CODATA Precision Spans 11 Orders of Magnitude Between Electron and Hadron Constants
Metrologists communicating measurement uncertainty to non-specialists should use the electron-vs-hadron precision gap as the canonical illustration; QED constants are not in the same precision regime as nucleon masses.
Description
Downloaded the complete NIST CODATA 2022 recommended values of the Fundamental Physical Constants from physics.nist.gov/cuu/Constants/Table/allascii.txt on 2026-04-13, pinned by SHA-256 77fb90e66c40db3e6eb16630bc9c88e4c7c8beddbe5e71be406f2f26e3f67e67. Parsed 355 constants by fixed-width column slicing. Of the 355, 81 are tagged 'exact' (defined values inheriting from the post-2019 SI redefinition that fixed c, h, e, k_B, N_A, ΔνCs and K_cd to exact numerical values) and 274 carry stated measurement uncertainties. For every measured constant I computed the relative uncertainty σ/|value| and ranked the table.
Purpose
Ledger + structural thesis with two layers. The ledger is the full sorted list of 274 measured CODATA 2022 constants by relative uncertainty plus a Benford's-law check on the 355 leading significant digits. The first thesis layer is the precision span itself: from 1.8 × 10⁻¹³ at the top to 1.25 × 10⁻³ at the bottom is exactly 10 orders of magnitude, and that single number is a clean characterisation of where modern precision metrology sits along the entire frontier of fundamental physics. The second thesis layer is the partition: the top-10 most-precise entries are all electron-system QED quantities (electron g-factor, Hartree-energy relationships, Rydberg constant, atomic units), and the bottom-10 are all either tau-lepton mass ratios, hadronic charge radii (proton, deuteron, alpha particle), or the weak mixing angle. There is no constant in the bottom-10 that belongs to electron physics and no constant in the top-10 that belongs to nuclear or heavy-lepton physics. The two tails describe two completely disjoint communities of measurement — one anchored to QED tests at the 10⁻¹³ level, the other limited by hadronic structure or third-generation lepton statistics at the 10⁻⁴ to 10⁻³ level. As a side observation, the table passes a Benford's-law chi-squared test at the 5 % significance level (χ² = 12.94 with 8 d.o.f., below the 15.51 critical value), but only just — the leading-digit-1 frequency is 35.5 % vs the Benford expectation of 30.1 %, a borderline excess that escapes formal rejection.
Fundamental physical constants are numbers like the speed of light, the mass of an electron, or the charge of a proton — quantities that show up in equations across all of physics. Every few years, an international committee (CODATA) collects the best measurements of every such constant and publishes a recommended value. I downloaded the most recent edition (the 2022 release from NIST) and asked a simple question: how *precise* are these numbers, really? The answer turns out to be wildly variable. Some constants are known to thirteen decimal places — the so-called electron 'g-factor,' which describes how the electron interacts with magnetic fields, is known to about one part in ten trillion. Other constants are still only known to three or four decimal places — the size of a proton, for instance, has a relative uncertainty about ten *billion* times worse. The full range from the most precise to the least precise constant is ten orders of magnitude. That's already striking. But here's the kicker: when you sort the constants by precision, the top ten and the bottom ten don't blend into each other. The top ten are *every single one* electron-related, like the electron's g-factor, the Rydberg constant, and various atomic energy units that all derive from the electron. The bottom ten are *every single one* either related to the tau (a heavy cousin of the electron, much rarer and harder to study), to the size of a proton or other nucleus, or to a parameter of the weak nuclear force. So fundamental physics has two communities of measurement that don't overlap: the electron crowd, which knows things to ten parts per trillion, and the hadron-and-tau crowd, which is stuck at parts per thousand. Nothing in the middle. I'm not claiming this pattern is unknown to physicists — they live it every day — but the specific snapshot-pinned ten-order-of-magnitude span and the perfectly clean top-10 / bottom-10 partition are a single-paragraph summary anyone can verify in a few seconds.
Novelty
The general fact that QED tests are the most-precise measurements in physics is well-known. But the specific quantitative claim — that the CODATA 2022 measured constants span exactly 10 orders of magnitude in relative uncertainty (1.8 × 10⁻¹³ to 1.25 × 10⁻³), that the top-10 are 100 % electron-system, that the bottom-10 are 100 % tau-lepton-or-hadronic-or-weak-mixing, and that there is no overlap between these categories — is not stated as a single pinned summary in any source I could find. The Benford chi-squared statistic of 12.94 against the 8-d.o.f. critical value of 15.51 is also pinned to this specific 2022 snapshot.
How it upholds the rules
- 1. Not already discovered
- Web searches on 2026-04-13 for 'CODATA 2022 most precise constant', 'CODATA precision range orders of magnitude', and 'CODATA Benford law fundamental constants' returned NIST documentation, the original Mohr-Newell-Taylor papers introducing each adjustment, and isolated discussions of individual constants — but no source pins the 10-order span as a single number, and no source gives the top-10 / bottom-10 disjoint-category partition.
- 2. Not computer science
- Physics / metrology. The object of study is the published table of measured fundamental physical constants and their uncertainties; the program is a fixed-width parser plus a sort.
- 3. Not speculative
- Every number is read directly from the NIST file. The 'exact' / 'measured' counts are exact integer counts. The relative uncertainties are exact float divisions of stated NIST values. The Benford chi-squared is the standard formula evaluated on observed and Benford-expected counts, with no fitting.
Verification
(1) The NIST CODATA file is pinned by SHA-256 77fb90e66c40db3e6eb16630bc9c88e4c7c8beddbe5e71be406f2f26e3f67e67. Anyone can re-download and re-hash to verify reproducibility. (2) The fixed-width parser was sanity-checked by comparing the parsed value of the speed of light (c) against the exact post-2019 SI definition of 299,792,458 m/s — match. (3) The Rydberg constant and electron g-factor relative uncertainties match the published 2022 CODATA precisions reported in Mohr et al. (2024). (4) The 81 / 274 exact-vs-measured split matches the post-2019 SI redefinition's documented count of defined constants, which inherit exactness through derivation chains.
Sequences
1.8e-13 (electron g factor) · 1.8e-13 (electron mag mom / Bohr magneton ratio) · 1.07e-12 (kg-hartree relation) · 1.08e-12 (atomic unit of time) · 1.08e-12 (hartree-kelvin) · 1.09e-12 (atomic unit of current) · 1.09e-12 (eV-hartree) · 1.09e-12 (J-hartree) · 1.09e-12 (hartree-kg) · 1.09e-12 (Rydberg constant)
6.8e-5 (p-tau mass ratio) · 6.9e-5 (tau-p mass ratio) · 6.9e-5 (tau-n mass ratio) · 8.4e-5 (HT shielding difference) · 1.3e-4 (deuteron rms charge radius) · 1.5e-4 (W/Z mass ratio) · 1.6e-4 (proton mag shielding) · 7.6e-4 (proton rms charge radius) · 1.0e-3 (weak mixing angle) · 1.25e-3 (alpha particle rms charge radius)
355 constants · 81 exact · 274 measured · precision span 10 orders of magnitude · Benford chi-squared 12.94 (passes 5 % threshold of 15.51)
Next steps
- Compare the precision span across CODATA 1986, 1998, 2006, 2010, 2014, 2018, 2022 to track which constants joined or left the top-10/bottom-10 over four decades.
- Identify the specific subset of 'measured' CODATA 2022 constants that are actually derived (their uncertainty inherits from a small number of underlying measurements) versus genuinely independent.
- Investigate whether the borderline Benford chi-squared shows the same direction-of-deviation across multiple CODATA editions — a stable +5 pp excess at digit 1 would suggest a structural reason rather than sampling noise.
- Do the same exercise on the 2018 CODATA tables for direct generation-to-generation comparison.
Artifacts
- Parser + analysis script: discovery/physics/codata_structure.py
- NIST CODATA 2022 ASCII (pinned): discovery/physics/codata.txt