A Self-Consistent Geometric Fixed Point Near the Inverse Fine-Structure Constant from a 6-Port Substrate Model

We show that a 6-port unitary scattering node on a simple cubic lattice at the site-percolation threshold $p_c = 0.3116$ admits a self-consistent geometric fixed point whose value lies numerically close to the inverse fine-structure constant $\alpha^{-1}$. Two geometric strands, coupled through the lattice function $n_c(R)$ (mean outward-bond count per surface node at sphere radius $R$), yield: Strand~A, a structured geometric ansatz for the dimensionless substrate impedance $\alpha^{-1} = 266.3/n_c(R)$ as a product of four geometric and symmetry factors; and Strand~B, a soliton equilibrium radius $R = 40.63\,a/(6 - n_c(R))$ from energy minimization with a null-transition correction derived conditional on a linear-response assumption. The self-consistency condition has a unique solution at $R^* = 10.019\,a$, giving a predicted value $136.93$, a deviation of $0.08\,\%$ from the experimental value $\alpha^{-1} = 137.036$. The construction introduces no continuous fit parameter and no numerical calibration to $\alpha$; the only empirical numerical substrate input is $p_c$. The principal result is the existence of a unique self-consistent fixed point near $\alpha^{-1}$ on a constrained discrete choice space, not the third-decimal-place agreement, which depends on several substantive model inputs and is not theory-bearing at that precision. Sensitivity checks show that the fixed-point output remains in the same numerical neighborhood under variations of lattice type, mixing rule, and sphere-center position, with the baseline choice giving the closest agreement; the best agreement is not robust to arbitrary substitutions. All six model assumptions are stated explicitly. This manuscript was drafted in May 2026 and is uploaded here in May 2026 as Version 1.0 of the preprint.