This work presents a unified structural analysis across multiple physical scales using a single invariant closure relation.The collection contains four completed studies demonstrating how the same structural hierarchy governs atomic organization, planetary distances, optical phenomena, and material color formation without introducing scale-specific laws or adjustable parameters. The first study establishes an exact cubic structural closure (Z³) in hydrogen-like atomic systems and identifies a precise break index in neutral multi-channel atoms, extractable directly from empirical ionization data.The second study tests whether the same cubic hierarchy appears in the Solar System by projecting the atomic sequence using a fixed projection factor, revealing non-random correspondence with observed planetary orbital distances.The third study reformulates classical optical phenomena—reflection, refraction, dispersion, birefringence, and polarization—as direct projection-layer consequences of structural equilibrium preservation.The fourth study identifies the structural origin of color in metals, demonstrating that color arises only in the collective material regime through phase-selective structural admissibility, and provides predictive classification without optical input data. All results are derived from a single invariant structural relation applied consistently across regimes.No curve fitting, tuning, probabilistic models, force laws, or reinterpretations of existing physical theories are introduced.The work is structural rather than dynamical and is presented as a closed, self-contained reference.
Danijus Kazlauskas (Fri,) studied this question.