The Perez Hourglass, a Trip Between Pascal, Fibonacci, Lichtenberg, Ramanujan and Pingala - Perez-Pingala Triangular Mock Theta Function: A Bridge from Vedic Meters to Ramanujan's Mock Thetas via the Perez Hourglass Fractal 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 Lichtenberg 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.

The Perez–Pingala theta function emerges as a novel mock theta-like series, uniquely linking the ancient Indian prosodic meters of Piṅgala (circa 300–200 BCE) and the Ṛig-Veda's signed chanda patterns to Ramanujan's 1920 mock theta functions. Its coefficients directly encode the signed syllabic counts from Vedic chant traditions, while its analytic properties inherit the Pascal-Fibonacci-Lichtenberg skeleton of the Perez Hourglass. This construction not only revives a 2,300-year-old mathematical intuition from Indian scholarship but also proposes a fault-tolerant quantum memory architecture grounded in golden-ratio harmonics, extending the hourglass's applications to modular forms and q-series in quantum information.