PLI Candidate Layer Supplementary Note: α's c₀ Coefficient and the Third-Order Coupling of Hexagonal Patterns

This candidate layer supplementary note addresses two technical problems left unresolved in the PLI Shell Candidate Equation v6.0. The first problem is the coefficient c₀ in the expression α = 1/(c₀·t_p·√N_eff). The note shows that the numerical value c₀ = 1 used in v6.0 is incompatible with the area element geometry of A3, because a two-dimensional shape with area l_p² cannot have perimeter equal to l_p. The minimum perimeter for area l_p² is about 3.545 l_p, corresponding to a circle. Candidate values of c₀ range from 3.54 for a circle to 4.56 for an equilateral triangle. The form of the α expression is not in conflict with A3; only the numerical value c₀ = 1 needs correction. The second problem is the third-order coupling v of the hexagonal pattern. An order-of-magnitude estimate gives v ≈ 0.018, which is very small compared with the saturation term g ≈ 0.4. The stability of the hexagonal pattern may not depend on the third-order coupling but on the saturation term. A complete amplitude equation analysis is required. The note also clarifies that the v6.0 equation produces two types of patterns under different initial conditions: concentric rings under center perturbation, and hexagonal patterns under global random initial conditions. The third-order coupling analysis applies only to the hexagonal pattern. The note does not change the form of the v6.0 equation. It points to a numerical correction of the c₀ value and lists open problems. 本候选层补充笔记处理 PLI 壳层候选方程 v6.0 中两个未解决的技术问题。 第一个问题是 α = 1/(c₀·t_p·√N_eff) 中的系数 c₀。笔记表明,v6.0 中使用的数值 c₀ = 1 与 A3 的面元几何不相容,因为面积为 l_p² 的二维形状,其周长不可能等于 l_p。面积 l_p² 的最小周长约为 3.545 l_p,对应圆形。c₀ 的候选值范围从圆形的 3.54 到等边三角形的 4.56。α 表达式的形式不与 A3 冲突,需要修正的只是数值 c₀ = 1。 第二个问题是六角斑图的三阶耦合 v。量级估算给出 v ≈ 0.018,与饱和项 g ≈ 0.4 相比非常小。六角斑图的稳定性可能不依赖三阶耦合,而依赖饱和项。需要完整的振幅方程分析。 笔记还澄清,v6.0 方程在不同初始条件下产生两种图案:中心扰动下出现同心环,全局随机初始条件下出现六角斑图。三阶耦合分析只适用于六角斑图。 笔记不改变 v6.0 方程的形式,但指出 c₀ 数值需要修正,并列出开放问题。