# Do atomic spectra share a musical correction factor?

21 September 2026 · HARMONIC-ELEMENTS-01 · Corrected and independently audited

## Result

**This catalog-wide test does not support a single shared multiplier that moves atomic lines closer to equal-tempered musical notes.** The fitted factor improved the training sample slightly but did not improve the held-out elements. This is a negative result for this particular mapping and objective, not a denial of genuine quantum structure in atomic spectra or a test of every possible harmonic model.

We captured **12,008 source entries** from all **99 elements** in the NIST Strong Lines handbook. **11,729 lines** passed the predefined scoring rules. Every line was lowered by exactly 40 octaves, using vacuum frequency and A4=440 Hz, as in the previous hydrogen study.

The best common factor fitted to 45 training elements was **0.982681487617**, equivalent to shifting the musical tuning reference from A4=440 Hz to **447.754440828 Hz**. It is a fitted choice of tuning, not a derived physical constant.

| Sample | Elements | Lines | Original mean error (cents) | With common factor | Improvement |
|---|---:|---:|---:|---:|---:|
| Training | 45 | 5,309 | 25.03109 | 24.77829 | +0.25281 |
| Held-out elements | 53 | 6,397 | 24.72406 | 24.74255 | **−0.01848** |
| Full catalog, including previously explored hydrogen | 99 | 11,729 | 24.81734 | 24.72886 | +0.08847 |

Lower error is better. Each element has equal total weight, with its lines equally weighted within that element. The full-catalog improvement includes the fitting data and must not be substituted for the held-out result.

The held-out improvement's descriptive 95% element-bootstrap interval was **−1.15808 to +1.24887 cents**, spanning zero. In the predeclared random-alignment comparison, **2,222 of 10,000** simulations aligned at least as closely as the corrected held-out sample; the plus-one reference fraction is **0.22228**. The predeclared success criterion failed.

## What “all” means here

This is the complete selected **Strong Lines handbook**, not the complete NIST Atomic Spectra Database and not every frequency matter can produce. [NIST describes the handbook](https://www.nist.gov/pml/handbook-basic-atomic-spectroscopic-data) as selected data for neutral and singly-ionized atoms, hydrogen through einsteinium (Z=1–99). Its landing page states the data content was last updated in November 2013; checking the website today does not make those underlying measurements new.

The [element index](https://physics.nist.gov/PhysRefData/Handbook/element_name.htm) and all individual tables were captured on 21 September 2026. These reference entries include compiled, blended, calculated and resolved-component values; they are not uniformly raw, independent measurements with uncertainties.

Scored species:

- **6,700 neutral-atom lines**, covering 99 elements.
- **5,029 singly-ionized lines**, covering 94 elements.

Excluded but preserved entries:

- **239** air wavelengths beyond the predefined 185–1690 nm source-air range for conversion.
- **28** entries lacking an assigned ion stage.
- **12** repeated wavelength entries within the same element, stage and medium.

The 19 elements absent from this handbook are Fm, Md, No, Lr, Rf, Db, Sg, Bh, Hs, Mt, Ds, Rg, Cn, Nh, Fl, Mc, Lv, Ts and Og. Missing from this source does not mean lacking a physical spectrum. All 118 symbols appear in `Element_By_Element.md` and `release_corrected/COVERAGE.json`, without invented substitutes.

This run does **not** cover molecules, liquids, solids, all isotopes, all charge states, thermal continua, every external-field condition or every possible transition. Solid/liquid/gas phase is not the same thing as an atom's ionization stage. A compound's spectrum is not generally just the sum of isolated constituent-atom lines.

## Calculation and hypothesis

For each eligible reference wavelength, air wavelengths were converted to vacuum using the standard-air Peck-Reeder formula documented by [NIST](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html). With c=299,792,458 m/s:

\[
f_{\mathrm{mapped}}=\frac{c}{\lambda_{\mathrm{vac}}2^{40}},\qquad
m=69+12\log_2(f_{\mathrm{mapped}}/440\,\mathrm{Hz}).
\]

The signed offset is 100 times the difference between m and its nearest integer, wrapped into [−50,50) cents. One semitone is 100 cents. “Perfect notes” here means **twelve-tone equal temperament**, a specified musical convention; it does not mean every possible pure-ratio harmonic tuning.

We fitted one common grid displacement delta by minimizing the average absolute wrapped residual over training elements. The corresponding frequency multiplier is 2^(−delta/1200). All 10,604 piecewise-linear breakpoints were evaluated. The chosen delta was **+30.245060616 cents**.

Hydrogen was excluded from training and held-out inference because it had already been explored. For the remaining elements, a fixed SHA-256 rule on their symbols assigned the split before scores were computed. No isotope, intensity, note assignment, element subset or tolerance was adjusted to make validation improve.

The main metric was mean absolute cents. The predefined ±10-cent descriptive match rate on validation rose from **19.9825% to 21.2170%**, even though mean error worsened. We retain this result but do not switch the success criterion after seeing it: improving one threshold count is not the same as reducing the overall error.

## Is there a common offset pattern?

Across the full catalog, offsets are broadly distributed rather than concentrated near one displacement. In ten equal-width bins, an element-balanced uniform phase distribution would put 10% in each. The observed percentages before applying the factor are:

| Offset bin (cents) | Element-balanced share |
|---|---:|
| −50 to −40 | 9.9495% |
| −40 to −30 | 9.8535% |
| −30 to −20 | 9.8062% |
| −20 to −10 | 9.7389% |
| −10 to 0 | 9.8295% |
| 0 to 10 | 9.8345% |
| 10 to 20 | 10.8253% |
| 20 to 30 | 10.2947% |
| 30 to 40 | 10.2425% |
| 40 to 50 | 9.6256% |

This is descriptive near-uniformity, not a proof of randomness or exact uniformity. Spectral families, multiplets, shared levels, selection effects and correlations remain present. `Element_By_Element.md` reports every covered element's score and a separately fitted, explicitly exploratory offset, including sparse cases where an apparent optimum is especially unstable.

The state summaries also do not support one consistently helpful factor:

| Catalog stage | Elements | Original error | Corrected error |
|---|---:|---:|---:|
| Neutral, I | 99 | 24.50939 | 24.87164 |
| Singly ionized, II | 94 | 25.00102 | 24.43894 |

These state summaries include training elements and are descriptive, not independent confirmation of a state-specific law. Weighting raw lines instead of elements also changes the apparent size of an improvement; both are retained in the JSON results.

## Why a single factor has a strict mathematical limit

For two frequencies f1 and f2, multiplying both by F leaves their ratio unchanged. If both corrected frequencies were exact equal-tempered notes, necessarily

\[
1200\log_2(f_2/f_1)\in100\mathbb Z.
\]

This is a necessary condition for exact alignment, regardless of F or the octave count. The two resolved [sodium D-line entries](https://physics.nist.gov/PhysRefData/Handbook/Tables/sodiumtable2.htm), 5889.950 and 5895.924 angstrom in air, have a vacuum-frequency separation of approximately **1.75503 cents**, not an integer multiple of 100 cents. No common multiplier can put both exactly on distinct equal-tempered notes, and distinct frequencies cannot become the same exact note under a nonzero common multiplier.

This counterexample rejects exact universal alignment, not the possibility of useful approximate musical sonification. Integer octave changes shift note number by whole octaves and leave the residual unchanged. For any finite spectrum, the average absolute residual across a uniformly shifted common note grid is exactly 25 cents.

## Reference comparison and statistical limits

The random-alignment reference independently rotates each held-out element's entire residual pattern by a uniform offset across one semitone. Its internal line spacings and correlations stay intact. The frozen training factor is not refitted during this validation comparison.

The 10,000-simulation mean was **25.00141 cents**, compared with the observed corrected validation score of 24.74255. The simulation's central 95% range was 24.33981–25.65764 cents. This reference fraction is **not** the probability that a physical theory is true or false, and the element bootstrap is not measurement uncertainty or a random sample of possible universes. The selected handbook is a convenience catalog, with differing spectral coverage across elements.

Even a significant catalog alignment would need a physical explanation and independent replication. An offset can be expressed equally well as a frequency multiplier or as a changed musical tuning reference; that alone does not identify a new force, dimension, or universal constant.

## Audit, correction and provenance

**Eleven unit tests passed.** The independent implementation imported none of the main analysis helpers. It:

- Reparsed **all 12,008 source rows** using a different tokenization rule.
- Recomputed **all 11,729 eligible wavelengths and offsets**, using a bisection air conversion and logarithmic remainder arithmetic.
- Checked the fitted minimum against a 10,000-point grid. Its minimum, 24.778288132 cents, agrees within the known grid resolution with the piecewise result, 24.778285417 cents.
- Reproduced the 10,000 bootstrap replicates and all 10,000 random-alignment decisions using a different circular-distance formula.
- Rechecked source hashes, element splits and neutral/ion aggregates.

The audit found a parser bug before final release: a linked **P** marker meaning “persistent line” was being treated as phosphorus. This affected 32 phosphorus entries; phosphorus belongs to the training split. Correcting that marker interpretation increased the eligible count from 11,697 to 11,729 and slightly changed the training-fitted factor. The original parser and preliminary results are preserved as `analyze_before_parser_fix.py` and `release/`, explicitly superseded by `analyze.py` and `release_corrected/`.

The data rules, objective, random seed, split and success criterion did not change. Validation was recalculated after a source-parsing correction; it is **not a second independent validation**. The conclusion did not change. Earlier messages' preliminary numerical values are superseded by this report.

Raw web-reader source text, line numbers and per-file hashes are included. Neon and platinum needed supplementary page captures; complete line-number coverage was checked before ingestion. No measurements were fabricated, no physical experiment was performed, and the Theory of Reality manuscript and other engines were not changed by this standalone study.

## Reproduce

Python 3.10+ and NumPy are required. This bundle is offline: no downloads or credentials are needed. From its directory:

```sh
python3 -m unittest -v test_analysis.py
python3 analyze.py --output-dir my_reproduction
python3 independent_audit.py
```

The analysis refuses an existing output directory. The audit reads the shipped `release_corrected/` and prints its verification; it preserves the saved audit record rather than overwriting it. `LINES.json` contains all source entries and their exclusions, not only successful matches. `PER_ELEMENT.json` contains all 118 coverage rows. The precision printed by software is arithmetic precision, not claimed physical precision of every input.

## Next test

Do not adopt the fitted multiplier. A useful expansion would freeze an observed-only sample from the larger NIST ASD, including higher charge states and explicit transition labels, before examining its note offsets. Use previously unused spectra for validation; the present validation set is now spent. Molecular/condensed-matter spectra need a separate declared dataset and physical conditions, not invented atomic compositions.

If a more specific harmonic hypothesis is proposed, it should predict particular frequency ratios or state-dependent shifts before looking at those data. That would be distinguishable from simply assigning each observed line its nearest musical note.