A Number-Theoretic Origin for the Fine Structure Constant from Gaussian Integer Flanking Norms

The fine structure constant alpha = 1/137.036... has been called the most mysterious number inphysics. Its reciprocal, approximately 137, has no derivation from first principles in the StandardModel. We present a construction in Gaussian integer arithmetic that produces the integer 137 asthe norm of a flanking element: N(11 + 4i) = 11^2 + 4^2 = 137, where 11 = Im((2+i)^3) and4 = Im((2+i)^2). We prove that 137 decomposes as 133 + 4, where 133 = dim(e_7) and4 = max_mark(E_7-hat), and that this decomposition is unique among all simple Lie algebras.We show that the flanking norm of any thin Gaussian prime a + i follows a quartic polynomialf(a) = 9a^4 - 2a^2 + 1, with 137 = f(2) being the first prime value. These are arithmetic results,verified by direct computation. No physical assumptions are required. The paper is self-containedand addresses only the question: is there a number-theoretic reason why the integer nearest to1/alpha is 137?