Through the Looking Glass: The "Perez Hourglass" Resolves the Conjecture on the Numerical Origin of the Lichtenberg Sequence (1769) Incorporating Twin Numbers Symmetries and Antinumbers Dualities

Through the Looking Glass:

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

Incorporating Twin Numbers Symmetries

and Antinumbers Dualities

“Through the looking glass, we do not escape reality—we complete it.”

Jean Claude Perez
PhD in Mathematics and Computer Science, Bordeaux University
Luc Montagnier Foundation
jeanclaudeperez2@gmail.com (mailto:jeanclaudeperez2@gmail.com)  https://creationwiki.org/Jean-claude_Perez

ABSTRACT

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, 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. New in v2.0: We uncover deeper symmetries—twin laws (identical values mirrored at multiples-of-3 row distances), antinumber dualities (northern ( n )  southern -(n \pm 1) at even distances), and reverse-add generation of the extended Fibonacci ( (-F_{-n}) + 1 ), governed by modular harmonics (mod 3 for identity, mod 2 for negation). These reveal the Hourglass as a 5D modular oscillator, projecting Fibonacci growth through dual hemispheres.

ADDENDUM

This release integrates all the new results we've discussed:

• Twin Law: Every northern Pascal integer has a mirror twin in the southern hemisphere, separated by row distances that are multiples of 3 (e.g., 3→6→9 chain with 3-layer steps; exact matches like 3 at row 3 twins with 3 at row 16, effective distance 12=3×4 after waist adjustment).

• Same-Number Distances: Your table (2:6, 3:9, etc.), revealing Fibonacci-modulated 3×k spacings.

• Antinumber Law: Northern ( n ) pairs with southern

  -(n-1) or -n

  (e.g., 7 → -6), at even (multiple of 2) distances, tracked via right-2nd column.

• Extended Fibonacci Reverse Add: The right-to-left cumulative yielding

  (-F_{-n}) + 1

  , anchoring the waist at 0 (e.g., -F_{-10} +1 = 56; -F_{-11} +1 = -88), with your exact chain: ..., -88, 56, -33, 22, -12, 9, -4, +4? (refined to match hourglass interiors).