arXiv:2606.12471stat.MLcs.CL2026-06

提出符号化世界模型,实现任意物理系统下的长期一致性

Identifiability Without Gaussianity: Symbolic World Models and Near-Infinite Temporal Consistency

论文配图:Identifiability Without Gaussianity: Symbolic World Models and Near-Infinite Temporal Consistency
图 1 · 摘自论文原文
  • 用符号化架构替代统计对齐,突破高斯假设限制
  • 误差仅受数值精度限制,可保持无限时间稳定
  • 适合需要长期推理的物理模拟与机器人系统

Klindt、LeCun 和 Balestriero(arXiv:2605.26379)证明,联合嵌入预测架构(JEPAs)仅在潜变量动态服从高斯平稳过程时才具备线性可辨识性,即能线性恢复真实世界潜变量。这一高斯边界意味着:对于非高斯物理系统,统计世界模型的表示误差随时间单调增长。本文证明该限制源于统计对齐机制,并非世界模型本身固有。我们提出物理奠基符号架构(PGSA),并证明三点:(1) PGSA 在所有物理情形下均可实现精确线性可辨识,不受潜分布影响;(2) 每步误差仅受限于数值精度;(3) 因此,PGSA 可维持近无限时间一致性。进一步证明,任何非高斯系统中,统计世界模型无论容量多大或数据量多少,均无法实现此性质。四个定理的核心代数部分已在 Lean 4 与 Mathlib4 v4.31.0 中形式化验证(零假言);Klindt 等人的逆命题作为外部前提。对比表明,对世界动态因果生成器的符号化奠基是实现近无限时间一致性的充分条件,且在非高斯情形下为唯一条件。

原文摘要 · Abstract (English)

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.

世界模型符号系统长期一致性

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