The Quantized Lattice: Derivation of the Atomic Mass Spectrum from Vacuum Stiffness and Topological Strain

[NOTICE]: This preprint has been superseded by updated publications as of January 2026. The theoretical framework and computational findings presented in this document have been refined, expanded, and separated into two distinct domains to ensure ontological clarity. Please cite the following authoritative version depending on your area of interest: Nuclear Physics, Isotope Stability, and the Nuclear Lattice Derivation:Refer to: Nuclear Crystallography: The Universal Tension-Driven Lattice (UTDL-N)Focus: The formal proof of the Geometric Efficiency Law (√2), the validation of Norman Cook's FCC Lattice, and the 0.14% RMS error derivation of the atomic mass spectrum.https://doi.org/10.5281/zenodo.18237972 UTDL FrameworkRefer to: The Universal Tension-Driven Lattice (UTDL): Unification of Gravity, Matter, and Quantum Topology via Dynamic Harmonic Resonancehttps://doi.org/10.5281/zenodo.17674761 Historical Context:This document (UTDLP) remains accessible for historical provenance. It details the initial identification of the Vacuum Stiffness Ratio (αfs) and the A1.01 scaling law, which served as the computational prototype for the finalized UTDL framework. Readers seeking the definitive mathematical proofs and the "Vacuum Stability Criterion" should refer to the updated manuscripts linked above. (Abstract of Superseded Preprint [see above])The Standard Model describes atomic mass as the sum of constituent nucleons minus a binding energy derived from semi-empirical liquid drop models. While accurate, this approach lacks a unified mechanical origin for the binding coefficients. This paper applies the Universal Tension-Driven Lattice (UTDL) framework to derive the atomic mass spectrum from the non-linear elastic properties of the vacuum itself. By defining the Fine Structure Constant (α_fs) as the vacuum's stiffness ratio, we identify a fundamental Strain Quantum E_λ ≈ 20.93 MeV. We demonstrate that the nuclear binding energy coefficients—traditionally fitted parameters—are actually quantized geometric multiples (1, 2, √2, 4α_fs) of this single vacuum modulus. Comprehensive computational validation against the full AME2020 nuclear chart (~3500 isotopes) yields a global RMS relative error of ~0.14%, with residuals quantizing into integer multiples of E_λ. This reveals a novel phenomenon: trapped Dynamic Harmonic Resonance modes governed by proton-neutron orthogonality (d_√2 = |N/Z - √2|), maximizing at low d_√2 and producing trimodal mode distributions corresponding to topological resonance classes. Nuclear shell effects and magic numbers thus emerge as Pythagorean geometric commensurability in the vacuum lattice, without ad-hoc corrections, unifying the periodic table with Planck-scale topology.