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 ℂG. The group algebra carries the natural inner product ⟨f, g⟩ = (1/|G|) Σₗ∈₆ f (x) g (x) *. The heat semigroup on ℂG 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: τG = 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: τG = 1/|G| = 1/48 This is derived from the group algebra structure of Oₕ alone, with no external input. Assumed: τₛpatial = 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 τₛpatial. To obtain τ = 1/|G|², this paper sets τₛpatial = τG = 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 = τG × τₛpatial = 1/|G|² = 1/2304 and substituting into the standard CW heat kernel trace K (τ) = Σₖ aₖ · 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: τG = 1/|G| is derived from the self-dual point of the Poisson summation formula on ℂOₕ. The identification τₛpatial = 1/|G| is an assumption. Closing this step — by deriving τₛpatial 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 b₁EM = 352/27 is derived in Part XXXI of the Core Framework v7. The master equation λ²−CA²λ+ (CA+1) ²=0 (Part XLII) shows all spectral predictions follow from CA=3 alone. https: //github. com/WebEnvy/UnifiedFoamFieldTheory/blob/main/UFFTCoreFrameworkᵥ7. 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 8π^5/2 = (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).