A radius inferred from light
The Proton
That Stayed Small
In 2010 muonic hydrogen put the proton 4.0 per cent smaller than the accepted value, 5.0 standard deviations away. Rebuild that claim from its measured energy, bring in ordinary hydrogen, and plant the old radius into the control’s own frequencies to see whether it could have told the difference. Then move both frequencies together and watch a precision check miss the entire shift, and follow the claim’s own either-or to the Rydberg frequency, which has since fallen by about 110 kHz.
A proton has no hard surface to put a ruler against. Its charge has a spread. That spread nudges the energy levels of any atom built around it, and the nudge leaves a mark in the frequency of the light the atom absorbs. Measure the light well enough, and you can read the size back out.
The original claim / at full strength
we find rₚ = 0.84184(67) fm, which differs by 5.0 standard deviations from the CODATA value of 0.8768(69) fm.Pohl and 31 coauthors, The size of the proton, Nature (2010). Mathematical typography converted to plain text and a reference numeral omitted; words and values unchanged.
Recomputed now from the measured energy
Computing…
Printed radius: 0.84184 ± 0.00067 fm. This is an inversion of the published transition energy, not a new fit of the resonance counts.
I / The claim, at its printed strength
Rebuild the original number.
Replace hydrogen’s electron with a negative muon, a particle about two hundred times heavier. The muon orbits far closer to the proton, so the atom becomes far more sensitive to how the proton’s charge is spread. The collaboration measured one transition in that atom with a pulsed laser, then inverted a theory formula to get a root-mean-square charge radius. This page does the same inversion, from the measured energy, with the formula they printed.
Published transition energy
206.2949 ± 0.0032 meV
The paper also prints 49,881.88(76) GHz. The energy belongs to the hyperfine component they measured, not to the pure Lamb splitting, so the formula below is theirs for that component. Paper, equation (1) and the measured-energy paragraph (checking copy).
E(r) = 209.9779 − 5.2262 r² + 0.0347 r³
Energy in meV, radius in fm. The first number carries a published theory uncertainty of 0.0049 meV. A femtometre is a millionth of a billionth of a metre.
What the inversion gives
Solving that formula for the radius at the measured energy gives Computing…, Computing…. Against the accepted value of the day, 0.8768 ± 0.0069 fm, that is a proton Computing… smaller, and Computing… away.
The page’s centre differs from the printed one by Computing…, well inside the spread produced by rounding the printed inputs (the check, below). Nothing here reproduces the photon counts or the resonance fit; the claim is rebuilt from the energy the collaboration reported.
II / Ordinary hydrogen, another observation
The electron lands beside it.
In 2026 a group at the Max Planck Institute of Quantum Optics in Garching published two measured transitions of ordinary hydrogen, from the 2S state to the two 6P fine-structure levels. The page combines those two measured frequencies into a centroid, then infers the radius with the measured 1S-2S transition held fixed as a constraint.
Computing…
Computing…Computing…Computing…
Inferred centroid
Computing…
Observed component, lower fine-structure level
Computing…
Observed component, upper fine-structure level
Computing…
centroid = ν₁/₂ / 3 + 2ν₃/₂ / 3 + HFS
HFS moves the result to the hyperfine centroid: −132,985.25 kHz. The components already carry the experiment’s other corrections, which the page does not apply again.
Published centroid: 730,690,248,610.79 kHz. Published radius: 0.8406 ± 0.0015 fm, printed as 0.0005 fm from QED and 0.0014 fm from the measurement. The engine’s split: Computing…. Maisenbacher and colleagues, equations (4) to (10) and Table 1.
The radius uses the paper’s full-theory intercept and a local leading finite-size response. Counterfactual radii are approximate. The current CODATA value is not held fixed to determine the answer.
The source CSV independently checks the published comparison: Computing… against CODATA 2014, printed as 5.5 σ. That CSV is a radius inventory, not the measured frequencies used in the inference above.
III / Run the control on the control
Give it the original difference.
A control that agrees with a claim is only worth something if it could have disagreed. So hand this one a world where the old radius is true, and then plant the whole 2010 contrast, 0.8768 → 0.84184 fm, into its own measured frequencies. The real record already reads small, so it is first reset, in a labelled copy, to the old radius.
Measured record
Computing…
Add Computing… to each frequency →
Synthetic old-radius baseline
Computing…
Add Computing… to each frequency →
2010 claim planted
Computing…Computing…
Same centroid. Same inverse equation.
What the unmodified control reported
Computing… Computing…
Each distance is read off the control’s own output: how far the centroid it recovered sits from the centroid the old radius predicts, divided by the published measurement and QED uncertainties in quadrature, Computing…. The recovered centroid moved by Computing….
Grade A injection at the published-frequency level. It trusts the reported frequency uncertainties and corrections. It does not test photon detection, scan selection, technical noise or the completeness of the systematic budget.
The rule, which is this page’s choice: the planted copy must sit at least Computing… standard uncertainties from the old radius, and the old-radius copy must not. A control that ignored its input would fail the second half. Anything else prints INCONCLUSIVE. This is a sensitivity margin, not an estimated probability of detection.
The recovered change in squared radius is Computing…, the claimed contrast itself. Counting the old estimate’s own uncertainty as well gives Computing…, the original claim’s significance again, just under five before rounding.
How much worse could the measurement have been?
The same plant, the same unmodified control and the same rule, with only the measurement uncertainty enlarged; the QED part stays at 0.18 kHz. The rule stops confirming beyond Computing… the published value, a measurement uncertainty of Computing…. This is a what-if about precision, not a statement about the experiment.
The injection changes the observations before inference. Component separation, component uncertainties, centroid uncertainty and correction assumptions are preserved. No radius answer is appended to any output. In a response linear in r², getting the planted radius back is guaranteed once the forward and inverse steps agree; what carries information is the separation against the published uncertainties, and the baseline that must stay unflagged.
IV / A precision check with a blind direction
Move both. The difference stays.
The same experiment checks the separation of its two transitions against theory, a separation that does not depend on the proton’s size. It is a good check. But a shift shared by both frequencies cancels from a subtraction, even a shift big enough to carry the whole puzzle.
This comparison tests the (uncorrelated) Doppler shift extrapolation and corrections for the light force, QI and dc-Stark shifts, with the last two of opposite sign for the two transitions.Maisenbacher and colleagues (2026), after equation (6). The authors’ own word is in the parentheses; this panel shows what the word implies.
This panel always edits copies of the measured electronic record. A shared shift adds (s, s). A separation shift adds (−2d/3, d/3). These are hypothetical edits, not evidence of an overlooked experimental fault.
Local radius domain: 0.80 to 0.90 fm. The negative end of the shared-shift slider reaches the refusal path.
This number has no sensitivity to a common shift in this local response.
Computing…
Can the separation check alone exclude these edits?
Computing…
The button adds Computing… to both measured frequencies. The inferred radius goes back to Computing…, the old value, while the separation moves by Computing… and its discrepancy stays at Computing…. The whole puzzle fits in the check’s blind spot.
What rules such a shift out is not this check. It would have to hide inside the paper’s systematic budget, where the entire applied experiment-specific correction to the centroid is 1.74 ± 0.48 kHz (Table 1): the shift is Computing… that correction and Computing… the centroid’s total uncertainty. The other line of defence is an independent laboratory measuring different transitions (below). This page reports those scales; it does not audit that budget.
The measured separation and the rounding difference
Subtracting the rounded published components gives Computing…, a residual against theory of Computing…. The paper prints the separation as 405,164.62 ± 0.97 kHz, the theory as 405,164.51 ± 0.01 kHz and their difference as 0.11 ± 0.97 kHz. The page’s value differs from the printed centre by Computing…, kept as a rounding discrepancy rather than tuned away.
The separation uncertainty is supplied by the authors. Treating the total component errors as independent would lose their correlations, so the page does not do it. Equation (6) and the surrounding discussion.
A sensitivity map you can take away
The page perturbs each observed component in turn, runs the unchanged inference, and obtains this response matrix in kHz per kHz:
Computing…
It also sweeps Computing… shared shifts and Computing… separation shifts. The largest numerical leakage into the supposedly blind output is Computing… for separation, and Computing… for centroid. The shared-shift sweep contains Computing… radius-domain refusals, kept in the export as refusals.
This matrix is the algebra of the two observables, one third and two thirds of the components for the centroid and their difference for the separation. Perturbing the records confirms that the code implements that algebra; it is not a property discovered in the data.
This map is the page’s analysis of the published measurements, within the declared local response. It does not establish exact radius-independence of every higher-order physical contribution.
we searched the Artificial Wasteland built index, the Nature source-data records and web searches for proton-radius interactive fine-structure and power-injection tools on 2026-09-22 and did not find an interactive reconstruction that places the 2010 radius contrast in the blind direction of the 2026 fine-structure check while also showing the radius-sensitive centroid.
V / A control that cannot decide
One scan cannot carry the verdict.
The paper’s illustrated scan releases fitted, corrected frequencies for atoms at different speeds. Fit a line and extrapolate to zero speed. Plant the same historical contrast in every frequency and fit again. The shift comes back, but a single scan is far too uncertain to tell the two radii apart, and the page says so.
Recovered shift: Computing…
Fitted slope: Computing…
χ² / degrees of freedom: Computing…
Included groups: Computing…
Weighted least squares, with inverse-variance weights. The supplied errors set the covariance. The page does not shrink the uncertainty because the residual scatter happens to be small.
An insensitive subset earns INCONCLUSIVE, even when the full experiment is precise. This subset exercise does not revise the published experiment or the dated scientific verdict.
VI / The claim’s own either-or
Something had to give. The paper said what.
The 2010 paper did not stop at a radius. It said what else would have to be true.
Our result implies that either the Rydberg constant has to be shifted by −110 kHz/c (4.9 standard deviations), or the calculations of the QED effects in atomic hydrogen or muonic hydrogen atoms are insufficient.Pohl and 31 coauthors (2010), abstract. Typography converted to plain text.
The Rydberg constant sets the scale of every hydrogen line. Ordinary hydrogen’s 1S-2S transition is measured so precisely that it pins a combination of the Rydberg constant and the proton radius. Shrink the radius and the Rydberg constant must move down with it, by an amount fixed by the same finite-size arithmetic the control uses.
The claim’s own Rydberg constant, against CODATA 2006
Computing… printed −110 kHz/c
The paper did more than name a shift. Its last discussion paragraph combines the measured hydrogen 1S-2S interval with Lamb shifts calculated from the muonic radius, and prints a new Rydberg constant: R = 10,973,731.568160(16) per m. CODATA 2006 had 10,973,731.568527(73) per m. Times the speed of light, the difference is Computing…. With both uncertainties in quadrature, Computing… and Computing…, that is Computing…, printed as 4.9 standard deviations; and the new value is Computing… more precise, printed as 4.6 times. Pohl and 31 coauthors (2010), final discussion paragraph.
An independent route, from the radius contrast alone: across 0.8768 → 0.84184 fm, at Computing… of squared radius, this page’s leading-order relation gives Computing…, the outlined diamond in the plot.
What happened next, against the claim’s own value
Hydrogen 1S-2S with 2S-6P, 2026: Computing…
Hydrogen 1S-2S with 2S-nS, 2026: Computing…
CODATA 2022, which includes muonic data: Computing…
Each distance counts the 2010 value’s own uncertainty in quadrature. The two 2026 values use ordinary hydrogen only.
And against CODATA 2006
Hydrogen 1S-2S with 2S-6P, 2026: Computing…
Hydrogen 1S-2S with 2S-nS, 2026: Computing…
CODATA 2022, which includes muonic data: Computing…
Each of the two hydrogen values sits within Computing… of what the same leading-order relation predicts for its own radius.
| Determination | As printed | Radius | Change from CODATA 2006, kHz (± this value’s own uncertainty) | Relation predicts, kHz | Difference, kHz | Includes muonic data |
|---|
Each change also carries CODATA 2006’s own Computing…; the ± in that column is only the listed value’s own uncertainty.
The other branch: was QED insufficient?
The 2026 paper tested that too, by comparing its measured centroid with the Standard Model prediction built from the muonic radius 0.84060 ± 0.00039 fm. From the frozen frequencies, the page gets a difference of Computing…; the paper prints 0.00 ± 0.53 kHz.
That comparison uses the paper’s full 0.23 kHz prediction uncertainty, which includes the muonic radius. The radius inference above uses only the QED part, so the muonic value is never counted as a prior.
A printed number that picks the right coefficient
Equation (10) prints the muonic radius’s share of the prediction uncertainty as 0.14 kHz. With the page’s positive 29/216 response the engine gets Computing…. Holding the Rydberg constant fixed instead would give Computing…, which does not round to the printed value. Combined with the QED part, the engine’s prediction uncertainty is Computing….
On these numbers, the dichotomy resolved on its first branch. Two hydrogen experiments in two laboratories put the Rydberg frequency Computing… below CODATA 2006, each within Computing… of the value the 2010 paper printed, and the QED prediction meets the measurement within its uncertainty. The relation here is leading order, and the one-to-three kHz remainders of the three later values are not decomposed here: they may reflect revisions of theory and data between adjustments, or the limits of this relation.
The check / visible assumptions
What has actually been reproduced?
In 2010 a measurement with muons said the proton is smaller than the accepted value, and the most precise later measurements with ordinary hydrogen agreed. This page recomputes those results from the published measurements and tests whether the later check could have caught the old size. It cannot re-run the original detectors or prove that every correction in those experiments was complete.
The original claim
Expected as printed: 0.84184 ± 0.00067 fm. Computed from the energy: Computing…, Computing…. Central-value gap: Computing….
Varying the energy and each coefficient by half its final printed unit gives Computing…. Varying the printed input uncertainties gives Computing….
Tolerances: 0.000030 fm for the centre, 0.000015 fm for its uncertainty and 0.05 σ for the historical comparison. They accommodate the printed rounding, not arbitrary disagreement.
The electronic inference
The computed centroid is Computing…; the radius is Computing…. Measurement uncertainty 0.48 kHz and QED uncertainty 0.18 kHz are published inputs, combined here in quadrature.
The paper’s total prediction uncertainty, 0.23 kHz, also includes uncertainty in the muonic reference radius. That part is not an independent electronic prior and is not added again.
The centroid error is not reconstructed from scan statistics or from independent component errors. This page tests the inference conditional on the authors’ uncertainty and correction model.
Original component estimates, with their published total errors:
730,690,111,486.30 ± 0.69 kHz
730,690,516,650.91 ± 0.66 kHz.
Why the Rydberg constant must move with the radius
For the leading finite-size contribution, an S level shifts by K r²/n³. The measured ground-state transition constrains a combination of radius and Rydberg constant. Eliminating the Rydberg constant changes both the size and the sign of the 2S-6P response.
B = [−1/8 − (8/27)(−7/8)] K = (29/216) K
The engine computes K = Computing… from the CODATA 2022 constants and its equation (50), and B = Computing…. Holding the Rydberg constant fixed would give Computing…, the wrong response for this inference, and would quietly import the global adjustment, muonic data included, into a control meant to be independent of it.
ν(r) = ν(ref) + B [r² − r(ref)²]
The theory origin is a coordinate convention for the curve. Translate the reference radius and the intercept together, and the inferred radius stays put. These recalculations use different theory origins and the same measured frequencies:
| Reference radius, fm | Intercept shift, kHz | Inferred radius, fm |
|---|
This table is what shows that the frequencies decide the answer. The paper evaluates its prediction at the muonic radius, which is also where the electronic result lands, so at the default origin the measured centroid sits exactly on the prediction and a control that ignored its frequencies would print the same radius. Move the origin and only a control that reads its frequencies finds its way back.
This is a leading finite-size approximation anchored to the paper’s full QED prediction. Counterfactual frequency shifts are displayed to a tenth of a kilohertz, counterfactual radii to four decimal places. Extra digits in the anchor and the checks document reproducibility, not new physical precision.
The measurement inventory, including its awkward metadata
Computing… rows from the source CSV, selected by label. The electronic values there have been re-evaluated with updated theory, so they are not necessarily the values the original papers printed. CODATA rows overlap their constituent experiments and must not be pooled as independent observations.
| Source label | Source year | Radius, fm | Standard uncertainty, fm | Theory source |
|---|
The row labelled “2S-6P (2026)” has a year field of 2025 and a blank DOI. Those bytes are preserved. Separate metadata records the article’s actual online publication date, 11 February 2026, and its DOI. This is a metadata correction, not a changed observation.
The row labelled “1S-3S (2018)” carries the DOI 10.1103/PhysRevLett.120.183003, which resolves to an unrelated article. The paper that row means is 10.1103/PhysRevLett.120.183001. The DOI is not shown in the table above and the CSV bytes are preserved.
Unchanged source CSV · Attributed frequency transcription · Rydberg constants, attributed
Make the model refuse
Enter a common shift outside the supported local radius range. The same engine rejects it rather than inventing a radius. Missing or duplicate component identities, non-finite frequencies and nonpositive standard uncertainties are refused too.
No request run.
The instrument also refuses an exact-QED claim and a radius CSV passed off as a frequency record. Its refusal to exclude a shared shift on separation agreement alone runs live in the blind-direction panel above.
Free choices this page made
- The detection rule, Computing… standard uncertainties, with the old-radius copy required not to trip it.
- In the precision what-if, only the measurement part of the centroid uncertainty is enlarged; the QED part is left as published.
- The two-standard-deviation cut used to name the partly discrepant hydrogen rows, with their uncertainties treated as independent.
- The local radius domain, 0.80 to 0.90 fm, outside which the leading-order response refuses to answer.
- The anchor tolerances above, set from the rounding of the printed inputs.
- Which historical comparator to show; the planted contrast is always the 2010 one.
- The leading-order Rydberg relation, which is not a CODATA adjustment and does not reproduce one to better than a few kHz.
What a null test would require
This page does not implement the original photon-count likelihood or resonance search. White noise passed through a radius inversion would not test how often that procedure finds a false resonance.
The condensed electronic experimental record is available on Zenodo under CC BY; the article says its analysis code is available from the corresponding author. This page uses published transition estimates and the illustrated scan’s corrected fits. It does not claim to have rerun the complete detection and selection pipeline.
Loading the frozen record and checking its fingerprints…
VII / The dated scientific record
The small scale stayed.
VINDICATED
As of 2026-09-22
VINDICATED applies to the small-radius scale, independently supported by electronic hydrogen. It does not preserve every digit of the original theory, establish a new force, or settle every scattering analysis.
Ordinary hydrogen puts the proton at 0.8406 fm. Planted with the 2010 contrast, its control lands 24.6 of its own standard uncertainties from the old radius, a sensitivity margin that falls to 4.96 σ once the old value’s uncertainty is counted, while its fine-structure check stays blind to the same shift.
Independent observable, overlapping authors
The electronic experiment reports 0.8406 ± 0.0015 fm. Pohl and Hänsch also belonged to the original muonic collaboration. Independence here means a distinct electronic observation; it does not mean disjoint personnel.
Having confirmed the muonic value of rₚ, we now explicitly compare the SM predictionMaisenbacher and colleagues (2026), opening the Standard Model test. This is the claimants’ side of the record as it stands.
Bullis and colleagues, at Colorado State University, independently report 0.8433 ± 0.0031 fm from different transitions. Both experiments support the small scale.
Not every hydrogen result agreed, and the control paper says so in its abstract:
values of rₚ from recent measurements of atomic hydrogen are partly discrepant with each other and with a more precise value from spectroscopy of muonic hydrogenMaisenbacher and colleagues (2026), abstract. Reference numerals omitted.
In the source inventory, Computing… hydrogen rows sit more than two standard deviations from the muonic value, treating the uncertainties as independent: Computing…. The verdict rests on the two most precise hydrogen determinations on this page, which agree with the small radius.
What remains outside the verdict
The CODATA 2022 adjustment recommends 0.84075 ± 0.00064 fm. It includes muonic measurements, so it is context, not another independent replication. It was published on 16 September 2025, before both electronic experiments used here, and it said:
The proton radius “puzzle” is not yet solved.CODATA 2022, overview of the 2022 adjustment (section I.B).
A resolution, based on very recent experiments, is stated. The proton radius puzzle is no more.Gerald A. Miller, The Proton Radius Puzzle, arXiv:2604.15546v2, abstract. A specialist review preprint, not another measurement.
The muonic number itself moved. The source inventory’s Computing… muonic row reads Computing…, Computing… from the 2010 value, Computing… the 2010 paper’s own uncertainty. The two share data and theory, so this is a revision, not an independent disagreement. The scale held; the last digits did not.
Electron-scattering analyses still limit any wider claim of universal agreement.
The selected control arrived Computing… calendar years after the original claim. Earlier corroboration existed; this interval is not a claim that the field had no evidence in between.
A verified correction that moves these electronic frequency determinations back to the former large radius, or independent measurements with controlled uncertainties establishing a reproducible difference between muonic and electronic radii, would change this verdict.
Apparatus / provenance
The record stays open.
The electronic article and its source files are CC BY 4.0; the authors are credited above. The frequency JSON is an attributed transcription, the illustrated-scan extract selects a named worksheet, and every plot is an original graphic computed here. The 2010 paper, the CODATA publications and the Bullis paper are not redistributed; their factual numbers are transcribed with attribution. Sources and terms.
Machine-readable claim record and checksums · Original energy transcription · Corrected single-scan frequency extract · Read the verifier