The hexagram as a spin chain
The Ising model on the six lines
Move the temperature and the sign of J and watch the Boltzmann probability concentrate on a few hexagrams.
The connection with the I Ching is a mathematical identity of structure (the same 64 states, the same transfer matrix), not a physical claim about the oracle. Ising (1925), building on Lenz's formulation of 1920, proved that in 1D there is no phase transition: this chain of 6 orders gradually on cooling, never all at once.
Read each hexagram as a chain of 6 spins: yang = +1, yin = −1. The energy is E = −J times the sum of products of neighbouring lines, and the probability of each hexagram is Boltzmann's, proportional to e^(−βE). The transfer matrix T = [[e^(βJ), e^(−βJ)], [e^(−βJ), e^(βJ)]] gives the partition function: the open chain is Z open = 1ᵀT⁵1 and the ring Z ring = Tr(T⁶).
dot size = Boltzmann probability · touch a hexagram for its energy
With the transfer matrix T (beta = 0.7, J = 1), the open-chain partition function is 1ᵀT⁵5 = 199.384322 and the ring one Tr(T⁶) = 262.456561; with beta at zero, Z = 64 and the distribution is uniform.
verificar_ising · source: Ising, 1925At low temperature with positive J, Qian and Kun dominate 50/50, and with negative J, Ji Ji and Wei Ji; the hard constraint (no adjacent yin-yin) reproduces F(8) on the chain and L(6) on the ring, with dominant eigenvalue phi, the same transfer matrix as the design-your-rule experiment.
verificar_isingThe model, formulated by Lenz in 1920, was solved by Ising in 1925 in one dimension, proving that in 1D there is no phase transition: the chain orders gradually on cooling.
The connection with the I Ching is a mathematical identity of structure, not a physical claim about the oracle.
The types of claim and the full bibliography (APA) are in Foundations.