Fibonacci in the hexagram
Counting with no two adjacent yin lines draws the sequence
Walk the ladder and let the colour explain the recurrence, or open the Venn: only Ji Ji and Wei Ji remain.
Original findingIt is a counting theorem, not a hidden code. The numerology circulating about Fibonacci and the I Ching (hand-picked offsets, trading retracement levels, the golden ratio hidden in the King Wen ordering) is left out as undemonstrable: here there are only sets that get counted and turn out to be a Fibonacci number. The only φ that appears is the real limit of those counts, not an imposed pattern.
As far as we know, this verified formulation over the hexagrams (the survivors as independent sets of the path P6 and of the cycle C6) is not published anywhere else. We say it with that exact humility: we have not found the source, not that it does not exist.
Every new line doubles the population (2, 4, 8, 16, 32, 64), but the figures with no two adjacent yin only grow like Fibonacci (2, 3, 5, 8, 13, 21). The colour shows why: those ending in yang come from the previous row, and those ending in yin come from two rows back. Blues plus greens of one row = the live ones of the two rows above: F(n+2) = F(n+1) + F(n), in plain sight.
Grey: figures with two adjacent yin (eliminated). The right-hand column compares survivors against the total population and their ratio between rows.
The 21 survivors per rule are the independent sets of P6; in the circular version, of C6. Everything is counted, nothing is hard-wired.
The hexagrams with no two consecutive yin (and, by symmetry, no two yang) are exactly 21 = F(8); counting by number of lines, the ladder is 2, 3, 5, 8, 13, 21 = F(n+2).
verificar_fibonacci · source: OEIS Foundation, s.f.In the circular version (line 6 a neighbour of line 1) the survivors are 18 = L(6), the Lucas number.
verificar_fibonacci · source: OEIS Foundation, s.f.The intersection of both rules (perfect alternation) is exactly Ji Ji and Wei Ji, and the breakdown of the 21 by number of yin is 1, 6, 10, 4 = C(7-k, k), the Fibonacci identity in Pascal’s triangle.
verificar_fibonacciA counting theorem, not a hidden code: the numerology of Fibonacci and the golden ratio in the I Ching is left out as undemonstrable.
The same count generalises with a 2x2 transfer matrix: the no-two-yin rule is one case and its dominant eigenvalue is phi.
verificar_fibonacciWeb search of the formulation (21 = F(8) with no two consecutive yin lines, the ladder F(n+2), Lucas in the circular version, the Ji Ji / Wei Ji intersection); only numerology was found (Fibonacci offsets on the bagua, the golden ratio in King Wen); no equivalent verified formulation.
The types of claim and the full bibliography (APA) are in Foundations.