Odd Zeta Values in QED: The Loop Order Chooses the Weight

Summary: The anomalous magnetic moment of the electron is a power series aₑ = Σₙ Aₙ(α/π)ⁿ whose coefficients are known exactly up to four loops. They contain ζ(3) from two loops on, ζ(5) from three loops on and ζ(7) at four loops. It is not usually asked why odd zeta values appear here at all, while in the Casimir effect, the blackbody law and the Bose gas the even values and the Gamma factors dominate. We make the arithmetic bookkeeping explicit. First, the loop order chooses the weight: in every known coefficient the transcendental weight is at most 2n−1, the bound is attained, and the top weight contains the single value ζ(2n−1). Since 2n−1 is odd, the new constant at each order is an odd zeta value; even values ζ(2k) are rational multiples of π²ᵏ and enter only at lower weight or in products. Second, beyond the weight only the prime 2 enters: all 115 rational coefficients of Laporta's four-loop result have denominators built from primes ≤ 7 = 2n−1, with two exceptions carrying the prime 11. Both multiply Cl₆(π/3), and we prove Cl₂ₘ(2π/3) = Cl₂ₘ(π/3)/(1 + 2^(1−2m)); for m = 3 the factor is 32/33, and in the basis Cl₆(2π/3) the 11 disappears. Third, in massless φ⁴ theory the same rule holds with weight 2ℓ−3, and the zig-zag graphs give a proved closed form: a Catalan number times ζ(2ℓ−3), with the Euler factor 1 − 2^(4−2ℓ) of the prime 2 exactly when ℓ is odd. The parity of the loop number thus decides between ζ and the alternating η, the Bose and the Fermi side of our earlier papers. The fine-structure constant is not derived; we record its present status, including the 5.5σ tension between the rubidium and caesium recoil measurements. The paper is written for number theorists: Section 2 explains the physics that is needed, from the magnetic moment and Feynman parameters to periods, weights and graph polynomials. No new physics is claimed; the contribution is the arithmetic reading of known coefficients and the checks in the companion code, which also recomputes the periods 6ζ(3) and 20ζ(5) of the first two wheel graphs from their Kirchhoff polynomials. A final section proposes, as the author's hypothesis, to read the loops not as vacuum fluctuations but as the independent cycles of the graphs, whose number fixes the weight. quantum electrodynamicsanomalous magnetic moment of the electronRiemann zeta functionodd zeta valuestranscendental weightmultiple zeta valuesperiodsFeynman integralsKirchhoff polynomialgraph polynomialsphi^4 theoryzig-zag graphsCatalan numbersClausen functionDirichlet eta functionEuler productfine-structure constant