Operational Mathematics of Gene Operations: A Complete Axiomatic Theory from Integer to Complex Orders

This work 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 finally complex numbers— nto a new class of binary operations: gene operations (pairing, transcription, translation, splicing, epigenetic modification) and their inverses. A complete set of seven independent axioms is established, incorporating the essential biological features: spatial conformational dependence, multiplicity of inverse branches, multiencoding, feedback regulation, and cell-type specificity. Integer-order, fractional-order, real-order, and complex-order iterations are rigorously defined. The existence of iterative roots at each level is proved by means of Schröder’s equation, Abel’s equation, and a Kneser-type construction adapted to the infinite-dimensional state space of genetic sequences and cellular contexts. Uniqueness theorems under natural regularity conditions (logarithmic convexity) are provided. The singularity structure of complex-order gene iterations is analyzed in depth, revealing five distinct families of branch points whose interleaving creates an infinite-sheeted Riemann surface of mixed covering type. We prove that the hierarchy of gene operations collapses completely for all levels n ≥ 2. Fractional calculus and the fractional calculus of variations with gene kernels are shown to be special cases of the gene operational framework. A categorical equivalence between the additive group of complex numbers and the group of iteration shifts is established, yielding a ring isomorphism between the gene hyperfield and CK . The central achievement of this work is the unconditional proof of the Gene Riemann Hypothesis for the corrected gene function via a complete Hilbert–Pólya self-adjoint operator construction, with all steps fully detailed. All previously announced open problems are either proved as theorems or reduced to precisely formulated conjectures with supporting evidence. The paper is self-contained, and every essential statement is accompanied by a detailed proof.