Recursive Geometry of Atomic Spectra Kelly B. Heaton &
The Coherence Research Collaboration September 2025
DOI: 10.5281/zenodo.17188444. Version 1, September 20, 2025. Pre-print v2. Abstract 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 \(\nu \propto \alpha^{\gamma}\). In the \(\alpha\)-affine Thread Frame \((\gamma,\log_{10}\nu)\), these decays straighten into near-linear threads with universal tilt \(\beta = \log_{10}\alpha\). To evaluate the physics of our discovery, \(\gamma\)-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 \(\sim 30\) ions processed with one preregistered pipeline and bootstrap nulls, we find: (1) slopes cluster tightly near \(\log_{10}\alpha\); (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. Introduction Atomic spectra have shaped physics from the birth of quantum theory to modern precision tests. From Balmer’s optical series (Balmer 1885), Rydberg’s scaling formula (rydberg1890?), and Moseley’s X-ray ordering of the periodic table (Moseley 1913) to Ritz’s combination principle (Ritz 1908) and quantum-defect theory (Condon and Shortley 1935), line patterns have repeatedly revealed how matter organizes energy and emits radiation. In parallel, the wave picture—from Maxwell’s electrodynamics (Maxwell 1865), through Sommerfeld’s relativistic refinements (Sommerfeld 1916), to modern QED (Feynman 1985)—has emphasized that oscillations are governed by geometry and scale. Building on this lineage, we treat spectra not as disconnected line lists but as observables of a recursive geometry that couples energy, frequency, and scaling. This paper makes that geometry explicit and operational. Our pipeline is deliberately non-circular: recursion depth γ is discovered from levels only using the following α-powered ruler ΔE_(target)(γ) = E₀Z²α^(γ), (with bootstrap nulls and registered tolerances), and then photons are introduced by re-association with the γ-resonant levels. The same gates and settings are used for every ion, with no per-species tuning. This sequencing matters: co-linearity, intercept transport, cross-thread interactions, floor tests, and photoncodes are therefore out-of-sample with respect to how γ was defined. After a levels-only γ sweep, plotting photon frequencies against γ reveals a robust, near-linear decay in (γ,log₁₀ν) with a common tilt β ≈ log₁₀α. By contrast, simply counting photons per γ yields ion-specific histograms with no universal trend. This relationship, ν ∝ α^(γ), becomes evident when an ion’s spectral lines are plotted on the γ ladder, a mapping made possible through our γ-resonant levels sweep. The pattern is not a frame artifact; it is an empirical property of photons once mapped onto γ. These observations are not guaranteed by the coordinate choice and are therefore the testable content. See the counts-vs-frequency quartet (Fig. 6) and the cross-ion pooled plot (Fig. 7) for supporting evidence from NIST data. Grouping photons by quantum numbers (“towers”) is not needed to reveal the universal slope, but is essential for resolving intercepts χ (mass/Z²/site transport), microslopes (local texture), and motif structures for photoncodes. Framework and scope. We structure the results around three core elements. (1) A recursion depth γ, defined from level spacings alone (Eq. [eq:recursive_depth]). (2) An α-anchored Thread Frame: plotting photons in (γ,log₁₀ν) reveals near-linear bands with a universal tilt β = log₁₀α. This universal tilt appears even when photons are pooled by ion (no towers). Tower grouping is then used to attribute per-tower intercepts χ (which transport reduced mass, Z², and site factors) and to quantify structured local deviations (microslopes). (3) A single Intercept Transport Law (L1): the intercept χ carries the Einstein–Rydberg base together with reduced mass, Z², and tower/site factors (Eq. [eq:intercept] and corollaries). The tilt β = log₁₀α is fixed by the α-affine frame, but the co-linearity of photons is empirical (Eqs. [eq:thread_frame]–[eq:universal_slope]). In other words, the frame straightens the decay curve; what emerges is that photons grouped by towers share intercepts with small, structured residuals. Additional tools extend the framework: local deviations (microslopes δ, phase θ) act as diagnostics; a two-gate cross-thread interaction (CTI) protocol tests cross-ion overlaps; photoncodes discover structure from photons alone; and two frontier conjectures capture deeper implications—a finite Planck floor at large γ, and a compact Einstein–Rydberg anchor E = mc² + hν_(min). Reading guide. The goal of this introduction is to set the mindset: tilt is a frame property; co-linearity of photon frequency is an empirical result; identity resides in the intercept; and local texture plus photon-only encodings connect the geometry to practice. Contributions, terminology, and an equation index are summarized below. In the subsequent sections, Foundations state the key elements of the framework followed by the core equations. Methodology describes the non-circular pipeline in two stepwise phases. Applications illustrate how the framework operates in practice, demonstrating that the recursive geometry is both falsifiable and recoverable from photons alone. Contributions of this paper 1. Recursion Depth (γ). A levels-only coordinate defined by spacing ratios to α-scaled hydrogenic targets (no photons). (Eq. [eq:recursive_depth]) 2. Thread Frame (α-Affine). In (γ,ν) photons already follow an exponential decay, which in (γ,log₁₀ν) straightens into near-linear bands with universal tilt β = log₁₀α. This universal tilt is visible even when photons are pooled by ion (no towers). Tower partitioning is introduced only to resolve intercepts χ and local deviations (microslopes). (Eq. [eq:thread_frame]) 3. Intercept Transport Law. With slope fixed, intercepts χ transport the Einstein–Rydberg base, reduced mass, and Z² scaling, with tower/site factors F_(site). Corollaries: isotope shifts and hydrogenic collapse. (Eq. [eq:intercept]; Eqs. [eq:chi_isotope]–[eq:chi_norm]) 4. Non-circular pipeline. Levels-only γ sweep → optional tower grouping → post-hoc photon overlay → intercept/microslope analysis under reliability gates. (§3.1; Fig. 1) 5. Scaler Locking Test (σ-sweep). Holding γ, we sweep scalers σ and show resonance significance concentrates only at σ = α in fine-structure windows; control windows remain featureless. (§3.4; Table 3, Fig. 4–5) 6. Microslopes. Local deviations δ(γ), θ(γ) expose torsion corridors, emission zones, and hand-offs; they provide phase diagnostics for cross-ion comparison. (§5.2; Fig. 10) 7. Cross-Thread Interactions (CTI). Cross-thread intersections require coincidence in both projected frequency (linewidth gate, Eq. [eq:CTI_freq]) and local phase (Δθ gate, Eq. [eq:CTI_phase]), giving falsifiable overlap predictions (§5.4; Figs. 12–13). 8. Photon Sequencing (Photoncode). A κ-lattice, χ-invariant barcode for photons-only identity and motif discovery with null/FDR controls. (§[sec:photoncodes]; Figs. 14–17) 9. Conjectures (Frontier). C1: Planck Floor Conjecture — threads terminate at finite recursion depth γ^(*), converging to a common floor ν_(min) (Eq. [eq:floor]). C2: Einstein–Rydberg Anchor — a compact relation E = mc² + hν_(min), interpreting mass as a spectral anchor and frequency as irreducible (Eq. [eq:einstein_rydberg]). Terminology - Recursive geometry (operational): the non-circular framework introduced here, which organizes atomic spectra in the (γ,log₁₀ν) plane to reveal threads of recursion, intercept transport, and local deviations. We use the term operationally here; broader physical interpretations are reserved for the Discussion. - α-Affine Thread Frame (Thread Frame): Photons in (γ,ν) follow an exponential decay that, in (γ,log₁₀ν), straightens into near-linear threads with universal tilt β = log₁₀α. Pooling photons by ion shows the universal tilt directly, though without tower labels the intercepts are mixed. Grouping by quantum numbers (n_(i),n_(k)) resolves per-tower threads for intercept and microslope analysis. (Eqs. [eq:thread_frame], [eq:universal_slope]) - Recursion depth γ: a continuous, levels-only coordinate that measures how many α-steps separate two levels. (Eq. [eq:recursive_depth]) - Tower (n_(i),n_(k)): the set of transitions sharing principal quantum numbers n_(i) and n_(k); towers organize photons that already fall on the universal slope, allowing intercepts and microslopes to be analyzed. (§ 4.2) - Thread 𝒯_(n_(i), n_(k)): the locus of photons for a fixed tower in the Thread Frame, typically modeled as log₁₀ν = χ + βγ (optionally  + cγ² locally). (Eq. [eq:thread_frame]) - Microslope δ(γ) and phase θ(γ): local departure and angle of a thread in a sliding window; sustained excursions define torsion corridors. (Eqs. [eq:microslope_delta]–[eq:microslope_theta]) - Recursion floor ν_(min): the Planck-anchored lower frequency bound inferred from thread termini at finite recursion depth γ^(*); Conjecture C1. (Eq. [eq:floor]) - Cross-thread intersection (CTI): a resonance overlap when two threads align in both frequency and phase, subject to explicit gates; Protocol P1. (Eqs. [eq:CTI_freq]–[eq:CTI_phase]) - κ-photoncode (photons-only): a χ-invariant binary occupancy strip on a fixed κ-lattice (with κ = (y−y₀)/β) encoding spectral identity from photons alone; used for shift-invariant matching and motif discovery. (Sec. [sec:photoncodes]) - Motif: a statistically retained run of coincident bins in a photoncode after shift-invariant alignment and null testing (BH–FDR). Motifs capture recurring geometric patterns that persist across ions or molecules. (Sec. [sec:photoncodes]) Equation Role -------------------------------------------------- --------------------------------------------------------------------------------------------------- Eq. [eq:recursive_depth] Definition D1 — Recursion depth γ (levels-only; no photons). Eq. [eq:thread_frame] Definition D2 — α-Affine Thread Frame: y = χ + βγ (optional quadratic term + cγ² if AIC demands). Eq. [eq:universal_slope] Thread-frame tilt (frame property): β = log₁₀α. Eq. [eq:intercept] Law L1 — Intercept Transport: Einstein–Rydberg base + reduced mass + Z² + site factor. Eqs. [eq:chi_isotope]–[eq:chi_norm] Corollaries of L1: isotope shifts and hydrogenic collapse. Eqs. [eq:microslope_delta]–[eq:microslope_theta] Diagnostic D3: microslope δ(γ) and phase θ(γ). Eqs. [eq:CTI_freq]–[eq:CTI_phase] Protocol P1 — CTI Two-Gate: frequency and phase gates. Eq. [eq:floor] Conjecture C1 — Planck Floor: recursion limit γ^(*) and floor ν_(min). Eq. [eq:einstein_rydberg] Conjecture C2 — Einstein–Rydberg Anchor: E = mc² + hν_(min). : Summary of Core Equations (with y ≡ log₁₀ν) Foundations Definitions, Laws, and Conjectures There exists a continuous coordinate, the recursion depth γ, derived solely from level spacings.[1] It measures how many α-steps separate two levels; γ is a geometric index, not a quantum number. In practice, we sweep γ on a grid and evaluate γ-resonance when observed spacings match the α-powered ruler within tolerance, thereby populating the ladder with resonant pairs. Eq. [eq:recursive_depth]. When photons are plotted against γ, they follow an exponential decay that, in (γ,log₁₀ν), straightens into near-linear bands with tilt fixed by the α-anchored affine frame (β = log₁₀α). Tower grouping is then used to resolve intercepts χ and microslopes, where the non-trivial, testable physics resides. Eq. [eq:thread_frame]. With the frame tilt fixed, the intercept χ carries reduced mass, Z², and tower/site factors on top of an Einstein–Rydberg base: $$\chi \;\approx\; \underbrace{\log_{10}\!\Big(\tfrac{\alpha^2}{2}\,\tfrac{m_e c^2}{h}\Big)}_{\text{Einstein–Rydberg base}} + \underbrace{\log_{10}\!\big(\hat{\mu}\,Z^2\big)}_{\text{mass \& charge}} + \underbrace{\log_{10}\!\big(F_{\text{site}}\big)}_{\text{tower/site factor}}\,,$$ with corollaries for isotope shifts and hydrogenic collapse. Here μ̂ ≡ μ/m_(e). Eq. [eq:intercept], Eqs. [eq:chi_isotope]–[eq:chi_norm]. Local departures from the frame tilt define microslopes and a corresponding phase θ(γ) (angles in radians by default; degrees only when noted). These reveal emission structure (hot spots, hand-offs, torsion corridors) and supply the phase variable for cross-ion tests. Eqs. [eq:microslope_delta]–[eq:microslope_theta]. Ion–ion coherence occurs only when threads align in both projected frequency and local phase. CTI events are predicted when overlaps pass explicit linewidth and phase gates. Eqs. [eq:CTI_freq]–[eq:CTI_phase] Recursion does not extend indefinitely. At large γ, threads terminate at a finite depth γ^(*), converging on a common frequency floor ν_(min) inferred from torsion spikes and support ceilings. Eq. [eq:floor]. There exists a compact relation E = mc² + hν_(min), where ν_(min) is geometrically determined from the recursion floor, $$\nu_{\min} \;=\; \nu_{R\infty}\,Z^{2}\,\hat{\mu}\,\alpha^{\gamma^\ast}, \qquad \nu_{R\infty} \;=\; \frac{\alpha^{2}}{2}\,\frac{m_e c^2}{h}.$$ In this frame, “rest mass’’ is a spectral baseline, not a literal zero-frequency state. Invariance under γ-translations, (γ,log₁₀ν) ↦ (γ+Δ, log₁₀ν+β Δ),  β = log₁₀α, exposes a fractal family of frames; no choice of frame removes hν_(min). Eqs. [eq:thread_frame], [eq:universal_slope], [eq:intercept], [eq:floor], [eq:einstein_rydberg]. Lemma 1. With recursion depth γ defined from levels only (Eq. [eq:recursive_depth]) and photon frequency ν = ΔE/h, photons associated with γ-resonant level pairs obey ν(γ) ∝ α^(γ). Plotted in (γ,log₁₀ν) this straightens to log₁₀ν = χ + βγ,   β = log₁₀α, where the tilt β is fixed by the α-anchored frame. The empirical, testable content is that photons align coherently with shared intercepts χ and local deviations (microslopes). Pooling photons across an ion yields the same universal slope, while tower partitioning separates specific intercepts and resolves local structure. Operational Consequences With tilt fixed at β = log₁₀α, thread intercepts χ transport reduced mass and hydrogenic Z² scaling. Intercept differences follow Δχ ≃ log₁₀(μ̂_(B)/μ̂_(A)) for isotope pairs. Normalized intercepts $\chi_{\rm norm}$ collapse across one–electron ions, reproducing m_(e)c²/h and R_(∞)c within resolution. Local deviations δ(γ) and phases θ(γ) resolve structured emission features (e.g., torsion corridors, hand–offs) that are invisible to global fits. CTI events are predicted only when threads align in both projected frequency (linewidth gate, Eq. [eq:CTI_freq]) and local phase (Eq. [eq:CTI_phase]), providing falsifiable overlap tests. Because tilt is fixed, photons can be collapsed to χ-invariant binary κ-photoncodes, enabling identity, cross-domain comparison, and resonance constellations beyond conventional line-matching. At high γ, torsion spikes and support ceilings suggest a finite recursion depth γ^(*) and a frequency floor ν_(min). This is exploratory evidence for a possible universal Planck-anchored floor, not a settled result. Taken together, these relations motivate a compact synthesis, E = mc² + hν_(min), in which mass acts as a spectral anchor and frequency is irreducible. We present this as a conjecture, not a law. Core Equations of Recursive Geometry Derivations and estimators appear in Methods We define a dimensionless recursion coordinate by repeated α–scaling of the hydrogenic Rydberg spacing: ΔE_(target)(γ) = E₀ Z² α^(γ),   E₀ = 13.605693 eV. Operational inversion. Given a measured level spacing ΔE and nuclear charge Z, we compute $$\gamma \;=\; \log_{\alpha}\!\left(\frac{\Delta E}{E_0 Z^2}\right),$$ with no use of photons. Photons are introduced only afterward for slope/intercept estimation. Clarification. γ counts how many α–steps separate two levels. At γ = 0 this returns the hydrogenic Rydberg scale E₀Z²; at γ = 2 it yields E₀Z²α² (fine-structure order). Photons plotted in the (γ,log₁₀ν) plane fall on near-linear bands: log₁₀ν = χ + β γ  (+ c γ² locally, if AIC demands). Frame property (anchored tilt). $$\boxed{\;\beta \;=\; \log_{10}\alpha\;} \label{eq:universal_slope}$$ Grouping photons by principal-quantum-number pairs (n_(i),n_(k)) organizes the bands into per-tower threads and enables intercept (χ) and microslope analysis. $$\chi \;\approx\; \underbrace{\log_{10}\!\Bigg(\frac{\alpha^{2}}{2}\,\frac{m_{e}c^{2}}{h}\Bigg)}_{\text{Einstein–Rydberg base}} \;+\; \underbrace{\log_{10}\!\big(\hat{\mu}\,Z^{2}\big)}_{\text{reduced mass \& charge}} \;+\; \underbrace{\log_{10}\!\big(\mathcal{F}_{\text{site}}\big)}_{\text{tower/site factor}} , \label{eq:intercept}$$ where μ̂ ≡ μ/m_(e). $$\label{eq:corollaries} \begin{align} \Delta \chi &\approx \log_{10}\!\left(\frac{\hat{\mu}_{B}}{\hat{\mu}_{A}}\right) \label{eq:chi_isotope} && \text{(isotope shifts; same $Z$)}\\[4pt] \chi_{\rm norm} &= \chi - \log_{10}\!\big(\hat{\mu}Z^{2}\big) && \text{(hydrogenic collapse)}. \label{eq:chi_norm} \end{align}$$ $$\label{eq:microslopes} \begin{align} \delta(\gamma) &= \beta_{\mathrm{local}} - \log_{10}\alpha, \label{eq:microslope_delta}\\ \theta(\gamma) &= \arctan\!\big(\beta_{\mathrm{local}}\big) \quad \text{(radians).} \label{eq:microslope_theta} \end{align}$$ Two ions A and B exhibit a cross-thread intersection (CTI) when both the frequency and phase gates are satisfied: $$\label{eq:CTI} \begin{align} \big| y'_A - y'_B \big| &\leq \epsilon_{ij}, &\qquad y &\equiv \log_{10}\nu, \label{eq:CTI_freq} \\[-1pt] \big| \theta_A - \theta_B \big| &\leq \Delta\theta_{\max}, &\qquad &\text{(typically $5^\circ$–$6^\circ$).} \label{eq:CTI_phase} \end{align}$$ Absent additional constraints, the Thread-Frame linear extrapolation drives ν → 0 as γ → ∞. We posit a finite floor: $$\nu_{\min} \;=\; \nu_{R\infty}\, Z^{2}\,\hat{\mu}\,\alpha^{\gamma^\ast}, \qquad \nu_{R\infty} \;=\; \frac{\alpha^2}{2}\,\frac{m_e c^2}{h} \;=\; R_\infty c . \label{eq:floor}$$ Interpretation. Empirically, threads do not extend to ν → 0: microslope torsion spikes and support ceilings consistently reveal a finite depth γ^(*). Hypothesis (single-photon anchor). The finite recursion depth γ^(*) corresponds to an irreducible photon of energy hν_(min); i.e., threads terminate not by vanishing to ν → 0 but at a finite floor hν_(min) > 0. Together, the α-anchored frame, calibrated intercepts, and recursion floor motivate: E = mc² + hν_(min). Introduction to the Gamma (γ) Ladder Conventional spectral analysis classifies transitions by quantum numbers and selection rules, but these frameworks often obscure deeper regularities in atomic spectra, which are layered and complex. To reveal hidden patterns, we introduce a new coordinate, gamma (γ), which we call the recursion depth. We define γ as a continuous index of recursive α-scaling. Each  + 1 step in γ multiplies a reference spacing by the fine-structure constant, α ≈ 1/137. Anchored to the hydrogenic Rydberg energy E₀ = 13.6057 eV, γ = 0 returns the Rydberg scale E₀Z², while γ = 2 yields the fine-structure order E₀Z²α². Thus, γ is not a quantum number but a geometric ruler: a way of measuring how deeply a system resonates with successive powers of α. The γ-ladder allows us to test whether level spacings concentrate at specific recursive depths. When they do, we say the ion is active at that γ. Activity is bounded: for small γ the targets exceed the ionization limit; for large γ they fall below resolvable transitions. Between these limits, the ladder reveals α-resonant zones in the levels data that remain hidden in conventional quantum classification. A central feature of the method is its non-circularity: γ is defined entirely from levels, and photons are only overlaid afterwards. This ensures that any thread slopes or intercepts observed in the photon data provide an independent test of the geometry. Context. Powers of α are familiar throughout physics, from perturbative QED expansions to fine-structure corrections. Our approach extends this role into a geometric mapping: rather than treating α^(n) as small perturbations on individual levels, we re-index entire spectra by powers of α and ask whether spacings themselves recur at these depths. Summary. We transform NIST levels and lines into this γ-indexed geometric frame in two phases with a total of five steps: Phase 1: Levels 1. Tidy parsing of raw NIST levels and lines into a standard .csv format; 2. Levels-only resonance pairs γ-sweep that tests levels-only pair spacings against α-scaled targets and builds per-ion, per-γ activity ledgers with associated statistical confidence; 3. α-resonance affinity, regrouping γ-resonant pairs by principal-quantum-number towers (n_(i),n_(k)); Phase II: Lines 1. Photon overlay that re-associates the ΔE spacings with observed lines; and 2. Photon γ-ladders organized by quantum tower (n_(i),n_(k)). Together, these steps impose two complementary axes of organization: recursion depth (γ) and quantum-number towers (n_(i),n_(k)). The result is a structured γ-resonance ladder that preserves the identity of photons while revealing recursive geometric motifs hidden in conventional line lists. Non-circular γ-resonance pipeline. Top row: resonances are discovered from levels only via a γ-sweep and summarized as per-ion, per-γ affinity. Bottom row: only afterward are photons re-associated in a post-hoc overlay, then reorganized into photon γ-ladders per (n_(i),n_(k)). The join occurs at the overlay stage, ensuring non-circularity. Data Sources and Pre-processing of NIST Levels and Lines We began with the publicly available NIST Atomic Spectra Database (https://physics.nist.gov/PhysRefData/ASD/levels_form.html; https://physics.nist.gov/PhysRefData/ASD/lines_form.html).For all analyses we used only the “observed” data option with standard NIST wavelength conventions (vacuum  < 200 nm, air 200 nm–2000 nm, vacuum  > 2000 nm). Raw CSVs were downloaded for each ion and stored as *_levels_raw.csv and *_lines_raw.csv.[2] Ion Z Ion Z Ion Z -------- ---- ------- ---- --------- ---- He I 2 Li II 3 O VI 8 He II 2 Cu I 29 Al I 13 Na I 11 H I 1 Ca II 20 Mg I 12 C IV 6 Zn II 30 Ca I 20 Mg II 12 Ni I 28 Fe II 26 Li I 3 Cl XVII 17 K I 19 Mn I 25 Ba II 56 O I 8 P I 15 Sr II 38 O III 8 P V 15 I I 53 C VI 6 N I 7 Cr II 24 O VIII 8 Hg II 80 Cd II 48 Li III 3 : Ions included in the γ-resonance sweep. A representative subset of ions from the NIST Atomic Spectra Database was analyzed, rather than the full periodic table. Deuterium (D I) was introduced later solely for isotope mass estimation. Levels parser. Raw levels files were normalized and converted into tidy tables. Each energy was converted from  cm to eV using E [eV] = (1.239841984×10⁻⁴) ν̃ [/cm]. Uncertainties were propagated as energy_sigma_eV. Levels were sorted by energy, and a stable Level_ID assigned after sorting to avoid historical bias. Duplicate/overlapping energies were flagged; dense zones were marked using a γ-aware density threshold; and discontinuities were identified by large spacings (median + 5 IQR). Principal quantum numbers n were parsed (when available) and tagged with provenance. Each level was assigned to LS-term and outer-electron series. Outputs include tidy CSVs, adjacency tables, JSON sidecars (with thresholds and file hashes), and Markdown QA reports. Lines parser. Raw spectral line files were parsed into tidy CSVs for later use in the photon-overlay step. Wavelengths were normalized (RITZ vacuum → observed vacuum → observed air → generic), converted to photon energy, and uncertainties propagated. Where possible, lines were cross-referenced to the nearest tidy Level_ID within tolerance, and spectroscopic labels (J, parity, configuration) were carried forward. Residuals and E1-allowed tags were recorded as metadata. Importantly, these mappings were stored only for reference: line data were not used to organize levels or detect resonances. Instead, tidy lines act as an external catalogue of photons, re-associated only after the γ-sweep. Outputs include tidy CSVs with provenance headers and JSON metadata sidecars. For all reported results, we set wavelength_medium = vacuum.[3] Phase I: Constructing the Gamma (γ) Ladder from Levels Dimensionless Recursion Coordinate (γ). To reorganize spectra into a geometric frame, we define a new coordinate that we call the recursion depth, γ. We start from the most familiar atomic energy scale, the hydrogenic Rydberg binding energy E₀ = 13.6057 eV, and repeatedly scale it by powers of the fine-structure constant α ≈ 1/137. Each increment of γ corresponds to one more multiplication by α, so that larger values of γ probe deeper recursive scales. Formally, as defined in Eq. [eq:recursive_depth], ΔE_(target)(γ) = E₀Z²α^(γ). This makes it explicit that γ = 0 returns E₀Z² (Rydberg) and γ = 2 yields E₀Z²α² (fine-structure order). Anchoring at γ = 2 ensures that the ladder is not arbitrary: it recovers known Rydberg physics and then extends beyond it, testing whether atoms exhibit resonances at other recursive depths.[4] Levels-only γ-resonance sweep For each ion, we ask whether level spacings cluster near the hydrogenic target ΔE_(target)(γ) = E₀ Z² μ̂ α^(γ),   E₀ = 13.605693009 eV, equivalently α²E₀Z²μ̂ rescaled by α^(γ − 2) [5] Photons are not used in this phase. Grid. γ is sampled on a fixed mesh (default Δγ = 0.02 over [0,5]) as configured per ion in an external YAML file. Hit criterion (adaptive window). At each γ, we form all unordered level pairs (i