M25c Algebra of g-ONS: Arithmetic Consequences: The Non-Fundamentality of C, BSD with Loaded Periods, Completed Zeta as Half-Branch Fourier, Operational Galois Theory, the Half-Branch Principle, Flux on the SC Cascade, and Roots in Cτ

This monograph develops the arithmetic and algebraic consequences of the g-ONS programme, including branch-based arithmetic, Operational Galois Theory, loaded BSD structures, and roots in the extended field Cₜau = C x (R/Z). The work proposes that the complex field C is not fundamental, but emerges from the real line together with the branch circle S¹ = R/Z. In this picture, the phantom unit i appears as a quarter-rotation of branch space rather than as a primitive axiom. The monograph reformulates BSD structures using loaded branch periods and interprets the completed Riemann zeta function as a half-branch Fourier transform. It further introduces Operational Galois Theory (OGT), where the solvability rank rho (f) = 1 + gₘin (Galf) /2 measures the minimal HC half-rank required for solving a polynomial. In this framework, quintics correspond to the Jacobi elliptic level R = 1. 5, sextics to the Heun level R = 2. 5, and degree-7 equations to the ISHE level R = 3. 5. The monograph concludes with the study of roots inside Cₜau, branch flux on the SC cascade, and the unifying Half-Branch Principle linking arithmetic, branch geometry, and operational algebra.