More Data Cannot Break a Symmetry: Identifiability by Design

Unsupervised representational alignment recovers a stimulus-by-stimulus correspondence from geometry alone, but the automorphism group of the stimulus geometry bounds what any such alignment can identify, before data exist. The obvious diagnostic for this degeneracy, the cheapest non-identity relabelling, ranks two published designs in the wrong order, because dense sampling creates near-duplicates whose transposition is nearly free. We turn this known invariance (Demetci et al., 2024) into a design-time diagnostic and intervention. In colour, where candidate geometries have closed form, we show that the failure is structural: sixty-four times the restart budget leaves a symmetric design unmoved while an asymmetric set at the same N recovers every time. Discriminating representational models and recovering a correspondence are essentially uncorrelated objectives (r = -0.02 over 3,000 subsets). Choosing nine colours by this diagnostic alone, without consulting any learned representation, moves all 93 model representations away from the degenerate point and cuts catastrophic alignment failures from 75% to 2% with the models, the layers, N and the solver all held fixed. The same risk arises wherever a regular design meets its candidate geometry's isometry group, including evenly spaced orientations, tones, or motion directions, and the check costs one function call before data collection.

When Does Context Routing Help? A Systematic Study of Multi-Modal Fusion in Time Series Forecasting

Multi-modal time series forecasting methods integrate auxiliary context into temporal predictions through increasingly sophisticated fusion mechanisms. A growing body of work reports substantial gains, yet it is often unclear whether they reflect genuine use of the context or incidental architectural effects. We ask a narrower, checkable question: when can auxiliary context help a forecaster at all? We identify two dataset-level conditions that must both hold: (1) the target is not dominated by a last-value shortcut (low autocorrelation rho_h), and (2) the context carries information about the target beyond history (non-zero conditional mutual information delta; when delta=0 no predictor can benefit---a distribution-free result). Through controlled experiments on MoME (a 14.3B-parameter mixture-of-experts model, 6 datasets, 10 seeds) and four additional fusion mechanisms implemented within a single-backbone testbed (5 datasets), we find that when both conditions hold, text-conditioned expert modulation contributes a sizeable MSE reduction; when either fails, the contribution collapses to the capacity floor of the modulation pathway and carries no context-attributable signal. We establish causality through two interventions: adding a shortcut to MoME suppresses routing contribution by 77-93% across 3 datasets; progressively corrupting context quality drives the context-specific benefit from +44% to negative. We validate the autocorrelation component of our diagnostic on 27 Monash Archive datasets. We provide a calibrated pre-training diagnostic that, on the datasets we test, yields no false positives in well-powered settings. We are explicit about the asymmetry of our evidence: the negative arm is broadly reliable, while the large positive magnitudes come from a single model family (MoME) and are corroborated only in direction by the testbed.

The Spectrum Is Not Enough: When Context Helps Time-Series Forecasting

A growing family of indices scores how predictable a series is from its spectrum. Practitioners increasingly read these scores as answering a different question: whether \emph{adding context}, a longer lookback, a retrieval plug-in, or a pretrained model, will help. These are not the same question. The value of context is a property of the operating point, not of the series. Any index built from the power spectrum is invariant under phase randomization, whereas the beyond-second-order value that retrieval and foundation models supply is not, because a phase-randomized series is asymptotically Gaussian. We state this as an impossibility result and isolate it with surrogate pairs that fix the spectrum and the marginal by construction. We then give a label-free, configuration-level diagnostic, the coverage deficit, whose principal term measures beyond-spectrum structure as the gain of analog over linear prediction. On seven benchmarks the prediction holds: window-keyed retrieval's value collapses across surrogate pairs (ECL median $+33\%\!\to\!-35\%$, $p{<}10^{-40}$) while every spectral index stays frozen; a foundation model's value splits into a surviving second-order part and a small beyond-linear margin that collapses; a longer linear window's value survives. Leave-one-dataset-out, the structure term predicts the sign of beyond-spectrum value where the spectral indices trail it, and the reverse holds for the second-order mechanism. We introduce no new forecaster; the contribution is the distinction, a controlled comparison, and a diagnostic for the deployment decision. Code: https://anonymous.4open.science/r/SINE.

When Are Neural Interaction Discoveries Real? Identifiability, Recoverability, and a Pre-Fit Diagnostic

When a neural time-series model reports that one variable modulates another's effect on a target, is the discovered interaction a property of the data or an artifact of model flexibility? We argue that this is fundamentally a question of identifiability, governed by the geometry of the observed input support rather than by the specific neural architecture. We study the problem in a multiplicative-gating extension of neural additive vector autoregression (GNAVAR), in which source contributions are modulated by other lagged variables. We show that representational capacity is not identifiability: dependent inputs induce leakage between edge-specific interaction terms, and low-dimensional support permits distinct interaction decompositions that agree on the observed data while differing elsewhere. We then prove a population identifiability theorem for normalized minimal GNAVAR decompositions under explicit support conditions, including settings with shared modulators. The theory yields a simple practitioner-facing diagnostic: the effective rank of the joint lag-block covariance predicts, before fitting, whether interaction recovery is feasible for a given candidate set. When the candidate set is unknown, a two-seed stability check provides a practical operational test. The same support condition organizes empirical outcomes into the three states predicted by the theory. Our results show that interaction recoverability depends on support geometry, that effective rank provides a practical pre-fit diagnostic, and that instability across independent fits is a characteristic signature of non-identifiable interaction discovery. The identifiability phenomenon, the support condition, and the instability signature are model-agnostic; GNAVAR is the vehicle that makes them provable.