We present a deterministic, matrix-free framework for deep representation learning based on geometric invariants and analytic Koopman–tangent projections in a weighted log-prime Hilbert space rather than stochastic optimization. The method constructs a representation in a single, fully parallelizable pass through the data, without iterative optimization or gradient descent. The core of the framework is a Goldilocks (Gamma) measure (a multiplicative–additive equilibrium measure minimizing energy in the log-prime basis) that defines a weighted Hilbert space, a log-prime orthogonal basis that yields a diagonal *surrogate* under the weighted spectral basis, rendering per-mode updates independent, and a variational principle that selects the optimal representation by balancing energy and harm to geometric primitives (polynomial moments and spectral curves defined by the stationary invariants of the Koopman operator). We demonstrate the framework on the Fashion-MNIST dataset, achieving classification accuracy comparable to a standard CNN (85-88%) with an estimated one to three orders of magnitude reduction in energy consumption. Orthogonality arises natively because log-prime channels are multiplicatively and rationally independent; no Gram–Schmidt is required. The framework is deterministic, interpretable, and provides a continuous trade-off between compression and accuracy, offering a deterministic alternative for specific regimes (spectral, geometric-invariant tasks).