Information Loss Is All You Need: $TC^0$ Pseudorandomness from Worst-Case Lattices

Can deterministic information loss yield post-quantum pseudorandomness with $TC^0$ evaluation? We show that deterministic rounding achieves this under quantum worst-case lattice hardness, with fixed polynomial parameters and no predetermined sample bound. We introduce learning with algebraic rounding (LAR), which applies an explicit integer rounding map to blocks of secret-dependent linear measurements and encodes the result in a lattice quotient. For one fixed family with polynomially bounded modulus and dimensions, deterministic labels on exactly uniform, secret-independent public matrices are computationally indistinguishable from independent uniform labels for every polynomial number of classical random examples. Our classical reduction from Gaussian LWE, combined with the quantum worst-case reduction to LWE, bases LAR on GapSVP and SIVP over general lattices at approximation factor $\widetilde{O}(q)$, where $q$ is the modulus. The proof uses source-independent affine transport: public affine maps sampled independently of the hidden errors transport the Gaussian error law close to uniform on the rounding map's zero-label fiber while preserving exactly uniform public matrices. For every fixed inverse-polynomial accuracy, we give a polynomial-time sampler for the same public parameter family. This yields the first post-quantum weak pseudorandom function with joint key-and-input evaluation in $TC^0$, secure against uniform quantum polynomial-time distinguishers given any polynomial number of classical uniform random examples. We also prove hardness of learning polynomial-size two-hidden-layer ReLU networks from noiseless uniform random examples, even for quantum polynomial-time improper learners.