Mathematical Foundations of Xiao Liu Ren: Algorithmic Time and Modulo Arithmetic
Authored • 2026-08-10Archival Record
The Six Palaces (*Xiao Liu Ren*) system is fundamentally a discrete mathematical mapping applied to temporal sequences. Rather than relying on continuous solar mechanics, it operates over a finite cyclic permutation group $\mathbb{Z}_6$.
## The Modular Sequence
The algorithm projects three temporal coordinates—lunar month ($M$), lunar day ($D$), and double-hour stem ($H$)—into an ordered state space:
$$\text{State}_1 = M \pmod 6$$
$$\text{State}_2 = (M + D - 1) \pmod 6$$
$$\text{State}_{\text{final}} = (M + D + H - 2) \pmod 6$$
Each state corresponds strictly to an archetypal attractor within traditional Daoist chronometry:
* $1 \equiv \text{Da An (Equilibrium)}$
* $2 \equiv \text{Liu Lian (Viscosity)}$
* $3 \equiv \text{Su Xi (Velocity)}$
* $4 \equiv \text{Chi Kou (Polarization)}$
* $5 \equiv \text{Xiao Ji (Convergence)}$
* $0 \equiv \text{Kong Wang (Entropy)}$
This algebraic structure ensures deterministic reversibility while maintaining sensitive dependence on initial boundary timestamps.