The Euler Product Bifurcation Theorem: Intrinsic Multiplicative Field Horizons and the Overhaul of Continuous Transcendental Approximations for ζ(s)

Abstract: This research presents the mathematical architecture of the Euler Product Bifurcation Theorem (EPBT), demonstrating that the classical monolithic Euler product splits into two interlocking, asymmetrical field horizons governed by prime index parity: the Plus Horizon and the Minus Horizon. By treating the multiplication domain as dual expansion and compression operators rather than a single continuous sequence, this framework demonstrates that the prime spectrum intrinsically regularizes its own pole boundaries from the inside out. High-precision numerical verification across a 3,000,000-point discrete prime lattice reveals a permanent topological curvature residue against classical additive baselines, which vanishes asymptotically as the scale dimension reaches higher integer states. The simultaneous intersection of these split operators yields a stable, characteristic quadratic field polynomial free from transcendental approximations, establishing a discrete gauge field framework that uncovers a deterministic structural equilibrium within the prime landscape.