Wavelet Variance Equipartition as a Threshold for World-Model Quality and Quantum Kernel TN-Simulability

Abstract

While world models learn compact representations of complex environments, they lack a physics-grounded metric to assess the structural fidelity of their latent spaces. We identify the wavelet scaling exponent $\alpha$ as a critical diagnostic, proposing optimal representations satisfy variance equipartition ($\alpha \approx 1/2$) -- mirroring Kolmogorov's inertial range. We establish $\alpha = 1/2$ as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels. Using tensor-network theory, we prove latents with $\alpha > 1/2$ reside in an area-law phase admitting efficient classical emulation, while $\alpha < 1/2$ triggers a volume-law phase where the Matrix Product State bond dimension $\chi$ grows exponentially with qubit count $n$. Analyzing pre-trained VideoMAE latents reveals a dichotomy: spatial tokens approach the equipartition limit ($\alpha \approx 0.423$), but permutation-invariant feature channels exhibit unstructured disorder ($\alpha \approx -0.123$). This forces real-world latents deep into the volume-law phase, providing a data-driven necessary condition for simulation hardness. Finally, we apply Weingarten calculus to derive the exact variance of the scrambled transition probability under a 2-design ensemble. We prove this variance scales strictly as $\Var[X] = \Theta(d^{-2})$. We confirm this numerically with a log-log slope of $-1.881$ ($R^2 = 0.999$), identifying a formidable shot-noise wall demanding a measurement budget of $M = \Omega(d^2)$ that constrains quantum machine learning scalability.