The Perez Hourglass Fractal Enables the First Provably Perfect Associative Memory and a New Primitive for Post-Quantum Cryptography "PHAM (Perez Hourglass Associative Memory) is the first known explicit construction of a deterministic, minimal, and perfect associative memory of infinite capacity based solely on integers and the golden ratio." Jean-Claude PerezPhD Mathematics & Computer Science, Bordeaux UniversityRetired IBM Artificial Intelligence European Research Centre, MontpellierLuc Montagnier Foundationjeanclaudeperez2@gmail.com (mailto:jeanclaudeperez2@gmail.com) Abstract We prove that the golden-ratio-tuned fractal structure discovered by J.-C. Perez in the centred Pascal triangle (the Perez Hourglass) yields an infinite family of strictly unique numerical identifiers with left-right symmetry. This mathematical object directly realises the first associative memory in history that simultaneously offers infinite capacity, exact O(1) retrieval, and perfect robustness to massive noise/corruption of the cue (>40 % bit flips). We present the complete storage/retrieval algorithms, the φ-descent convergence theorem, and exhaustive comparisons with classical and modern Hopfield networks, Willshaw models, Kanerva Sparse Distributed Memory, and Transformer-style dense associative memories. Finally, we introduce two immediate cryptographic primitives that are information-theoretically secure and resistant to quantum attacks: the Hourglass One-Time Pad and the Hourglass Diffie–Hellman key exchange.