From A1 to the Shell Equation: The Turing Instability of the Geometric Constant a

本文从 PLI 框架的唯一公理 A1 出发,推导壳层方程中的几何系数 a = π/(3√3),并证明壳层方程在均匀态处满足图灵不稳定性条件。图灵条件给出相变点 a_c = 0.5766。数值模拟给出相变落在 a ∈ [0.57, 0.58]。PLI 的 a = 0.6046 落在斑图区内,距相变点 0.028。数值生长阶段给出的最不稳定波数为 k = 8,与理论预测 k_FFT = 8.98 相差 10.9%,差异源于有限格点离散化。a 从 A1 的推导包含三处读法,本文显式标出。本文不主张 PLI 是已完成的物理理论。 We derive the geometric coefficient a = π/(3√3) in the shell equation from A1, the single axiom of PLI, and prove the shell equation satisfies the Turing instability condition around the uniform state. The Turing condition gives the critical point a_c = 0.5766. Numerical simulation gives the transition in a ∈ [0.57, 0.58]. PLI's a = 0.6046 falls inside the patterned region, 0.028 from the critical point. The growth-stage numerical most unstable wavenumber is k = 8, versus the theoretical prediction k_FFT = 8.98, a 10.9% deviation due to finite-lattice discretization. The derivation of a from A1 contains three readings, explicitly marked. We do not claim PLI is a completed physical theory. Accompanying code: pli_shell.py.