# Does the nearly uniform spectrum-offset histogram hide a cylinder?

21 September 2026 · HARMONIC-CIRCLE-02 · Exploratory catalog reanalysis

## Outcome

**The near-uniform pooled distribution is confirmed, but these checks do not support a shared cylinder-like winding law.** This is a bounded negative result for the explicitly tested phase statistics and linear winding rule, not a proof that atomic spectra have no structure or that every geometric model fails.

The unchanged source is **11,729 eligible reference lines from 99 elements**, hydrogen through einsteinium, in the earlier NIST Strong Lines study. Each element receives equal total weight. The ten 10-cent bins again contain **9.6256%–10.8253%** each. These are percentages of an element-balanced catalog, not ten-percent steps in physical frequency.

We fixed the new statistics, four controls, seeds, and model grid before computing this follow-up's results. The underlying data and pooled histogram had already been inspected. This is **not fresh validation**, not a blinded experiment, and not new measurements.

## Four controlled comparisons

There were **9,999 reference simulations per comparison**. A reference fraction uses `(1 + tail count) / 10,000`, never zero. The four-comparison adjustment was fixed as `min(1, 4p)`.

| Question | Observed statistic | Reference result | Four-comparison adjusted fraction |
|---|---:|---:|---:|
| Is the pooled pattern unusually flat or concentrated? | 0.000230816 | Two-sided 0.9116 | 1.0000 |
| Do individual elements differ more than line-label shuffles? | 0.0254771 | Upper-tail 0.0304 | 0.1216 |
| Does that excess remain when nearby lines are grouped? | 0.0221739 | Upper-tail 0.9101 | 1.0000 |
| Does the training-fitted winding predict other elements? | 0.0100044 | Upper-tail 0.2653 | 1.0000 |

These are conditional randomization comparisons, not probabilities that a cylinder exists, that a theory is true, or that the observations are random. None passes the fixed four-comparison 5% threshold. In particular, the raw 0.0304 line-level result cannot be promoted to a discovery while omitting the other tests or the block sensitivity.

### 1. Flatness is ordinary under a pattern-preserving control

Let `theta = 2*pi*d/100`, where `d` is the signed note offset in cents. For element `e`, form its first five circular moments:

\[
a_{e,h}=\frac{1}{n_e}\sum_j e^{ih\theta_{e,j}},\qquad h=1,\ldots,5.
\]

The pooled score is

\[
A=\frac15\sum_{h=1}^5\left|\frac1{99}\sum_e a_{e,h}\right|^2.
\]

Each control rotates an element's **whole phase pattern** by one random angle. This preserves its internal line spacings and correlations; it does not replace every line with an independent random draw. The reference mean was **0.000257911**, with central 95% range **0.0000872734–0.000509025**. The observed score sits comfortably inside it. **4,557/9,999** controls were at least as flat, and **5,442/9,999** were at least as concentrated.

This statistic checks the first five Fourier components, not every possible departure from uniformity. A non-rejection does not establish randomness. The ordinary first-moment concentration was 0.018249 after pooling, versus a mean of 0.128793 for individual elements; finite sample sizes alone contribute to this difference, so the latter number is not itself evidence of hidden organization.

### 2. Individual-element variation needs a dependence control

The between-element score is

\[
B=\frac1{99}\sum_e\frac15\sum_{h=1}^5|a_{e,h}-\bar a_h|^2.
\]

The line-level reference shuffles phase vectors only within the same **ionization stage and wavelength octave**, retaining each element's line counts and coarse spectral coverage. **303/9,999** controls had at least as much heterogeneity as observed. Its raw fraction is 0.0304, but the adjusted fraction is 0.1216.

Nearby spectral lines are not necessarily independent pieces of evidence. The frozen sensitivity check joined consecutive lines separated by at most **5 cents of raw frequency spacing**, within the same element, ion stage and wavelength octave. It yielded **6,907 near-line blocks**. Each block contributes its mean phase-moment vector, with equal block weight inside an element. We shuffled complete block vectors among compatible slots.

Now **9,100/9,999** controls had at least as much heterogeneity as observed, fraction **0.9101**. The excess seen in the line-level comparison does not survive this control.

The block comparison changes the observational unit and its weighting. It therefore does **not** prove that all variation was caused by duplicate lines. A 5-cent chained-neighbor block is only a proximity proxy, not a documented quantum multiplet. Shared energy levels, correlations between blocks, uneven source coverage, and selection of strong lines remain unresolved. These shuffles are not substitutes for a physical atomic-structure null model.

### 3. The best fitted helix does not predict convincingly

We specified one simple relationship, using atomic number as a coordinate not derived from frequency:

\[
\theta_{\mathrm{pred}}(Z)=\phi+2\pi qZ\pmod{2\pi}.
\]

On the original **45 training elements**, we tested exactly 512 slopes, `q=j/512`, including zero. The selected model was:

- `q = 0.125` turns per atomic-number step: one turn every eight elements.
- `phi = 1.861987369` radians.
- Training coherence: **0.0561412**, on a scale whose maximum is 1.

That model was written to `FROZEN_WINDING.json` before computing its score on the original **53 validation elements**. Hydrogen was excluded from both sets. The validation score was **0.0100044**. Permuting entire validation-element phase patterns over the same atomic numbers produced at least this score in **2,652/9,999** cases, giving **0.2653**.

The eight-element training winner is **not an established periodic law**. Searching 512 candidate slopes can produce an appealing winner, and it failed to distinguish itself on the validation-label comparison. We did not select another slope after seeing validation, add nonlinear curves, change the angle definition, or move to a more flattering subset. The earlier validation set is already used data; this split supplies an internal predictive check, not independent external confirmation.

## What the cylinder does—and does not—mean mathematically

Since `d` is wrapped modulo 100 cents, its endpoints are identified. This gives a circle **by definition of the coordinate**. Set

\[
(x,y,z)=(r\cos\theta,r\sin\theta,Z).
\]

For any chosen positive radius `r`, the data then lie on a cylinder. All radii work equally well because no spatial length was measured in this mapping. Consequently, constructing this plot cannot identify a physical cylinder, an extra spatial dimension, or a frequency-generating mechanism.

A flat pooled histogram also does not rule out ordered structure: 99 hypothetical groups with phase `2*pi*e/99` each have maximal within-group coherence while their pooled first five moments are exactly zero. This follows from the finite geometric-series identity and is checked numerically in the tests. At the other extreme, broad unstructured groups can have the same pooled appearance. The pooled histogram alone cannot choose between these constructions.

Using log-frequency as the cylinder's height while calculating its angle from that same log-frequency would impose a winding relationship algebraically. We avoided that circular argument by using atomic number, then asking whether a training-fitted rule predicted the unfit element set.

Ten equal-width bins do not force equal counts. But **ten bins each near ten percent is the expected shape of a roughly uniform distribution**, not a new factor of ten. Ten cents corresponds to the frequency ratio `2^(10/1200)`, approximately **1.005793**, a 0.5793% change.

## Scope, sources and evidence status

The source remains the [NIST Handbook of Basic Atomic Spectroscopic Data](https://www.nist.gov/pml/handbook-basic-atomic-spectroscopic-data), whose current page was checked on 21 September 2026 and identifies the underlying data-content update as **November 2013**. It is a selection for neutral and singly-ionized species H–Es, not every frequency, all 118 elements, molecules, or solid/liquid/gas states. The saved source study records exclusions, source pages, air-to-vacuum conversion, and the corrected phosphorus parser. Those files are unchanged.

Nonzero Monte Carlo reference fractions follow the caution established by [Phipson and Smyth (2010)](https://arxiv.org/abs/1603.05766). This is an established statistical safeguard, not a newly discovered physical method. The arbitrary musical reference remains A4=440 Hz with twelve-tone equal temperament and a 40-octave shift. Integer octave shifts do not change the wrapped offsets.

Evidence categories:

- **Written mathematics:** periodic-coordinate construction, arbitrary radius, and the finite-sum counterexample.
- **Finite computation on compiled reference data:** the four scores and shuffled comparisons.
- **Synthetic checks:** known uniform, antipodal, hidden-group and injected-helix examples.
- **New measured evidence:** none.
- **Physical cylindrical structure, extra dimensions, a universal tuning factor, theological identification:** not established by these calculations.

## Verification and reproducibility

**Ten unit tests passed.** A separate numerical implementation imported none of the main analysis helpers. It recomputed all **11,729** offsets from the saved vacuum wavelengths, rebuilt all **6,907** proximity blocks, rechecked every one of the **512** candidate slopes, reproduced all **39,996** simulated reference statistics, and matched every tail count. This is independent calculation on the same data, not independent measurements.

The complete numerical audit and maximum discrepancies are in `results/AUDIT.json`. The protocol SHA-256 is `28341dca50cecb27214a8f2bb1b6cbf0f0b0c0858f44afb32f64611036138e40`; source line-data SHA-256 is `ca8b243698d57e2a1e50f0eb1e357712490c29aa7655c5b8fdec778240eff825`. The source study is preserved byte-for-byte inside `source_study/`.

Run offline with Python and NumPy; Matplotlib is needed only to regenerate the figure:

```sh
python3 -m unittest -v test_circle.py
OPENBLAS_NUM_THREADS=1 python3 analyze_circle.py --output reproduction
OPENBLAS_NUM_THREADS=1 python3 independent_circle_audit.py
python3 render_circle.py
```

The analysis refuses an existing output directory. The audit verifies shipped `results/` and does not replace an existing audit record. `results/PER_ELEMENT.json` retains all 99 elements and their neutral/ionized descriptive breakdowns. `Element_Phase_Map.png` shows the complete 99-element, ten-bin table as an unfolded cylinder; no lines or elements are omitted to improve its appearance.

## Next test

Obtain a **previously unused, transition-labeled observed-line sample**, with actual multiplet/energy-level families and wavelength uncertainties. Fix one predicted relationship and a family-level holdout before looking at its phases. That can distinguish a reproducible element-dependent effect from strong-line selection, neighboring-line overcounting, and the geometry we imposed when wrapping the note offsets. Do not promote the fitted eight-element slope or modify the theory's physical claims on this result.