Recursive Geometry of Atomic Spectra

Atomic spectra reveal hidden regularities when reorganized in a recursive geometry. Our method is non-circular: (1) we fix a recursion coordinate gamma by defining an alpha-powered ruler that is anchored to the Rydberg scale; (2) evaluate level spacings statistically for resonance with this ruler; and (3) overlay photons on their gamma-resonant levels only afterwards.

We find empirically that, when plotted by (gamma, nu), photon frequencies decay as nu ∝ alpha^gamma. In the alpha-affine Thread Frame (gamma, log10 nu), these decays straighten into near-linear threads with universal tilt beta = log10 alpha. Gamma-resonant transitions are grouped by principal-quantum-number towers (n_i, n_k), and tower partitioning is used to resolve intercepts chi (carrying reduced mass, Z^2, and site factors) and local deviations (microslopes).

Across ~30 ions processed with one preregistered pipeline and bootstrap nulls, we find: (1) slopes cluster tightly near log10 alpha; (2) intercepts enable isotope calibration and hydrogenic collapse; (3) a sigma-sweep recovers alpha only in fine-structure windows; (4) microslopes reveal torsion corridors and support ceilings; and (5) cross-thread interactions (CTI) are falsifiable by phase and linewidth gates.

We also introduce photoncodes — chi-invariant binary sequences on a fixed kappa-lattice — showing that recursive structure is recoverable from photons alone. Finally, many ions exhibit terminal photons approaching a common geometric envelope, motivating the conjecture:

E = mc^2 + h * nu_min,

with h * nu_min as a putative single-photon anchor. We present this as a falsifiable synthesis: a reproducible reorganization of spectra in which photons themselves reveal recursive geometry, independent of the gamma construction.