The Network ζ-Operator: A Complex Analysis Encoder from Number Theory to General Networks

Abstract Starting from the information pixel axiom system (L0 layer), this paper rigorously proves that Axiom 4 (local coupling) possesses a unique inevitable operator form in the complex parameter space—the network ζ-operator Z_G(s). The entire derivation proceeds from the axioms: the antisymmetry of Axiom 1 (distinguishability) and Axiom 4 (local coupling) jointly force the coupling kernel to adopt a complex representation—complex numbers are not an introduced mathematical tool, but the unique linear representation of antisymmetry; this complex coupling uniquely determines the global product form under the axiom compatibility condition; the Holomorphic Guarantee Theorem subsequently proves that this form must be a holomorphic function. On this basis, this paper establishes a zero-point theory for general networks: it is proved that the zeros of a general network ζ-operator admit no explicit analytic solution (negative theorem), a result that conversely establishes the mathematical inevitability of the Hexad rule composition engine as the unique alternative for zero searching; a rigorous correspondence between zeros and topological holes is established—zero positions are the complex analytic projections of topological holes, zero density equals topological charge density, and zero statistics are determined by phase type; a zero critical band theorem is given, naturally positioning the critical line Re(s)=1/2 of the Riemann Hypothesis as a special case of the universal theorem on the prime network. Thus, the network ζ-operator becomes a unified complex analysis encoder from number-theoretic networks to general networks: all objects of dynamic number theory (Riemann ζ-function, Dirichlet L-function, Epstein ζ-function, Ihara ζ-function, etc.) are special cases under the zero-phase constraint; the activation of the phase degree of freedom allows this encoder to transcend the limitations of number theory and become a universal tool for characterizing the topological emergence of arbitrary networks. The network ζ-operator and the Hexad constitute a duality of static encoding and dynamic evolution: the ζ-operator maps network topology to a holomorphic function (structural encoding), while the Hexad searches for zeros in the information density landscape of the holomorphic function (dynamical encoding). Together they open the complete closed loop of "encoding → computation → proof" in generativist mathematics; for the closed-loop verification paradigm, see relevant literature. This paper, together with Topology as a Bridge, constitutes the complete axiom system of the L0 layer: the former proves that topology is the unique bridge (existence), and this paper provides the encoder form of the bridge (constructiveness) along with its zero-analysis theory. Keywords: information pixel axioms; local coupling; antisymmetry; complex number emergence; network ζ-operator; Holomorphic Guarantee Theorem; zero theory; topological holes; complex analysis encoder; structural encoding; duality of static encoding and dynamic evolution