Modular Operational Mathematics: Extending the Iteration Count of Modular Transformations and Their Inverses to the Complex Domain

This paper systematically transplants the core methodology of Operational Mathematics---the extension of the repetition count of fundamental operations from natural numbers to integers, rational numbers, real numbers, and ultimately complex numbers---onto a new class of binary operations: the modular transformation operation ∘ₙM(a,b) based on the action of the modular group Γ = PSL(2,Z) on the extended upper half-plane, and its inverse operation ∘ₙ^M⁻¹(a,b). A complete axiomatic system of seven independent axioms is established; integer-order, fractional-order, real-order, and complex-order iterations are rigorously defined; and the existence of iterative roots at each level is proved by means of the generalized Schr\"oder--B\"ottcher equation (for elliptic fixed points), the Abel equation (for parabolic fixed points), and a suitably adapted Kneser construction on the fundamental domain F. Uniqueness theorems under the modular regularity condition (log j-convexity) are provided. The singularity structure of complex-order modular iterations is analyzed in depth, revealing a fundamentally novel phenomenon: the simultaneous presence of logarithmic branch points (arising from the parabolic cusps) and algebraic branch points of order $k=2,3$ (arising from the elliptic fixed points i and ρ = e2π i/3). The negative real axis is shown to be a natural boundary, and the Riemann surface is an infinite-sheeted branched covering with mixed monodromy group being a free product of Z and Zₖ factors. Furthermore, a fundamental structural discovery is rigorously proved: the modular operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base operations at level $n=1$ and the collapsed family at level $n=2$. Fractional calculus and the fractional calculus of variations with modular invariant kernels are shown to be special cases of the modular operational framework. A categorical duality between the mathematics of numbers and the mathematics of modular operations is established, yielding a field isomorphism between the modular hyperfield and the complex numbers, including a p-adic extension. The connection between modular iteration values and the arithmetic of modular curves is explored: the unconditional transcendence of non-integer rational iterates is proved; the modular zeta function is constructed with explicit Gamma factor and functional equation; and the Modular Riemann Hypothesis is proved---all non-trivial zeros lie on the critical line (s)=1/2. The residue at $s=1$ is identified as the inverse translation length of the hyperbolic element, establishing a direct link with the hyperbolic geometry of the modular curve and with the Birch--Swinnerton-Dyer conjecture. High-precision numerical algorithms with rigorous error bounds are developed, and their polynomial-time computational complexity is established, thereby settling the previously open question of efficient computability. All theoretical predictions are verified on a concrete hyperbolic element of trace $3$.