Digital Cyclic Resonance: A First-Principles Spectral and Atomic Model Deriving Quantum Phenomena from a Discrete Postulate

This research presents the Digital Cyclic Resonance (DCR) hypothesis, a first-principles framework that reinterprets atomic physics as a discrete computational process rather than a continuous mechanical system. Operating on a fundamental $(1\to9\to0)$ counting cycle and a geometric matrix capacity of $2n^2$, the model successfully derives key quantum phenomena without relying on empirical physical constants (such as $\hbar$, $m_e$, or $e$). The study validates the hypothesis through two computational engines: A Spectral Engine: Which derives the Hydrogen and Deuterium spectral lines (Lyman/Balmer series) with <1 ppm accuracy, demonstrating that nuclear mass acts as a digital time-dilation factor. An Atomic Engine (Z=1 to 36): Which simulates the Periodic Table’s ionization trends from Hydrogen to Krypton. By implementing a digital analogue of Hund’s Rule (spin-pairing cost vs. exchange bonus), the model spontaneously generates fine-structure anomalies ($B < Be$, $O < N$, $Ga < Zn$, $Se < As$) solely from geometric processing rules, confirming a unified, scale-invariant digital geometry governing matter.