Full Derivation of Atomic Periodicity from Time–Scalar Field Theory

We derive the structural organization of the periodic table directly from Time–Scalar Field Theory (TSFT) without introducing empirical shell-ordering rules, phenomenological filling principles, externally imposed Coulomb structure, or independently postulated quantum-mechanical operator axioms. The derivation proceeds from the scalar-time fieldΘ = Θ (x^μ), together with the globally constrained temporal-efficiency geometry established in previous stages of the TSFT program. Earlier work demonstrated that localized scalar-time backgrounds generate asymptotic fluctuation operators possessing inverse-radial leading structure and subleading coherence-binding deformations. Subsequent closure analysis showed that the subleading spectral coefficient is not arbitrary but is fixed by temporal-efficiency partition geometry through the globally constrained scalar-time potential. Beginning from the fully constrained scalar-time potential, V (Θ) = V_∗ (e²Θ − 1 − 2Θ − 2Θ²), we derive the asymptotic scalar-time fluctuation operator governing admissible bound coherence modes. The resulting radial spectral problem produces discrete principal-shell structure together with a coherence-binding correction determined uniquely by scalar-time curvature hierarchy and internal temporal allocation. We then derive the exact threshold inequalities governing cross-shell inversions between admissible subshell sectors. These inequalities generate the observed ordering hierarchy of atomic structure, including the emergence of the s, p, d, and f block organization of the periodic table. The resulting ordering structure arises as a consequence of stability-selected scalar-time coherence geometry rather than empirical Madelung-type insertion. The derivation establishes the following structural progression: Θ (x^μ) → V (Θ) → temporal-efficiency closure → spectral fluctuation structure → shell degeneracy → subshell splitting → cross-shell No phenomenological shell-ordering rules are introduced. No externally postulated Coulomb potential is assumed. No independent Hilbert-space quantum structure is inserted. Atomic periodic organization emerges as the unique admissible coherence-ordering phase generated by globally constrained scalar-time closure dynamics.