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Forecast Collapse in Time-Series Foundation Models

Shu Wan, Miles Ma, Hank Zhu, Guangqi Liu, Stephen Wang, Qingsong Wen, Huan Liu

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 100

Intelligence

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

Last extracted: 8/17/2026, 5:03:40 AM

Summary

The paper identifies 'forecast collapse' in Time-Series Foundation Models (TSFMs), where predictions for low-predictability targets like equity returns become flat and lose cross-sectional ranking accuracy. This is attributed to low target predictability limiting forecast amplitude and per-series training objectives failing to capture cross-series structure. The authors propose 'CalibRank', a composite objective balancing calibration (MSE) and ranking (IC), which significantly improves cross-sectional correlation without inflating forecast amplitude, validated on the Finance1K dataset.

Entities (8)

Finance1K · dataset · 99%Forecast Collapse · phenomenon · 98%CalibRank · methodobjective · 97%Time Series Foundation Models · model-class · 95%Information Coefficient · metric · 92%Mean Squared Error · loss-function · 90%Chronos · model · 85%TimesFM · model · 85%

Relation Signals (8)

CalibRank → improves → Cross-Sectional Correlation

confidence 96% · On Finance1K, CalibRank nearly triples cross-sectional correlation

Forecast Collapse → affects → Time Series Foundation Models

confidence 95% · We investigate forecast collapse across time-series foundation models (TSFMs)... and find that it is closely tied to target predictability.

Finance1K → usedfor → CalibRank

confidence 95% · On Finance1K, CalibRank nearly triples cross-sectional correlation

CalibRank → balances → Calibration

confidence 94% · CalibRank, a simple objective that balances calibration and ranking.

CalibRank → balances → Ranking

confidence 94% · CalibRank, a simple objective that balances calibration and ranking.

Forecast Collapse → causedby → Low Predictability

confidence 93% · low predictability limits the amplitude of calibrated point forecasts

Forecast Collapse → causedby → Per-Series Objectives

confidence 93% · per-series objectives leave cross-series structure unidentified

Mean Squared Error → leadsto → Flat Predictions

confidence 90% · optimizing squared error leads to flat predictions

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Abstract

Abstract:When forecasting hourly returns for 1,000 US equities, we observe an unexpected phenomenon: predictions become nearly flat and show poor stock ranking, as measured by cross-sectional correlation. We call this forecast collapse. Surprisingly, the phenomenon largely disappears when forecasting trading volume under the same setting. We investigate forecast collapse across time-series foundation models (TSFMs), twelve deep-learning forecasting models, and 97 public benchmark configurations, and find that it is closely tied to target predictability. We identify two distinct reasons behind it: low predictability limits the amplitude of calibrated point forecasts, while per-series objectives leave cross-series structure unidentified. These findings reveal a calibration-ranking tradeoff: optimizing squared error leads to flat predictions, whereas directly optimizing cross-sectional correlation improves ranking but can inflate forecast amplitude by more than an order of magnitude. To address this tradeoff, we introduce CalibRank, a simple objective that balances calibration and ranking. On Finance1K, CalibRank nearly triples cross-sectional correlation while keeping amplitude close to the target, and improves correlation on all tested models. Our results reveal a blind spot in conventional time-series evaluation: per-series metrics can hide failures in cross-series structure needed by downstream decisions.

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Forecast Collapse in Time-Series Foundation Models Shu Wan 1,2,∗ , Miles Ma 1,∗ , Hank Zhu 1 , Guangqi Liu 1 , Stephen Wang 1 , Qingsong Wen 3 , Huan Liu 2 1 Abel AI Lab, 2 Arizona State University, 3 University of Oxford ∗ Equal contribution When forecasting hourly returns for 1,000 US equities, we observe an unexpected phenomenon: predictions become nearly flat and show poor stock ranking, as measured by cross-sectional correlation. We call this forecast collapse. Surprisingly, the phenomenon largely disappears when forecasting trading volume under the same setting. We investigate forecast collapse across time-series foundation models (TSFMs), twelve deep-learning forecasting models, and 97 public benchmark configurations, and find that it is closely tied to target predictability. We identify two distinct reasons behind it: low predictability limits the amplitude of calibrated point forecasts, while per-series objectives leave cross-series structure unidentified. These findings reveal a calibration-ranking tradeoff: optimizing squared error leads to flat predictions, whereas directly optimizing cross-sectional correlation improves ranking but can inflate forecast amplitude by more than an order of magnitude. To address this tradeoff, we introduce CalibRank, a simple objective that balances calibration and ranking. On Finance1K, CalibRank nearly triples cross-sectional correlation while keeping amplitude close to the target, and improves correlation on all tested models. Our results reveal a blind spot in conventional time-series evaluation: per-series metrics can hide failures in cross-series structure needed by downstream decisions. Date: August 17, 2026 Data: https://huggingface.co/datasets/abel-lab/finance1k Correspondence: Shu Wan swan@asu.edu, Stephen Wang stephen@abel.ai 1 Introduction Time series foundation models (TSFMs) (Liang et al., 2024) are large, pre-trained models designed to forecast across diverse domains with little or no dataset-specific tuning. Financial markets provide a particularly challenging stress test. Equity returns have notoriously low signal-to-noise ratios, yet many financial decisions depend not only on the accuracy of individual forecasts but also on their relative rank across many assets, measured by the correlation between predicted and realized values at each timestamp, commonly known as the information coefficient (IC). When we apply off-the-shelf TSFMs (Das et al., 2024; Ansari et al., 2024) to hourly US equity returns, something unexpected happens: the predictions are almost flat near zero, and are poorly correlated with the actual stock ranking (Figure 1(b)). This finding echoes a recent large-scale evaluation (Rahimikia et al., 2025) on equity excess returns, which finds that TSFMs are weak both zero-shot and after fine-tuning. We refer to this combination of low forecast amplitude and poor cross-sectional structure as forecast collapse. Surprisingly, forecast collapse largely disappears when we change the target from equity returns to traded volume on the same stock panel while keeping the forecasting protocol fixed. The contrast suggests that forecast collapse is not simply a failure of a particular model or implementation. Instead, it raises a more fundamental question: Why does the same forecasting pipeline collapse for one target but not for another? We provide a theoretical explanation for forecast collapse through two distinct mechanisms. First, low target predictability limits the amplitude of calibrated point forecasts: when little of the target variation is 1 arXiv:2608.14106v1 [cs.LG] 14 Aug 2026 −0.025 0.000 0.025 QLYS MSE 0.03x IC 25.63x 98% off scale CalibRank 1.49x 5% off scale −0.025 0.000 0.025 RAIL 0.01x21.10x 100% off scale 1.38x 4% off scale 0510152025303540 −0.025 0.000 0.025 MCRB 0.02x 0510152025303540 Trading step 23.22x 100% off scale 0510152025303540 1.54x (a) Realized returnMSE forecastIC forecastCalibRank forecast Realized ±1 sd 10 −3 10 −2 10 −1 10 0 MSE (lower is better; log scale) −0.05 0.00 0.05 0.10 0.15 IC (higher is better) MSE IC CalibRank (others) Pareto frontier Autoformer λ sweep TimesFM Chronos (b) 10 −2 10 −1 10 0 Achieved R 2 (log scale) 0.0 0.2 0.4 0.6 0.8 1.0 Raw amplitude ratio Finance1K Chronos (97) TimesFM (97) (c) Figure 1 Forecast collapse and the calibration-ranking tradeoff. (a) Realized and predicted returns for three equities under MSE, IC, and CalibRank; labels report raw amplitude relative to the target. (b) MSE against cross-sectional IC across forecasting models and objectives; lines mark the empirical Pareto frontier and the Autoformerλsweep. (c) Raw forecast amplitude against achievedR 2 for TimesFM and Chronos across 97 GIFT-Eval configurations (Aksu et al., 2024). Finance1K is far less predictable than any of them. predictable, calibrated forecasts necessarily shrink relative to the realized target. Second, per-series objectives leave cross-series structure unidentified: forecasts with the same per-series risk can have different cross-sectional correlations. We formalize the first mechanism through an exact relation between best-scaled amplitude, correlation, and predictable variance, and the second through an invariance result for per-series risk. These two results lead to a calibration-ranking tradeoff. A per-series calibration objective, such as mean squared error (MSE), scores each forecast against its own target but does not directly optimize cross-sectional ranking. Under low predictability, the resulting calibrated point forecast can have low amplitude. Rescaling can change forecast amplitude but cannot recover missing ranking information. In contrast, directly optimizing IC improves ranking but does not constrain forecast scale; in our experiments, this produces forecasts over 20 times the target amplitude (Figure 1(a)). This tradeoff suggests that calibration and cross-sectional ranking should be considered jointly rather than evaluated through either objective alone. Based on this analysis, we introduce CalibRank, which combines MSE with cross-sectional correlation, withλcontrolling their relative importance. We test these theoretical claims using Finance1K, a panel of hourly observations for 1,000 US equities with aligned return and traded-volume targets. Its matched targets allow us to compare a low-predictability target with a more predictable target while keeping the forecasting setting fixed. We use synthetic data with known signal and noise to verify the theoretical relations directly, and varyλto trace the calibration-ranking tradeoff predicted by the analysis. We then test whether the same behavior holds across twelve forecasting architectures and examine the relation between forecast amplitude and achieved predictability for TimesFM and Chronos across 97 public GIFT-Eval configurations (Figure 1(c)). These experiments serve as tests of the 2 theoretical claims and of CalibRank as a practical way to balance calibration and cross-sectional ranking. More broadly, forecast collapse exposes a limitation of conventional time-series evaluation. When a multivariate panel is evaluated as independent series, per-series accuracy can remain informative while the cross-series structure used by downstream decisions is never measured. Forecasting benchmarks should therefore preserve and evaluate this structure when it matters to the downstream task. Our contributions are: • We identify forecast collapse, low forecast amplitude together with poor cross-sectional structure, and show a sharp target contrast between low-predictability stock returns and higher-predictability traded volume under a matched forecasting protocol. • We provide a theoretical explanation for forecast collapse by characterizing two distinct mechanisms. Predictable variance limits the amplitude of calibrated point forecasts, while per-series risk leaves cross- series coupling unidentified. We derive the exact relation between best-scaled amplitude, forecast-target correlation, and predictable variance. • We characterize the resulting calibration-ranking tradeoff and introduce CalibRank, a simple objective that balances calibration and cross-sectional ranking. We validate the predicted tradeoff across twelve forecasting models and show that CalibRank improves cross-sectional correlation for every model while maintaining control of forecast scale. • We release Finance1K, an hourly panel of 1,000 US equities with aligned return and volume targets, which supports controlled comparisons of target predictability and training objectives under the same forecasting protocol. 2 Related Work Ceilings and collapse. Green et al. (2026) studies how mean-optimal forecasting under-disperses the marginal law, which is what a raw amplitude below one records. Andreoletti (2026) independently documents low-amplitude transformer forecasts on returns. Our analysis relates best-scaled amplitude to a declared predictability ceiling and separately measures cross-sectional ordering. Zhang (2026) bounds out-of-sampleR 2 through hit rates, while ours uses predictable variance. Related variance-share limits appear in psychometrics and ecological forecasting (Shrout and Fleiss, 1979; Pennekamp et al., 2019). Classical measurement-error theory supplies the attenuation identity used in our derivation (Fuller, 1987; Carroll et al., 2006). Here it connects a familiar statistical limit to best-scaled forecast amplitude and separates that quantity from raw model scale. Related ideas in other domains. Image restoration has shown that distortion and perceptual quality can trade off along a frontier (Blau and Michaeli, 2018; Freirich et al., 2021), motivating structural criteria beyond pixelwiseL 2 (Wang et al., 2004; Wang and Bovik, 2009). This is a useful precedent for evaluating more than pointwise error, but the quantities differ: its perception index is a divergence between marginal laws, whereas IC measures dependence between forecasts and targets. Deterministic weather models provide a second parallel: training toward a conditional mean can produce smooth, low-amplitude fields, which has motivated generative ensembles that preserve variability (Price et al., 2025). Regression and ranking objectives. Composite regression-ranking losses are established in stock prediction and information retrieval. Feng et al. (2019) combines squared error with a pairwise hinge, Lin et al. (2026) optimizes rank correlation, and Yang et al. (2020) provides correlation losses in a forecasting toolkit. Differentiable ranking operators are also well developed (Blondel et al., 2020; Swezey et al., 2021). Bai et al. (2023) shows that carefully designed listwise objectives can reduce conflicts between regression and ranking. These results motivate stronger composites that may improve on the simple scalarization evaluated here. Section 3.2 places this comparison in the current TSFM landscape. Fifteen surveyed models disclose a forecasting loss, four use a squared-error-like loss, and the remainder use quantile, likelihood, cross-entropy, or flow-matching objectives. None of the eighteen sourced training objectives scores a whole cross-section 3 at once. The issue is therefore broader than MSE: point forecasts have different calibration properties, but per-series scoring leaves the same multivariate axis absent. Joint and foundation-model forecasting. Probabilistic forecasters such as GPVar and TACTiS model richer dependence directly (Salinas et al., 2019; Drouin et al., 2022). CalibRank addresses the narrower cross- sectional functional the downstream decision uses. Current TSFMs span autoregressive, masked-reconstruction, and patch-based objectives (Das et al., 2024; Ansari et al., 2024; Woo et al., 2024; Goswami et al., 2024; Ekambaram et al., 2024; Rasul et al., 2023), yet the sourced objectives in our survey do not score the cross-section jointly. The competitiveness of simple forecasting baselines (Zeng et al., 2023) further supports evaluating objectives and protocols independently of architectural scale. 3 Preliminaries This section fixes the objects that are easy to conflate in panel forecasting: the information available before a prediction, the predictable component of the target, the point forecast a model outputs, and the cross-sectional functional evaluated downstream. It then places these objects against the training objectives used by current time-series foundation models. 3.1 Forecasting panel and evaluation unit At forecast timestampt, theNseries form one cross-section. A model uses the history beforetto produce one point forecastˆy it for each realized targety it . We collect the targets and forecasts into vectors and define temporal reporting from timestamp-level statistics: y t := (y 1t ,...,y Nt ) ⊤ , b y t := (ˆy 1t ,..., ˆy Nt ) ⊤ , g t := g( b y t , y t ), g := 1 T T X t=1 g t . (1) Thus the cross-section at one timestamp is the evaluation unit, and timestamps are the replication unit. In this paper,g t includes cross-sectional correlation, raw amplitude, and per-timestamp squared error. For a complete panel, averaging per-timestamp squared error is equivalent to averaging all scalar errors. Correlation is different: pooling all (i,t) pairs would mix temporal and cross-sectional variation and would no longer score the ordering available to a decision made at timestamp t. The notation also separates three layers of the argument. The decompositiony it =μ it +ε it describes a target-side predictability limit under the declared information set. The model errorδ it =ˆy it − μ it describes alignment with that predictable component. Finally,IC t , raw amplitude, and best-scaled amplitude describe different properties of the forecast itself. A forecast can therefore have weak scale, weak ordering, or both; the paper does not treat these as the same failure. Table 1 groups the recurring symbols by these roles. Subscripts are suppressed when a statement concerns a generic forecast-target pair. 3.2 Training objectives of time-series foundation models Table 2 audits the objective stated in each source paper. The survey covers nineteen models and eighteen sourced objectives; TimeGPT names no loss function and is omitted from the table. Fifteen models disclose a forecasting loss, of which four are squared-error-like. The entries excluded from that denominator train masked reconstruction or fit no loss at all. Three objectives carry more than one term: Time-MoE adds mixture-of-experts load balancing, TSPulse a cross-entropy term on the log-magnitude spectrum, and Toto a Cauchy robust point term. The table marks both cases. The table asks two questions of each objective. The per-series columns record which conditional quantity of a single series it identifies. A scoring function is consistent for a functional when that functional minimizes its expected score (Gneiting, 2011). Squared error and its robust surrogates pin the conditional mean. A pinball grid pins the levels it contains, so it delivers the median exactly when 0.5 lies on the grid, which it does for all six quantile-trained models. Likelihood, cross-entropy, flow-matching, and in-context prior fitting fit a 4 Panel, signal, and evaluationForecast behavior and optimization SymbolDefinition and roleSymbolDefinition and role i,t,N,T,LSeries/time indices; numbers of series and evaluated timestamps; context length. ρAlignment corr(ˆy,μ) with the predictable component. y t ,by t Realized and forecast cross-sections at timestamp t. IC t , ICWithin-timestamp Pearson correlation and its signed temporal mean. F t−1 Information available to every admissible forecast before t. e A t ,A ⋆ Raw output scale and best-scaled amplitude; only the latter obeys the amplitude-correlation identity. g t , gTimestamp-level statistic and temporal average in (1). s ⋆ MSE-optimal slope for a centered forecast. μ it ,ε it ,σ 2 i Conditional mean, innovation, and irreducible variance E[ε 2 it ]. L,λComposite objective and calibration-ranking tradeoff. δ it ,Q i Error ˆy it − μ it and reducible error Q i = E[δ 2 it ]. B,MBatch timestamps and scalar errors; M = N|B| for a complete panel. SNR,R 2 max , IC max Signal-to-noise ratio, forecastable variance share, and correlation ceiling under F t−1 . c t Predictability proxy used only by the weighted ablation. Table 1 Notation. Quantities indexed bytare computed within a cross-section before temporal averaging. Raw amplitude e A t is distinct from best-scaled amplitude A ⋆ . predictive distribution, off which the mean, the median, and arbitrary quantiles can all be read. Likelihood and cross-entropy are strictly proper scoring rules (Gneiting and Raftery, 2007); flow matching regresses a velocity field rather than a value, so the same quantities come from its draws. The last column asks whether the objective scores a whole cross-section at once, and it answers the same way for every row: in the sourced formulations, the training score breaks into a sum over individual series or forecast tokens, with no term that depends on two series together, so nothing in it scores the cross-section of(1). The families therefore differ widely in what they identify about a single series and agree completely in what they leave out. The survey is about scope, not performance. Forecast quality is itself multi-axis, maximizing sharpness subject to calibration (Gneiting et al., 2007), and a distributional objective can represent conditional uncertainty even when a single point forecast has low amplitude. Once that forecast is used by a cross-sectional decision, however, its output scale and within-timestamp ordering remain separate quantities that must be measured. Section 4 formalizes the resulting predictability ceiling and coupling gap, and Section 6 introduces the objective used to score the missing functional. 4 Why Forecasts Collapse Call an objective pointwise when its population risk is a sum of per-series, per-timestamp terms,R(ˆy) = P i,t E[ℓ(ˆy it ,y it )]. Squared error, absolute error, quantile, likelihood, and cross-entropy losses commonly have this form. Low predictability constrains the amplitude of a calibrated point forecast. Pointwise scoring creates a separate identification gap for cross-series dependence. 4.1 Predictability and best-scaled amplitude All quantities in this subsection use the same sampling distribution. For the Finance1K analysis, the distribution is one timestamp’s cross-section; empirical metrics are averaged only after the per-timestamp quantities are computed. We suppress the timestamp index below. Let y = μ + ε, μ = E[y |F t−1 ],E[ε|F t−1 ] = 0,(2) whereF t−1 is the information available to every competing forecast. The law of total variance gives, with SNR := Var(μ)/E[Var(y |F t−1 )], R 2 max := Var(μ) Var(y) = SNR 1 + SNR ,IC max := p R 2 max .(3) 5 Table 2 Training objectives of surveyed TSFMs. No sourced objective scores the cross-section jointly. † Not counted among the fifteen disclosed forecasting losses. ‡ Multi-term objective. Identified per seriesScored jointly ModelLossMean Median Quantiles Dist. Cross-section TimesFM (Das et al., 2024)Squared error✓× TTM (Ekambaram et al., 2024)Squared error✓× Timer (Liu et al., 2024b)Squared error✓× Time-MoE (Shi et al., 2025)Squared error ‡ ✓× MOMENT † (Goswami et al., 2024)Squared error✓× TSPulse † (Ekambaram et al., 2026) Squared error ‡ ✓× Chronos-Bolt (Amazon Science, 2024) Quantile✓× Chronos-2 (Ansari et al., 2025)Quantile✓× TiRex (Auer et al., 2025)Quantile✓× Kairos (Feng et al., 2025)Quantile✓× YingLong (Wang et al., 2025)Quantile✓× FlowState (Graf et al., 2026)Quantile✓× Moirai (Woo et al., 2024)Likelihood✓× Lag-Llama (Rasul et al., 2023)Likelihood✓× Toto (Cohen et al., 2025)Likelihood ‡ ✓× Chronos (Ansari et al., 2024)Cross-entropy✓× Sundial (Liu et al., 2025)Flow matching✓× TabPFN-TS † (Hoo et al., 2026)Prior fit✓× This is the forecastable share of variance under the declared information set. For anF t−1 -measurable forecast ˆy, write ρ = corr(ˆy,μ). Orthogonality of ε to F t−1 yields corr(ˆy,y) = ρ IC max .(4) This is the standard attenuation identity (Fuller, 1987; Carroll et al., 2006). For a centered forecast-target pair, the scalar minimizing E[(sˆy− y) 2 ] is s ⋆ = Cov(ˆy,y)/ Var(ˆy). Proposition 4.1 (Amplitude after optimal rescaling). For any centered (ˆy,y) with Var(ˆy) > 0, A ⋆ := sd(s ⋆ ˆy) sd(y) =|corr(ˆy,y)|.(5) If, in addition, (2) holds and ˆy is F t−1 -measurable, then A ⋆ ≤ IC max . Proof sketch.Substitutings ⋆ givessd(s ⋆ ˆy)/sd(y) =|Cov(ˆy,y)|/(sd(ˆy)sd(y)). Under the additional conditions, the bound follows from (4) and |ρ|≤ 1. The equality is the ordinary-least-squares slope identity in standard-deviation units. It applies per timestamp asA ⋆ t =|IC t |. The experiments report two different descriptive quantities: signed meanIC t and the raw amplitude e A t := sd i (ˆy it ) sd i (y it ) .(6) No identity equates e A t withIC t . Squared-error training approaches the optimal scaling only to the extent that the fitted model approaches its population optimum. The distinction also clarifies what the proposition predicts. It constrains the amplitude after the best scalar 6 recalibration of a given forecast. A model may output any raw scale before recalibration, including the large values produced by a pure correlation loss. Comparing raw amplitude with correlation can diagnose residual scale error, but a ratio of their time averages is not an estimate ofs ⋆ t : the slope is timestamp-specific and involves a ratio of covariance to variance. For pretrained models with different objectives, raw amplitude is an empirical diagnostic rather than the left-hand side of (5). Section 8 tests these empirical implications on Finance1K, synthetic panels, and public foundation-model benchmarks. The amplitude argument also extends beyond squared-error training. Of the nineteen foundation models surveyed in Section 3.2, fifteen disclose a forecasting objective and only four are squared-error-like. Proposition 4.2 (Mean and median point forecasts). Supposey |F t−1 =μ+σZfor a fixed distribution Zsymmetric about zero with finite mean, and a scaleσcommon across series. The conditional mean and median equal μ, so their optimally scaled forecasts satisfy (5). Proof sketch.Symmetry about zero and a finite mean giveE[Z] = 0, and zero is a median ofZ. Both forecasts are therefore μ, and Theorem 4.1 applies to the pair (μ,y). Section 5 gives the details. The result concerns a point forecast. A well-specified predictive distribution can retain the target’s conditional spread even when its mean or median is small. Series-varying scale and off-median quantiles require separate treatment, given in Section 5. The two conditions are distinct. Low predictability is the condition for amplitude attenuation, rather than a complete explanation of forecast collapse. A low-signal univariate target can produce a small but well- calibrated point forecast without any cross-sectional question. Conversely, a highly predictable multivariate target can have well-scaled marginal forecasts while its joint ordering is wrong. The pattern in Figure 1(a) combines the two: weak predictable variance limits the calibrated amplitude, and per-series scoring leaves the cross-series relationship underdetermined. 4.2 Per-series risk leaves coupling unidentified Amplitude is one part of collapse. Cross-sectional ordering depends on the joint behavior of all series at one timestamp. Proposition 4.3 (Insensitivity to cross-series coupling). LetRbe pointwise. ThenRis a functional of the per-coordinate joint lawsLaw(ˆy it ,y it )alone. It is unchanged when those pairs are recoupled across iwithout changing any per-coordinate law, whereasIC t can change. Hence equal-risk forecasters can have different cross-sectional correlation. Proof sketch.Every summand ofR(ˆy) = P i,t E[ℓ(ˆy it ,y it )] is an expectation of a function of the single pair (ˆy it ,y it ), soRonly reads the per-coordinate laws. Replacing the panel’s joint law by another one under which every pair keeps its law therefore leavesRfixed. Cross-sectional correlation is computed from the two vectors (ˆy it ) i and (y it ) i at one timestamp, so it depends on exactly the coupling that such a replacement is free to move. Theorem 5.4 carries out the replacement explicitly. A two-series Gaussian construction holds squared-error risk at 1.40 while changingE[IC t ] from 0.371 to 0.032; the explicit covariance construction and admissibility conditions appear in Section 5. This is an invariance result. With sufficient capacity, the population squared-error minimizerˆy=μcan order the cross-section well. The proposition says that per-series risk alone does not identify the coupling, which motivates measuring and rewarding the cross-sectional functional used downstream. The language of coupling is useful because marginal accuracy does not determine a multivariate forecast. By Sklar’s theorem (Sklar, 1959), a joint law separates into marginal distributions and a copula carrying 7 dependence. Pointwise evaluation probes the per-series forecast-target laws; it does not fully determine the dependence among forecast-target pairs across series. CalibRank does not estimate a copula. It rewards one observable functional of that dependence, the per-timestamp cross-sectional correlation used by the downstream decision. Models that require calibrated scenarios or tail dependence need a richer joint objective. Implication for optimization. The population MSE solutionE[y |F t−1 ] can rank the cross-section as well as the information set allows. The invariance result applies when optimization selects among finite-model solutions with similar marginal risk. Optimization, regularization, pretraining, and shared representations choose among those solutions without a guarantee on the downstream functional. A cross-sectional loss supplies that preference explicitly. The empirical comparisons show that it improves ordering under the matched protocol studied here. 5 Formal Foundations and Proofs This section carries the assumptions, the full proofs of the results stated in Section 4, and the explicit construction behind Theorem 4.3. The classical direction law closes the section as an illustrative consequence. Only the assumptions used throughout are collected here; one needed by a single result is stated in the subsection that proves it, so each result carries its own hypotheses where they are used. 5.1 Assumptions A1 Conditional-mean decomposition: μ = E[y |F t−1 ] and ε = y− μ, so E[ε|F t−1 ] = 0. A2 Admissible forecast: ˆy is F t−1 -measurable. Therefore δ := ˆy− μ is also F t−1 -measurable. A3 Weak signal: SNR ≪ 1, which holds for hourly returns and not for volume. Every claim below is also checked on a synthetic panel that satisfies(2)directly by drawingμ,ε, andδwith prescribed variances, so the quantities are exact ground truth rather than estimates and no market data enters. 5.2 Additivity of squared error Proposition 5.1 (Additive decomposition). Under A1 and A2,MSE=E[(ˆy − y) 2 ] =Q+σ 2 with Q = E[δ 2 ] and σ 2 = E[ε 2 ]. Proof.Step 1, the residual carries no signal. By A1 the target splits asy=μ+ε, and measuring the forecast against the same conditional mean,δ:=ˆy− μ, writes it asˆy=μ+δ. Subtracting,μappears in both and cancels: ˆy− y = (μ + δ)− (μ + ε) = δ− ε.(7) This is the point of the decomposition. Whatever the forecast knows about the conditional mean has already been spent; what is left is the model’s own error δ set against the unpredictable part ε. Step 2, square and expand. Squaring (7) and taking expectations gives three terms: MSE = E (δ− ε) 2 = E[δ 2 ]− 2 E[δε] + E[ε 2 ] = E[δ 2 ]− 2 Cov(δ,ε) + E[ε 2 ]. (8) The last line replaces the raw cross moment by a covariance, which is legitimate becauseE[ε] =E E [ε| F t−1 ] = 0 by the tower property, so the two agree. 8 Step 3, the cross term vanishes. Here A2 does the work. BecauseδisF t−1 -measurable it passes out of the inner conditional expectation as a constant: E[δε] = E[δ E[ε|F t−1 ]] = 0. (9) A forecast that used information outsideF t−1 would break exactly here:δcould then correlate withε, and MSE would pick up a cross term of either sign. Step 4, collect. With the cross term gone,MSE=E[δ 2 ] +E[ε 2 ] =Q+σ 2 . Both parts are nonnegative and onlyQresponds to the forecast, soσ 2 is a floor no admissible forecast can get under. The decomposition is an exact identity, not an approximation, under the declared information set. This is the classical reducible and irreducible split of the bias-variance decomposition (Geman et al., 1992; Hastie et al., 2009). Root mean squared error is a monotone transform and ranks identically. The next result needs one further condition on the cross-section, used nowhere else. Assumption for Theorem 5.2 A4 Heteroscedastic dispersion: Var i (σ 2 i )≫ Var i (Q i ) (Bollerslev, 1986). Corollary 5.2 (Squared-error rankings can be dominated by irreducible noise). Under A4, if Var i (Q i )≪ Var i (σ 2 i ) across series, then corr i (MSE i ,σ 2 i )→ 1. Proof.Step 1. By Theorem 5.1 each series satisfiesMSE i =Q i +σ 2 i , so the quantity in question is the correlation of a sum with one of its own two parts. Bilinearity splits the numerator accordingly: Cov i (MSE i ,σ 2 i ) = Cov i (Q i ,σ 2 i ) + Var i (σ 2 i ).(10) A series is therefore ranked by squared error partly on its model error and partly on its noise level, and the second term is present no matter how the model behaves. Step 2. Write v := Var i (Q i ) Var i (σ 2 i ) ,(11) which is well defined because A4 makes the denominator positive, and small precisely when A4 holds. It measures how much the reducible part varies across series relative to the irreducible part. Step 3. Cauchy-Schwarz applied to the two centered cross-sectional variables gives|Cov i (Q i ,σ 2 i )| ≤ sd i (Q i )sd i (σ 2 i ) = √ v Var i (σ 2 i ). The covariance is thusO( √ v) once measured in units ofVar i (σ 2 i ), and the same bound controls the denominator throughVar i (MSE i ) =Var i (σ 2 i ) 1 + 2Cov i (Q i ,σ 2 i )/ Var i (σ 2 i ) +v . Step 4. Dividing(10)by the two standard deviations and factoringVar i (σ 2 i ) out of both puts every term on one scale: corr i (MSE i ,σ 2 i ) = Cov i (Q i ,σ 2 i ) + Var i (σ 2 i ) p Var i (Q i + σ 2 i ) Var i (σ 2 i ) = 1 + O( √ v) p 1 + O( √ v) + v ,(12) which tends to 1 as v → 0. Step 5. The approach isO( √ v), notO(v). A small dispersion inQ i still reaches the numerator through its covariance withσ 2 i , which carries one power ofsd i (Q i ), whereas its own variance carries two. The gap therefore closes slowly: drivingvdown by a factor of four only halves it, so a squared-error ranking stays close to a noise ranking well after the model errors have been made comparable. Thus heterogeneous noise can dominate a cross-sectional squared-error ranking of series. This diagnostic follows the tradition of metric-pathology analyses (Hyndman and Koehler, 2006). 9 5.3 Attenuation and the amplitude identity Proposition 5.3 (Attenuation). Under A1 and A2,IC=ρ p SNR/(1 + SNR), whereSNR= Var(μ)/E[Var(y |F t−1 )]. Proof.Step 1. Splity=μ+εand use bilinearity. The second piece vanishes:ˆyisF t−1 -measurable by A2, soE[ˆyε] =E ˆy E[ε|F t−1 ] = 0 by A1, andE[ε] = 0 by the same assumption, so the covariance is the product moment. Hence Cov(ˆy,y) = Cov(ˆy,μ) + Cov(ˆy,ε) = Cov(ˆy,μ). (13) A forecast can only covary with the target through the predictable part. No amount of skill lets it reach ε. Step 2. The law of total variance givesVar(y) =Var(μ) +E[Var(y |F t−1 )], so the target’s spread is the predictable spread plus the average conditional spread. Only the first is available for a forecast to match. Step 3. Dividing(13)by the two standard deviations and insertingsd(μ) top and bottom separates the two effects: IC = Cov(ˆy,y) sd(ˆy) sd(y) = Cov(ˆy,μ) sd(ˆy) sd(y) = corr(ˆy,μ) |z ρ · sd(μ) sd(y) | z IC max . (14) The first factor is forecast skill relative to the conditional mean, and a model can move it. The second is fixed by the target and the declared information set, and no model can. Step 4. Substituting Step 2 intoIC max and dividing numerator and denominator byE[Var(y | F t−1 )] turns the ceiling into the stated function of the signal-to-noise ratio: IC max = s Var(μ) Var(μ) + E[Var(y |F t−1 )] = r SNR 1 + SNR ,(15) soIC=ρ p SNR/(1 + SNR). Under A3 the ceiling behaves like √ SNR, and the square root matters: it leaves the ceiling far above the signal-to-noise ratio itself, soSNR= 0.01 still permitsIC max ≈0.10. The multiplicative factor is the reliability ratio of errors-in-variables regression (Fuller, 1987; Carroll et al., 2006), with roots in Spearman (1904). The amplitude identity is a statement about slopes through the origin, so it needs one condition that the results above do not. Assumption for Theorem 4.1 A5 Centering: variables are centered, so the rescaling is a slope through the origin. Proof of Theorem 4.1.Step 1. Expand the objective ins. Under A5 both variables are centered, so second moments are variances: E (sˆy− y) 2 = s 2 Var(ˆy)− 2s Cov(ˆy,y) + Var(y).(16) The right-hand side is a parabola inswhose leading coefficientVar(ˆy) is strictly positive, so it is convex and its stationary point is the unique minimizer. Setting the derivative 2s Var(ˆy)−2Cov(ˆy,y) to zero 10 gives s ⋆ = Cov(ˆy,y) Var(ˆy) .(17) This is the least-squares slope ofyonˆy, sos ⋆ is the single best post-hoc rescaling available to any user of the forecast: it is what a practitioner would recover by regressing realized values on predictions. Step 2. Scaling multiplies the standard deviation by|s ⋆ |, and one factor ofsd(ˆy) then cancels against the variance: A ⋆ = sd(s ⋆ ˆy) sd(y) = |s ⋆ | sd(ˆy) sd(y) = |Cov(ˆy,y)| Var(ˆy) · sd(ˆy) sd(y) = |Cov(ˆy,y)| sd(ˆy) sd(y) =|corr(ˆy,y)|. (18) No model has been used: this holds for any centered pair withVar(ˆy)>0. The identity is worth stating in words. Once a forecast has been given its best possible scaling, the amplitude it retains is not a separate property to be tuned; it is exactly its correlation with the target. A forecast cannot be both weakly correlated and correctly scaled. Step 3. Here the conditional-mean decomposition enters. By Theorem 5.3 the correlation factors as corr(ˆy,y) = ρ IC max , and |ρ|≤ 1 because ρ is itself a correlation. Substituting into Step 2, A ⋆ =|corr(ˆy,y)| =|ρ| IC max ≤ IC max .(19) The ceiling depends only on the target and the declared information set, so it binds every admissible forecast at once. Improving the model moves |ρ| toward one and nothing else. The identity and the bound require different assumptions Step 2 is an identity about least squares and holds universally. Step 3 is where predictability enters through the conditional-mean decomposition. The best-scaled amplitude identity remains valid for any centered forecast-target pair, while the ceiling depends on the declared information set. Centering and attribution The equality is the classical ordinary-least-squares slope identity written in standard-deviation units (Fuller, 1987; Carroll et al., 2006). With an intercept, or with variables centered as in A5, a constant offset inδis absorbed. A slope fit through the origin on uncentered variables has a different minimizer. Thus a best-scaled centered forecast with weak correlation necessarily has weak amplitude. 5.4 Mean and median point forecasts This subsection is the only place the conditional law is given a shape. Assumption for Theorem 4.2 A6 Location family symmetric about zero, with finite mean and a common conditional scale. Proof of Theorem 4.2.Setup. Under A6 the conditional law ofygivenF t−1 is a location-scale family, y=μ+σZ, whereZis symmetric about zero with finite mean andσ >0 is the common conditional scale. Both candidate forecasts are read off this one law. 11 Step 1. Symmetry meansZand−Zhave the same law, soE[Z] =E[−Z] =−E[Z], and the finite mean in A6 makes this subtraction legitimate, forcingE[Z] = 0. SinceμandσareF t−1 -measurable, linearity gives E[y |F t−1 ] = μ + σE[Z] = μ. Step 2. The same symmetry givesPr(Z ≤0) =Pr(Z ≥0), and these two probabilities sum to at least one because together they cover the line, so each is at least 1 2 . That is the definition of zero being a median ofZ. The mapz 7→ μ+σzis strictly increasing forσ >0, and a strictly increasing map carries medians to medians, so q 1/2 (y |F t−1 ) = μ + σ· 0 = μ. Step 3. The two forecasts therefore coincide: both are the sameF t−1 -measurable variableμ. Nothing distinguishes a mean-optimal from a median-optimal point forecast here, so choosing absolute error over squared error cannot escape the conclusion. Applying Theorem 4.1 to the centered pair (μ,y) gives A ⋆ =|corr(μ,y)|. The bound is attained by these forecasts The forecastˆy=μhasρ=corr(μ,μ) = 1, so Theorem 5.3 turns the inequality of Theorem 4.1 into the equalityA ⋆ =IC max . The best case for a point forecast under A6 is the correlation ceiling itself, and A3 is what makes that ceiling small. The common conditional scale in A6 does no work atτ= 1 2 , since a series-varying σ i leaves both forecasts at μ i ; it binds away from the median. Off-median quantiles under a series-varying scale If the conditional scale varies across series, the location-scale form givesq τ (y i |F t−1 ) =μ i +σ i q τ (Z). For τ ̸= 1 2 the termσ i q τ (Z) is not constant acrossi, so the forecast is a blend ofμ i andσ i and Theorem 4.2 does not apply. This is the same ingredient as Theorem 5.2, where cross-sectional variation inσ i drives a ranking. Its effect on ordering depends on the relative dispersion and dependence of μ i and σ i . 5.5 Insensitivity to cross-series structure Proof of Theorem 4.3. Step 1. A pointwise risk is a sum of per-coordinate terms, R(ˆy) = X i,t E ℓ(ˆy it ,y it ) ,(20) and each summand is the expectation of a function of the single pair (ˆy it ,y it ). Whenever it exists it is therefore an integral ofℓagainstLaw(ˆy it ,y it ) and against nothing else. Summing overiandtadds such terms without introducing any new dependence, soRis a functional of the collection of pair laws Law(ˆy it ,y it )alone. Squared error, absolute error, quantile loss and every other per-observation loss enter here on the same footing; the argument uses only the shape of (20). Step 2. Call a panel lawP ′ a recoupling ofPwhen every pair (ˆy it ,y it ) has the same law under both. Step 1 then givesR(P) =R(P ′ ) for any recoupling, becauseRnever queries the joint behavior of two distinct coordinates. The recouplings of a givenPare exactly the joint laws having those pair laws as prescribed margins. That set collapses to a single point only in degenerate cases; in general it is large, and Theorem 5.4 constructs a one-parameter family of its members explicitly. Risk is thus constant on a set of panel laws that is typically infinite. Step 3. Cross-sectional correlation is computed from the two vectors (ˆy it ) i and (y it ) i observed at a single timestamp, so its law depends on the joint distribution acrossi. In particular it depends on the cross termsCov(ˆy it ,y jt ) fori̸=j, which describe how one series’ forecast relates to another series’ outcome, and which no pair law records. Those are precisely the quantities a recoupling is free to move while Step 2 holds R fixed. 12 Step 4. Combining the two, a recoupling can changeIC t while leavingRunchanged, so equal risk does not determine cross-sectional correlation and two forecasters with identical pointwise risk can rank the cross-section differently. Theorem 5.4 exhibits this concretely: the risk stays at 1.40 exactly whileE[IC t ] ranges over an order of magnitude. Example 5.4 (Fixed risk, varying cross-sectional correlation). TakeN= 2 and let (ˆy 1 ,y 1 , ˆy 2 ,y 2 ) be centered joint Gaussian with unit variances,corr(ˆy i ,y i ) =rfor both coordinates,corr(ˆy 1 , ˆy 2 ) = corr(y 1 ,y 2 ) = 0, and one free cross parameter g = Cov(ˆy 1 ,y 2 ) = Cov(ˆy 2 ,y 1 ). Each coordinate’s joint law is (ˆy i ,y i ) bivariate normal with correlationr, and is therefore the same for every admissibleg. Squared-error risk depends only on that law, sinceE[(ˆy i − y i ) 2 ] = 2(1− r), so it is constant in g. The cross-sectional correlation is not. With two series it is the sign of (ˆy 1 − ˆy 2 )(y 1 − y 2 ), and corr(ˆy 1 − ˆy 2 , y 1 − y 2 ) = 2r− 2g 2 = r− g,(21) soE[IC t ] moves withgwhile the risk does not. Atr= 0.3, sweepinggover−0.25,0,0.25leaves the squared-error risk at 2(1− r) = 1.40 exactly whileE[IC t ] = 2 π arcsin(r− g) takes the values 0.371, 0.194, and 0.032: an order of magnitude, at identical loss. Interpretation With enough capacity, the population MSE minimizer isˆy=μand can order the cross-section as well as the information set allows. The proposition establishes invariance across equal-risk couplings: pointwise risk alone does not select the one with betterIC. A cross-sectional term resolves that indifference for its chosen functional; alternative objectives may do so with a different number or form of terms. 5.6 The direction law The Gaussian direction law translates a correlation into the hit rate often reported in financial forecasting. Lemma 5.5 (Gaussian orthant probability). For a standard bivariate normal pair (U,V) with correlation r, Pr(U > 0,V > 0) = 1 4 + 1 2π arcsinr. This is Sheppard’s orthant identity (Sheppard, 1899); see also Kruskal (1958). His rotational-invariance argument runs as follows. Proof of Theorem 5.5.Step 1. WriteV=rU+ √ 1− r 2 Wwith (U,W) a pair of independent standard normals. This reproduces the required moments, sinceVar(V) =r 2 + (1− r 2 ) = 1 andCov(U,V) = r Var(U) =r, so (U,V) has the law in the statement and the probability may be computed in the (U,W) plane instead. Step 2. The law of (U,W) is a standard bivariate normal with independent coordinates, whose density depends only on∥(u,w)∥. It is therefore rotationally invariant, and the probability of any cone with apex at the origin is its opening angle divided by 2π. The problem reduces to measuring one angle. Step 3. In the (U,W) plane the eventU >0, V >0is the intersection of two half-planes through the origin, with inward normalsn 1 = (1,0) andn 2 = (r, √ 1− r 2 ). Two half-planes whose normals meet at angle θ cut out a wedge of angle π− θ, and here cosθ = n 1 · n 2 = r, so θ = arccosr. Hence Pr(U > 0,V > 0) = π− arccosr 2π = 1 4 + 1 2π arcsinr,(22) using arccosr = π 2 − arcsinr. 13 Step 4. Three reference values confirm the sign convention:r= 0 returns the quadrant probability 1 4 , r = 1 returns 1 2 because the two events coincide, and r =−1 returns 0 because they are disjoint. Turning the lemma into a statement about a forecast and its target requires distributional structure that nothing earlier in the section assumed. Assumption for Theorem 5.6 A7 Joint Gaussianity of the forecast and the target, used only for the exact direction law. Proposition 5.6 (Direction law). Under A7 with zero means, Pr(sign ˆy = signy) = 1 2 + 1 π arcsin(IC). Proof.Step 1. Dividing by a positive constant leaves signs alone, sosign ˆy=sign(ˆy/sd(ˆy)) and likewise fory. Replace the pair by (ˆy/sd(ˆy), y/sd(y)), which under A7 with zero means is a standard bivariate normal pair whose correlation is IC. Nothing about the event has changed. Step 2. Signs agree exactly onˆy >0,y >0∪ˆy <0,y <0, a disjoint union. The tiesˆy= 0∪y= 0 have probability zero, since each is a line in a plane carrying a density, so they may be ignored. Step 3. A centered Gaussian law is invariant underz 7→−z, which maps the second event onto the first, so the two carry equal probability and the total is twice either one. Theorem 5.5 withr=ICthen evaluates it: Pr(sign ˆy = signy) = 2 Pr(ˆy > 0,y > 0) = 1 2 + 1 π arcsin IC.(23) AtIC= 0.05 this is a hit rate of 51.6%, which is the sense in which a correlation that looks negligible is still economically meaningful, and equally the sense in which a hit rate near one half is consistent with real skill. 6 Calibrated Cross-Sectional Ranking Proposition 4.3 motivates scoring the cross-sectional functional directly while retaining a calibration anchor. Our version, CalibRank, combines squared error with differentiable per-timestamp Pearson correlation: L = 1 M X i,t (ˆy it − y it ) 2 − λ |B| X t∈B c IC t .(24) The first term fixes scale; the second rewards ordering at each timestamp. Other calibration and ranking surrogates fit the same construction, as detailed in Section 6.1. Sweepingλtraces the calibration-ranking frontier in Figure 1(b). 6.1 Relation to Existing Objectives Theorem 4.3 motivates a two-term objective while leaving the specific choice open. Writing the objective as L=C(ˆy,y;w)− λS(ˆy,y;w), there are four independent axes. Table 3 places published alternatives and the evaluated CalibRank variants in this space. The endpoints of theλrange correspond to the two single-term objectives, which are trained separately and reported as comparators. Huber calibration is a natural extension because robust fidelity terms can reduce the influence of large target innovations. 14 AxisChoicesPrior artOurs Calibration C Squared error, absolute error, Huber, quantileFeng et al. (2019) squared error Squared error structure SPearson, Spearman, pairwise hinge, listwiseFeng et al. (2019) hinge; Lin et al. (2026), Yang et al. (2020) rank correlation differentiable Pearson Weight wUniform, theory-derived c t , learned, inverse-variance Kendall et al. (2018) learned Uniform; c t variant Tradeoff λFixed, sweptFeng et al. (2019) fixed Swept, 7 points Table 3 Calibration-ranking objective design space. 7 Experimental Design 7.1 Finance1K Finance1K contains hourly observations for 1,000 US equities over 28,510 steps running from 2015 to early 2026. Its two aligned targets are the closing-price log return,log(p t /p t−1 ), and the log-volume change,log(v t /v t−1 ). The split is chronological: the first 19,957 steps, ending in late 2022, form the training set, and the following 8,553 form the test set. From the previous 96 hourly observations of one series, the task is to predict its target at the next step. Evaluation uses every fourth test step. The calendar grid contains 684 weekend and holiday evaluation steps with zero cross-sectional spread; all reported IC values use the remaining 1,455 trading hours. 7.2 Forecasters and matched training protocol The controlled reference encoder maps a length-L=96 context window of one series to a next-step scalar forecast for that same series: the window is split into 6 patches of length 16, each linearly embedded to widthd=128 with a learned positional term, encoded by a 3-layer Transformer (4 heads, Gaussian error linear unit activations, dropout 0.1), mean-pooled over patches, and passed through a LayerNorm+linear head (≈0.38M parameters). In the weighted CalibRank variant, the scalar proxyc t is concatenated to the pooled representation before the head. This controlled encoder is used only to isolate objective effects and is not proposed as a new forecasting architecture. Both loss terms are weighted byc t = q [ SNR t /(1 + [ SNR t ), computed from a 24-hour trailing volatility window and normalized on training statistics. The main controlled comparison uses the uniform objective in(24)without this input. A matched ablation finds no detectable contribution from the weighting. For the twelve standard trainable families, a shared two-layer scalar mixer applies the same conditioning without modifying the backbone. All supervised models receive the same univariate history for each series and share parameters across series. We train with AdamW at learning rate 3×10 −4 , weight decay 10 −4 , batch size 24 complete cross-sections, and 1,500 steps over seeds 11, 23, and 42. The objective is the only change in the controlled comparison. Foundation-model fine-tuning uses learning rate 10 −6 and one timestamp per batch. We report squared error, raw amplitude from(6), and Pearson IC computed within each trading hour and then averaged over hours. Computing IC after pooling all (i,t) pairs would mix temporal and cross-sectional variation and answer a different question. 7.3 Synthetic verification design The real targets do not reveal their conditional means, so neither best-scaled amplitude nor the predictability ceiling can be observed directly. We therefore generate panels with known signal, noise, and model error. The sweep spans two decades of signal-to-noise ratio and three levels of forecaster skill, with fitting and evaluation on disjoint draws. Section 8 reports the quantitative verification. 15 SNR σ med σ disp q frac q disp 10 −2 10 −1 10 0 Raw parameter value 0.0120.012 0.85 0.50.5 0.25 0.3 0.22 0.6 0.7 (a) CloseVolume 10 −2 10 −1 10 0 SNR 0.0 0.2 0.4 0.6 0.8 1.0 Amplitude (b) ρ= 1.0 ρ= 0.7 ρ= 0.4 10 −2 10 −1 10 0 SNR 0.0 0.2 0.4 0.6 0.8 1.0 IC (c) 10 −2 10 −1 10 0 10 1 Raw amplitude MSE IC CR 0.0274 25.3 1.73 (a) Close amplitude 0.0500.0750.1000.125 Trading IC MSE IC CR 0.046 0.124 0.126 (b) Close ordering 0.10.5 1 λ 10 −1 10 0 (c) Amplitude 0.10.5 1 λ 0.06 0.08 0.10 0.12 (d) IC 0.10.5 1 λ 10 −3 (e) MSE <.05 n=6 .05-.30 n=26 >=.30 n=65 0.0 0.2 0.4 0.6 0.8 Median amplitude (f) Foundation-model strata Chronos TimesFM 0.000.050.100.150.200.25 Held-out decision return MSE IC CR 0.052 0.143 0.170 (g) Decision return MSEICCR 0.1 0.2 0.3 0.4 0.5 Trading IC (h) Target contrast Close Volume Figure 2 Synthetic verification of the predictability mechanism. (a) Parameters for the low-SNR close and higher-SNR volume regimes. (b) Raw amplitude and (c) cross-sectional IC across SNR and forecaster-skill levels. Markers show three-seed means, bars one standard deviation, and dashed curves the analytical predictions. Generator specification. ForNseries overTtime steps, under the common-SNR design, the per-series noise scale is lognormal, the signal variance scales with it, and the model-error scale sets Q i and ρ i : σ i = exp logσ med +N (0,σ 2 disp ) ,v i = SNR· σ 2 i ,(25) μ ti =N (0, 1)· √ v i ,ε ti =N (0, 1)· σ i ,(26) τ i = √ q frac v i · exp N (0,q 2 disp ) ,δ ti =N (0, 1)· τ i ,(27) withy=μ+ε,ˆy=μ+δ, andδ ⊥ ε(satisfying Assumption A2). ThenQ i =τ 2 i andρ i = p v i /(v i + τ 2 i ). Each experiment draws an independent random number generator (RNG) stream keyed by experiment name from the global seed 11, so results are reproducible regardless of call order. Synthetic predictor. Each experiment usesˆy=μ+δ, built by adding controlled independent errorδto the true signalμ, so its signal alignmentρis set by the single knobq frac (withρ= 1 atq frac = 0). Known skill and noise make the propositions directly testable against ground truth. Real-model results appear in Section 9. Locked synthetic parameters. Figure 2(a) sets close as the low-SNR regime: its signal variance is only 1.2% of the noise variance and its larger dispersion creates heterogeneous noise floors across series. Volume has a 20×larger SNR and more homogeneous noise. This contrast isolates the predicted transition from MSE-IC decoupling to partial coupling without changing the generator or evaluation metrics. 7.4 Additional evaluation protocols Held-out decision return. The training objective uses Pearson IC, so we also evaluate a downstream functional that is absent from the loss. At each timestamp we subtract the cross-sectional mean from the forecast, so the resulting weights sum to zero, and divide by their total absolute size, so they sum to one in magnitude. Applying those weights to the realized returns gives a return per timestamp. The reported statistic is its mean divided by its standard deviation, the Sharpe ratio of that decision rule. These archived runs use the weighted controlled-encoder protocol shared by the cross-architecture experiment. Post-hoc amplitude sweep. The headline controlled comparison uses the uniform objective at the pre- specifiedλ= 1. Separately, we examine the earlier weighted-objective variant across six values ofλon the evaluation steps. The sweep is post hoc and does not select the headline test result. Computing infrastructure. Every run executes on a single NVIDIA RTX Pro 6000 with one visible CUDA device, and the OpenMP, MKL, OpenBLAS, and NumExpr thread counts are pinned to one before any numerical import so that a seeded run repeats exactly. 16 SNR σ med σ disp q frac q disp 10 −2 10 −1 10 0 Raw parameter value 0.0120.012 0.85 0.50.5 0.25 0.3 0.22 0.6 0.7 (a) CloseVolume 10 −2 10 −1 10 0 SNR 0.0 0.2 0.4 0.6 0.8 1.0 Amplitude (b) ρ= 1.0 ρ= 0.7 ρ= 0.4 10 −2 10 −1 10 0 SNR 0.0 0.2 0.4 0.6 0.8 1.0 IC (c) 10 −2 10 −1 10 0 10 1 Raw amplitude MSE IC CR 0.0274 25.3 1.73 (a) Close amplitude 0.0500.0750.1000.125 Trading IC MSE IC CR 0.046 0.124 0.126 (b) Close ordering 0.10.5 1 λ 10 −1 10 0 (c) Amplitude 0.10.5 1 λ 0.06 0.08 0.10 0.12 (d) IC 0.10.5 1 λ 10 −3 (e) MSE <.05 n=6 .05-.30 n=26 >=.30 n=65 0.0 0.2 0.4 0.6 0.8 Median amplitude (f) Foundation-model strata Chronos TimesFM 0.000.050.100.150.200.25 Held-out decision return MSE IC CR 0.052 0.143 0.170 (g) Decision return MSEICCR 0.1 0.2 0.3 0.4 0.5 Trading IC (h) Target contrast Close Volume Figure 3 Empirical evidence for forecast collapse and CalibRank. (a,b) Raw amplitude and cross-sectional IC under MSE, IC, and CalibRank. (c–e) The calibration-ranking tradeoff asλvaries. (f) TSFM amplitude across achieved-R 2 strata. (g) Held-out decision return. (h) Return against volume as a target contrast. Error bars are one standard deviation over three seeds where available. 8 Empirical Results We first isolate the objective on the controlled encoder, then test the amplitude identity synthetically and across public foundation-model benchmarks. The final comparisons examine a held-out decision functional, the weighting ablation, and the high-predictability volume target. Comprehensive cross-architecture evidence follows in Section 9. 8.1 Controlled objective comparison on Finance1K TargetObjective e A→ 1IC↑MSE↓ Close returnMSE0.027± 0.0050.046± 0.003(4.30± 0.13)× 10 −4 IC25.260± 1.1140.124± 0.000(4.22± 2.84)× 10 −1 CalibRank1.836± 0.1180.126± 0.002(1.43± 0.06)× 10 −3 CalibRank (weighted)1.726± 0.1060.126± 0.000(1.29± 0.12)× 10 −3 Volume change MSE0.391± 0.0110.530± 0.003(3.19± 0.02)× 10 0 IC0.233± 0.0130.550± 0.004(7.08± 0.25)× 10 0 CalibRank0.391± 0.0090.538± 0.002(3.18± 0.01)× 10 0 CalibRank (weighted)0.393± 0.0060.542± 0.002(3.14± 0.01)× 10 0 Table 4 Objective comparison on the reference backbone. Raw amplitude and IC are stated on trading rows, squared error on the full evaluation grid. CalibRank is the uniform objective of(24)atλ= 1; the weighted row is the per-timestamp weighted variant. Mean ± one standard deviation over three seeds. Table 4 and Figure 3(a,b) expose both halves of the result. MSE produces a nearly flat forecast, at 0.027 of target amplitude and IC 0.046. IC training raises IC to 0.124 but leaves scale unidentified, reaching 25×the target amplitude. Uniform CalibRank at the prespecifiedλ= 1 reaches the same ordering, IC 0.126, while keeping raw amplitude within a factor of two of the target. Pure ranking incurs over two orders of magnitude more squared error than CalibRank. 17 For the MSE row, the ratio of the reported mean IC to mean raw amplitude is 0.046/0.027 = 1.68. This is a descriptive scale gap rather thans ⋆ t : the plotted entries average timestamp-level quantities, whereass ⋆ t is a ratio of covariance to variance at each timestamp. Under a common-slope approximation, the ratio says that an additional 1.7× rescaling would be optimal. The observed collapse therefore includes finite-fit scale error beyond the best-scaled attenuation in Proposition 4.1. These objectives answer different questions. MSE shows the forecast selected by a calibration objective when predictable variance is small. IC verifies that the architecture and inputs can recover substantially more ordering, while its raw amplitude demonstrates the scale indeterminacy of correlation. The composite shows that the ordering gain does not require accepting that indeterminacy. It is an operating point on a frontier; the downstream cost of amplitude error determines which point is useful. 8.2 The calibration-ranking frontier The seven nonnegative values ofλin Figure 1(b) are evaluated at three seeds. For the cross-architecture records, IC is restated on trading hours while MSE remains on the full evaluation grid; MSE has no uniform restatement factor. The first nonzero ranking weight produces most of the IC gain; larger weights mainly increase amplitude and squared error. The single-term MSE and IC objectives are trained separately and appear at the two extremes. The full MSE range is retained on the log-scaled axis, including the high-error IC-only points on the right. The red line connects the empirically non-dominated points across all backbone objectives. The cyan line traces the seven-point Autoformer sweep, while the remaining points show matched MSE and CalibRank runs for the other eleven backbones and the zero-shot or fine-tuned foundation-model baselines. Red therefore identifies the best observed calibration-ranking tradeoffs regardless of architecture; cyan holds the architecture fixed and isolates movement caused by the objective. Autoformer is highlighted because its seven mean MSE values are strictly ordered and span 3.05×, keeping all operating points legible. The twelve-backbone endpoint movements appear in Table 6; the plotted sweep includes all seven points and their three-seed uncertainty. All four red points come from Non-stationary Transformer. Their near- vertical shape is consistent with its stationarization preserving forecast scale while the ranking term changes cross-sectional order. Post-hoc weighted-objective sweep. The headline controlled comparison uses the uniform objective at the prespecifiedλ= 1. Separately, we examine the earlier weighted-objective variant across six values ofλon the evaluation steps. Figure 3(c–e) reports trading-hour raw amplitude, IC, and squared error, averaged over three seeds. This sweep is post hoc and does not select the headline test result. IC reaches 0.1246 byλ= 0.1 and changes by less than 0.002 throughλ= 1, while raw amplitude continues to rise. Atλ= 0.1, the forecast reaches 87.5% of target amplitude with about half the squared error of theλ= 1 setting. The measured points bracket unit amplitude betweenλ= 0.1 and 0.25; linear interpolation places the crossing nearλ= 0.17. This descriptive crossing illustrates the validation criterion in Section 10; selecting an operating point for deployment still requires a validation split taken by date. Figure 3(c–e) makes this asymmetry visible directly: IC is nearly horizontal after the first nonzero weight, whereas amplitude and MSE continue to grow. 8.3 Synthetic verification Across the synthetic grid,(5)predicts held-out best-scaled amplitude to within 0.0015 and correlation to within 0.0021. This checks the quantitative identity under controlled conditions; the GIFT-Eval scan in Figure 3(f) supplies complementary evidence from pretrained models on real benchmark data. Figure 2(b,c) plots the sweep, which spans two decades of signal-to-noise ratio and three levels of forecaster skill. It is fitted and evaluated on disjoint halves, so the amplitude ratio is a genuine out-of-sample measurement rather than an in-sample slope. At the low end of the sweep even a perfect ranker retains under a tenth of the target’s amplitude. 18 8.4 Foundation-model evidence TimesFM and Chronos are evaluated zero-shot on Finance1K and across the 97 GIFT-Eval configurations. The benchmark analysis uses each model’s own point forecast after one fixed stationarization rule. We compare raw amplitude with theR 2 achieved by a fixed set of fitted baselines on the same configuration. Because that set may miss available structure, a large value shows that a target is predictable, while a small value cannot show it is unpredictable. The association in Figure 3(f) is therefore a cross-dataset regularity; the exact bound remains a statement about best-scaled forecasts under a declared information set. For an empirical summary, we divide achievedR 2 into three descriptive strata using the conventions 0.30 and 0.05. Figure 3(f) reports their median raw amplitude across all 97 observations. Raw amplitude follows achieved predictability empirically. We scan all 97 GIFT-Eval configurations (Aksu et al., 2024) under a fixed stationarization rule. The fitted baselines attain medianR 2 = 0.56, while the same procedure attains 0.0044 on Finance1K close returns. These achieved values lower-bound the predictability available to a richer forecaster; a small value therefore does not prove the ceiling is low. Figure 3(f) compares those achieved values with the raw amplitude output by two pretrained models. Amplitude correlates with √ R 2 at 0.88 for Chronos and 0.87 for TimesFM. In the 65 configurations with achieved R 2 ≥0.30, both models retain about three quarters of target amplitude. In the six configurations with achievedR 2 <0.05, they retain under six percent. The cutoffs are descriptive strata, not derived thresholds. This result establishes a strong empirical association. It does not test the upper bound in Proposition 4.1. 8.5 Decision metric, ablation, and negative control As an out-of-objective check, we turn each forecast into a decision: subtract the cross-sectional mean so the weights sum to zero, scale them to unit total size, and apply them to the realized returns. In the archived matched controlled runs, the held-out decision return rises from 0.052±0.010 under MSE to 0.143±0.029 under IC and 0.170±0.030 under weighted CalibRank. This functional is absent from(24); its definition is given in Section 7.4. The separation in Figure 3(g) is important because this return is not optimized directly: the composite objective improves the downstream functional after the loss has already balanced amplitude and ordering. A matched ablation replaces the theory-motivated per-timestamp weighting in the original implementation with uniform weights. Weighted minus uniform IC is−0.0006 on close and +0.0014 on volume, against weighted-arm seed ranges of 0.0019 and 0.0050, respectively; raw amplitude is similar. The gain therefore comes from scoring cross-sectional structure. We use the simpler uniform objective in(24); the complete cross-architecture ablation appears in Section 9. Negative control. Figure 3(h) shows the target contrast. On volume changes from the same panel, MSE reaches ( e A, IC ) = (0.391,0.530) and uniform CalibRank reaches (0.391,0.538) (Table 4). The near-identical amplitude and modest IC difference contrast with close returns under the same split and protocol. When more variation is predictable, MSE does not produce the near-zero raw amplitude seen in Figure 3(a). Because MSE already recovers strong ordering on this target, the cross-sectional term has little remaining IC to recover; the small gain is the expected negative-control result. 9 Comprehensive Cross-Architecture Evidence This section reports every aggregate in the unified empirical protocol. All supervised entries use seeds 11,23,42 and report mean±one standard deviation. Table 5 compares the three training objectives, while Table 6 separates the conditioning ∆IC contrast from theλ= 0 and 1 endpoints. Together, the two tables and Figure 1(b) report the objective comparison, the complete conditioning matrix, and the seven-point sweeps over all twelve backbones. 19 Table 5 Objective comparison across twelve forecasting models. Mean±one standard deviation over three seeds. IC is evaluated on trading hours; MSE uses the full evaluation grid. Backbone MSE lossIC lossCalibRank IC↑MSE↓IC↑MSE↓IC↑MSE↓ DLinear (Zeng et al., 2023)0.096± 0.024(4.12± 0.03)× 10 −4 0.129± 0.000 (1.82± 1.58)× 10 −2 0.127± 0.004 (5.20± 1.84)× 10 −4 PatchTST (Nie et al., 2023)0.082± 0.016(4.79± 1.10)× 10 −4 0.123± 0.007 (4.41± 0.85)× 10 −1 0.122± 0.008 (5.02± 0.32)× 10 −4 iTransformer (Liu et al., 2024a)0.091± 0.002(4.33± 0.08)× 10 −4 0.129± 0.002 (2.35± 0.38)× 10 −1 0.127± 0.004 (5.53± 0.60)× 10 −4 TimeMixer (Wang et al., 2024)0.110± 0.003(4.14± 0.03)× 10 −4 0.129± 0.000 (1.07± 0.22)× 10 −2 0.127± 0.003 (5.24± 0.81)× 10 −4 Autoformer (Wu et al., 2021)0.021± 0.007(4.30± 0.09)× 10 −4 0.096± 0.008 (3.58± 0.63)× 10 −1 0.105± 0.003 (7.86± 2.31)× 10 −4 Transformer (Vaswani et al., 2017) −0.011± 0.023 (4.26± 0.14)× 10 −4 0.121± 0.001 (4.45± 0.47)× 10 −1 0.120± 0.002 (5.91± 0.50)× 10 −4 Informer (Zhou et al., 2021)−0.004± 0.017 (4.18± 0.02)× 10 −4 0.117± 0.002 (4.64± 3.91)× 10 −1 0.105± 0.005 (8.26± 1.17)× 10 −4 FEDformer (Zhou et al., 2022)0.024± 0.013(5.43± 1.19)× 10 −4 0.120± 0.001(1.18± 0.95)× 10 0 0.117± 0.004 (5.77± 1.89)× 10 −4 Nonstat. Transformer (Liu et al., 2022)0.118± 0.008(4.09± 0.01)× 10 −4 0.148± 0.001 (5.27± 0.14)× 10 −4 0.147± 0.002 (4.26± 0.28)× 10 −4 Reformer (Kitaev et al., 2020)0.062± 0.029(4.21± 0.10)× 10 −4 0.119± 0.001 (1.50± 0.28)× 10 −1 0.121± 0.002 (5.82± 1.17)× 10 −4 LightTS (Zhang et al., 2022)0.092± 0.009(4.12± 0.00)× 10 −4 0.128± 0.001(2.63± 4.43)× 10 0 0.128± 0.001 (5.03± 1.04)× 10 −4 TSMixer (Chen et al., 2023)0.062± 0.010(4.15± 0.03)× 10 −4 0.131± 0.000 (4.11± 1.13)× 10 −3 0.130± 0.001 (5.35± 1.43)× 10 −4 9.1 Matched objectives across backbones Figure 1(b) distinguishes the red empirical Pareto frontier from the cyan Autoformer sweep. The red line reports the best observed tradeoffs over the full backbone set. The cyan path keeps the architecture fixed, so its movement isolates the effect of changingλ. We repeat the matched MSE and CalibRank comparison across recurrent, convolutional, mixing, linear, and attention-based architectures under the same context, optimizer, schedule, and three-seed protocol. IC values in Table 5 use trading hours; MSE values use the full evaluation grid, matching the archived cross-architecture records. Table 5 reports the result. Under squared error the backbones span a wide range of IC, but replacing MSE with CalibRank improves every one. The mean IC rises from 0.062 to 0.123, while mean squared error changes from 4.3×10 −4 to 5.8×10 −4 . The latter is over 800 times smaller than the IC-only mean of 4.9×10 −1 . The consistent within-backbone gain makes architecture an unlikely explanation for the ordering failure. 9.2 Conditioning ablation and target contrast What the ablation shows. Table 6 reports conditioning-minus-no-conditioning ∆IC for all twelve backbones and both targets, together with every ∆MSE and seed standard deviation. The close deltas straddle zero, while volume deltas are more often favorable. This is expected rather than contradictory: hourly close volatility has little lagged predictability, so its proxy input is often noisy; volume persistence makes the same input more informative. The matched comparison separates this conditioning limitation from the core CalibRank coupling, whose weighted objective is unchanged in the ablation. Why volume looks different from close. The held-out prediction trajectories show that the MSE forecasts have more visible variation than on close because volume log-change has substantially higher SNR: its conditional mean occupies a larger fraction of realized variance. Pure IC predictions still show avoidable scale drift because correlation does not identify amplitude. CalibRank tracks the realized scale more closely by retaining a calibration anchor, and the seed ribbons show that the distinction is stable across initializations rather than caused by a selected trajectory. 9.2.1 No-conditioning ablation by model and target For each backbone, we compute the mean and standard deviation of CalibRank minus its matched no- conditioning run. Positive ∆IC and negative ∆MSE are favorable. The aggregate pattern is that the conditioning channel raises close IC in only 4/12 backbones, but raises volume IC in 10/12 and lowers volume MSE in 9/12. Several close MSE improvements coexist with negligible or negative IC changes because calibration can benefit from scale information even when that information does not locate cross-sectional ranking skill. Large standard deviations for Informer and Autoformer caution against interpreting isolated signs as universal gains. 20 Table 6 Conditioning ablation and calibration-ranking sweep. Conditioning columns report CalibRank minus no-conditioning ∆IC; sweep columns report the close-target endpoints atλ= 0 andλ= 1. Mean±one standard deviation over three seeds. Backbone Conditioning ablationClose-targetλ sweep Close ∆IC↑ Volume ∆IC↑IC(0)↑IC(1)↑MSE(0)↓MSE(1)↓ DLinear−0.001± 0.003+0.003± 0.0040.072± 0.0160.127± 0.004 (4.74± 1.02)× 10 −4 (5.20± 1.84)× 10 −4 PatchTST+0.006± 0.002+0.000± 0.0010.049± 0.0450.122± 0.008 (4.18± 0.03)× 10 −4 (5.02± 0.32)× 10 −4 iTransformer+0.002± 0.001+0.001± 0.0010.092± 0.0130.127± 0.004 (4.15± 0.03)× 10 −4 (5.53± 0.60)× 10 −4 TimeMixer−0.001± 0.002+0.002± 0.0000.114± 0.0050.127± 0.003 (4.11± 0.01)× 10 −4 (5.24± 0.81)× 10 −4 Autoformer+0.006± 0.004 −0.004± 0.0110.041± 0.0340.105± 0.003 (4.17± 0.02)× 10 −4 (7.86± 2.31)× 10 −4 Transformer−0.002± 0.002+0.006± 0.0040.006± 0.0130.120± 0.002 (4.37± 0.24)× 10 −4 (5.91± 0.50)× 10 −4 Informer−0.023± 0.047+0.001± 0.003 −0.003± 0.005 0.105± 0.005 (4.36± 0.21)× 10 −4 (8.26± 1.17)× 10 −4 FEDformer−0.001± 0.004+0.001± 0.0030.051± 0.0480.117± 0.004 (4.17± 0.02)× 10 −4 (5.77± 1.89)× 10 −4 Nonstat. Transformer −0.000± 0.000+0.002± 0.0020.059± 0.0610.147± 0.002 (4.15± 0.04)× 10 −4 (4.26± 0.28)× 10 −4 Reformer−0.002± 0.001+0.004± 0.0020.012± 0.0320.121± 0.002 (4.62± 0.43)× 10 −4 (5.82± 1.17)× 10 −4 LightTS+0.000± 0.000 −0.001± 0.0060.049± 0.0110.128± 0.001 (4.18± 0.02)× 10 −4 (5.03± 1.04)× 10 −4 TSMixer−0.000± 0.001+0.001± 0.0010.060± 0.0310.130± 0.001 (4.16± 0.01)× 10 −4 (5.35± 1.43)× 10 −4 9.3λ-sweep by model A similar pattern across model families. The endpoint columns of Table 6, together with the seven-point sweeps, show the same pattern across backbones: the first nonzero ranking weight produces almost the entire correlation gain, after which correlation plateaus while squared error and amplitude keep rising. Differences in curvature reflect backbone stability, and Informer has visibly larger seed dispersion. The table keeps the endpoints readable, while Figure 1(b) displays the intermediate operating points. Moving fromλ= 0 toλ= 1 raises close IC for every backbone; the median gain is 0.0489 at a median 1.280×MSE multiplier. Most models obtain nearly all of that ranking gain byλ= 0.25–0.5, while larger weights mainly increase MSE. This saturation occurs because once cross-sectional correlation saturates, stronger correlation pressure adds little and continues relaxing scale discipline. Non-stationary Transformer is the clearest favorable case: its ranking gain is obtained with almost no MSE penalty, suggesting that its normalization already supplies part of the calibration control. 10 Discussion Account for target predictability in evaluation. The same correlation can be weak on a predictable target and strong on an unpredictable one. A predictability ceiling makes that distinction explicit, provided it is estimated under a declared information set and at the same aggregation level as the reported score. In a panel, this means comparing per-timestamp cross-sectional IC with per-timestamp ceilings before averaging. We use the pooled identity only to characterize pooled amplitude attenuation and do not convert it into a percentage of the time-averaged cross-sectional IC. Evaluate amplitude and joint structure. GIFT-Eval scores one series at a time. Although 43 of its 97 configurations are multivariate, each is flattened before scoring, and none of its eleven metrics measures raw amplitude or cross-sectional structure. A complete evaluation should report e A, a cross-sectional dependence score, and ordinary per-series error. The first identifies attenuation or overshoot. The second asks whether the forecast preserves relationships used by a downstream decision. Neither can be reconstructed from a scalar average of per-series errors. A multivariate benchmark can add these measurements without replacing its existing metrics. It must preserve channel grouping and timestamps through evaluation, compute the ordering score within each cross-section, and average only afterward. It should also state whether amplitude is measured on raw output or after a fitted calibration map. These choices prevent pooled correlation from mixing temporal and cross-sectional effects and prevent post-hoc rescaling from hiding what the model outputs. 21 Evaluate the missing cross-series functional. Finance is a clear testbed because portfolio selection uses the ordering of a cross-section, but the issue is broader. In environmental sensor networks, clinical monitoring, energy systems, and traffic forecasting, decisions depend on which channels move together or which deteriorate first. Benchmarks for these settings should retain their multivariate structure through scoring. Depending on the application, the relevant relationship may be correlation, concordance, tail dependence, or a learned graph. The target determines which metric belongs in the protocol. A portfolio selector uses rank or correlation; an energy dispatcher may require calibrated joint tails; a clinical alert system may care about which channels cross thresholds together. There is no universal dependence score. The requirement is that evaluation retain the relationship the decision uses instead of flattening the problem into independent series. Where to expect forecast collapse. The analysis gives two diagnostics rather than a universal claim about TSFMs. First, estimate how much variance a reasonable set of baselines can recover under the same information set. Low achievedR 2 is a warning sign for amplitude attenuation, although it cannot prove that the true ceiling is low. Second, identify whether the downstream task needs a relationship across channels that the training and benchmark losses score. The strongest failure should appear when both conditions hold: little predictable variance and an important multivariate functional absent from the objective. Short-horizon returns are one instance; demand residuals after strong seasonal adjustment, local sensor innovations, and differences between closely related measurements are other plausible targets. The same diagnostics make the claim falsifiable. If a target clearly has high predictable variance and a well-fitted MSE model still produces a nearly flat forecast, the predictability explanation is insufficient and optimization, regularization, or preprocessing becomes the leading suspect. If a pointwise model preserves the relevant cross-sectional structure across architectures and seeds, then the coupling invariance has not produced an empirical failure in that setting. Conversely, a cross-sectional loss should improve the targeted functional under a matched protocol, as it does for every backbone in Table 5. These checks separate a property of the statistical problem from a defect in a particular implementation. Choose the operating point on validation data. Increasingλinitially recovers most of the available ordering and then primarily increases amplitude and squared error. This creates a practical selection rule: chooseλon a validation split taken by date to meet a declared amplitude tolerance, then report the untouched test result. The present paper reports the prespecifiedλ= 1 setting and treats the seven-point sweep as a measured frontier, not as test-set model selection. The chosen operating point is application-dependent: a ranking system may tolerate amplitude error, whereas a portfolio or physical forecast may not. Implications for model selection. Per-series error remains necessary: a model with good ordering can still produce unusable magnitudes, as the IC-only point in Figure 3(a) shows. The reverse is also true: an apparently calibrated marginal forecast may carry little cross-sectional discrimination. Reporting both axes changes model selection from a single leaderboard rank into a constrained choice. A practitioner can set an admissible calibration range and maximize structure within it, or set a minimum ordering score and minimize error. The frontier communicates the available tradeoff without asserting that one scalar weight is correct for every application. 11 Conclusion and Limitations Forecast collapse combines an amplitude failure with an ordering failure. Predictability limits the amplitude of a best-scaled point forecast. Separately, per-series risk does not identify cross-series coupling, even though cross-sectional decisions depend on that coupling. Finance1K shows both patterns: MSE forecasts are nearly flat, IC-only forecasts are badly miscalibrated, and CalibRank improves cross-sectional correlation while retaining a calibration anchor. TimesFM and Chronos show a parallel association between raw amplitude and achieved predictability across 97 public benchmark configurations. Evaluation should preserve and score multivariate structure as well as per-series error. 22 Practical consequence. A small forecast is not automatically a weak model: it can be the correctly calibrated response to a target with little predictable variance. Nor does a low per-series error guarantee that the forecast preserves the relationships a decision needs. The useful audit is therefore two-dimensional. Measure what the model outputs before calibration, and measure the relevant structure at the level where the decision is made. If amplitude is attenuated in line with predictability, changing architecture alone should not remove it. If structure is missing, adding a matched cross-sectional term should matter more than rescaling the output. This separation turns "flat forecasts" from a visual symptom into two testable questions about information and objective design. Limitations. The two parts of the argument have different empirical reach. The amplitude identity is exact for best-scaled forecasts and is checked in simulation; the raw outputs of two pretrained models show the predicted association across a large public benchmark. AchievedR 2 from a fitted set of baselines is a lower bound on available predictability, however, so the GIFT-Eval experiment does not estimate the true ceiling or test the inequality in Theorem 4.1. It shows where amplitude collapse appears in practice. The cross-sectional diagnosis and remedy are demonstrated on one real panel. The twelve-backbone experiment reduces the chance that the result is tied to one architecture, but does not establish the same ordering failure in another domain. Finance1K is stratified by volatility and requires long coverage. It is useful for testing predictability and cross-sectional ranking, but it does not support a market-wide performance claim. The Finance1K release package provides the aligned panels, exact chronological split, ticker order, validity masks, and checksums needed to reproduce the data contract. CalibRank targets one dependence functional, per-timestamp Pearson correlation. It does not estimate a joint distribution, and applications concerned with rank, tails, or calibrated scenarios may require a different cross-sectional term. The frontier is measured for a simple scalarization and is not fundamental; regression- compatible listwise objectives may dominate it. The remedy also requires a meaningful cross-section at each timestamp and does not directly apply to a single series. Finally, aggregation matters under drift. Squared error pools by linearity, whereas IC is a ratio and can show Simpson effects when pooled over time (Robinson, 1950). A ceiling computed from average predictability is also Jensen-loose because the square-root map in(3)is concave. Our empirical IC is therefore computed at each timestamp and averaged, and we avoid comparing it with a pooled ceiling percentage. Time-varying predictability changes where attenuation is strongest, while the per-timestamp identity in Proposition 4.1 remains unchanged. 12 Generative AI Usage We disclose all uses of generative AI in the preparation of this work. The authors remain responsible for all of its content. Generative AI assistants were used in two ways: to write and refactor code, including the experiment harness, the figure and table renderers, and the validation suite; and for writing assistance, including drafting, editing, and improving clarity. They were not used to generate research ideas, to develop the theory or its proofs, to design the experiments, or to interpret results. All AI-assisted text and code was reviewed and edited by the authors, who accept responsibility for its accuracy and integrity. References Taha Aksu, Gerald Woo, Juncheng Liu, Xu Liu, Chenghao Liu, Silvio Savarese, Caiming Xiong, and Doyen Sahoo. GIFT-Eval: A benchmark for general time series forecasting model evaluation. arXiv preprint arXiv:2410.10393, 2024. Amazon Science. Chronos-bolt: Fast and accurate pretrained time series forecasting models. Model re- lease,amazon-science/chronos-forecastingandamazon/chronos-bolt-base, 2024.https://github.com/ 23 amazon-science/chronos-forecasting. No accompanying paper; objective read from the released implemen- tation and configuration. Pierre Andreoletti. Forecast collapse of transformer-based models under squared loss in financial time series, 2026. https://arxiv.org/abs/2604.00064. Abdul Fatir Ansari, Lorenzo Stella, Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Oleksandr Shchur, Syama Sundar Rangapuram, Sebastian Pineda Arango, Shubham Kapoor, Jasper Zschiegner, Danielle C. Maddix, Hao Wang, Michael W. Mahoney, Kari Torkkola, Andrew Gordon Wilson, Michael Bohlke-Schneider, and Yuyang Wang. Chronos: Learning the language of time series. Transactions on Machine Learning Research (TMLR), 2024. arXiv:2403.07815. Abdul Fatir Ansari, Oleksandr Shchur, Jaris Küken, Andreas Auer, Boran Han, Pedro Mercado, Syama Sundar Rangapuram, Huibin Shen, Lorenzo Stella, Xiyuan Zhang, et al. Chronos-2: From univariate to universal forecasting, 2025. https://arxiv.org/abs/2510.15821. Andreas Auer, Patrick Podest, Daniel Klotz, Sebastian Böck, Günter Klambauer, and Sepp Hochreiter. Tirex: Zero-shot forecasting across long and short horizons with enhanced in-context learning. In Advances in Neural Information Processing Systems (NeurIPS), 2025. https://arxiv.org/abs/2505.23719. Aijun Bai, Rolf Jagerman, Zhen Qin, Le Yan, Pratyush Kar, Bing-Rong Lin, Xuanhui Wang, Michael Bendersky, and Marc Najork. Regression compatible listwise objectives for calibrated ranking with binary relevance. In Proceedings of the 32nd ACM International Conference on Information and Knowledge Management (CIKM), pages 4502–4508, 2023. doi: 10.1145/3583780.3614712. Yochai Blau and Tomer Michaeli. The perception-distortion tradeoff. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 6228–6237, 2018. Mathieu Blondel, Olivier Teboul, Quentin Berthet, and Josip Djolonga. Fast differentiable sorting and ranking. In Proceedings of the 37th International Conference on Machine Learning (ICML), 2020. arXiv:2002.08871. Tim Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3):307–327, 1986. doi: 10.1016/0304-4076(86)90063-1. Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu. Measurement Error in Nonlinear Models: A Modern Perspective. Chapman and Hall/CRC, 2 edition, 2006. ISBN 9781584886334. doi: 10.1201/9781420010138. Si-An Chen, Chun-Liang Li, Nate Yoder, Sercan O. Arik, and Tomas Pfister. TSMixer: An all-MLP architecture for time series forecasting. Transactions on Machine Learning Research, 2023. Ben Cohen, Emaad Khwaja, Youssef Doubli, Salahidine Lemaachi, Chris Lettieri, Charles Masson, Hugo Miccinilli, Elise Ramé, Qiqi Ren, Afshin Rostamizadeh, et al. This time is different: An observabil- ity perspective on time series foundation models. In Advances in Neural Information Processing Sys- tems, volume 38, pages 50907–50951, 2025.https://proceedings.neurips.c/paper_files/paper/2025/hash/ 48ea942b60e6c52b50ca90e4a40d8ac2-Abstract-Conference.html. Abhimanyu Das, Weihao Kong, Rajat Sen, and Yichen Zhou. A decoder-only foundation model for time-series forecasting. In Proceedings of the 41st International Conference on Machine Learning (ICML), 2024. arXiv:2310.10688. Alexandre Drouin, Étienne Marcotte, and Nicolas Chapados. TACTiS: Transformer-attentional copulas for time series. In Proceedings of the 39th International Conference on Machine Learning, 2022. Vijay Ekambaram, Arindam Jati, Pankaj Dayama, Sumanta Mukherjee, Nam H. Nguyen, Wesley M. Gifford, Chandra Reddy, and Jayant Kalagnanam. Tiny time mixers (TTMs): Fast pre-trained models for enhanced zero/few-shot forecasting of multivariate time series. In Advances in Neural Information Processing Systems (NeurIPS 2024), 2024. arXiv:2401.03955. Vijay Ekambaram, Subodh Kumar, Arindam Jati, Sumanta Mukherjee, Tomoya Sakai, Pankaj Dayama, Wesley M. Gifford, and Jayant Kalagnanam. TSPulse: Tiny pre-trained models with disentangled representations for rapid time-series analysis. In International Conference on Learning Representations (ICLR), 2026.https://arxiv.org/ abs/2505.13033. Fuli Feng, Xiangnan He, Xiang Wang, Cheng Luo, Yiqun Liu, and Tat-Seng Chua. Temporal relational ranking for stock prediction. ACM Transactions on Information Systems, 37(2):27:1–27:30, March 2019. doi: 10.1145/3309547. 24 Kun Feng, Shaocheng Lan, Yuchen Fang, Wenchao He, Sihan Lu, Shuqi Gu, Lintao Ma, Xingyu Lu, and Kan Ren. Kairos: Toward adaptive and parameter-efficient time series foundation models, 2025.https://arxiv.org/abs/2509.25826. Dror Freirich, Tomer Michaeli, and Ron Meir. A theory of the distortion-perception tradeoff in Wasserstein space. In Advances in Neural Information Processing Systems (NeurIPS), 2021. Wayne A. Fuller. Measurement Error Models. Wiley Series in Probability and Statistics. Wiley, New York, 1987. ISBN 9780471861874. doi: 10.1002/9780470316665. Stuart Geman, Elie Bienenstock, and René Doursat. Neural networks and the bias/variance dilemma. Neural Computation, 4(1):1–58, 1992. doi: 10.1162/neco.1992.4.1.1. Tilmann Gneiting. Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494): 746–762, 2011. doi: 10.1198/jasa.2011.r10138. Tilmann Gneiting and Adrian E. Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477):359–378, 2007. doi: 10.1198/016214506000001437. Tilmann Gneiting, Fadoua Balabdaoui, and Adrian E. Raftery. Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society Series B: Statistical Methodology, 69(2):243–268, 2007. doi: 10.1111/j. 1467-9868.2007.00587.x. Mononito Goswami, Konrad Szafer, Arjun Choudhry, Yifu Cai, Shuo Li, and Artur Dubrawski. MOMENT: A family of open time-series foundation models. In International Conference on Machine Learning (ICML), 2024. arXiv:2402.03885. Lars Graf, Thomas Ortner, Stanisław Woźniak, and Angeliki Pantazi. FlowState: Sampling-rate-equivariant time-series forecasting. In Proceedings of the 43rd International Conference on Machine Learning (ICML), volume 306 of Proceedings of Machine Learning Research, 2026. https://arxiv.org/abs/2508.05287. Riku Green, Zahraa S. Abdallah, and Telmo M. Silva Filho. Expectations vs. realities: The cost of MSE-optimal forecasting under conditional uncertainty. In Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), Research Track, 2026. Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York, 2nd edition, 2009. doi: 10.1007/978-0-387-84858-7. Shi Bin Hoo, Samuel Müller, David Salinas, and Frank Hutter. From tables to time: Extending TabPFN-v2 to time series forecasting. Transactions on Machine Learning Research, 2026. ISSN 2835-8856.https://openreview.net/ forum?id=KIkQj8VOUY. Rob J. Hyndman and Anne B. Koehler. Another look at measures of forecast accuracy. International Journal of Forecasting, 22(4):679–688, 2006. doi: 10.1016/j.ijforecast.2006.03.001. Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. Nikita Kitaev, Łukasz Kaiser, and Anselm Levskaya. Reformer: The efficient transformer. In International Conference on Learning Representations (ICLR), 2020. https://arxiv.org/abs/2001.04451. William H. Kruskal. Ordinal measures of association. Journal of the American Statistical Association, 53(284):814–861, 1958. doi: 10.1080/01621459.1958.10501481. Yuxuan Liang, Haomin Wen, Yuqi Nie, Yushan Jiang, Ming Jin, Dongjin Song, Shirui Pan, and Qingsong Wen. Foundation models for time series analysis: A tutorial and survey. In Proceedings of the 30th ACM SIGKDD conference on knowledge discovery and data mining, pages 6555–6565, 2024. Yan Lin, Yihong Su, and Yi Yang. Lambdarankic: Directly optimizing rank ic for financial prediction, 2026. https://arxiv.org/abs/2605.00501. Yong Liu, Haixu Wu, Jianmin Wang, and Mingsheng Long.Non-stationary transformers: Ex- ploring the stationarity in time series forecasting.In Advances in Neural Information Pro- cessing Systems, volume 35, 2022.https://proceedings.neurips.c/paper_files/paper/2022/hash/ 4054556fcaa934b0bf76da52cf4f92cb-Abstract-Conference.html. Yong Liu, Tengge Hu, Haoran Zhang, Haixu Wu, Shiyu Wang, Lintao Ma, and Mingsheng Long. iTransformer: Inverted transformers are effective for time series forecasting. In International Conference on Learning Representations (ICLR), 2024a. https://arxiv.org/abs/2310.06625. 25 Yong Liu, Haoran Zhang, Chenyu Li, Xiangdong Huang, Jianmin Wang, and Mingsheng Long. Timer: Generative pre-trained transformers are large time series models. In Proceedings of the 41st International Conference on Machine Learning (ICML), 2024b. https://arxiv.org/abs/2402.02368. Yong Liu, Guo Qin, Zhiyuan Shi, Zhi Chen, Caiyin Yang, Xiangdong Huang, Jianmin Wang, and Mingsheng Long. Sundial: A family of highly capable time series foundation models. In International Conference on Machine Learning (ICML), Oral, 2025. https://arxiv.org/abs/2502.00816. Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. In International Conference on Learning Representations (ICLR), 2023. Frank Pennekamp, Alison C. Iles, Joshua Garland, Georgina Brennan, Ulrich Brose, Ursula Gaedke, Ute Jacob, Pavel Kratina, Blake Matthews, Stephan Munch, Mark Novak, Gian Marco Palamara, Björn C. Rall, Benjamin Rosenbaum, Andrea Tabi, Colette Ward, Richard Williams, Hao Ye, and Owen L. Petchey. The intrinsic predictability of ecological time series and its potential to guide forecasting. Ecological Monographs, 89(2):e01359, 2019. doi: 10.1002/ecm.1359. Ilan Price, Alvaro Sanchez-Gonzalez, Ferran Alet, Tom R. Andersson, Andrew El-Kadi, Dominic Masters, Timo Ewalds, Jacklynn Stott, Shakir Mohamed, Peter Battaglia, Remi Lam, and Matthew Willson. Probabilistic weather forecasting with machine learning. Nature, 637:84–90, 2025. doi: 10.1038/s41586-024-08252-9. Eghbal Rahimikia, Hao Ni, and Weiguan Wang. Re(Visiting) time series foundation models in finance. arXiv:2511.18578 [q-fin.CP], 2025. Kashif Rasul, Arjun Ashok, Andrew Robert Williams, Hena Ghonia, Rishika Bhagwatkar, Arian Khorasani, Mo- hammad Javad Darvishi Bayazi, George Adamopoulos, Roland Riachi, Nadhir Hassen, Marin Biloš, Sahil Garg, Anderson Schneider, Nicolas Chapados, Alexandre Drouin, Valentina Zantedeschi, Yuriy Nevmyvaka, and Irina Rish. Lag-llama: Towards foundation models for probabilistic time series forecasting. arXiv preprint arXiv:2310.08278, 2023. W. S. Robinson. Ecological correlations and the behavior of individuals. American Sociological Review, 15(3):351–357, 1950. ISSN 0003-1224. doi: 10.2307/2087176. David Salinas, Michael Bohlke-Schneider, Laurent Callot, Roberto Medico, and Jan Gasthaus. High-dimensional multivariate forecasting with low-rank gaussian copula processes. In Advances in Neural Information Pro- cessing Systems, volume 32, pages 6824–6834, 2019.https://proceedings.neurips.c/paper/2019/hash/ 0b105cf1504c4e241fcc6d519ea962fb-Abstract.html. W. F. Sheppard. Iii. on the application of the theory of error to cases of normal distribution and normal correlation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 192:101–167, 1899. doi: 10.1098/rsta.1899.0003. Xiaoming Shi, Shiyu Wang, Yuqi Nie, Dianqi Li, Zhou Ye, Qingsong Wen, and Ming Jin. Time-moe: Billion-scale time series foundation models with mixture of experts. In The Thirteenth International Conference on Learning Representations (ICLR), 2025. arXiv:2409.16040. Patrick E. Shrout and Joseph L. Fleiss. Intraclass correlations: Uses in assessing rater reliability. Psychological Bulletin, 86(2):420–428, 1979. doi: 10.1037/0033-2909.86.2.420. Abe Sklar. Fonctions de répartition àndimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris, 8:229–231, 1959. Charles Spearman. The proof and measurement of association between two things. The American Journal of Psychology, 15(1):72–101, 1904. doi: 10.2307/1412159. Robin Swezey, Aditya Grover, Bruno Charron, and Stefano Ermon. Pirank: Scalable learning to rank via differentiable sorting. In Advances in Neural Information Processing Systems, volume 34, 2021. doi: 10.48550/arXiv.2012.06731. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, volume 30, 2017. doi: 10.48550/arXiv.1706.03762. https://arxiv.org/abs/1706.03762. Shiyu Wang, Haixu Wu, Xiaoming Shi, Tengge Hu, Huakun Luo, Lintao Ma, James Y. Zhang, and Jun Zhou. Timemixer: Decomposable multiscale mixing for time series forecasting. In International Conference on Learning Representations (ICLR), 2024. Xue Wang, Tian Zhou, Jinyang Gao, Bolin Ding, and Jingren Zhou. Output scaling: YingLong-delayed chain of thought in a large pretrained time series forecasting model, 2025. https://arxiv.org/abs/2506.11029. 26 Zhou Wang and Alan C. Bovik. Mean squared error: Love it or leave it? a new look at signal fidelity measures. IEEE Signal Processing Magazine, 26(1):98–117, 2009. doi: 10.1109/MSP.2008.930649. Zhou Wang, Alan C. Bovik, Hamid R. Sheikh, and Eero P. Simoncelli. Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4):600–612, 2004. doi: 10.1109/TIP.2003.819861. Gerald Woo, Chenghao Liu, Akshat Kumar, Caiming Xiong, Silvio Savarese, and Doyen Sahoo. Unified training of universal time series forecasting transformers. In Proceedings of the 41st International Conference on Machine Learning (ICML), volume 235 of Proceedings of Machine Learning Research, pages 53140–53164. PMLR, 2024. https://proceedings.mlr.press/v235/woo24a.html. Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer: Decomposition transformers with auto- correlation for long-term series forecasting. In Advances in Neural Information Processing Systems, volume 34, 2021. doi: 10.48550/arXiv.2106.13008. Xiao Yang, Weiqing Liu, Dong Zhou, Jiang Bian, and Tie-Yan Liu. Qlib: An ai-oriented quantitative investment platform, 2020. https://arxiv.org/abs/2009.11189. Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pages 11121–11128, 2023. doi: 10.1609/aaai.v37i9.26317. Cheng Zhang. A nontrivial upper bound on the out-of-sampler 2 in return forecasting, 2026.https://arxiv.org/ abs/2602.07841. Tianping Zhang, Yizhuo Zhang, Wei Cao, Jiang Bian, Xiaohan Yi, Shun Zheng, and Jian Li. Less is more: Fast multivariate time series forecasting with light sampling-oriented MLP structures. arXiv preprint arXiv:2207.01186, 2022. Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 11106–11115, 2021. doi: 10.1609/aaai.v35i12.17325. Tian Zhou, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin. FEDformer: Frequency enhanced decomposed transformer for long-term series forecasting. In Proceedings of the 39th International Conference on Machine Learning (ICML), 2022. 27

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