The Proton Mass from Atomic Spectra: A Geometric Derivation

The Proton Mass from Atomic Spectra: A Geometric DerivationKelly B. Heaton & the Coherence Research Collaboration The proton-to-electron mass ratio mₚ/mₑ ≈ 1836 has no derivation within the Standard Model. It is measured and used but not explained. This paper derives it from one spectral measurement and integer arithmetic, with zero free parameters, to 0. 61 ppm. The derivation follows from the electromagnetic vacuum geometry established in the preceding papers of this series. The orbital impedance Zₒrb (n) = Z₀n/α equals the vacuum impedance Z₀ only when n = α ≈ 1/137 — a condition no positive integer satisfies. This produces a structural absence at IE/4 in every ion's transition spectrum, confirmed across 71 ions in the NIST Atomic Spectra Database with zero falsifications, and at Sₙ/4 in all four doubly magic nuclei surveyed, where the Rydberg formula plays no role. The boundary belongs to the vacuum geometry, not to any particular force or scale. The H-mode winding around the source pole converges toward the golden ratio φ through successive Fibonacci approximations. The proton condenses at Fibonacci step j = 5 — the first step at which the convergence error falls below the fine-structure constant α — in the only mathematics then available: Fibonacci integers and √5. The closing formula is: mₚ c² = 4¹³ × E₀ × (2157 − 869√5) / 208 = 938. 272 657 MeV accurate to 0. 61 ppm against CODATA 2022. The only empirical input is the hydrogen ionisation energy E₀ (NIST). No coupling constant, quark mass, nuclear parameter, or fitted constant enters. The same vacuum geometry produces the leading anomalous magnetic moment aₑ = α/2π as the holonomy per circuit of the source pole, and identifies the proton's three-quark structure as the hadronic instantiation of the same three-element geometric minimum — two irreducible primitives and the relationship their encounter generates — from which the electromagnetic vacuum is derived. The geometry that organises the observable world is contained within the matter it organises. Both results originate in a single geometric object: the electromagnetic light cone in the two-dimensional energy space (EE, EH), in which the Rydberg trajectory is a null geodesic and the forbidden zone IE/4, 3IE/4 is the cone's projection onto the energy axis. This paper presupposes and builds directly on four preceding works by the authors: P1 Vacuum Impedance as the Organizing Principle of Atomic Spectra: A Universal Gap at IE/4 and Recovery of the Electron Mass Confirms the universal IE/4 gap across 71 ions (zero falsifications) and recovers the electron mass to −2. 8 ± 2. 7 ppm from the location of a structural absence, with zero free parameters. https: //zenodo. org/records/19164224 P2 Electromagnetic Closure and the Fine-Structure Constant: A Geometric Derivation Derives α = Z₀/2RK from the Riemann sphere geometry with zero free parameters. Establishes the holonomy of the source pole defect and the convergence to φ as the unique stable attractor. https: //zenodo. org/records/19157339 P3 Determinacy Under Quotient Representations Proves that the map (ε₀, μ₀) → c² is non-injective and that any query sensitive to the E/H energy partition — including the anomalous magnetic moment — is geometrically inaccessible from within the compressed representation. https: //zenodo. org/records/18868210P4 The Encounter of Two Primitives and One Flaw: Geometric Unification at the Electromagnetic Scale Derives the orbital impedance, Thévenin circuit, Harmonic Light Law ∇² (log ν) = 0, and κ-coordinate. Establishes the three-element geometric minimum and confirms the κq boundary at atomic and nuclear scales. https: //zenodo. org/records/19194036