r_d × f_π = ℏc(1 − α): A Four-Constant Identity Linking Nuclear, Chiral, Quantum, and Electromagnetic Physics

Version 4.3 (5 April 2026): Errata update. Five corrections documented in Paper4_v43_ERRATA.pdf: (1) f_π precision claim corrected from 0.009% to 0.17 ± 0.15% (PDG 2024); (2) F_π vs f_π convention made explicit; (3) Author location corrected from Auckland, New Zealand to Warrnambool, Australia; (4) Equation (7) deviation corrected from 0.05% to 0.72%; (5) Proton charge radius updated to Maisenbacher et al. 2026 value (0.8406 ± 0.0015 fm). We report that the product of the deuteron charge radius r_d and the pion decay constant f_π satisfies r_d × f_π = ℏc(1 − α) to 0.17 ± 0.15%, where α = 1/137.036 is the fine structure constant. This four-constant identity connects nuclear structure (r_d), chiral symmetry breaking (f_π), quantum mechanics (ℏc), and electromagnetism (α) with zero free parameters. The predicted r_d = ℏc(1 − α)/f_π = 2.12759 fm lies 0.7σ from the CODATA 2018 value 2.12778 ± 0.00027 fm. Among twelve nuclear charge radii, only the deuteron satisfies this relation, consistent with its unique status as the only nucleus whose binding energy is far below the pion mass. The α correction is physically motivated: pion mass electromagnetic splitting (Dashen's theorem) and proton form factor corrections each contribute O(α) terms that sum to α × ℏc/f_π within 0.3%. A five-quantity sum rule with zero free parameters provides the theoretical framework. After look-elsewhere correction over 336 trial expressions, the agreement remains significant (p = 0.013). Falsifiable predictions include the neutron charge radius and quark-mass sensitivity testable by lattice QCD. Three progressively tighter descriptions: • r_d × f_π = ℏc → 0.721% (raw observation) • r* × f_π = ℏc → 0.065% (EM+rel corrections removed) • r_d × f_π = ℏc(1 − α) → 0.17 ± 0.15% (fine structure constant, PDG 2024 f_π) This is Paper 4 in the Planck Lattice series. Previous papers: • Paper 1 (Neural Oscillations): DOI 10.5281/zenodo.19267675 • Paper 2 (Cross-Domain Survey): DOI 10.5281/zenodo.19312741 • Paper 3 (Driver Analysis): DOI 10.5281/zenodo.19312939 All scripts and data are provided as supplementary material under CC BY 4.0.