Atoms and Elements: The Topological Origin of Periodicity — Volume I of the Axiomatization of Chemistry within the Global-Realism Framework

This manuscript is the opening volume of a systematic program that seeks to axiomatize chemistry on the foundation of the global-realist physical ontology. Beginning from three fundamental physical axioms — the reality of the spacetime vacuum as a dynamical substrate, the microscopic reality of quantum objects as localized entities with unique histories, and the persistent coupling between quanta and the vacuum field — we redefine the atom as a composite spacetime-topological configuration consisting of a highly topologically complex nuclear soliton together with self-consistent standing-wave modes of the electronic disturbance field. The electron is treated as a finite-size Hopf soliton with intrinsic chirality and topological rigidity, and the atomic nucleus as a multi-nucleon topological composite stabilized by strong-channel locking. From this ontological basis, the manuscript derives electronic shells and quantum numbers as structural labels of stable disturbance modes rather than as externally imposed rules. The Pauli exclusion principle is reinterpreted as the macroscopic expression of topological repulsion among identical electron solitons. The Aufbau principle, Hund's rules, and orbital energy-level ordering emerge as variational consequences of total-energy minimization under topological constraints. Atomic spectra are reinterpreted as frequency fingerprints produced when atomic disturbance modes reorganize and release or absorb energy through radiative channels of the vacuum substrate. The periodic table is reconstructed not as an empirical chart but as the classification image of a spectral-ranking map generated by the atomic number Z. Periods arise from the opening and closure of frontier-mode clusters, families from the isomorphism of valence-mode sets, and blocks from the angular-momentum content of the active frontier sector. Periodic trends in atomic radius, ionization energy, electron affinity, and electronegativity are derived from the joint competition among effective nuclear charge, penetration, screening, exchange stabilization, and pairing repulsion. The special behavior of transition metals, lanthanides, and heavy p-block elements is traced to frontier near-degeneracy, differential screening, and relativistic contraction, respectively. The physical limits of the periodic table are analyzed through the joint criteria of electronic criticality and nuclear stability. Building on the periodic data, the manuscript defines the first chemically operative layer of reaction theory: admissible reaction paths, effective activation costs, memory-assisted rate functionals, and zeroth-order branching fractions are constructed from previously defined atomic threshold quantities. Collective chemistry is further introduced through coarse free-energy functionals, which yield phase-coexistence conditions, spinodal instability criteria, interfacial evolution laws, nucleation barriers, electrolyte screening laws, and adsorption isotherms — all obtained as continuum descendants of the same physical and chemical inputs. As Volume I, the present work establishes the atomic foundation of chemistry together with its zeroth-order reaction-theoretic interface and a first collective-matter layer. It does not yet axiomatize higher-order multicenter bonding, full reactive hydrodynamics, or chemically specific condensed phases in crystalline detail — these are reserved for subsequent volumes. The principal claim of this volume is that the most fundamental objects of chemistry — atoms, elements, electronic shells, chemical periodicity, spectra, and the threshold-level laws of chemical transformation — can be returned to a unified spacetime-vacuum ontology and derived within a single non-circular logical chain.