Universal Operational Equation for Locating the Zeros of the Zeta Function

This preprint presents the Universal Operational Equation for Locating the Zeros of the Riemann Zeta Function, based on an arithmetic-geometric and computational framework that combines discrete angular states, integer parameters, Cantor pairing, the critical-line representation, and Newton iteration for the numerical localization of nontrivial zeros of the Riemann zeta function. The nontrivial zeros are represented in the form ρn=12+iγn,ρ_n=12+iγ_n, with the critical line defined by Re⁡(s)=12.Re(s)=12. The construction begins with five discrete angular states: k1=0.0900,k_1=0.0900, k2=0.1067,k_2=0.1067, k3=0.1101,k_3=0.1101, k4=0.1122,k_4=0.1122, and k∞=ln⁡26≈0.1155245300933242.k_∞=ln2/6 ≈0.1155245300933242. For the original formation, the operational relation is γmodelo=p kj,{γmodelo=p\,k_j}, where p is an integer parameter and k_j is one of the five angular states. The associated initial point on the critical line is s0=12+iγmodelo.{s_0=12+iγmodelo}. The five representative tests of the original formation produced the following correspondences: 1765(0.0900)=158.8500→ρ58,1765(0.0900)=158.8500 → ρ₅₈, with Newton refinement to γ58=158.84998817142049872,γ₅₈=158.84998817142049872, and absolute difference ∣Δγ∣≈1.18286×10−5.|Δγ|≈1.18286×10⁻⁵. The second test gives 3458(0.1067)=368.9686→ρ182,3458(0.1067)=368.9686 → ρ₁₈₂, with refined ordinate γ182=368.96843809573438989,γ₁₈₂=368.96843809573438989, and ∣Δγ∣≈1.61904×10−4.|Δγ|≈1.61904×10⁻⁴. The third test gives 4562(0.1101)=502.2762→ρ272,4562(0.1101)=502.2762 → ρ₂₇₂, with refined ordinate γ272=502.27627032711823318,γ₂₇₂=502.27627032711823318, and ∣Δγ∣≈7.03271×10−5.|Δγ|≈7.03271×10⁻⁵. The fourth test gives 5060(0.1122)=567.7320→ρ318,5060(0.1122)=567.7320 → ρ₃₁₈, with refined ordinate γ318=567.73175790117693745,γ₃₁₈=567.73175790117693745, and ∣Δγ∣≈2.42099×10−4.|Δγ|≈2.42099×10⁻⁴. The fifth test uses the limiting angular state k∞=ln⁡26,k_∞=ln2/6, with 3170ln⁡26≈366.2127603958378→ρ180,3170ln2/6 ≈366.2127603958378 → ρ₁₈₀, and Newton refinement to γ180=366.21271028833131686,γ₁₈₀=366.21271028833131686, with ∣Δγ∣≈5.01075×10−5.|Δγ|≈5.01075×10⁻⁵. In all five cases, the Newton operator is sm+1=sm−ζ(sm)ζ′(sm).{ sₘ₊₁ = s_m-ζ(s_m)/ζ'(s_m) }. The five original tests therefore follow the operational chain p→p kj→γmodelo→12+iγmodelo→Newton→12+iγn.{ p → p\,k_j → γmodelo → 12+iγmodelo → Newton → 12+iγ_n }. The original formation is subsequently extended by a discrete diagonal construction based on the Cantor pairing function C(p,j)=(p+j)(p+j+1)2+j.{ C(p,j)=(p+j)(p+j+1)/2+j }. This transformation does not replace the five original tests; it constitutes the subsequent diagonal formation of the framework. The original relation γ=p kjγ=p\,k_j is therefore distinguished from the later relation γC(p,j)=C(p,j)kj.{ γ_C(p,j)=C(p,j)k_j }. For the diagonal formation, the parameter range is 433≤p≤5874,433≤ p≤5874, with j∈{0,1,2,3,4}.j∈\{0,1,2,3,4\}. The interval contains 5874−433+1=54425874-433+1=5442 integer values of p. With five angular states, the resulting discrete configuration space contains 5442×5=272105442×5=27210 combinations. For every combination (p,j), the diagonal integer is defined by P=C(p,j),{P=C(p,j)}, followed by γC=P kj=C(p,j)kj,{γ_C=P\,k_j=C(p,j)k_j}, and the initial point s0(p,j)=12+iC(p,j)kj.{ s_0(p,j) = 12+iC(p,j)k_j }. The same Newton operator is then applied individually to the resulting initial points: sm+1(p,j)=sm(p,j)−ζ(sm(p,j))ζ′(sm(p,j)).sₘ₊₁(p,j) = s_m(p,j) - {ζ(s_m(p,j))} {ζ'(s_m(p,j))}. When convergence occurs, the final state is represented as sm(p,j)→ρn=12+iγn.{ s_m(p,j) → ρ_n = 12+iγ_n }. The complete diagonal operational chain is therefore (p,j)→C(p,j)→C(p,j)kj→s0→Newton→ρn.{ (p,j) → C(p,j) → C(p,j)k_j → s_0 → Newton → ρ_n }. The first block of the diagonal construction illustrates the transformation explicitly. For example, C(433,0)=93961,C(433,0)=93961, giving γC=93961(0.0900)=8456.49.γ_C=93961(0.0900)=8456.49. For j=1, C(433,1)=94396,C(433,1)=94396, and γC=94396(0.1067)=10072.0172.γ_C=94396(0.1067)=10072.0172. For j=2, C(433,2)=94832,C(433,2)=94832, giving γC=94832(0.1101)=10440.9432.γ_C=94832(0.1101)=10440.9432. For j=3, C(433,3)=95269,C(433,3)=95269, giving γC=95269(0.1122)=10688.1618.γ_C=95269(0.1122)=10688.1618. For the limiting state, C(433,4)=95707,C(433,4)=95707, and γC=95707ln⁡26=11056.5062016418….γ_C = 95707ln2/6 = 11056.5062016418…. Thus, the diagonal formation transforms each pair (p,j) into a single Cantor integer P, couples that integer to its corresponding angular state k_j, and uses the resulting ordinate as the initial height for the Newton construction. The complete formation contains 27,210 operational trajectories.The angular states are associated with directions through ϑj=arctan⁡(kj),{ ϑ_j=(k_j) }, so that the arithmetic parameter, angular state, angular direction, spectral ordinate, and complex initial point form a unified arithmetic-geometric representation. The computational framework therefore combines two successive constructions: Original formation: p kj{ Original formation: p\,k_j } and Diagonal formation: C(p,j)kj.{ Diagonal formation: C(p,j)k_j }. The first construction establishes the five representative numerical correspondences. The second organizes the interval 433≤ p≤5874 into a five-state Cantor-diagonal configuration of 27,210 cases. The distinction between these two layers is an integral part of the formulation. For the numerical treatment of the critical line, the Hardy Z(t) function is used in the standard representation Z(t)=eiϑ(t)ζ(12+it),Z(t)=eiϑ(t) ζ(12+it), where the zeros of Z(t) correspond to zeros of the zeta function on the critical line. The computational procedure consequently combines: Cantor+Angular States+Hardy Z(t)+Newton-Raphson.{ Cantor + Angular States + Hardy Z(t) + Newton-Raphson }. The Cantor pairing provides the discrete addressing transformation in the diagonal formation; the angular states provide the coupling factors; the Hardy function supplies the real-valued representation associated with the critical line; and Newton iteration performs the numerical refinement of the resulting complex initial points. The complete data structure records the original integer p, state j, Cantor integer P, angular state k_j, angular coordinate ϑ_j, generated ordinate γ_C, initial point s_0, number of Newton iterations, final state, identified zero index n, refined ordinate γ_n, and the signed and absolute ordinate differences. The framework therefore establishes the following complete operational architecture: (p,j)→{p kjoriginal formationC(p,j)kjdiagonal formation→12+iγ→Newton→ρn.{ (p,j) → {cases} p\,k_j & original formation\\ C(p,j)k_j & diagonal formation {cases} → 12+iγ → Newton → ρ_n }. The reported five original tests reproduce, through the stated numerical procedure, the spectral positions associated with the nontrivial zeros ρ58,ρ182,ρ272,ρ318,ρ180,ρ₅₈, ρ₁₈₂, ρ₂₇₂, ρ₃₁₈, ρ₁₈₀, all with real part Re⁡(ρn)=12.Re(ρ_n)=12. The results constitute a computational and structural investigation of the proposed arithmetic-geometric framework for generating and refining candidate ordinates associated with nontrivial zeros of the Riemann zeta function. The numerical correspondences reported here are presented as results of the proposed procedure and do not, by themselves, constitute a rigorous proof of the Riemann Hypothesis.