536 0.430 0.484 0.383 0.598 0.465 0.500 0.441 0.481 0.382 0.596 0.466 0.492 0.443 AdaptiveFixed 0.202Fixed 1.0 Figure 9:Gate ablation on overall FEV and five task types. Improvement supports layer- wise allocation of cross-variate interaction. Metric Real Only Synthetic Only No-Joint Full Recipe FEV Skill Score↑0.4431 0.47100.47250.4887 GIFT MASE↓0.6998 0.68840.68190.6806 GIFT CRPS↓0.4850 0.47410.47220.4688 TIME MASE↓0.6579 0.64380.63790.6372 TIME CRPS↓0.5526 0.54560.53590.5363 Table 1:Data recipe ablation. Red bold and blue underline mark the best and second-best scores in each row. Joint generation benefits most aggregates. Adaptive gates.We compare learned layer-wise coefficientsα ℓ with two uniform alternatives. Fixed 0.202uses the mean gate value from the stable checkpoint, preserving a representative in- teraction strength while removing layer-wise variation;Fixed 1.0applies unattenuated cross-variate updates. Adaptive gates achieve FEV SQL skill0.4887, versus0.4836and0.4814, improving three of five task types. The learnedα ℓ generally increase with depth, suggesting stronger cross-variate integration in deeper layers. Together, this depth-dependent pattern and the advantage over mean- matched scaling support allocating interaction strength across layers, allowing relational primitives to be integrated progressively with temporal representations. Data construction.We compareReal Only,Synthetic Only,No-Joint, and theFull Recipe. The first two restrict generation to their respective source pools. No-Joint includes both sources but ex- cludes joint real–synthetic generation; the full recipe additionally composes them through shared relation operators. The full recipe improves FEV skill and both GIFT-Eval metrics over all three alternatives and achieves the lowest TIME MASE. Its FEV skill exceeds No-Joint by0.0162, sug- gesting benefits beyond source diversity alone. By combining observed dynamics with controlled relational primitives, joint generation broadens multivariate compositions, supporting a unified data construction pipeline rather than independent real and synthetic training collections. 5 CONCLUSION We present Timer-M1, a multivariate time series foundation model that learns with temporal and relational primitives. Its primitive-based data pipeline combines observed dynamics with con- trolled generation to construct diverse multivariate samples. Episode-based pretraining exposes these primitives under varied variate roles and observation availability, while gated two-dimensional Transformer blocks integrate temporal and cross-variate information. Evaluations on three public benchmarks and controlled primitive tasks demonstrate competitive forecasting performance and strengths in temporal extrapolation and relational forecasting. These results support aligning data construction, forecasting episodes, and model architecture around shared primitives. Future work will expand real-data coverage and primitive compositions to better represent irregular dynamics and partially observed dependencies in complex real-world systems. On the architectural side, explor- ing input-dependent cross-variate interaction and more efficient long-context modeling may further improve how foundation models learn and apply primitives across domains and task settings. 9 AIUSE STATEMENT Generative AI tools were used only for translation, drafting and revising portions of the manuscript, language polishing, literature search, reference formatting, and figure preparation. They were not used for developing theoretical models or conceptual frameworks, designing research methodol- ogy or experiments, writing proofs, implementing experimental methods, generating or processing datasets, conducting data analysis, or interpreting results, and no substantive AI-generated sugges- tions regarding mathematical assumptions, derivations, proofs, or research hypotheses were adopted. 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A.1 TEMPORAL ANDRELATIONALPRIMITIVES Temporal primitives describe evolution within a series. Periodic and trend components may appear alone or together with stochastic variation; autoregressive dependence provides a further source of temporal continuity. Relational primitives describe how multiple variates evolve together. Source reuse introduces shared drivers, signed mixing produces positive or negative responses, and temporal offsets create delayed effects. These families are complementary: temporal primitives determine the evolution of individual sources, while relational primitives determine how their information is shared across variates. For example, a periodic source can produce responses with different signs, scales, or delays without changing its underlying temporal pattern. The resulting dependencies are specified during construction rather than inferred as causal mechanisms from real observations. Table 2:Operator families in primitive-based data construction. The taxonomy follows the temporal and relational primitives introduced in the main text. Examples describe mechanisms rather than a fixed generation recipe. FamilyPrimitiveRepresentative construction Temporal PeriodicityOscillatory components and their superposition TrendGradual growth, decline, and shared drift Stochastic dependenceNoise and autoregressive dynamics Regime changesPiecewise changes in level or evolution Relational Signed responsesPositive and negative source mixing Delayed responsesTemporal offsets between source and response Shared driversMultiple responses derived from common sources CointegrationShared nonstationary components Event effectsResponses to discrete-valued inputs TransformsInversion, clipping, resampling, and discrete mappings A.2 CONSTRUCTION ANDOBSERVATION Following Equation 1, source series are selected from real observations or generated temporal primitives, combined through relation operators, and transformed into aligned multivariate records. For source arrayξ∈R κ×τ , a schematic linear response is η i (t) = X j∈J i w ij ξ j (t−δ ij ), i= 1,...,ν,(8) whereJ i ⊆1,...,κselects the sources contributing to responsei,w ij controls their signed contribution, andδ ij denotes a temporal lag. Nonlinear relation operators replace the linear com- bination with a mapping of the selected sources. Reusing a source across responses introduces shared variation. The transform operatorA ψ then changes the observed forms through opera- tions such as rescaling or clipping. For branches retaining source channels, the resulting record isΞ=A ψ ([ξ;η]); response-only branches omitξ. Transformations that modify time coordinates are coordinated where alignment must be preserved. Original multivariate observations can also enter directly, retaining their existing dependencies. A multivariate record specifies aligned values, whereas a forecasting episode additionally specifies their roles and availability. After channel selection and temporal sampling, letX∈R C×(L+H) contain the episode’s target and covariate values. Following the main text,M obs ∈0,1 C×(L+H) 12 denotes the observation mask, whileM loss ∈0,1 C y ×H selects supervised target-future entries. The masked values are defined entrywise as X obs c,t = X c,t , M obs c,t = 1, 0, M obs c,t = 0, M loss ih = 1,if targetiat horizonhis valid, 0,otherwise. (9) Here,cindexes all episode variates,tindexes history and forecast positions,iindexes target variates, andhindexes forecast positions. Normalization uses observed-history statistics, and hidden entries remain zero in the values supplied to the patch embedder. The observation mask accompanies these values so that hidden entries remain distinguishable from observed zeros. Target futures are withheld from the input but may contribute to supervision; known-future covariates remain visible where permitted. Different role assignments therefore expose the same primitives through different forecasting conditions without changing the underlying record. The record and scalar-position counts in Figure3describe stored data before episode sampling. Records, channel-wise series, and scalar positions are distinct counting units. If recordrcontains C r channels of lengthT r , their respective counts are N records = X r 1, N series = X r C r , N positions = X r C r T r .(10) These quantities characterize stored data volume rather than independent forecasting tasks or con- sumed training tokens. Episode sampling can reuse a record under different horizons and observa- tion roles. B IMPLEMENTATIONDETAILS B.1 MODELCONFIGURATION Timer-M1 uses the same gated two-dimensional Transformer backbone across task settings. Table3 summarizes its dimensions. Temporal attention operates along patches within each variate, and variate attention operates across variates at each patch position. The residual order is temporal attention, feed-forward transformation, and gated variate attention. The prediction head outputs marginal quantiles, with the median used for point forecasts. Table 3:Model configuration. The gate contributes one learned scalar per layer; the total parameter count includes the embedding and prediction head. ComponentConfiguration Parameters120,998,412 (approximately 121M) Transformer layers12 Representation / feed-forward width768 / 3072 Attention heads / head dimension12 / 64 Patch length16 observations Prediction quantiles0.1,0.2,...,0.9 Temporal position encodingRotary positional embeddings Gate parameterizationα ℓ =σ(a ℓ ); one scalar per layer Gate initializationa ℓ =−4.6, givingα ℓ ≈0.01 ForCvariates,Npatches, and widthD, factorized attention requiresO(CN 2 D+NC 2 D)oper- ations per layer, rather thanO(C 2 N 2 D)for attention over all flattened tokens. Linear projections and feed-forward transformations additionally requireO(CND 2 )operations. Gating adds a scalar multiplication to each variate-attention residual update. The same patch embedder and prediction head are shared across variates, while episode boundaries determine which cross-variate interactions are permitted. This separates the number and roles of input variates from the learned representation width. B.2 LAYER-WISEGATES Figure 10shows the learned coefficientsα ℓ =σ(a ℓ )of the evaluated model. Each coefficient scales the variate-attention residual update after temporal attention and the feed-forward transformation. It 13 is shared across tokens in a layer and remains fixed during inference; it is not a sample-dependent attention score. The coefficients generally increase with depth, with larger updates in later layers and nonmonotonic local variation. This pattern is consistent with first forming temporal representations and subsequently giving greater weight to cross-variate integration, although coefficient magnitudes alone do not measure the information carried by either branch. The fixed value0.202is the rounded mean of twelve gate coefficients from an earlier stable model. It preserves a representative interaction scale while removing layer-specific allocation; the other fixed control,1.0, uses the unscaled variate-attention update. As shown in Table11, the learned gates improve overall FEV skill. 123456789101112 Layer 0.150 0.175 0.200 0.225 0.250 0.275 Gate coefficient α ℓ Timer-M1Reference mean (0.202) Figure 10:Layer-wise allocation of cross-variate interaction. The learned coefficients generally rise with depth, while local deviations allow different layers to assign different weights to variate- attention updates. The dashed line marks the earlier stable-model mean used by the fixed-0.202 control, not the mean of the displayed curve. Together with the gate ablation, the profile motivates adaptive allocation rather than a uniform residual scale. B.3 EPISODES ANDPACKING Episodes share a forecast origin and specify target, past-only covariate, and known-future covariate roles. Target futures are hidden; available future covariates remain visible. Observation masks determine which values enter the model, while supervision masks select valid target-future entries for the loss. Padding is excluded from valid observations without disabling the forecast tokens that must produce predictions. We formulate episode batching as a capacity-constrained bin-packing problem and use a greedy packing heuristic. Episodes remain intact, and distinct attention groups prevent information ex- change between packed episodes. Packing changes how episodes share a batch, not their permitted observations or prediction targets. The packing objective improves batch utilization while preserv- ing episode semantics; the greedy procedure does not require an exact solution of the bin-packing problem. GroupMix.GroupMix extends episode construction by changing cross-variate visibility during training. Starting from separate episode groups, a selected group is associated with additional vari- ates from other groups in the packed batch. In the group-level form, the selected groups share an attention group, permitting interaction among their visible variates. A finer-grained form intro- duces only a subset of external variates. Sequence values, variate roles, and supervision targets are retained: the operation modifies the attention grouping rather than interpolating series or mixing forecast labels. Observation and supervision masks continue to enforce the original information constraints. In particular, hidden target futures are not made visible by a group reassignment, and padded entries do not become observations. Unrelated external variates can consequently act as distractors, providing a way to vary the context in which temporal and relational primitives must be recognized. GroupMix is a training-time extension; ordinary episode grouping is used at inference. It is not enabled for the results reported here, so its potential regularization effect is distinct from the measured gate and data-construction ablations. 14 B.4 PUBLICBENCHMARKPROTOCOLS We follow the official task definitions and scoring conventions of FEV, TIME, and GIFT- Eval (Shchur et al.,2025;Qiao et al.,2026;Aksu et al.,2024). Forecast origins, prediction horizons, and permitted observations are determined by the benchmark tasks. FEV distinguishes historical covariates from future-known covariates; only inputs available at the forecast origin are supplied. Scores use the complete matched configuration sets reported in the main text. Public reference scores are fixed comparison snapshots, not a continuously updated ranking. The official online evaluation documentation was checked on September 26, 2026. 1 Compliance with task definitions and metrics is distinct from training-data eligibility. No task-specific param- eter updates are performed during evaluation. Training-data exposure is not matched across the compared methods. For modelmand configurationj, letE m,j denote the benchmark error andE SN,j the Seasonal Naive error. For a configuration groupG, the relative aggregate is R m,G = exp 0 @ 1 |G| X j∈G log E m,j E SN,j 1 A .(11) FEV clips each relative error to[0.01,100]before aggregation and reports skill1−R m,G . We report MASE and SQL skill separately. TIME and GIFT-Eval report normalized MASE and quantile-based CRPS. Higher skill and lower normalized error indicate better forecasts. Missing matched results remain unavailable rather than being assigned zero error. B.5 PRIMITIVEEVALUATIONPROTOCOL The controlled evaluation separates temporal extrapolation, relational forecasting, and input reliabil- ity. Each test fixes a generated trajectory and forecast origin, then specifies the observations available to each model. All compared models receive the same target values and permitted covariates for a given setting. Future targets are used only for scoring. The displayed groups are defined by their mechanism, not by the model attaining the smallest error, and retain all scoring records within each selected group. Temporal extrapolation.The elementary-function tests useL= 256historical observations and H= 128forecast steps. Periodicity covers sinusoidal and nonsinusoidal waves, multiple frequen- cies, amplitude modulation, frequency drift, and narrow pulses. Trend tests include constant, in- creasing, decreasing, nonlinear, and piecewise trajectories. The shift group includes a level change visible in history and changes in level, slope, amplitude, frequency, or phase introduced at the fore- cast boundary. The latter have no advance indication in the supplied history and are interpreted as stress tests, not as evidence that a model can anticipate an unobserved intervention. Table 4connects these mechanisms to the result groups. Relational forecasting.Structured tasks use the sameL= 256,H= 128window and construct multiple responses from shared sources. Signed, delayed, phase-shifted, scale-varying, cointegrated, and shared-regime relations probe different ways of transferring information across variates. The joint-target settings hide all target futures; promotion-driven settings additionally supply the event covariate over its permitted interval. Each target is scored separately before group aggregation. These cross-model comparisons assess forecasting with the specified inputs; they are not, by them- selves, estimates of the gain from joint rather than independent prediction. Input reliability.These tests useL= 512andH= 128and hold the clean future fixed while changing the available history or auxiliary inputs. Missing-history settings include random removal at10%,30%, and50%, internal gaps of 32 or 96 steps, and missing suffixes of 16 or 64 steps. Internal gaps use common linear interpolation; missing suffixes use forward filling, without con- sulting future targets. Corruption tests perturb the observed history; irrelevant-input tests introduce unrelated or misleading context. Event tests alter the supplied event indicators while retaining the same target trajectory. Variate-order tests permute the same input channels and restore predictions to the original target order before scoring. Thus, missingness results concern forecasting after the specified common preprocessing, rather than a separate imputation task. 1 FEV,TIME, andGIFT-Eval. 15 Table 4:Displayed primitive evaluation groups and their controlled conditions. The counts of scor- ing records and model-specific results appear in Tables8–10. All groups report MSE; NMAE addi- tionally supports the comparisons discussed in the text. CapabilityGroupControlled pattern or observation condition TemporalPeriodicPeriod, waveform, modulation, and gradual frequency change TrendDirection, growth shape, and observed slope change ShiftObserved level change and unannounced boundary changes RelationalSign; scaleShared dynamics with opposite signs or different scales Lag; phaseDelayed responses and phase-shifted periodic targets CointegrationMultiple responses sharing a nonstationary source Promotion; shiftEvent-driven responses and shared regime changes ReliabilityIrrelevant contextUninformative or misleading portions of historical con- text Irrelevant targetsAdditional unrelated variates in the joint target set Missing event inputsIncomplete or incorrect future event information Target corruptionNoise and outlying values in the observed target history Target missingnessRandom gaps, internal blocks, and missing recent history Variate orderPermutation of input channels with target identities re- stored Metrics and aggregation.For scoring recordr, letΩ r contain its evaluated forecast positions and letˆy r,i andy r,i denote the point forecast and ground truth. We compute MSE r = 1 |Ω r | X i∈Ω r (ˆy r,i −y r,i ) 2 ,NMAE r = P i∈Ω r |ˆy r,i −y r,i | |Ω r |s r ,(12) wheres r is a fixed reference scale shared by the models for that record. The displayed short-window temporal and relational groups use the range of the complete generated trajectory, including history and future. For a nearly constant trajectory, the denominator is the larger of its mean absolute value and one. Reliability tests use the range of the clean historical reference with a positive floor of10 −6 . These quantities normalize evaluation only and are not provided to the forecasting model. Group scores are arithmetic means over their scoring records, with one record per evaluated target and condition. The countnis therefore not necessarily the number of independent generated trajectories. Rankings are computed before rounding; red bold and blue underline indicate the lowest and second- lowest distinct errors, respectively. MSE retains the effect of absolute error scale, whereas NMAE facilitates comparison between tra- jectories of different amplitudes. The two metrics are not interchangeable, and no cross-capability composite score is inferred from the selected groups. Unsupported input interfaces and unavailable metrics are left unranked rather than assigned an artificial penalty. Where quantiles are available, showcase shading spans the 10th to 90th percentiles; these are marginal prediction intervals, not confidence intervals for the estimated mean. Examples illustrate individual mechanisms and do not determine the numerical aggregates. 16 C FULLRESULTS C.1 PUBLICBENCHMARKS Table5provides the numerical aggregates behind the public-benchmark comparisons. Scores retain the original benchmark direction and normalization. The comparisons cover 100 FEV, 98 TIME, and 97 GIFT-Eval configurations. Training-data eligibility is described in AppendixB.4; numerical ranks refer to the displayed methods rather than a claim of official zero-shot leaderboard placement. Table 5:Aggregate benchmark results. FEV reports skill (↑); TIME and GIFT-Eval report normal- ized errors (↓). Red bold and blue underline mark the best and second-best displayed scores per column. Dashes indicate unavailable matched results. Training exposure differs across methods (AppendixB.4). Model FEV MASE FEV SQL TIME MASE TIME CRPS GIFT MASE GIFT CRPS Timer-M10.37710.48870.63720.53630.68060.4688 TimesFM-30.37420.48660.63980.53630.66680.4557 Chronos-20.35500.47280.66200.55630.69780.4854 TiRex-20.33740.4550–0.69730.4781 Toto 2.0 (2.5B)0.32540.44420.64190.53940.69560.4759 TimesFM-2.50.35620.46680.66860.56740.70500.4903 Timer-S1–0.70210.58860.69340.4853 Sundial0.25220.33870.75760.66340.75020.5590 Moirai 2.00.28030.40250.70300.58850.72810.5164 TiRex0.30070.42680.68310.57310.71580.4885 Chronos-Bolt0.26520.38890.73280.62000.80760.5743 FEV evaluates heterogeneous variate roles, TIME emphasizes task-specific forecasting configura- tions, and GIFT-Eval spans diverse temporal resolutions and forecast lengths. Reporting the metrics separately avoids combining point and distributional accuracy into an arbitrary composite score. 17 C.2 FEV TASKTYPES The five groups partition all FEV configurations. Univariate and multivariate groups have no dy- namic covariates; the remaining groups distinguish past-only inputs, known-future inputs, and their combination. Covariate tasks may contain multiple target variates. Each group uses the same geo- metric aggregation as the overall benchmark, rather than an arithmetic average of the displayed skill values. Table 6:FEV MASE skill by task type. Groups are mutually exclusive and include all 100 configu- rations; higher is better. Best and second-best values are marked within each row. Task typenTimer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Univariate320.29680.29980.27990.26290.2653 Multivariate260.42660.42520.40170.38890.4366 Past-only120.42560.39330.36550.39690.3426 Known-future180.44810.45490.43230.39790.3332 Known + past120.29880.29230.30290.24650.1710 Table 7:FEV SQL skill by task type. Groups are mutually exclusive and include all 100 configura- tions; higher is better. Best and second-best values are marked within each row. Task typenTimer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Univariate320.38280.38470.37030.35380.3563 Multivariate260.59540.59530.57950.56680.6033 Past-only120.46680.43820.42390.43870.3950 Known-future180.53630.54580.52570.49710.4344 Known + past12 0.42950.41710.42540.37650.3022 The breakdown retains both point and probabilistic scores to distinguish temporal forecasting from the use of additional cross-variate information. In particular, separating past-only and known-future inputs distinguishes inference from related history from conditioning on observations available over the forecast interval. This organization follows the variate roles used to construct forecasting episodes, while keeping the evaluation tasks fixed across models. Overall FEV skill is recomputed from all configuration-level errors, not from an unweighted mean of the five group scores, because group sizes differ. 18 C.3 PRIMITIVEGROUPS Tables8–10expand the groups displayed in Figure8. They retain every record within each displayed group and all available compared models. These groups are a selected view of the diagnostic suite, not its exhaustive inventory; no overall rank over the entire suite is inferred from them. Window lengths, observation conditions, and metric definitions are specified in AppendixB.5. The countn denotes scoring records, which may include multiple views of the same generated task. Table 8:Temporal extrapolation groups corresponding to Figure8. Mean squared error is averaged over the complete records of each displayed group; lower is better. GroupnTimer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Shift66.42787.48306.84906.90926.4418 Periodic130.02710.68030.09190.07350.2300 Trend100.13120.02990.45220.45120.0083 Table 9:Relational forecasting groups corresponding to Figure8. Mean squared error is averaged over the complete records of each displayed group; lower is better. GroupnTimer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Sign20.00210.00360.00770.01000.0073 Cointegration20.24770.39532.26271.39304.7848 Promotion2 0.30080.30421.04240.5567– Lag20.00120.00050.01850.00740.0049 Phase20.0000530.0000430.00020.00870.0010 Scale2 0.06880.09550.86902.29714.7189 Shift2 0.10600.20680.25930.31290.4098 Table 10:Input reliability groups corresponding to Figure8. Mean squared error is averaged over the complete records of each displayed group; lower is better. GroupnTimer-M1TimesFM-3Chronos-2TiRex-2Toto 2.0 Irrelevant context70.36810.25880.25540.25850.2451 Irrelevant targets30.0035–0.01670.06330.0055 Missing event inputs75.79175.75943.91302.2120– Target corruption2112.09382.608123.98020.22603.8370 Target missingness210.19570.14770.72310.21840.3595 Variate order7 0.0034–0.02210.06230.0055 Dashes denote unsupported inputs or unavailable squared-error results, not failed forecasts assigned an artificial penalty. TimesFM-3 uses the fixed configuration for these controlled tests; public- benchmark scores follow their own evaluation settings. Selected examples in Appendix Dillustrate the observed temporal patterns and relations without changing the numerical groups above. 19 C.4 ABLATIONRESULTS Gate controls are independently trained with fixed residual multipliers, retaining variate attention and the same residual order. Fixed 0.202 uses the unrounded stable-reference mean, approximately 0.201676, while Fixed 1.0 uses the full variate-attention update. Adaptive gates are learned sepa- rately at each layer and remain fixed during inference. These comparisons are not inference-only interventions on a common checkpoint. Table 11:Gate ablation on FEV overall and five task types. Both metrics report skill scores. Red bold and underline indicate the best and second-best results within each column and metric. Adap- tive gates achieve the highest overall skill, with the clearest advantage in known-future forecasting. GateOverallUnivariateMultivariatePast-onlyKnown-futureKnown + past MASE Skill↑ Adaptive0.37710.29680.42660.42560.44810.2988 Fixed 0.2020.37140.29690.42990.42660.40840.3081 Fixed 1.00.36800.29630.42600.42380.39790.3104 SQL Skill↑ Adaptive0.48870.38280.59540.46680.53630.4295 Fixed 0.2020.48360.38260.59810.46510.50020.4406 Fixed 1.00.48140.38190.59560.46590.49200.4426 Table 12:Data-construction ablations across three benchmarks. Recipe names refer to the source- generation routes under comparison; shared benchmark-associated training data are unchanged across variants. Recipe FEV MASE↑ FEV SQL↑ TIME MASE↓ TIME CRPS↓ GIFT MASE↓ GIFT CRPS↓ Real Only0.31960.44310.65790.55260.69980.4850 Synthetic Only0.35680.47100.64380.54560.68840.4741 No-Joint0.35780.47250.63790.53590.68190.4722 Full Recipe0.37710.48870.63720.53630.68060.4688 Real Only and Synthetic Only restrict the source-generation routes under comparison; No-Joint retains both source types but removes corpus-derived and mixed-source joint generation. Shared GIFT-Eval and TIME training collections remain unchanged across the four variants. Consequently, these names describe the controlled routes rather than the exclusive provenance of all training ob- servations. The recipe comparisons assess inclusion of generation routes and do not isolate each individual operator. 20 D PRIMITIVEEVALUATIONSHOWCASE The following pages present 24 illustrative forecasts, eight for each capability. Examples cover distinct temporal patterns, cross-variate relations, and observation conditions, with an emphasis on mechanisms discussed in the main text. They are selected examples rather than a random sample for estimating overall success rates. All model curves within a panel use the same target and forecast origin. The dashed boundary separates history from the forecast interval; shading denotes Timer- M1’s marginal 10–90% prediction quantiles, not confidence intervals for its mean prediction. −1 0 1 (a) Long-period continuation −250−200−150−100−50050100 Relative time 0.00 0.01 Abs. error −1 0 1 (b) Short-period continuation −250−200−150−100−50050100 Relative time 0.00 0.01 Abs. error −2 0 2 (c) Frequency drift −250−200−150−100−50050100 Relative time 0 1 Abs. error −1 0 1 (d) Amplitude modulation −250−200−150−100−50050100 Relative time 0.00 0.25 Abs. error −1 0 1 (e) Multiple frequencies −250−200−150−100−50050100 Relative time 0.0 0.1 Abs. error 0 1 2 3 (f) Periodic pulses −250−200−150−100−50050100 Relative time 0.0 0.5 Abs. error 2 4 (g) Observed level shift −250−200−150−100−50050100 Relative time 0.00 0.25 Abs. error 0 10 20 (h) Observed slope change −250−200−150−100−50050100 Relative time 0.0 2.5 Abs. error Ground truth Timer-M1 TimesFM-3 Chronos-2 TiRex-2 Toto 2.0 (2.5B) Figure 11:Temporal extrapolation across periodicity, modulation, frequency drift, and observed regime changes. Lower strips show Timer-M1’s absolute forecast error. Each panel uses its own value scale. 21 0 1 (a) Shared dynamics across scales −250−200−150−100−50050100 Relative time 0 100 Targets −1 0 1 (b) Signed target relations −250−200−150−100−50050100 Relative time −1 0 1 Targets −1 0 1 (c) Delayed target response −250−200−150−100−50050100 Relative time −1 0 1 Targets −1 0 1 (d) Phase-shifted targets −250−200−150−100−50050100 Relative time −1 0 1 Targets 0 5 10 15 (e) Cointegrated targets −250−200−150−100−50050100 Relative time 0 5 Targets −2 0 2 (f) Shared regime change −250−200−150−100−50050100 Relative time −2 0 2 Targets 2 4 6 (g) Promotion-driven target I −250−200−150−100−50050100 Relative time 2.5 5.0 Targets 4 6 (h) Promotion-driven target I −250−200−150−100−50050100 Relative time 2.5 5.0 Targets Ground truth Timer-M1 TimesFM-3 Chronos-2 TiRex-2 Toto 2.0 (2.5B) Figure 12:Relational forecasting with shared, signed, delayed, phase-shifted, cointegrated, and event-driven dynamics. Lower strips show available related inputs or target histories; they are not additional forecast targets unless specified by the task. 22 −1 0 1 (a) Random missing history −500−400−300−200−1000100 Relative time 0.0 0.2 Abs. error −1 0 1 2 (b) Missing middle block −500−400−300−200−1000100 Relative time 0.0 0.5 Abs. error −1 0 1 (c) Missing recent history −500−400−300−200−1000100 Relative time 0.0 2.5 Abs. error 2.5 5.0 7.5 10.0 (d) Missing recent trend −500−400−300−200−1000100 Relative time 0 1 Abs. error 2 4 6 8 (e) False-positive future events −500−400−300−200−1000100 Relative time 0 5 Abs. error 2 4 6 8 (f) Incomplete event observations −500−400−300−200−1000100 Relative time 0 5 Abs. error 2 4 6 8 (g) Additional irrelevant inputs −500−400−300−200−1000100 Relative time 0 1 Abs. error 0 10 20 (h) Historical outliers −500−400−300−200−1000100 Relative time 0.00 0.25 Abs. error Ground truth Timer-M1 TimesFM-3 Chronos-2 TiRex-2 Toto 2.0 (2.5B) Figure 13:Input reliability under missing histories, imperfect event information, irrelevant inputs, and historical outliers. Lower strips show Timer-M1’s absolute forecast error. 23