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Training binary or quantized neural networks is inherently challenging due to the discrete nature of the problem and the high dimensionality of modern models. But imagine the existence of a meta-distribution from which one could easily sample binary neural networks with controllable accuracy—and the sampling process did not involve any training in the binary/quantized domain. We present a novel randomized framework that makes this possible. In contrast to traditional quantization and hyperdimensional computing methods, our approach models binary representations as points on the Hamming cube, embedding data directly into binary space. At the core of our method is Grothendieck’s lemma, which enables the construction of a new class of network layers—Grothendieck layers—that emulate standard neural layers in the floating-point domain. Each of these layers admits a randomized binary counterpart—an embedded hyperdimensional network—whose accuracy closely mirrors that of the original, with rigorous theoretical guarantees derived from the concentration of measure in high dimensions. Empirically, these sampled binary networks match the performance of convolutional neural networks while being amenable to efficient deployment on edge devices. This framework introduces a principled and scalable pathway for designing robust binary networks without requiring any discrete optimization during training.