Is the QED Running a Slice of Geometric Modes? A Zero-Fit Fingerprint Test

This paper uses a zero-fit inversion to test the proposition that the α running curve computed from the geometric modes of the original paper [1] is isomorphic to the QED 1–5 loop α running curve. The α sequence computed in [1] in order of the mode index L (L = 1–330) is inverted to the energy scale μ using standard β-function coefficients from the literature (consistent with the experimental running); this paper examines the segment-by-segment shape isomorphism between ln(μ) and the geometric-side ln(u), where u is the first nonzero root of j_L (ground state, L = 1–330). Since the inversion is reversible, this test is equivalent to an indirect verification of the original proposition. The segmental residuals of the last 5 segments are ≤ 2%, and the variation of each metric across 1–5 loops is < 1.5%. Five classes of fingerprint metrics (DTW, derivative, inflection point, curvature, segmental residual) and three steepness metrics (front-segment slope ratio, sagitta-to-chord ratio relative difference, crossing point) all indicate that the two sequences are isomorphic in shape, with the 5-loop case serving as supplementary verification. The results support, at the methodological level defined in §1.3, the shape-level validity of the original proposition; this paper does not claim mechanistic identity (shape isomorphism is used in a weak sense; see §1.1). The author is also the proposer of that proposition and does not claim third-party independence; the validity of the verification rests on the reproducibility of the computational process, not on the identity of the verifier.