Through the Looking Glass: The "Perez Hourglass" Resolves the Conjecture on the Numerical Origin of the Lichtenberg Sequence (1769)

ABSTRACTThe 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, which separates positive and negative integers. Similarly, while it is well-known that the Fibonacci numbers emerge mysteriously from Pascal's triangle, the existence of a symmetrical 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.Here, we demonstrate that the Perez Hourglass not only unifies these elements but also provides a geometric resolution to a 256-year-old conjecture: the numerical origin of the Lichtenberg sequence (OEIS A000975), first noted by Georg Christoph Lichtenberg in 1769 in connection with the Chinese Rings puzzle. By counting non-border positive (or negative) elements in each row of the difference-based southern hemisphere of the Hourglass (excluding the two border 1s), the sequence 1, 2, 5, 10, 21, 42, 85, ... emerges precisely.