The algebraic cores of four theorems are formalized in Lean 4 with Mathlib, with no sorry placeholders; the Klindt et al. converse is taken as an external premise.
Abstract
Klindt, LeCun, and Balestriero (arXiv:2605.26379) proved that Joint-Embedding Predictive Architectures (JEPAs) achieve linear identifiability, the linear recovery of the world's true latent variables, if and only if the world's latent dynamics follow a Gaussian, stationary process. This Gaussian boundary implies a fundamental limit on temporal consistency: for any non-Gaussian physical system, the representation error of a statistical World Model grows monotonically with time. We prove that this limit is an artifact of the statistical alignment mechanism, not a property of World Models in general. We introduce the Physics-Grounded Symbolic Architecture (PGSA) and prove three results: (1) a PGSA achieves exact linear identifiability for all physical regimes, regardless of the latent distribution; (2) the per-step error of a PGSA is bounded by numerical precision alone; and (3) as a direct consequence, a PGSA maintains temporal consistency for an unbounded number of transitions, a property we term near-infinite temporal consistency. We further prove that statistical World Models cannot achieve this property for any non-Gaussian system, regardless of model capacity or the volume of training data. The algebraic cores of four of the theorems are formalized in Lean 4 with Mathlib4 v4.31.0 (zero sorry placeholders); the Klindt et al. converse is taken as an external premise. The contrast establishes that symbolic grounding in the causal generator of the world's dynamics is the sufficient condition and, in non-Gaussian regimes, the only condition for near-infinite temporal consistency.
Problem
Klindt, LeCun and Balestriero showed that JEPA world models achieve linear identifiability only for Gaussian, stationary latent dynamics. For non-Gaussian physical systems, this implies that representation error grows over time. The paper asks whether this temporal-consistency limit applies to world models in general or only to statistical alignment.
Approach
The authors define a Physics-Grounded Symbolic Architecture (PGSA): an Atom Registry of deterministic physical-law functions, a typed state graph, and a causal basis that composes atoms to reproduce the world generator. They prove exact identifiability and per-step error bounded by machine precision. A representation-bias propagation lemma shows that statistical models diverge on non-Gaussian systems, and an approximate result covers incomplete registries. The algebraic cores of four theorems are machine-checked in Lean 4 with Mathlib, and seven simulated physical systems plus solver-based engineering systems are analyzed.
Figure 1 : The Gaussian Assumption Fails for Physical World Models: a Direct Comparison. Each row shows a physical system’s true latent distribution (World z ), the nonlinearly mixed observation (Data x=g(z) ), and the representation recovered by a LeJEPA-style model ( f(x) ). Top row (Gaussian): the guarantee of Klindt et al. [ 1 ] holds: the recovered representation is a linear rotation of the t
Results
PGSA achieves exact identifiability regardless of the latent distribution, with a horizon of about 4.5×10^13 steps for non-chaotic systems. Statistical models have finite horizons on all non-Gaussian systems, and the measured linear identifiability is about R²≈0.90 for non-chaotic systems.
Figure 2 : Temporal error growth by World Model paradigm. The PGSA error stays bounded by machine precision, while statistical models diverge. Simulation steps on the x -axis; temporal error \epsilon_{t} (log scale) on the y -axis.Figure 4 : PGSA temporal consistency horizon T^{*}_{\mathrm{PGSA}} . Bar height is the horizon in simulation steps (log scale); the equivalent real-world time is labelled above each bar. For the five non-chaotic systems the horizon is the machine-precision bound \delta/\mu\approx 4.5\times 10^{13} steps (identical bars); the wall-clock times differ only because each system is integrated at a diffe
