Geometric Reconstruction of Prime Structure and Unified Analysis of Number-Theoretic Propositions via Discrete Primitive Growth Space

The fundamental structural disconnect between the multiplicative definition of prime numbers and the additive nature of core number-theoretic problems has long been a persistent limitation in elementary number theory. In this work, we establish an axiomatic system of discrete primitive growth space, which provides a unified geometric framework for both additive and multiplicative operations on integers. We redefine primes as irreducible anchor elements in the hierarchical expansion of discrete space, and introduce three core mechanisms: hierarchical growth operators, orthogonal dimension expansion rules, and structural collapse mappings for composite numbers. We rigorously prove that this system is bidirectionally equivalent and isomorphic to classical Peano arithmetic and the fundamental theorem of arithmetic, making it a higher-dimensional reconstruction of number-theoretic foundations fully compatible with all existing classical results. Within this axiomatic framework, core results such as the infinitude of primes, the decreasing density of primes, the existence of arbitrarily long consecutive composite sequences, and the fundamental theorem of arithmetic can be derived directly from the axioms without complex analytic estimations. Furthermore, the framework yields constructive structural interpretations for the binary Goldbach problem and the twin prime conjecture. This work serves as the pure mathematical axiomatic foundation for a companion paper on discrete spacetime modified gravity, where the framework is applied to gravitational and cosmological studies with multi-scale observational validations across 72 orders of magnitude from nucleons to superclusters. This work provides a geometric axiomatic framework for the unified characterization of integer additive and multiplicative structures, and offers a new analytical approach for prime conjecture research and discrete topology studies.