Deriving the Fine Structure Constant from Geometry

This work presents a direct geometric derivation of the Fine Structure Constant. Starting from the circular constant πππ, the construction builds a dimensional action coefficient A0=π+π2+4π3.A_0=π+π^2+4π^3.A0=π+π2+4π3. From that, three coefficients are fixed: b1=124,b2=38π3,c4=π12.b_1=1/24, b_2=3/8π^3, c_4=π/12.b1=241,b2=8π33,c4=12π. These define the quartic balance equation P(x)=1−A0x+b1x2+b2x3−c4x4.P(x)=1-A_0x+b_1x^2+b_2x^3-c_4x^4.P(x)=1−A0x+b1x2+b2x3−c4x4. The polynomial has exactly one positive real root. The root is... α=0.0072973525643199214730777403670980468956159640240703…α_{} = 0.0072973525643199214730777403670980468956159640240703…α=0.0072973525643199214730777403670980468956159640240703… with reciprocal α−1=137.0359991772160237478893219008909381598430809109487…α_{}⁻¹ = 137.0359991772160237478893219008909381598430809109487…α−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.