Mathematical Reconstruction of Hetu and Luoshu: A Pure Algebraic Annealing Approach to Demystify the I Ching

For centuries, the I Ching (Book of Changes) and its numerical diagrams, Hetu and Luoshu, have been largely relegated to the realm of mysticism and philosophy by Western academia. Early computational studies attempting to simulate these diagrams often relied on physical analogies (e.g., cellular automata with preset gravitational centers), which fundamentally fail to prove their mathematical nature. To address this, we propose a pure algebraic annealing engine that completely eliminates physical spatial coordinates. By strictly adopting the Southern Song dynasty’s standard (Zhu Xi’s Zhouyi Benyi) configurations, we model Luoshu as a strong magic-square constraint (all lines sum to 15) and Hetu as a congruence modulo constraint (opposing differences equal 5). Experimental results demonstrate that within the 9! state space, both structures act as distinct algebraic attractors, achieving 100% convergence under ideal conditions. Parameter robustness scanning reveals that under ±25% noise perturbation, Hetu maintains a 100% convergence rate, while Luoshu achieves 86%. Furthermore, we establish an isomorphism between modern dynamical annealing and ancient Chinese algebraic deduction, proving that the I Ching is not mysticism, but an ancient, highly rigorous algorithm and topological system.