Paper 83: A Demonstration that a Dimensionless Constant Consistent with the Fine Structure Constant Emerges as a Stable, Universal Fixed Point of a Many-Body Dynamical System.

This paper tests whether a simple, fully declared network model can produce a sharp, repeatable transition and a stable large-size (“large-n”) limit for a dimensionless quantity derived from that transition. Each node carries a unit quaternion state on SU(2) ≃ S³ and interacts through a neighbor mean field. A gate turns leakage on only when local alignment falls beyond a fixed threshold angle. We sweep the coupling strength κ and measure the mean closed (leak-suppressed) fraction, then extract a critical coupling κ* using a declared half-baseline crossing rule. From κ* we compute a derived inverse scale xeff−1=κ∗sin⁡(Δϕ)/ζx_{\mathrm{eff}}^{-1}=\kappa^*\sin(\Delta\phi)/\zetaxeff−1=κ∗sin(Δϕ)/ζ, where Δϕ\Delta\phiΔϕ is a declared drive increment and ζ\zetaζ is a declared leakage scale. Across fixed-degree topologies (ER fixed-k, ring, Watts–Strogatz), we observe a narrow transition band and find that the derived inverse scale approaches a stable plateau as n increases. By n ≳ 30,000, seed scatter and topology dependence are small under the declared protocol, and the plateau lies near the physical α−1\alpha^{-1}α−1 scale for the primary configuration (Δϕ=π/4,ζ≈0.037)(\Delta\phi=\pi/4,\zeta\approx0.037)(Δϕ=π/4,ζ≈0.037). A bounded-degree feasibility run at n = 300,000 remains consistent with the same asymptotic band. We also perform a Lane B′ discretization stress test at n = 40,000 that enforces estimator validity via a coarse scan followed by dense refinement only when a crossing exists. In this run, κ* is well-posed for Δϕ=π/4\Delta\phi=\pi/4Δϕ=π/4 and π/8\pi/8π/8 but becomes ill-posed (“no crossing”) for π/16\pi/16π/16, indicating that Δϕ\Delta\phiΔϕ acts as a genuine regulator. Finally, an attempted internal derivation of ζ\zetaζ using the criticality-matching condition m‾(κ∗;ζ)=e0\overline{m}(\kappa^*;\zeta)=e_0m(κ∗;ζ)=e0 yields no solutions in the tested bracket, clarifying that the present work establishes a robust transition and plateau under a declared mapping, not a first-principles derivation of ζ\zetaζ or α\alphaα.