A Structural Correspondence Between the Alpha-Ladder Formula and K3 Geometry via Fibonacci RCFT

We establish a scoped structural correspondence between every coefficient appearing in the Alpha-Ladder formula G = (φ²/2)(1 + 3 α_EM² + (φ/2) α_EM³) α_EM²¹ ℏc / m_e², φ = (1+√5)/2, and invariants of the K3 surface together with the G₂ level 1 Wess–Zumino–Witten model (the Fibonacci RCFT). The central mathematical result is a proof that the K3 worldsheet CFT at the (1)⁶ Gepner point contains a Fibonacci conformal/topological sector, realised through the conformal embedding chain G2,1 ⊂ SO(7)₁ ⊂ SO(12)₁. We emphasise that this constitutes a geometric reinterpretation of the formula's coefficients, not a dynamical derivation. A first-principles computation that produces the functional form from a K3 compactification remains an open problem. In particular, topology supplies the integer b₂(K3) − 1 = 21 but does not explain why each independent direction should contribute exactly one power of α_EM. The post-hoc dictionary maps: the exponent 21 to b₂(K3) − 1 (flux quanta minus one tadpole constraint); the golden ratio φ to the quantum dimension d_τ of the Fibonacci anyon τ (verified exactly via the Verlinde formula); the prefactor φ²/2 to d_τ²/nE₈ with nE₈ = 2 (E₈ summands in the K3 lattice); and the correction coefficients 3 and φ/2 to n_U = 3 (hyperbolic pairs) and d_τ/nE₈. These are exact algebraic identifications after the formula is given; they are not yet forced by a dynamical calculation. The conformal embedding is verified both structurally (via the Schellekens–Warner classification, GKO coset theorem, and embedding-index computation) and numerically (by extracting the G2,1 branching functions from the exact free-fermion SO(7)₁ characters to q¹⁰⁰, confirming non-negative integer coefficients at every checked order). Evaluated at CODATA 2018 values, the formula reproduces the measured G to −0.31 ppm (0.01σ), consistent with observation but not decisive given the current 22.5 ppm uncertainty in G.