Prime-Coordinate Analysis: Foundations, Measure Theory, Operator Structure, and Arithmetic Geometry

Abstract This work develops a systematic framework for studying positive integers through their prime-exponent coordinates. Writing n=∏ppvp(n),n=p∏pvp(n), an integer is represented as a finitely supported vector indexed by the primes. This representation separates three distinct forms of arithmetic information: labeled prime support, exponent data, and archimedean height. The topology generated by prime-support layers is determined exactly. It is a countable topological sum of indiscrete support fibres, with a discrete Kolmogorov quotient indexed by finite prime sets. This description yields precise results concerning separation, compactness, convergence, continuous functions, Borel measures, pseudometrics, and support-preserving transformations. The fine arithmetic structure is studied using uniform and Dirichlet probability measures. Fibre disintegration, conditional expectation, entropy, Fisher information, concentration estimates, Erdős–Kac limits, mod-Poisson convergence, and finite-coordinate boundary laws are developed. Multiplication by a fixed integer is identified with conditioning on divisibility under the Dirichlet ensemble. On the Hilbert space of arithmetic functions, prime multiplication becomes a commuting family of unilateral shifts, while Dirichlet convolution becomes an analytic Toeplitz representation. The Bohr transform identifies this theory with Hardy spaces on the infinite torus. Height, multiplicity, and support observables generate explicit semigroups, trace formulae, spectral zeta functions, Fredholm determinants, and Gibbs states. The logarithmic-height Hamiltonian has compact resolvent and partition function ζ(s)ζ(s), while the exact role of the archimedean gamma factor and the limitations of proposed Hilbert–Pólya interpretations are made explicit. The framework is extended to harmonic analysis on Q>0×Q>0×, adelic and pp-adic analysis, modular and automorphic forms, Diophantine geometry, schemes, Frobenius actions, étale and crystalline cohomology, global LL-functions, analytic continuation, functional equations, and arithmetic dynamics. Established theorems are carefully separated from conjectural Langlands, random-matrix, and spectral interpretations. Unlabeled exponent partitions produce a second quotient structure. Ferrers conjugation, shape fibres, symmetric Dirichlet masses, partition Hamiltonians, reversible shape dynamics, and exact information decompositions are obtained. Prime-coordinate automorphisms are classified: multiplicative and order automorphisms are precisely prime permutations, whereas preservation of logarithmic height forces arithmetic rigidity. Finally, completely additive prime functions Fa(n)=∑papvp(n)Fa(n)=p∑apvp(n) are developed as linear observables of the exponent lattice. Their Dirichlet transforms, Gibbs laws, convolution derivations, kernel semigroups, affine Hilbert bases, zero-sector operators, heat traces, Fredholm determinants, congruence projections, and spectral decompositions are calculated. This produces a unified but scope-controlled language connecting multiplicative number theory, probability, operator theory, harmonic analysis, arithmetic geometry, and dynamical systems. Keywords Prime coordinates; prime factorization; arithmetic topology; support fibres; additive arithmetic functions; Dirichlet convolution; Dirichlet series; Riemann zeta function; Euler products; Gibbs measures; operator theory; Hardy spaces; Bohr transform; adelic analysis; pp-adic analysis; modular forms; automorphic forms; arithmetic geometry; étale cohomology; Frobenius; global LL-functions; arithmetic dynamics; partition transposition; spectral zeta functions; zero-layer semigroups.