Wissner-Gross's Candidate for the Fine-Structure Constant: An Arithmetic Analysis: What Is Forced, What Is Chosen, and Where It Meets the Zeta Side

Summary: A. D. Wissner-Gross has proposed the closed form α_c = (79 + 48√2 + 42√3 + 3√6)/31104 for the fine-structure constant at zero momentum transfer; its inverse 137.035 999 189 6… lies 0.6σ from the CODATA 2022 value. The number is obtained by counting 172 "records" of a qutrit–qubit link read twice. We analyse the construction as a piece of arithmetic and separate what is forced from what is chosen. The field is ℚ(ζ₂₄)⁺ = ℚ(√2, √3), and the four weights 1, √2, √3, √6 are the four characters of its Galois group (ℤ/24)^×/{±1}: if the two survival signs of a record are read as the action of σ₇ and σ₁₁, the weight is forced up to a rational factor (proved, and checked in Lean 4). The numerator is not Galois invariant, so it can never be a sum over a Galois-stable set of phases. A single trace formula over the orbits of the exchange of the two reads produces all four counts, for every pair of primes (D, d); the rational part 79 = (169 − 11)/2 is the number of free orbits, and the denominator is N = #{weight-2 elements of (ℤ/3)² × (ℤ/2)²} · |(ℤ/6)²|² = 24 · 36². The sign clause that produces the √6 sector exists only because the commutation phase −1 occurs only modulo 2. On the zeta side, W = Σ_{d|6} m_d d^(−3/2) is a finite Dirichlet polynomial at s = 3/2, not an Euler product, and Γ(3/2)ζ(3/2)·W is the density moment of a mixture of ideal Bose gases; read as species of mass m/d in chemical equilibrium, the ratio W/N is independent of temperature and fugacity, and ζ(3/2) cancels. We also report what does not carry: the continuation to level 30, phase counts, natural Clifford representations, finite-temperature mode sums, Euler and Fermi factors as replacements of the counts, counts from L-values, and running towards M_Z. None of the analysis derives α; it locates exactly three arithmetic inputs that remain assumptions. The paper is written for number theorists and explains the physical notions it uses. Every number is reproduced by the companion script (115 checks, 0 failures), which imports Wissner-Gross's record enumerator unchanged; the Lean file compiles with the Lean 4 core library without sorry.