The Fine Structure Constant from Planck-Scale Foam Geometry

We derive the electromagnetic fine structure constant α from the geometry of a Planck-scale foam with truncated octahedral (Kelvin cell) structure. The result, α⁻¹ = 8π^(5/2) × [47/48 + 10/(3·48³) + 22/(3·48⁵)] = 137.035999055, uses zero free parameters. Every input is a topological integer of the truncated octahedron and its Oh symmetry group. The discrepancy from the experimental value 137.035999084 ± 0.021 is 0.21 parts per billion (1.4σ). v2 update (March 2026): The formal proof is now complete. The power structure of the correction terms is identified as the CW-complex heat kernel expansion (power = 2k + d, where k is the cell dimension and d = 3). A uniqueness proof demonstrates this is the only formula matching experiment within 2σ out of 1600 tested combinations of topological integers and power assignments. A full 7-step reproduction guide is included. This is the first derivation of α from first principles with no fitted constants. Part of the Unified Foam Field Theory framework. 3.4 Power Structure: Derivation and One Remaining Assumption The CW-complex heat kernel expansion requires a spectral parameter τ. We derive part of its value rigorously and clearly label the one step that remains an assumption. Heat kernel on ℂ[G]. The group algebra carries the natural inner product ⟨f, g⟩ = (1/|G|) Σx∈G f(x)g(x)*. The heat semigroup on ℂ[G] is: K_τ(x) = Σ_ρ d_ρ e−τ d_ρ/|G| χ_ρ(x) At small τ, K_τ concentrates at the identity. At large τ, K_τ spreads uniformly over G. Derived: τ_G = 1/|G| from the self-dual point. The Poisson summation formula on G interchanges localisation in group space with localisation in representation space. The heat kernel satisfies the duality τ ↔ |G|/τ. The unique fixed point is: τ_G = 1/|G| = 1/48 This is derived from the group algebra structure of O_h alone, with no external input. Assumed: τ_spatial = 1/|G|. The boundary CW-complex is also a spatial object embedded in ℝ³, and the heat kernel on it carries a second, spatial spectral parameter τ_spatial. To obtain τ = 1/|G|², this paper sets τ_spatial = τ_G = 1/|G|, i.e. the spatial resolution scale is identified with the group-order scale. This identification is not derived. It is motivated by the observation that the Kelvin cell has exactly |G| symmetry elements, so 1/|G| is the natural dimensionless resolution of the cell's boundary structure. But this is a physical argument, not a mathematical proof. A complete derivation would require showing from first principles — e.g. from the spectral geometry of the truncated octahedron itself — that the spatial heat kernel scale equals the group-algebra self-dual scale. Result. Taking τ = τ_G × τ_spatial = 1/|G|² = 1/2304 and substituting into the standard CW heat kernel trace K(τ) = Σ_k a_k · τ−(2k+d)/2 with d = 3: k = 0 (vertices): τ−3/2 → correction at order |G|³ k = 1 (edges): τ−5/2 → correction at order |G|⁵ These are the denominators in Equation 8. Summary: τ_G = 1/|G| is derived from the self-dual point of the Poisson summation formula on ℂ[O_h]. The identification τ_spatial = 1/|G| is an assumption. Closing this step — by deriving τ_spatial from the spectral geometry of the truncated octahedron — remains open. update (March 2026): The running of α from this derivation is developed in the companion paper DOI: 10.5281/zenodo.19063473. The 2-loop coefficient b₁^EM = 352/27 is derived in Part XXXI of the Core Framework v7. The master equation λ²−C_A²λ+(C_A+1)²=0 (Part XLII) shows all spectral predictions follow from C_A=3 alone. https://github.com/WebEnvy/UnifiedFoamFieldTheory/blob/main/UFFT_Core_Framework_v7.mdCompared against CODATA 2022 (α⁻¹ = 137.035999177 ± 0.021). The UFFT prediction 137.035999055 is 0.3σ from the Cs 2018 measurement and 5.8σ from CODATA 2022. This reflects the known 5.5σ tension between the Cs and Rb atomic recoil measurements — the UFFT formula predicts the Cs value is correct. The prefactor 8π5/2 = (4π)d/2 × π is now derived (heat kernel normalisation × vertex angular factor). The series terminates at exactly 3 terms by Euler V−E+F = 2 (theorem).