# At / Fr harmonic follow-up

Analysis date: 21 September 2026. Protocol: **AT-FR-ALIGNMENT-03**.

**The original narrow alignment does not generalize as a common offset across the additional transitions examined.** One additional At entry has a nominal offset of +22.118469 cents, but its sensitivity envelope crosses the fixed target boundary. The four new Fr transitions have offsets −1.813563, +14.005259, +18.245693 and +26.252356 cents. A shared correction that brings one of these values to zero cannot bring all four to zero.

This is a calculation using published spectroscopic measurements. It is not a new laboratory experiment, proof of randomness, or a test of every transition or material state.

## Frozen question and calculation

The old chart contained only two At lines and four Fr lines. A cluster selected after looking at that chart spanned **+21.075090665 to +22.247354466 cents**. That exact interval and the older broader **+20 to +30-cent** histogram bin were fixed before calculating the new offsets. No tuning constant was fitted here.

For vacuum photon wavenumber \(\tilde\nu\) in cm⁻¹:

\[
f=299792458\times100\tilde\nu,\qquad
f_{40}=f/2^{40},\qquad
d=1200\log_2(f_{40}/440)-100\operatorname{round}\!\left[12\log_2(f_{40}/440)\right].
\]

Here a cent is 1/100 of an equal-tempered semitone, and the offset is in [−50,50). The tie convention is fixed in code. Optical frequencies are transposed numerically; these are not sound waves emitted by atoms.

**Exact mathematical check:** lowering a frequency by any integer number of octaves subtracts an integer multiple of 1200 cents. It leaves its nearest-note offset unchanged. Thus 40 octaves is useful for audibility, but cannot create a special alignment. Changing the A4 reference rotates all offsets equally; it does not remove their pairwise differences modulo 100 cents.

## Additional source data

1. **At-199:** Teigelhöfer's experimental thesis, dated 18 September 2012, supplies ten observed second-step branches and 45 third-step entries, the latter identified by checked cells in Table 5.4. We reconstruct photon wavenumbers by subtracting the reported lower energy from the reported total excitation energy **only for observed branches**. The total energy alone is not the scanned photon frequency. [Primary thesis](https://mspace.lib.umanitoba.ca/bitstream/handle/1993/8866/Teigelhoefer_Andrea.pdf).
2. **Fr-221:** all four new 7s → 9p/10p measured centroids in Table 1 of Lassègues et al., arXiv:2604.12467v1, posted 14 April 2026. The old 8p benchmark is excluded. The experimental column is used; semi-empirical and theoretical values are excluded. This source is a preprint. [Primary experiment](https://arxiv.org/html/2604.12467v1).
3. A directly reported **At-205** second-step wavenumber, 12571.5(3) cm⁻¹, gives **−32.294596 ± 0.041313 cents** after propagation. It is an isotope measurement of a branch already represented in the thesis, so it is a cross-check, not another distinct transition. [Rothe et al., 14 May 2013](https://www.nature.com/articles/ncomms2819).

The NIST 2007 Fr compilation explicitly says its Table 4 wavelengths are calculated from compiled energy levels; that table also draws on semi-empirical extrapolation. It was not relabeled as a large new measured-line sample. [NIST source](https://www.nist.gov/system/files/documents/srd/jpcrd362007497p.pdf).

## Results

| Element | Additional entries analyzed | Strict count after source audit | Nominal hits in fixed narrow interval | Nominal hits in +20 to +30 cents |
|---|---:|---:|---:|---:|
| At | 55 | 53 | 1 | 3 |
| Fr | 4 | 4 | 0 | 1 |

The two withheld At entries do not hit either interval. The narrow At hit is the observed branch from **57157.1 to 70130.0 cm⁻¹**, giving a photon wavenumber of **12972.9 cm⁻¹**, **770.837669 nm**, and **353.718640 Hz** after transposition. Its nearest note is **F4**, offset **+22.118469 cents**.

The At source lacks a complete transition-error covariance matrix. Using a declared ±2 cm⁻¹ sensitivity envelope on the photon wavenumber gives approximately **21.852 to 22.385 cents**, overlapping the target's upper edge. This envelope is **not** a published confidence interval or a guaranteed error bound. Therefore the one nominal hit is not a securely resolved match to the narrow interval. Another At entry at +19.884968 cents overlaps the broader bin's lower edge under the same sensitivity check.

| Fr transition | Measured wavenumber (cm⁻¹) | Transposed frequency (Hz) | Nearest note | Offset (cents) |
|---|---:|---:|---|---:|
| 7s → 9p₁/₂ | 27111.238(5) | 739.214072 | F♯5 | −1.813563 |
| 7s → 9p₃/₂ | 27360.097(5) | 745.999453 | F♯5 | +14.005259 |
| 7s → 10p₁/₂ | 29058.1(6)[8] | 792.297144 | G5 | +18.245693 |
| 7s → 10p₃/₂ | 29192.8(8)[8] | 795.969869 | G5 | +26.252356 |

Parentheses are statistical and brackets systematic uncertainties in the last digits. Propagated statistical uncertainties are approximately 0.000319, 0.000316, 0.035747 and 0.047443 cents. The latter two systematic contributions are 0.047663 and 0.047443 cents. All four are well outside the fixed narrow target even when the listed components are added for a sensitivity check. Shared calibration errors are not treated as independent replication.

![Additional At and Fr offsets](At_Fr_Followup.png)

## Source audit and preserved corrections

- The frozen 59-entry input set and initial raw counts remain unchanged. A later source-quality audit found that the two last At entries, 72906.7 and 72907.8 cm⁻¹ from the same lower state, correspond to the region discussed as a possible hyperfine doublet in Section 5.3.2/Figure 5.19(c). Their separation is about 33 GHz, consistent with the reported 32.7(9) GHz. Both are withheld from strict electronic-transition counts until assigned. No fictitious centroid was supplied. Their offsets are −33.439842 and −33.304803 cents.
- The thesis has small internal labeling differences: 57267.5 versus 57267.8 cm⁻¹, and 57277.0/58805.3 versus 57276.7/58805.0 in different sections. Values from the corresponding section/table were retained. The 57267.5 alternative was explicitly recomputed. The ten stated vacuum diagram wavelengths cross-check the second-step calculations to within 0.030 cents; rounding and label variations were not hidden.
- A preliminary automatic check of PDF checkbox positions failed because splitting text removed the first-line column offset. The parser was corrected, without changing the frozen observations; it then matched all 45 Table 5.4 branches. The table was also visually inspected.
- Independent 70-digit Decimal calculations reproduce all **59** offsets to better than **2 × 10⁻¹³ cents**, and the 45 Table 5.4 branch assignments were checked with a separate positional extraction. This verifies the arithmetic/transcription, not experimental accuracy.

## Interpretation and next test

One nominal match among the larger At set, together with four differing Fr offsets, supplies **no supported universal displacement factor**. It does not establish that atoms are random or that all harmonic relations are absent. Atomic lines share energy levels, the original cluster was chosen after inspection, and these published subsets are selective. We therefore do not manufacture a binomial significance by calling them independent random draws.

The next useful test is a **specific transition-to-transition prediction**: fix a dimensionless frequency-ratio relation and its tolerance for named transitions before inspecting additional measurements; then test another unused electronic series with proper treatment of fine/hyperfine and isotope structure. Without such a rule, searching for another cluster would only restart the selection process.

## Reproduction

Run `python3 calculate.py`, then `python3 audit.py`. Only Python 3.11+ standard-library features are needed for the calculation and arithmetic audit. `python3 build_report.py` additionally needs matplotlib. Source URLs and hashes are recorded; full third-party papers are not redistributed. If `sources/Teigelhoefer2012.txt` is available from `pdftotext -layout` on the linked thesis, `audit.py` also repeats the 45-branch PDF text check; otherwise it explicitly reports that check as unavailable. The included AUDIT.json records the completed source check for this run.

`INPUTS.json` records every selected entry, source, uncertainty qualification, exclusion and auxiliary measurement. `AUDIT_DECISIONS.json` preserves the later doublet finding. Original study artifacts and the Theory of Reality paper were not changed by this focused follow-up.

## All calculated entries

The transition energies below identify the lower and upper states; the separate photon-wavenumber column is what enters the frequency calculation.

| Atom | Transition | Photon cm⁻¹ | Vacuum nm | Frequency / 2⁴⁰ (Hz) | Note | Offset (cents) | Status |
|---|---|---:|---:|---:|---|---:|---|
| At-199 | 46233.7 -> 57157.1 cm^-1 | 10923.400 | 915.465881 | 297.837044 | D4 | +24.423618 | outside narrow target |
| At-199 | 46233.7 -> 57277.0 cm^-1 | 11043.300 | 905.526428 | 301.106234 | D4 | +43.322863 | outside narrow target |
| At-199 | 46233.7 -> 58805.3 cm^-1 | 12571.600 | 795.443698 | 342.776808 | F4 | -32.280825 | outside narrow target |
| At-199 | 44549.2 -> 57157.1 cm^-1 | 12607.900 | 793.153499 | 343.766563 | F4 | -27.289158 | outside narrow target |
| At-199 | 44549.2 -> 57267.8 cm^-1 | 12718.600 | 786.250059 | 346.784905 | F4 | -12.154905 | outside narrow target |
| At-199 | 44549.2 -> 58805.3 cm^-1 | 14256.100 | 701.454114 | 388.706327 | G4 | -14.587521 | outside narrow target |
| At-199 | 44549.2 -> 69615.1 cm^-1 | 25065.900 | 398.948372 | 683.445958 | F5 | -37.611512 | outside narrow target |
| At-199 | 46233.7 -> 70055.4 cm^-1 | 23821.700 | 419.785322 | 649.521644 | E5 | -25.751108 | outside narrow target |
| At-199 | 46233.7 -> 71376.6 cm^-1 | 25142.900 | 397.726595 | 685.545437 | F5 | -32.301482 | outside narrow target |
| At-199 | 46233.7 -> 71708.7 cm^-1 | 25475.000 | 392.541708 | 694.600464 | F5 | -9.584179 | outside narrow target |
| At-199 | 57157.1 -> 68253.8 cm^-1 | 11096.700 | 901.168816 | 302.562236 | D#4 | -48.325911 | outside narrow target |
| At-199 | 57157.1 -> 68469.3 cm^-1 | 11312.200 | 884.001344 | 308.438052 | D#4 | -15.027316 | outside narrow target |
| At-199 | 57276.7 -> 68778.7 cm^-1 | 11502.000 | 869.414015 | 313.613132 | D#4 | +13.778936 | outside narrow target |
| At-199 | 57157.1 -> 69179.5 cm^-1 | 12022.400 | 831.780676 | 327.802340 | E4 | -9.612844 | outside narrow target |
| At-199 | 57276.7 -> 69505.7 cm^-1 | 12229.000 | 817.728351 | 333.435489 | E4 | +19.884968 | outside narrow target |
| At-199 | 57157.1 -> 69969.0 cm^-1 | 12811.900 | 780.524356 | 349.328820 | F4 | +0.498576 | outside narrow target |
| At-199 | 57267.8 -> 69969.0 cm^-1 | 12701.200 | 787.327182 | 346.310477 | F4 | -14.524985 | outside narrow target |
| At-199 | 57157.1 -> 70036.5 cm^-1 | 12879.400 | 776.433685 | 351.169272 | F4 | +9.595708 | outside narrow target |
| At-199 | 57267.8 -> 70036.5 cm^-1 | 12768.700 | 783.165083 | 348.150930 | F4 | -5.348775 | outside narrow target |
| At-199 | 58805.0 -> 70036.5 cm^-1 | 11231.500 | 890.353025 | 306.237688 | D#4 | -27.422015 | outside narrow target |
| At-199 | 57157.1 -> 70048.4 cm^-1 | 12891.300 | 775.716956 | 351.493737 | F4 | +11.194553 | outside narrow target |
| At-199 | 57276.7 -> 70048.4 cm^-1 | 12771.700 | 782.981122 | 348.232728 | F4 | -4.942071 | outside narrow target |
| At-199 | 58805.0 -> 70048.4 cm^-1 | 11243.400 | 889.410676 | 306.562153 | D#4 | -25.588709 | outside narrow target |
| At-199 | 57267.8 -> 70059.7 cm^-1 | 12791.900 | 781.744698 | 348.783500 | F4 | -2.206076 | outside narrow target |
| At-199 | 57276.7 -> 70076.4 cm^-1 | 12799.700 | 781.268311 | 348.996175 | F4 | -1.150759 | outside narrow target |
| At-199 | 58805.0 -> 70076.4 cm^-1 | 11271.400 | 887.201235 | 307.325600 | D#4 | -21.282690 | outside narrow target |
| At-199 | 57157.1 -> 70130.0 cm^-1 | 12972.900 | 770.837669 | 353.718640 | F4 | +22.118469 | nominal narrow hit; boundary overlap |
| At-199 | 57267.8 -> 70130.0 cm^-1 | 12862.200 | 777.471972 | 350.700298 | F4 | +7.282158 | outside narrow target |
| At-199 | 57276.7 -> 70303.7 cm^-1 | 13027.000 | 767.636447 | 355.193729 | F4 | +29.323104 | outside narrow target |
| At-199 | 58805.0 -> 70303.7 cm^-1 | 11498.700 | 869.663527 | 313.523154 | D#4 | +13.282163 | outside narrow target |
| At-199 | 57157.1 -> 70431.7 cm^-1 | 13274.600 | 753.318367 | 361.944782 | F#4 | -38.080688 | outside narrow target |
| At-199 | 57276.7 -> 70431.7 cm^-1 | 13155.000 | 760.167237 | 358.683772 | F4 | +46.250743 | outside narrow target |
| At-199 | 58805.0 -> 70431.7 cm^-1 | 11626.700 | 860.089277 | 317.013198 | D#4 | +32.447256 | outside narrow target |
| At-199 | 57157.1 -> 70467.9 cm^-1 | 13310.800 | 751.269646 | 362.931810 | F#4 | -33.366016 | outside narrow target |
| At-199 | 58805.0 -> 70467.9 cm^-1 | 11662.900 | 857.419681 | 318.000226 | D#4 | +37.829119 | outside narrow target |
| At-199 | 57157.1 -> 70745.0 cm^-1 | 13587.900 | 735.948896 | 370.487209 | F#4 | +2.304252 | outside narrow target |
| At-199 | 58805.0 -> 70745.0 cm^-1 | 11940.000 | 837.520938 | 325.555625 | E4 | -21.519351 | outside narrow target |
| At-199 | 57276.7 -> 70754.9 cm^-1 | 13478.200 | 741.938835 | 367.496132 | F#4 | -11.729348 | outside narrow target |
| At-199 | 58805.0 -> 71515.0 cm^-1 | 12710.000 | 786.782061 | 346.550418 | F4 | -13.325919 | outside narrow target |
| At-199 | 58805.0 -> 71548.7 cm^-1 | 12743.700 | 784.701460 | 347.469281 | F4 | -8.741703 | outside narrow target |
| At-199 | 58805.0 -> 71657.3 cm^-1 | 12852.300 | 778.070851 | 350.430365 | F4 | +5.949119 | outside narrow target |
| At-199 | 58805.0 -> 71668.2 cm^-1 | 12863.200 | 777.411531 | 350.727564 | F4 | +7.416752 | outside narrow target |
| At-199 | 58805.0 -> 71677.3 cm^-1 | 12872.300 | 776.861944 | 350.975684 | F4 | +8.641071 | outside narrow target |
| At-199 | 58805.0 -> 71721.0 cm^-1 | 12916.000 | 774.233509 | 352.167207 | F4 | +14.508460 | outside narrow target |
| At-199 | 58805.0 -> 71883.3 cm^-1 | 13078.300 | 764.625372 | 356.592473 | F4 | +36.127272 | outside narrow target |
| At-199 | 58805.0 -> 71925.5 cm^-1 | 13120.500 | 762.166076 | 357.743097 | F4 | +41.704485 | outside narrow target |
| At-199 | 58805.0 -> 72095.3 cm^-1 | 13290.300 | 752.428463 | 362.372858 | F#4 | -36.034350 | outside narrow target |
| At-199 | 58805.0 -> 72097.9 cm^-1 | 13292.900 | 752.281293 | 362.443749 | F#4 | -35.695699 | outside narrow target |
| At-199 | 58805.0 -> 72149.9 cm^-1 | 13344.900 | 749.349939 | 363.861579 | F#4 | -28.936561 | outside narrow target |
| At-199 | 58805.0 -> 72627.3 cm^-1 | 13822.300 | 723.468598 | 376.878351 | F#4 | +31.914481 | outside narrow target |
| At-199 | 58805.0 -> 72744.5 cm^-1 | 13939.500 | 717.385846 | 380.073922 | F#4 | +46.531821 | outside narrow target |
| At-199 | 58805.0 -> 72765.9 cm^-1 | 13960.900 | 716.286199 | 380.657414 | F#4 | +49.187584 | outside narrow target |
| At-199 | 58805.0 -> 72797.5 cm^-1 | 13992.500 | 714.668572 | 381.519018 | G4 | -46.898258 | outside narrow target |
| At-199 | 58805.0 -> 72906.7 cm^-1 | 14101.700 | 709.134360 | 384.496462 | G4 | -33.439842 | possible hyperfine component |
| At-199 | 58805.0 -> 72907.8 cm^-1 | 14102.800 | 709.079048 | 384.526454 | G4 | -33.304803 | possible hyperfine component |
| Fr-221 | 7s1/2 -> 9p1/2 | 27111.238 | 368.850733 | 739.214072 | F#5 | -1.813563 | outside narrow target |
| Fr-221 | 7s1/2 -> 9p3/2 | 27360.097 | 365.495780 | 745.999453 | F#5 | +14.005259 | outside narrow target |
| Fr-221 | 7s1/2 -> 10p1/2 | 29058.100 | 344.138123 | 792.297144 | G5 | +18.245693 | outside narrow target |
| Fr-221 | 7s1/2 -> 10p3/2 | 29192.800 | 342.550218 | 795.969869 | G5 | +26.252356 | outside narrow target |