Are Atoms Quantum Gravitational Compact Objects? - A Unified Approach to Atomic Radii, Discrete Energy Transitions and the Heptad Shell Model

String theory has long pursued a mechanism to compactify its extra dimensions into the observable physical constants of our universe, yet the vast landscape of ~10⁵⁰⁰ possible vacua remains unresolved. Our 4G Model Solution: The 4G Model introduces four interaction-dependent scalar gravitational constants (GN, Ge, Gn, Gw) as the practical bridge, deriving a fundamental 33 pm interaction length—the geometric mean of nuclear and electromagnetic gravity. This scale constrains atomic structure and nuclear radii via the empirical A¹/³ scaling law. Validation: The 4G Model’s fundamental scaling law (A¹/³ × 33 pm) predicts Carbon’s covalent radius at 75.6 pm-matching the experimental 75 pm value to under 1% error without the proposed common correction term. This precise agreement without corrections suggests that the geometric mean of nuclear and electromagnetic gravity [√(Gn*Ge)] may play a key role in atomic structure. Broader Deviations Contextualized: Secondary deviations in other groups stem from Z-dependent quantum screening of Ge and Gn, not flaws in the underlying scale- paralleling Bohr model successes for hydrogen before Sommerfeld’s fine-structure refinements addressed relativistic effects. Theoretical Confirmation: This selective precision affirms 4G’s unification: atomic radii emerge directly from string-like compactification geometry, with screening as tunable perturbations. Carbon’s validation anchors the model as a working hypothesis, indicating that gravitational constants could play a significant role in constraining chemistry at the 33 pm scale. Extension: Finally, by applying the proton’s charge‑mass “dual discreteness formalism,” we propose that atoms can be interpreted as quantum gravitational compact objects within this framework. These are structured into a hierarchy of 7 fundamental shells, dictated by the stability condition , n=1,2,3.., Z/Root(A) . Light magic numbers emerge from the integer values of Z/Root(A_stable) , while heavy magic numbers correspond to the half-integer form, [Z/Root(A_stable)+0.5].