Atomic Harmonics and the Shell Rule (2, 8, 18, 32 …): Exploratory Derivations from Kansūron Overlap Equations

The Kansūron modification of Euler’s identity — the K-wave identity —
reinterprets natural numbers as rotating vectors whose overlaps generate deterministic
structures on the time axis. When applied to atomic structure, these overlaps reproduce
the observed electron shell sequence as natural resonance plateaus. The result is not
a final proof, but an initial conceptual derivation that situates shell filling within a
universal harmonic law. This work is presented as a first recorded attempt to connect
Kansūron number-theoretic framework with atomic physics, establishing historical
traceability for future refinements. .... v1.0.1 adds DOI and typography fixes