A Conditional Trace-Capacity Formula for the Infrared Fine-Structure Constant in a Compact Randers–Finsler U(1) Substrate

We formulate a conditional structural theorem for the low-energy fine-structure constant in a compact Randers–Finsler U(1) substrate model. The construction is deliberately stated in terms of finite-dimensional response spaces, trace capacities, and compact closure constraints, rather than geometric metaphors. A four-dimensional Randers response space is extended by one scalar anisotropy response, giving an effective substrate rank˜5. Electromagnetic curvature is represented by the off-shell two-form space Λ2(Tx∗M), of dimension˜6. The activated local response space is therefore K = Hom(Hs∗ub, Λ2(Tx∗M))∼= Hsub⊗ Λ2(Tx∗M) with dim K = 30, and its quadratic comparison algebra has dimension D = dim End(K) = 900. Under explicit structural assumptions for the closed substrate normalization, the comparison zero-mode trace, compact closure subtraction, Randers transverse stiffness, and an operational endpoint half-cell, the infrared operational coupling is 7 13 208 1 7 1 47 1 π2 αl−ab1=1 +++6−15·9002 3 15 900 14369 46 2 · 900 The result is presented as a conditional trace formula: the arithmetic follows exactly from the stated structural lemmas, while the physical burden lies in independently establishing those lemmas within the substrate dynamics. The truncation ladder of this dressing expansion contains distinct metrological plateaus at the CODATA˜2018 and CODATA˜2022 recommended values of α−1, with no coefficient adjusted to match either. A controlled remainder bound shows that corrections beyond third order are suppressed below current experimental resolution.