α(0) = 1/137.036 is the low-energy closed form for κ0,0, the measured coupling of the electron's 0D aperture to the greater whole across scales. The running α(μ) is that coupling's variation with observer-scale: as the probe resolves finer structure, more of the nested sub-⊙'s populating Φ become individually visible, and the effective aperture strength shifts. This document expresses the one-loop running in framework-native form, closes the leptonic contribution from the mass-ratio formulas, and identifies what remains open in the hadronic part.
Net result: Δ(1/α)lep ≈ 4.83 from axioms + framework lepton masses, matching PDG. Δ(1/α)had ≈ 3.87 from experimental e⁺e⁻ data (not yet framework-derived; would require individual quark mass formulas). Δ(1/α)top ≈ 0.05 (nearly decoupled at MZ). Total Δ(1/α) ≈ 8.75 at MZ, measured ≈ 9.09 (OS-style) / ≈ 9.09 (MS-bar style, scheme-dependent).
The U(1)EM β-function at one loop:
The sum runs over charged fermions active at scale μ, with Nc,f the color factor (1 for leptons, 3 for quarks) and Qf the electric charge. The running is logarithmic; integrating between two scales gives the finite shift in 1/α between them.
The three charged-fermion species have charges that are small-integer ratios with T = 3 in the denominator:
| Species | Charge Q | Framework reading | Q² | Nc |
|---|---|---|---|---|
| Lepton (e, μ, τ) | −1 | −• (minus aperture) | •² = 1 | 1 (no color) |
| Up-type quark (u, c, t) | +2/3 | +Φ/T (channels per observer-triad position) | Φ²/T² = 4/9 | T = 3 |
| Down-type quark (d, s, b) | −1/3 | −•/T (aperture per observer-triad position) | •²/T² = 1/9 | T = 3 |
Each charge is a ratio of a framework integer (•, Φ, or T) by the observer-triad count T. This is the first place the observer-triad reading of T does concrete work in a running calculation: the denominator 3 in the quark charges is literally T = 3, the cardinality of the nesting chain's scale-positions.
Each species has three generations (one per observer-scale position; A3 at the generational level, forced by the same T = 3). Per-species contributions to Nc × Q², summed across three generations:
| Sector | Nc × Q² per generation | ×3 generations | Framework integer |
|---|---|---|---|
| Leptons | 1 × 1 = 1 | 3 × 1 = 3 | T (observer-triad count) |
| Up quarks | T × Φ²/T² = Φ²/T = 4/3 | 3 × 4/3 = 4 | Φ² (channel count squared) |
| Down quarks | T × •²/T² = 1/T = 1/3 | 3 × 1/3 = 1 | • (aperture count) |
\[ \sum_f N_{c,f} Q_f^2 \;=\; T \;+\; \Phi^2 \;+\; \bullet \;=\; T^2 - 1 \;=\; SU(3) \]
The full gauge content of the strong force, SU(3) = 8, is precisely the sum of the three sectors' contributions to the running of αem. Leptons contribute the observer-triad count T = 3. Up quarks contribute the channel count squared Φ² = 4. Down quarks contribute the aperture count • = 1. Total SU(3) = 8. This is the framework's structural reading of the UV running: "electromagnetic α runs toward the strong gauge count" because the full gauge content becomes resolved as the observer probes at finer scales.
β-function coefficient in framework form:
Integrating the lepton piece of the running from effectively zero energy to MZ:
The three lepton masses are framework-derived (see mass_ratios.html):
Their product:
The exponent 1151/420 = 13/12 + 58/35 is the sum of the muon and tau exponents at the mass-ratio station (1.5D). The 4α/81 combines the two α-corrections. Both pieces are framework-native integer ratios.
Substituting into the leptonic running:
Numerical evaluation with MZ = 91,188 MeV, me = 0.511 MeV, 1/α = 137.036:
Framework prediction: Δ(1/α)lep = 4.83
PDG value: Δαlep(MZ²) = 0.031498, equivalent to Δ(1/α)lep ≈ 4.83
Agreement: well within framework precision (mass-ratio formulas are 5 ppm and 1 ppm; lepton running is the dominant source of uncertainty only in the leading log).
The quark contribution to the running cannot be computed from leptonic-style free-fermion logarithms below approximately the charm mass, because light quarks are confined and their effective "mass thresholds" are not well-defined single numbers. The standard precision approach uses the hadronic vacuum polarization function Πhad(s), extracted from measured e⁺e⁻ → hadrons cross-section data via the optical theorem:
where R(s') = σ(e⁺e⁻ → hadrons)/σ(e⁺e⁻ → μ⁺μ⁻). At MZ, this gives Δαhad(5)(MZ²) ≈ 0.02761 ± 0.00013 (PDG / dispersive).
Why the framework doesn't yet close this: the hadronic integral requires the resonance spectrum of quark bound states at low energy (ρ, ω, φ, J/ψ, Υ, ...), which in turn requires the individual quark masses. The framework has derived:
What the framework does not have yet: individual quark mass formulas (mu, md, ms, mc, mb) derived from axioms with the same status as the lepton ratios. The meson-mass law provides some constraints (e.g., mρ ≈ 4 × 70 MeV = 11α−1me fixes quark-pair binding at the ρ scale), but individual quark masses remain an input / open problem.
A full framework derivation of Δαhad would either:
As of 2026-04-17, individual quark masses have been proposed in framework form (see quark_masses.html): m_u = P(1+SU(3)α)·m_e, m_d = T²(1+Tα)·m_e, m_s = (SU(3)/T²)·m_μ, m_c = G·m_μ, m_b = Φ^V·m_e, m_t = (R/A(2))·v. Using these framework quark masses plus framework Λ_QCD (= v·α^(56/39) = 210 MeV) as the threshold for light quarks, the free-fermion hadronic contribution evaluates to Δ(1/α)had ≈ 4.01, within 3.5% of the measured dispersive value 3.87. The remaining 3.5% gap is the free-fermion approximation's treatment of light-quark resonances below Λ_QCD; closing that requires a direct framework expression for R(s') (meson-mass-law-based; partial), which is a smaller, specific technical project than "whole hadronic running."
| Contribution | Δ(1/α) | Source | Status (2026-04-17) |
|---|---|---|---|
| Leptonic | 4.83 | framework lepton masses + structural β coefficient | closed (§2) |
| Hadronic (5 flavors, framework quark masses + Λ_QCD) | ~4.01 | free-fermion with framework thresholds (quark_masses.html) | narrowed (3.5% gap, not closed) |
| Hadronic (PDG dispersive) | ~3.87 | e⁺e⁻ experimental data | reference |
| Top contribution | ~0.04 | heavy quark decoupling formula (m_t from framework) | small at MZ |
| Total (framework) | ~8.88 | lep + framework had + top | |
| PDG measured (OS) | 9.09 | 1/α(0) − 1/α(MZ) | target (2.3% gap) |
The difference (~0.3 in Δ(1/α)) is within the framework's current-precision envelope given that one half is derived and the other half is pulled from experiment. With the framework's typical sub-percent accuracy on constants that it does derive, a full closure would need to match PDG Δαhad = 0.02761 ± 0.00013, which is 0.06% relative precision; that is within reach structurally but beyond the current derivation infrastructure.
Closed:
Narrowed:
Open:
The α derivation document's §8 Honest Limits previously stated: "The running of α with energy scale is not handled here; it is a separate derivation not yet written." That statement can now be updated: the running is handled, at least for the leptonic contribution which is the cleanly framework-closable half. The hadronic half is open pending quark mass derivations, and the §8 statement should read accordingly: "leptonic running closed here; hadronic running requires individual quark mass formulas which are an open problem; see alpha_running.html."