Conservative multifractal cascade prior

Construct a differentiable hierarchical latent tensor for an image of side length \(2^J\). Start with root mass \(m_0=1\); at each level split every active node into \(b=4\) spatial children. Sample or learn a positive vector \(r\in\mathbb R_+^4\), normalize it as \(p_i=r_i/\sum_k r_k\), and assign child masses \(m_{child_i}=m_{parent}p_i\). Enforce the conservative structure by using one simplex vector per parent and never sampling child multipliers independently. Convert level-\(J\) masses into one or more image-like latent channels, optionally apply a learned 1x1 convolution, and feed the multiscale tensor to a UNet, VAE decoder, or transformer patch encoder. Expose \(\Delta\alpha\) by interpolating between a uniform vector and a positive geometric ladder \(p_i\propto r^{i-1}\), while avoiding zero weights. During training, vary the width and condition the network on \(\Delta\alpha\). Add a small spectrum-consistency loss: estimate \(Z_q(j)=\sum_c m_{j,c}^q\) at each level and fit the slope of \(-\log_b Z_q(j)\) across levels to the analytic \(\tau(q)\). The formulas provide the exact spectrum and conservation constraint; decoder architecture and width schedule are empirical. First test on synthetic 2D cascades and fractional-noise textures, then CelebA-64 or microscopy data, comparing Gaussian latent noise against the cascade prior at equal parameter count. Success is lower multiscale moment error and better Euler-density curves at equal FID, with total mass remaining one at every hierarchy level.