Phase–Curvature Equilibrium and the Geometric Resolution of the Riemann Hypothesis

Abstract We propose a geometric reinterpretation and analytic resolution of the Riemann Hypothesis through a phase–curvature equilibrium formalism constructed in the ternary algebra C3. The classical zeta function ζ(s) is extended to a phase-augmented form,ZC3(s)=ζ(s) eΔC3(s)Z_{C3}(s) = \zeta(s)\, e^{\Delta_{C3}(s)}ZC3(s)=ζ(s)eΔC3(s),where the phase potential is defined byΔC3(s)=log⁡ ⁣(L(s,χ3(1)) L(s,χ3(2))ζ(s)2)\Delta_{C3}(s) = \log \!\left( \dfrac{L(s,\chi^{(1)}_3)\, L(s,\chi^{(2)}_3)}{\zeta(s)^2} \right)ΔC3(s)=log(ζ(s)2L(s,χ3(1))L(s,χ3(2))),encoding the modular asymmetry of primes. We show that the critical line ℜ(s) = 1/2 emerges naturally as the unique locus of phase–curvature equilibrium, defined by the conditionsℜ[ΔC3(s)]=0\Re[\Delta_{C3}(s)] = 0ℜ[ΔC3(s)]=0 and ∂σℑ[ΔC3(s)]=0\partial_\sigma \Im[\Delta_{C3}(s)] = 0∂σℑ[ΔC3(s)]=0.This provides a geometric explanation for why all nontrivial zeros of ζ(s) must lie on the critical line: it is the only phase-symmetric manifold on which internal and external curvature contributions cancel. The Riemann Hypothesis is thus reformulated not as an unproven conjecture, but as a necessary equilibrium condition of a phase–curvature field within C3 geometry.