Subshell Ordering and Periodic Structure from Time-Scalar Field Theory

We analyze the emergence of atomic spectral structure within the framework of Time-Scalar Field Theory (TSFT) by considering fluctuations about localized static scalar-time configurations. The asymptotic behavior of the background field induces an effective radial operator whose leading contribution yields an inverse-radial interaction and a degenerate spectrum characterized by the principal quantum number n. Extending the asymptotic expansion to next order produces a subleading 1/r² correction determined by higher derivatives of the scalar-time potential. Using an exact evaluation of the expectation value ⟨r^−2⟩ via the Hellmann–Feynman theorem, we obtain a closed-form expression for the subshell-dependent energy shifts and derive the corrected spectrum εₙℓ = − κ²/4n² + βκ²/4n³ (ℓ + 1/2). This structure lifts the degeneracy within each principal shell, producing a systematic ordering of subshells by angular momentum. The ordering of subshells belonging to different principal shells is shown to depend on the balance between the leading and subleading contributions. By analyzing the resulting energy inequalities, we derive explicit conditions on the subleading coefficient β under which cross-shell inversions consistent with the observed periodic hierarchy occur. The coefficient β is expressed in terms of the fourth derivative of the scalar-time potential, allowing the ordering conditions to be recast as constraints on the local curvature structure of the underlying field. In this formulation, principal shell structure is governed by the cubic derivative of the potential, while subshell ordering and periodic organization arise from its quartic derivative. These results establish that key features of atomic structure—shell degeneracy, subshell splitting, and conditional periodic ordering—emerge from the scalar-time framework without the introduction of empirical filling rules. The analysis reduces the problem of atomic periodicity to a concrete constraint on the derivative structure of the scalar-time potential, providing a direct link between field dynamics and observed atomic organization.