A Numerical Grammar of the Periodic Table: Deriving 118 Elements from Ten Ontological Constants via N=Z+2

This manuscript presents a deterministic grammatical framework that reinterprets the periodic table of elements through the numerical index N = Z + 2. By incorporating the photon (N = 1) and neutron (N = 2) as foundational precursors to hydrogen (N = 3), the system establishes that all 118 known elements can be compositionally derived from exactly ten indecomposable ontological constants. Utilizing three mathematically structured rule sets—multiplicative factorization via a Composite-Prime Expansion Rule, prime number decomposition via Iterative Agency Extraction, and reflexive loops for the 11-series—the grammar generates a closed sequence of 120 ontological positions. The framework maps these numerical derivations directly to empirically measured chemical properties, demonstrating a natural, logical termination of the periodic table at N = 120 (oganesson) while providing explicit, testable falsification criteria for theoretical elements beyond Z = 118.