The Euclidean–Basal Architecture Arithmetic, Surreal Geometry, Operators and Zeta Structures

This monograph develops a unified Euclidean–basal architecture connecting arithmetic, positional representation, generating functions, operator structures, surreal geometry, and the Riemann zeta function. The central organizing principle is dominant-scale division: an object is represented by successively extracting its maximal visible scale, recording the corresponding coefficient, subtracting it, and continuing with the lower-order remainder. Euclidean division appears as the discrete arithmetic form of this principle, with quotient, residue, carry, and normalization providing the fundamental local mechanism. Part I develops the Euclidean–basal correspondence for real numbers and establishes the transport from digit expansions to generating functions and operators. Part II extends the construction to the surreal setting through omnific support, dominant scales, Conway-type normal forms, levelwise ordered structures, truncation topology, and transfinite analysis. Part III applies prime-basal valuation to zeta structures. Prime-base decompositions are transported through Mellin substitution, Euler factors, Riemann completion, logarithmic differentiation, and Hadamard–Jensen residual reading. Within this framework, the manuscript derives a direct transport theorem in which the nonstructural divisor of the completed zeta object is reduced to the live critical channel supported at the centered radial coordinate u=0u=0, corresponding to Re⁡(s)=1/2\operatorname{Re}(s)=1/2. The classical Riemann hypothesis is consequently obtained as the level-zero instance of the broader Euclidean–valuative and levelwise construction developed in the work. The conceptual lineage of the manuscript is rooted principally in Euclidean division, Cantor's transfinite hierarchy, Conway's surreal numbers, Robinson's extension of analysis beyond the ordinary real continuum, and Riemann's analytic treatment of the zeta function.