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

We introduce the Wristband Gaussian Loss, a deterministic batch regularizer for learning standard-Gaussian representations without per-step Gaussian reference sampling, KL penalties or an inner transport solve. Gaussianity is equivalent to joint uniformity of direction and CDF-transformed radius on a sphere–interval product. The full reflected-Neumann population energy has a unique Gaussian target, and its excess equals a squared maximum mean discrepancy (MMD), with analytic proofs and an accompanying Lean 4 formalization. Independent Gaussian blocks compose, and joint Gaussian codes support block resampling. GPU-friendly three-image pairwise and truncated spectral repulsion cost $O(N^{2}d)$ and $O(NdK)$, respectively, for $K$ radial modes and fixed angular truncation. Optional radial and moment penalties and Gaussian-null calibration balance the finite-batch training terms. At $N=8192$, $d=128$, $K=16$, three B300 input seeds give a $1.86\pm 0.06\times$ spectral-over-pairwise speedup for full-loss forward/backward evaluation. Five-seed direct tests compare Gaussianization with analytic MMD, exact empirical assignment, refined Sinkhorn, and SIGReg. On learning representations: CIFAR-100, Shapes3D, and conditional-resampling experiments evaluate the trade-off between Gaussian representation structure and task performance. In the primary CIFAR-100 cohort, Wristband has lower held-out projector MMD${}^2$ than analytic MMD, with mean linear-probe accuracies of 29.56% and 30.88%, respectively.