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 αEM² + (φ/2) αEM³) αEM²¹ ℏc / mₑ², 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 αEM. 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/n₄䃘 with n₄䃘 = 2 (E₈ summands in the K3 lattice) ; and the correction coefficients 3 and φ/2 to nU = 3 (hyperbolic pairs) and d_τ/n₄䃘. 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.