The Fine-Structure Constant as a Self-Energy Fixed Point: A Forced Construction and the Measurement That Decides

A construction containing no physical input and no measured value outputs two numbers from one quadratic: its physical root 1/alpha_S = 137.035 999 074 948 275 854, and the harmonic sum of that quadratic's two roots, 1/alpha_D = 137.035 999 167 273 924 798 (equivalently its no-closure limit), separated by exactly alpha_S L^4 = 9.2326 x 10^-8 in 1/alpha (92.33 ppb). The electron g-2 determination — the most precisely measured quantity in physics — gives 1/alpha = 137.035 999 166(15): agreement with alpha_D to eleven significant figures, at 0.085 sigma. Both statements are checkable and neither is conditional on anything argued here. Every structural choice in the construction — kernel, domain, threshold, exponent, branch, and each term of the assembly — is pinned by a stated theorem or textbook physical fact that never references the output, and every named deformation — slot, exponent, branch, physics — is priced at the end; none lands near. The construction's signature is convergence: the same constant is characterized three independent ways, the same closure carries five equivalent faces, and the same fixed-point object is met again, unmodified, inside the quantum harmonic oscillator, the heat trace of a shifted line Laplacian, and the Bargmann–Fock space (Part IV) — one foundation, many walks. The closure condition is shown to be a level set of the inversion invariant u + u^-1, from which it follows that the predicted methodological split between recoil and g-2 extractions of the fine-structure constant equals, exactly, the inverse coupling of the construction's second fixed point; the sorting of the two extraction classes onto the two roots is itself derived, conditional on the paper's one identification and on nothing else. Three routes reach the same constant — a geometric construction carrying no physics, a physical argument carrying no geometry, and an inversion of the measured coupling; the first two have disjoint inputs, and the third is independent of the defining equation it recovers rather than of the other two (scope in §12.1). One identification remains open — that the construction's coordinate is quantum electrodynamics' vacuum polarization; it is not small, this paper does not claim it, and it is provably not resolvable by perturbation theory, hence a measurement. The abandoned identifications and an erratum against the author's own published proof are recorded in Part VIII; numbered falsification thresholds with hard numerical cut-lines are stated in §1, including the next-generation recoil measurement that decides. The verification script printed in Appendix F is included in this record as verify_alpha.py, together with its certified output. Developed with AI assistance (Anthropic's Claude) for high-precision verification, literature search, and drafting; all original content and structural decisions are the author's, who takes responsibility for all claims.