Through the Looking Glass: The "Perez Hourglass" Resolves the 256-Year Lichtenberg Conjecture via Evenness, Twin Symmetries, and a 5D Modular Oscillator

The Fibonacci sequence is universally recognized as a cornerstone of harmony in mathematics and nature. Yet, few are aware of its "extended Fibonacci sequence"—a mirror-like palindrome centered on the pivotal ZERO, separating positive and negative integers. Similarly, while Fibonacci numbers emerge mysteriously from Pascal's triangle, the existence of a symmetrical antimatter counterpart—a mirrored Pascal triangle with the number ONE as its axis of symmetry—has remained unexplored. This unified structure forms an hourglass shape, first named the "Perez Hourglass" in 1997. From this construct emerges the extended Fibonacci sequence, revealing a consistent "digital antimatter" counterpart to both Pascal's triangle and the Fibonacci numbers. Superimposing the northern (Pascal) and southern (antimatter) hemispheres yields sum and difference triangles whose entries are universally EVEN (Theorem). Twin laws (identical values every 3 rows, mod 3) and antinumber dualities (negation every 2 rows, mod 2; northern n ↔ southern −(n±1) at even distances) govern reverse-add generation of the extended Fibonacci (−F₋ₙ + 1) via modular harmonics. These symmetries elevate the Hourglass to a 5D modular oscillator, projecting Fibonacci growth across dual realms. By counting non-border positive interior elements in each row of the difference-based southern hemisphere (excluding the two border 1s), the Lichtenberg sequence (OEIS A000975): 1, 2, 5, 10, 21, 42, 85, … emerges precisely—resolving a 256-year-old conjecture first noted by Georg Christoph Lichtenberg in 1769 in connection with the Chinese Rings puzzle. v3.0 proves universal evenness for all rows, formalizes the 5D oscillator, and derives the Lichtenberg recurrence Lₙ = 2Lₙ₋₁ + Lₙ₋₃ directly from twin/modular harmonics.

We demonstrate that the hourglass superposition of the Northern (Pascal/matter) and Southern (antimatter) hemispheres enforces a universal parity alignment:

Every even element (0) faces another even element (0), and every odd element (1) faces another odd element (1) across the equatorial mirror.

This Universal Evenness Theorem emerges directly from the binomial structure of Pascal’s triangle and its inverted antimatter twin, producing self-similar Sierpiński fractal patterns in parity (mod 2). We further propose a physical interpretation in electronics (+ = 1, − = 0) and condensed matter.

This work unifies Fibonacci duality, modular Pascal geometry, and historical conjecture into a coherent antimatter-number theory, with the Lichtenberg sequence as its measurable heartbeat.