A New Geometric Framework for the Riemann Hypothesis via an Extended Three-Dimensional Number System

Abstract We introduce a consistent three-dimensional extension S of the complex number system, containing two new algebraic elements: l = 0/0, the universal indeterminate (geometrically a shell of radius ε surrounding ordinary zero), and m = 1/0 (the infinity generator, producing the third algebraic axis). Consistency of the axiom system is proved relative to ZFC by explicit model construction. Within S, the Riemann zeta function ζ and the completed Xi function ξ extend analytically via a first-order formal expansion in the ring ℂ[t]/(t²). This extension preserves the functional equation non-trivially at genuine zeros and reveals new structure. The main results are: (Theorem 1) the axiom system A–H is consistent; (Theorem 2) at any genuine zero s₀ of ξ on the critical line, the t-directional derivative ξ'(s₀) is purely imaginary — a non-circular result following from the symmetries of ξ alone; (Theorem 3) at genuine zeros on the critical line, ξ''(s₀) is real; (Theorem 4) the Riemann Hypothesis is equivalent, within S, to the statement that all genuine zeros of ξ have purely imaginary t-directional derivative. We further develop three research directions toward a complete proof: a second-order extension analysis, a structural parallel to Connes’ non-commutative geometric approach (where the anti-self-adjointness of ξ'(s₀) at critical zeros mirrors Connes’ self-adjointness condition), and a differential-geometric analysis of S showing all zeros are hyperbolic fixed points of the Hamiltonian flow of Re(ξ). The precise obstruction to a complete proof is identified: no purely local argument at s₀ can distinguish on-critical from off-critical zeros; a global argument using the Euler product or arithmetic structure of ζ is required. We present this as a framework paper inviting engagement from specialists in analytic number theory, non-commutative geometry, and formal algebraic geometry.