Prime factorization identifies the multiplicative monoid of positive integers with the free commutative monoid\: N₁ N₀^ (P), (vₚ (n) ) . volume develops the structures that arise when this identification is used consistently as an arithmetic coordinate system. The construction has three levels: the exponent monoid, its signed group completion, and its idempotent support reflection. Morphisms are represented by column-finite nonnegative integer matrices; necessary and sufficient conditions characterize their injectivity, surjectivity, divisibility reflection, extension to product completions, and metric behavior. For Krull domains, the height-one divisor lattice replaces the free exponent monoid, with units as the kernel and the divisor class group as the exact obstruction to global realization. Positive weights on prime axes produce an intrinsic path geometry_ (m, n) =ₚₚ|vₚ (m) -vₚ (n) |. intervals, medians, convex sets, gates, completions, and properness criteria are determined coordinatewise. For \ (ₚ= p\), _ (m, n) =lcm (m, n) (m, n), \#B_ (R) = eR. , the metric volume entropy equals \ (1\), and the orbit Poincaré series is \ ( (s) \). The vertex horofunction and Roller compactifications are identified with \ ( (N₀\\) ^ P\). A product conformal measure under prime translations is supported on finite arithmetic points precisely in the Euler-product convergence region and on the boundary in the divergence region. Divisor intervals become finite products of chains, so their incidence, order-topological, spectral, resistance, and metric invariants are recovered from their one-prime factors. The same coordinate object admits compatible probabilistic, operator-theoretic, and harmonic realizations. Weighted height operators have compact resolvent exactly when their weight sublevels are finite, while their Gibbs operators are trace class exactly when\ₚ e^-ₚ<. this region, the partition function is\ₚ (1-e^-ₚ) ^-1, the exponent coordinates are independent geometric random variables. Taking \ (ₚ= p\) gives the Dirichlet Gibbs law and the prime-axis tensor factorization of \ (² (N₁) \). Group completion yields Fourier analysis on the infinite prime torus and transforms Dirichlet convolution into multiplication. Integral exponent matrices act simultaneously on lattices, split algebraic tori, closed-geodesic classes, and Fourier frequencies, with Smith normal form determining their common discrete invariants. Prime coordinates linearize multiplication but not addition. Addition is therefore treated through valuations, residue initial forms, Newton polygons, and Hensel lifting. For a polynomial over a discretely valued field, its valuation is the tropical minimum plus the valuation of the normalized initial form; strict increase occurs exactly on the residue-cancellation locus. This local decomposition is combined with congruence methods, height bounds, support stratification, and exact lattice computation to obtain constructive criteria for bounded Diophantine systems. At unramified places, semisimple Galois data and cuspidal \ (GLₙ\) -automorphic data map to a common Euler skeleton. Established reconstruction theorems make both maps faithful on the stated categories, while equality of their global images remains the substantive correspondence problem. The resulting theory is an arithmetic-internal comparison framework. It specifies a common prime-indexed core, determines when information is preserved or lost under each realization, and separates formal coordinate reformulation from genuinely arithmetic input. It provides exact interfaces among multiplicative, local, spectral, probabilistic, geometric, and computational methods without erasing their distinct hypotheses and obstructions. Keywords Prime-coordinate arithmetic; geodesic number-theoretic geometry; valuation vectors; arithmetic monoids; divisor lattices; metric number theory; Euler products; zeta functions; arithmetic probability; spectral theory; infinite prime torus; \ (p\) -adic analysis; local–global principles; Diophantine equations; arithmetic geometry.
Kianming(Jianming) Wang (Fri,) studied this question.