The Perez Hourglass, a Trip Between Pascal, Fibonacci, Lichtenberg, and Ramanujan: Perspectives in Quantum Computers, Cryptography, and Associative Memory

Abstract

More than three decades after the discovery of self-organizing neural networks governed by the golden ratio (Perez, 1988, 1991, 1997), a remarkable fractal structure—the “Perez Hourglass”—emerges from Pascal’s triangle through recursive parity filtering and infinite-dimensional folding (Perez, 2025a–e). This exact, self-similar hourglass, indexed as OEIS A000975, constitutes the digital incarnation of the Fibonacci sequence, the topological antimatter of Sierpiński’s fractal triangle, and a direct bridge to Ramanujan’s continued fractions, nested radicals, and modular forms via golden-ratio harmonics
\phi = (1+\sqrt{5})/2

. We prove that the Perez Hourglass simultaneously enables:
1. A distance-3 Fibonacci-valued CSS quantum error-correcting code [[F_{2n+1}, 1, F_n]] surpassing the Bravyi–Poulin–Terhal bound;
2. Native golden-phase gates
e^{i\pi/\phi^2}
and
e^{i\pi \phi^2}
;
3. Magic-state distillation with O(log log N) overhead;
4. Fractal anyon protection analogous to Haah codes and quantum gravity time fractals;
5. Post-quantum cryptography based on the hardness of decoding random Fibonacci-coded linear systems;
6. The first perfect associative memory in history—Perez Hourglass Associative Memory (PHAM)—with storage capacity
\sim \phi^n
and one-shot perfect retrieval.

Main reference is
Perez, J. C. (2025). Seven Exceptional Properties of the "Perez Hourglass": Perspectives toward New Types of Artificial Intelligence and Quantum Computers. Appendix: A Topological Blueprint for Fault-Tolerant Quantum Computing, Post-Quantum Cryptology, and Golden-Ratio Associative Memory. Zenodo. https://doi.org/10.5281/zenodo.178300941. 

Keywords
fractal quantum computing, golden ratio, topological quantum error correction, post-quantum cryptography, associative memory, Fibonacci coding, CSS codes, magic-state distillation.