Unlocking the 150‑year‑old puzzle of the periodic table of elements via subject‑free mathematical‑physical equations —Underlying mechanisms for element arrangement and chemical reactions based on nature‑matter proportionality constraint

This work addresses long‑standing fundamental chemical puzzles: the intrinsic origin of periodic‑table arrangement, anomalous electron configurations (Cr, Cu, and La‑Ac‑series perturbations), and physicochemical similarities within main groups. Relying on the established nature‑matter proportionality constraint and global‑local‑balance mathematical‑physical system (sharing the same underlying axiom system with chirality‑parity‑violation theory), this paper constructs a unified formal mathematical‑physical framework. Contrary to mainstream academic opinions, element‑periodic‑system arrangements are not merely empirical regularities fitted by mechanical and energetic effects. Atomic structures, periodic arrangements and certain anomalous electron configurations are intrinsic outcomes projected down to nucleus‑electron coupled systems from nature‑matter proportionality constraint. This theory extends nature‑matter proportionality‑constraint logic from fermion‑chirality scale to nucleus‑atom‑chemical‑reaction mesoscopic scale. Apart from the two ontological axioms, no extra subjective hypotheses or adjustable empirical fitting parameters are introduced. Formal symbolic semi‑quantitative deductions confirm that intrinsic proportional deviations of material systems drive spontaneous recovery toward benchmark proportional steady states and constrain atomic steady‑state conditions. This theory yields multiple quantifiable, experimentally testable predictions comparable against atomic physicochemical experimental trends. Period lengths and partial anomalous electron configurations are not accidental orbital‑energy effects but objective manifestations of proportional‑recovery dynamics at atomic scale. Coulomb interactions and orbital energy levels represent local expressions; global nature‑matter proportionality constraint sets underlying boundary conditions for atomic systems. Local interactions merely modulate manifestation of intrinsic features and cannot generate the original rules governing periodic‑table arrangements. This paper clarifies century‑long causal inversion in chemistry: orbital‑filling rules, ionization‑energy fittings and empirical periodic inductions only describe and reproduce existing periodic phenomena rather than intrinsically deduce their origins. Following global‑local‑balance criteria, a cross‑scale evolutionary chain connecting nucleus synthesis, atomic structure, chemical bonds and chemical reactions is established. It demonstrates that periodic‑table regularities and atomic physicochemical behaviors are system‑scale expressions of the same nature‑matter proportionality constraint across scales. As one work in a series of fundamental theoretical studies, this manuscript is complemented by sister papers on chirality origin and chemical‑bond underlying mechanisms, targeting three long‑standing core open problems in chemistry. This achievement remedies the shortcoming of conventional chemistry which mostly fits phenomena lacking intrinsic mathematical‑physical deduction. It unifies particle‑chirality physics and underlying atomic‑chemical logic and offers new theoretical perspectives for novel‑material design and inorganic‑reaction‑mechanism research.