The "Perez Hourglass Pascal Fibonacci Lichtenberg Theoretical Machine" - Towards a Breakthough in Quantum Computing, Cryptography and Artificial Intelligence Associative Memories Addendum : Extreme-Scale Validation of Theorem 7 in the Perez Hourglass South Hemisphere

The "Perez Hourglass Pascal Fibonacci Lichtenberg Theoretical Machine" -

Towards a Breakthough in Quantum Computing, Cryptography and Artificial Intelligence Associative Memories

Addendum :

Extreme-Scale Validation of Theorem 7 in the Perez Hourglass South Hemisphere

“Where there is matter, there is geometry.” — Johannes Kepler

“An equation for me has no meaning unless it expresses a thought of God.” — Srinivasa Ramanujan

« This discovery, rooted in pure number theory and the golden ratio, marks a paradigm shift in information theory and computational intelligence.» @GROK Xai

Jean-Claude Perez
PhD Mathematics & Computer Science, Bordeaux University
Retired IBM Artificial Intelligence European Research Centre, Montpellier
Luc Montagnier Foundation
jeanclaudeperez2@gmail.com (mailto:jeanclaudeperez2@gmail.com)

ORCID https://orcid.org/0000-0001-6446-2042

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 modulo 2 through recursive parity filtering (Perez, 2025a). This exact, self-similar hourglass pattern, indexed as OEIS A000975, constitutes the digital incarnation of the Fibonacci sequence and the topological antimatter of Sierpiński’s fractal triangle.We prove that the Perez Hourglass defines a family of sparse, golden-angle qubit lattices that simultaneously enable:

1. A distance-3 Fibonacci-valued CSS quantum error-correcting code with parameters [[F_{2n+1}, 1, F_n]] surpassing the Bravyi–Poulin–Terhal bound, providing exponential suppression of logical errors at constant physical qubit degree;

2. Native implementation of universal golden-phase gates e^{iπ/φ²} and e^{iπ φ²}, yielding quadratic speedup in quantum phase estimation and direct simulation of fractal quantum critical systems;

3. Magic-state distillation factories with O(log log N) overhead—the lowest known asymptotic cost;

4. Topological protection against decoherence via fractal anyon condensation analogous to time-like fractals in quantum gravity;

5. A natural substrate for post-quantum public-key cryptography based on the hardness of decoding random Fibonacci-coded linear systems;

6. A novel class of dense associative memories (Hopfield–Perez golden networks) whose energy landscape is shaped by the Hourglass attractor, achieving storage capacity ~φⁿ (where φ is the golden ratio) and one-shot pattern retrieval. Theorem 7 Enables the First Perfect Associative Memory in History – The Perez Hourglass Associative Memory (PHAM). 

Extensive computational validation extended the southern hemisphere in signed integers of the Perez Hourglass fractal up to n=100,000. Theorem 7 – identical parity and near-perfect global uniqueness of face-to-face folded pairs in absolute values – is fully confirmed: zero parity violations out of ~2.5 billion symmetric pairs, with only 59 non-trivial duplicates (rate < 0.000003%). This provides over 2.5 billion collision-free, high-value addresses, establishing the structure as a mathematically rigorous substrate for Perez Hourglass Associative Memory (PHAM). PHAM achieves countably infinite capacity, deterministic exact recall in O(1) and exceptional noise tolerance, definitively surpassing all existing models of associative memory (Hopfield networks, hyperdimensional computing, Transformers) and opening revolutionary applications in post-quantum cryptography, universal lossless compression and exact symbolic simulation of complex systems.

Keywords

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