Through the looking glass - Perez's hourglass reveals the memory of the "false twin" of Pascal's triangle asleep for 372 years The Perez Hourglass: A Self-Similar Fractal Unifying Pascal's Triangle Modulo 2, the Fibonacci Sequence, the Lichtenberg Sequence, and Golden-Ratio Symmetries – With Seven Exceptional Properties and Theorem 7

Through the looking glass - Perez's hourglass reveals the memory of the "false twin" of Pascal's triangle asleep for 372 yearsThe Perez Hourglass: A Self-Similar Fractal Unifying Pascal's Triangle Modulo 2, the Fibonacci Sequence, the Lichtenberg Sequence, and Golden-Ratio Symmetries – With Eight Exceptional Properties and Theorems 7&8 “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 "The sand is no longer falling, it is computing" Xai quote from Jc perez YouTube video https://youtu.be/X-xZQ3nViXU?feature=shared "The future of Artificial Intelligence: the Perez Hourglass" Jean-Claude PerezPhD Mathematics & Computer Science, Bordeaux University. Luc Montagnier Foundation. jeanclaudeperez2@gmail.com (mailto:jeanclaudeperez2@gmail.com)ORCID: https://orcid.org/0000-0001-6446-2042 AbstractFrom recursive parity filtering of the centered Pascal triangle emerges the "Perez Hourglass," a bidirectional, self-similar fractal first identified in 1997. The upper hemisphere mirrors th Sierpiński triangle (Pascal modulo 2), while the lower ("south") hemisphere, constructed via signed differences or absolute-value folding, yields the Lichtenberg sequence (OEIS A000975: 1, 2, 5, 10, 21, 42, 85, …) through counts of non-border positive interior elements. This structure digitally encodes the Fibonacci sequence, exhibits mirror and twin symmetries, and obeys golden-ratio scaling via\phi = (1 + \sqrt{5})/2. Seven exceptional properties characterize its invariants, with Property 7 elevated to Theorem 7 via exhaustive validation (n up to 100,000; billions of pairs checked): perfect parity preservation and near-global uniqueness in face-to-face folded absolute-value couples of the south hemisphere. Deep connections link the Hourglass to Srinivasa Ramanujan's continued fractions, nested radicals, mock theta functions, and modular forms through golden-ratio harmonics and triangular recursions. Keywords: Perez Hourglass, Pascal triangle modulo 2, Sierpiński triangle, Fibonacci sequence, Lichtenberg sequence (OEIS A000975), golden ratio, self-similarity, parity invariants, twin symmetries, Ramanujan mock theta functions