The Wristband Gaussian Loss: Deterministic, Composable Latents via a Sphere-Interval Decomposition

We present the Wristband Gaussian Loss, a deterministic batch loss for Gaussianizing point embeddings without sampling, KL terms, or iterative transport. Each x ∈ ℝd is mapped to a direction u = x/|x| and a CDF-transformed radius t = Fchi2d(|x|2) on the wristband Sd-1 × [0,1]. We prove (and machine-verify in Lean~4) that for d ≥ 2 the pushforward wristband map equals σd-1 otimes Unif[0,1] iff the source is N(0, Id), and that the Neumann-reflected wristband repulsion energy is uniquely minimized at the uniform target. We compute this reflected-kernel objective in two ways: a nearest three-image pairwise truncation at O(N2 d), and a spectral Neumann path joining angular and radial Mercer modes (spherical-harmonic and cosine) at O(N d K), with empirically matched gradients. A 1D Wasserstein radial term and a moment penalty serve as finite-sample accelerators with the same optimum, and Monte-Carlo null calibration turns the components into a single standardized statistic. We evaluate direct point-cloud Gaussianization with a calibrated barycentric W2 score: a deterministic Gaussian reference batch is built by recursive Hungarian averaging, with each method reported as a z-score against same-size Gaussian batches. On the axis-uniform X benchmark, Wristband is competitive in 2D and gives the best 10D score. On a harder radial--angular-copula impostor whose Gaussian radial and angular marginals are correct but dependent, Wristband gives the best 10D and 128D scores. Coupled with learnable-key Euclidean attention and exact invertible flows, the resulting Deterministic Gaussian Autoencoder delivers a Gaussian-latent interface for counterfactual sampling with independent factors and a context/residual construction for dependent factors.