Deriving the Fine Structure Constant from Geometry

This work presents a direct geometric derivation of the Fine Structure Constant. Starting from the circular constant ππ, the construction builds a dimensional action coefficient A0=π+π2+4π3. A₀=+²+4³. A0=π+π2+4π3. From that, three coefficients are fixed: b1=124, b2=38π3, c4=π12. b₁=124, b₂=38³, c₄=12. b1=241, b2=8π33, c4=12π. These define the quartic balance equation P (x) =1−A0x+b1x2+b2x3−c4x4. P (x) =1-A₀x+b₁x²+b₂x³-c₄x⁴. P (x) =1−A0x+b1x2+b2x3−c4x4. The polynomial has exactly one positive real root. The root is. . . α=0. 0072973525643199214730777403670980468956159640240703…_ = 0. 0072973525643199214730777403670980468956159640240703α=0. 0072973525643199214730777403670980468956159640240703… with reciprocal α−1=137. 0359991772160237478893219008909381598430809109487…_^-1 = 137. 0359991772160237478893219008909381598430809109487α−1=137. 0359991772160237478893219008909381598430809109487… The Geometric derivation is within the measured fine-structure constant within current CODATA uncertainty. No measured fine-structure value is used as an input. The result follows from the geometric coefficient chain and the unique positive root of the quartic.