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

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