# Aspden's Numbers at a Glance: Digits Right and Standard Deviations, Set Beside QED and the Standard Model

**Every number of Aspden's tested so far, with how many digits he got right, how many digits the measurement pins down, how far off he is in standard deviations, and what mainstream theory gives for the same number**

By Doug Westmoreland (project lead: direction and review) and Claude, Anthropic (research and analysis) · Version 1.0, 31 October 2026. Draft 0.4 added "Where the digits come from", which traces each prediction, Aspden's and the Standard Model's, through the stages that built it; draft 0.5 revises it after an independent check (one labelling rule for both sides, the Standard Model's own choices counted, several of Aspden's rows relabelled). Draft 0.2 added the QED and Standard Model column and compared both electron-anomaly rows with the direct measurement. Draft 0.3 scores QED and the Standard Model exactly as Aspden is scored, adds earlier mainstream values that missed, applies the corrections from an independent check of the mainstream values (Opus, extra effort), and moves the particle measurements to PDG 2026 · Covers the working paper (v2.4), Part 1 (v1.5) and Part 2 (v1.0) · Every number is printed by `scorecard.py`, which also writes the table as `scorecard.csv` for sorting

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## Why two measures

Quoting only the number of standard deviations hides how close a number is. Quoting only the digits hides whether the remaining difference is real. Both are needed.

- **Digits right** says how close Aspden came. His 1/α, 137.035915, has the first seven digits of the measured 137.035999177 right.
- **Standard deviations (σ) off** says whether what is left over is larger than the measurement error. The measurement of 1/α now pins down about ten digits, with an uncertainty (1σ) of 0.000000021. Aspden's eighth digit is wrong, and the remaining gap of 0.000084 is 4,000 times that uncertainty, so the formula as it stands cannot be exactly right.
- **Both together.** A number can have many digits right and still be ruled out (1/α), or few digits right and be consistent (his neutral-pion lifetime: one digit, 1.8σ, because the measurement itself pins down fewer than two digits).

## The yardstick: QED and the Standard Model, scored the same way

The QED / Standard Model column, and a separate table further on, put mainstream physics through the same test as Aspden. Mainstream physics here means quantum electrodynamics (QED), the rest of the Standard Model, and lattice QCD, which computes strong-force quantities numerically.

- **Scored exactly as Aspden is.** Digits right and σ off are worked out the same way for both. σ off divides by the measurement's uncertainty alone, as it does for Aspden. Measured this way, any approximate value fails once the measurements become precise, whoever made it.
- **And with its own error bar.** Mainstream predictions come with a stated uncertainty; Aspden's formulas do not. A second σ figure adds the theory's own uncertainty, which is how physicists normally judge a prediction. "Digits it claims" is log₁₀(value ÷ the theory's own 1σ): how many digits the theory says can be trusted. Lattice QCD claims fewer than 2 digits for most hadron quantities, so it fails the first test badly and passes the second.
- **The Standard Model does not try to predict 20 of the 48 numbers.** It has about two dozen free parameters that are measured and put in: α, the masses of the electron, muon and tau, the quark masses and others. Lattice calculations also fix their quark masses to the charged-pion, kaon and J/ψ masses. Those rows are marked *input*. Aspden tried to derive many of them. Gravity, and so G, is outside the Standard Model altogether.
- **Earlier mainstream values are scored too.** Aspden's table includes his earlier versions. The Standard Model's table includes earlier or alternative mainstream values: the hydrogen Lamb shift with the proton radius accepted before 2018, the 2020 muon-anomaly prediction, the electron anomaly with α from cesium, the older He⁺ theory and the leading-order neutral-pion lifetime.

## How to read the table

- **Measured today** and **1σ today**: the best current value and its standard uncertainty, the "sigma". A correct theory should land within about 2σ.
- **Aspden**: his value, as printed or as his formula gives it (notes below say which).
- **Digits right**: the number of leading significant digits that are the same in his value and the measured value, also counting digits that round to the same figures (so 3683 against 3686.097 counts 3, and 0.9999 against 1.0001 would count 4). Never more than the digits he gave.
- **Digits measured, then → now**: log₁₀(value ÷ 1σ), how many digits the measurement pins down. "Then" is the measurement he compared with, or the measurement of the year he published; "—" where none is recorded. If his digits right reach the digits measured then, he matched everything that was known when he wrote.
- **Off**: his value minus the measured value, as a fraction of the measured value (ppm = parts per million, ppb = parts per billion).
- **σ off**: his value minus the measured value, divided by today's 1σ. A minus sign means he is low.
- **Our correction**: where this analysis has a corrected value, with its own digits right, off and σ off.
- **QED / Standard Model**: the mainstream value with its own uncertainty in the last digits (80.3558(61) means 80.3558 ± 0.0061), its digits right, its σ off scored as for Aspden and, in brackets, its σ off with its own uncertainty added. The label says where it comes from: *QED*; *SM* (the electroweak Standard Model fit, or chiral perturbation theory built on the Standard Model); *lattice* (lattice QCD). Where there is no prediction, the label says why: *input* (a free parameter, or a value the calculations are tuned to); *not in SM* (gravity); *fitted to it* (nuclear-force models are fitted to this number); *not scored* (no usable published value).

## The table

**Constants (working paper)**

| ID | Quantity | Year | Measured today | 1σ today | Aspden | Digits right | Digits measured, then → now | Off | σ off | Our correction | QED / Standard Model |
|---|---|---|---|---|---|---|---|---|---|---|---|
| W1 | 1/α, inverse fine-structure constant | 1972 | 137.035999177 | 0.000000021 | 137.035915 | 7 | 6.1 → 9.8 | −0.614 ppm | −4,010 |  | input |
| W2 | 1/α, his 1969 version | 1969 | 137.035999177 | 0.000000021 | 137.038 | 5 | 5.8 → 9.8 | +14.6 ppm | 95,300 |  | input |
| W3 | Rydberg energy hcR∞, eV (from his α) | 1972 | 13.605693122990 | 0.000000000015 | 13.6057099 | 6 | — → 12.0 | +1.23 ppm | 1.1 × 10⁶ |  | input |
| W4 | proton/electron mass ratio | 1975 | 1836.152673426 | 0.000000032 | 1836.152316 | 7 | 6.4 → 11.0 | −0.195 ppm | −11,200 | 1836.152673243 (10 digits, −0.1 ppb, −5.7σ) | 1832(65): 3 digits, −1.3 × 10⁸σ (−0.1σ with its own error) lattice |
| W5 | G, 10⁻¹¹ m³ kg⁻¹ s⁻² | 1966–2003 | 6.67430 | 0.00015 | 6.67216 | 3 | 3.2 → 4.6 | −321 ppm | −14 | 6.67408 (4 digits, −33 ppm, −1.5σ) | not in SM |
| W6 | deuteron magnetic moment, μN | 1986 | 0.8574382335 | 0.0000000022 | 0.8574389111 | 6 | — → 8.6 | +0.79 ppm | 308 | 0.8574385122 (6 digits, +0.325 ppm, 127σ) | fitted to it |
| W7 | (mₙ − mₚ)/mₑ | 1986 | 2.530988574 | 0.00000074 | 2.5309419 | 5 | 4.9 → 6.5 | −18.4 ppm | −63 |  | 2.95(55): 1 digit, 566,000σ (0.8σ with its own error) lattice |
| W8 | hydrogen Lamb shift, MHz | 1983 | 1057.8298 | 0.0032 | 1057.8479 | 5 | 5.1 → 5.5 | +17.1 ppm | 5.7 |  | 1057.8322(4): 6 digits, 0.8σ (0.7σ with its own error) QED |
| W9 | He⁺ Lamb shift, MHz (his formula, scaled by us) | 1983 | 14041.13 | 0.17 | 16692.4 | 1 | — → 4.9 | +18.9% | 15,600 |  | 14041.517(43): 5 digits, 2.3σ (2.2σ with its own error) QED |
| W10 | muon lifetime from the 1.875 cluster, μs | 2003 | 2.1969811 | 0.0000022 | 2.19908 | 3 | — → 6.0 | +955 ppm | 954 | 3.69445 (0 digits, +68.2%, 681,000σ) | input |
| W11 | electron anomaly (g − 2)/2, cavity model | 1983 | 0.00115965218059 | 0.00000000000013 | 0.00115964650448 | 5 | 7.5 → 10.0 | −4.89 ppm | −43,700 |  | 0.00115965218023(9): 10 digits, −2.8σ (−2.3σ with its own error) QED |

**Units (Part 1)**

| ID | Quantity | Year | Measured today | 1σ today | Aspden | Digits right | Digits measured, then → now | Off | σ off | Our correction | QED / Standard Model |
|---|---|---|---|---|---|---|---|---|---|---|---|
| U08 | cell energy E₀ against two muon masses, mₑ | 1980 | 413.5365654 | 0.0000092 | 412.6658 | 2 | 5.9 → 7.7 | −0.211% | −94,600 |  | input |

**Particles (Part 2)**

| ID | Quantity | Year | Measured today | 1σ today | Aspden | Digits right | Digits measured, then → now | Off | σ off | Our correction | QED / Standard Model |
|---|---|---|---|---|---|---|---|---|---|---|---|
| P01 | muon/electron mass ratio | 1983 | 206.7682827 | 0.0000046 | 206.7683078 | 7 | 6.3 → 7.7 | +0.121 ppm | 5.5 |  | input |
| P02 | muon/electron mass ratio, first model | 1983 | 206.7682827 | 0.0000046 | 206.7687 | 6 | 5.9 → 7.7 | +2.02 ppm | 91 |  | input |
| P03 | muon/electron mass ratio, 1980 | 1980 | 206.7682827 | 0.0000046 | 206.7688 | 6 | 5.9 → 7.7 | +2.5 ppm | 112 |  | input |
| P04 | muon anomaly (g − 2)/2 | 1984 | 0.001165920715 | 0.000000000145 | 0.001165918 | 6 | 5.1 → 6.9 | −2.33 ppm | −19 |  | 0.00116592033(62): 7 digits, −2.7σ (−0.6σ with its own error) SM |
| P05 | muon anomaly, first model | 1983 | 0.001165920715 | 0.000000000145 | 0.00116589 | 5 | 4.6 → 6.9 | −26.3 ppm | −212 |  | 0.00116592033(62): 7 digits, −2.7σ (−0.6σ with its own error) SM |
| P06 | electron anomaly, with his gravitational term | 1981 | 0.00115965218059 | 0.00000000000013 | 0.001159652200 | 8 | 7.5 → 10.0 | +0.0167 ppm | 149 |  | 0.00115965218023(9): 10 digits, −2.8σ (−2.3σ with its own error) QED |
| P07 | tau mass, 3^(7/12) × proton, MeV | 1986–89 | 1776.93 | 0.09 | 1780.94 | 3 | 2.8 → 4.3 | +0.226% | 45 |  | input |
| P08 | tau mass, his 1996 revision, MeV | 1996 | 1776.93 | 0.09 | 1776.51 | 4 | 3.6 → 4.3 | −236 ppm | −4.7 |  | input |
| P09 | tau mass, his 1998 revision, MeV | 1998 | 1776.93 | 0.09 | 1777.02 | 4 | 4.9 → 4.3 | +50.6 ppm | 1.0 |  | input |
| P10 | charged pion mass, MeV | 1989 | 139.57039 | 0.00018 | 139.5678 | 5 | 5.1 → 5.9 | −18.6 ppm | −14 |  | input |
| P11 | charged pion mass, 1980, MeV | 1980 | 139.57039 | 0.00018 | 139.5672 | 5 | 4.8 → 5.9 | −22.9 ppm | −18 |  | input |
| P12 | neutral pion mass (43e quark), MeV | 1980 | 134.9768 | 0.0005 | 134.959 | 4 | 4.3 → 5.4 | −132 ppm | −36 |  | 135.036(60): 2 digits, 118σ (1.0σ with its own error) lattice |
| P13 | neutral pion mass, his preferred lower end, MeV | 1989 | 134.9768 | 0.0005 | 134.960 | 4 | 4.6 → 5.4 | −124 ppm | −34 |  | 135.036(60): 2 digits, 118σ (1.0σ with its own error) lattice |
| P13b | neutral pion mass, upper end of his range, MeV | 1989 | 134.9768 | 0.0005 | 134.976 | 6 | 4.6 → 5.4 | −5.93 ppm | −1.6 |  | 135.036(60): 2 digits, 118σ (1.0σ with its own error) lattice |
| P14 | neutral pion lifetime, 10⁻¹⁷ s | 1989 | 8.43 | 0.13 | 8.2 | 1 | 1.3 → 1.8 | −2.73% | −1.8 |  | 8.04(11): 1 digit, −3.0σ (−2.3σ with its own error) SM |
| P15 | charged pion lifetime, 10⁻⁸ s | 1982 | 2.6033 | 0.0005 | 2.61 | 2 | 2.0 → 3.7 | +0.257% | 13 |  | 2.618(33): 2 digits, 29σ (0.4σ with its own error) lattice |
| P16 | charged kaon mass, MeV | 1989 | 493.677 | 0.015 | 493.633 | 4 | 4.5 → 4.5 | −89.1 ppm | −2.9 |  | input |
| P17 | neutral kaon mass, MeV | 1989 | 497.611 | 0.013 | 497.557 | 4 | — → 4.6 | −109 ppm | −4.2 |  | input |
| P18 | J/ψ mass, MeV | 1978 | 3096.900 | 0.006 | 3095 | 3 | — → 5.7 | −614 ppm | −317 |  | input |
| P19 | ψ(2S) mass, MeV | 1978 | 3686.097 | 0.011 | 3683 | 3 | — → 5.5 | −840 ppm | −282 |  | not scored |
| P20 | Λ mass, 2^(1/4) × proton, MeV | 1986 | 1115.683 | 0.006 | 1115.80 | 4 | — → 5.3 | +105 ppm | 20 |  | 1114(16): 3 digits, −281σ (−0.1σ with its own error) lattice |
| P21 | Ω⁻ mass, MeV | 1986 | 1672.45 | 0.29 | 1674 | 3 | — → 3.8 | +927 ppm | 5.3 |  | 1676(25): 3 digits, 12σ (0.1σ with its own error) lattice |
| P22 | neutron mean life, s | 1986 | 878.3 | 0.4 | 898 | 1 | — → 3.3 | +2.24% | 49 |  | 894(12): 1 digit, 39σ (1.3σ with its own error) SM |
| P23 | neutron mean life, 1981, s | 1981 | 878.3 | 0.4 | 921.8 | 1 | 1.8 → 3.3 | +4.95% | 109 |  | 894(12): 1 digit, 39σ (1.3σ with its own error) SM |
| P24 | neutron magnetic moment (size), μN | 1986 | 1.91304276 | 0.00000045 | 1.913043478 | 7 | 6.5 → 6.6 | +0.375 ppm | 1.6 |  | 1.893(70): 2 digits, −44,500σ (−0.3σ with its own error) lattice |
| P25 | proton magnetic moment, μN | 1988 | 2.79284734463 | 0.00000000082 | 2.792847367 | 8 | 7.6 → 9.5 | +8 ppb | 27 |  | 2.739(66): 2 digits, −6.6 × 10⁷σ (−0.8σ with its own error) lattice |
| P26 | deuteron binding energy, MeV | 1980 | 2.224566362 | 0.00000037 | 2.22464 | 5 | 4.7 → 6.8 | +33.1 ppm | 199 |  | fitted to it |
| P27 | W boson mass, GeV | 1983 | 80.3625 | 0.0077 | 82.03 | 1 | 1.5 → 4.0 | +2.07% | 217 |  | 80.3558(61): 4 digits, −0.9σ (−0.7σ with its own error) SM |
| P28 | Z boson mass, 1984, GeV | 1984 | 91.1879 | 0.0020 | 96.86 | 1 | — → 4.7 | +6.22% | 2,840 |  | 91.1938(69): 4 digits, 3.0σ (0.8σ with its own error) SM |
| P29 | Z boson mass, 1986, GeV | 1986 | 91.1879 | 0.0020 | 93.79 | 1 | 1.6 → 4.7 | +2.85% | 1,300 |  | 91.1938(69): 4 digits, 3.0σ (0.8σ with its own error) SM |
| P30 | Z boson mass, 1989, GeV | 1989 | 91.1879 | 0.0020 | 92.59 | 1 | 1.7 → 4.7 | +1.54% | 701 |  | 91.1938(69): 4 digits, 3.0σ (0.8σ with its own error) SM |
| P31 | Z boson mass, his 1997 revision, GeV | 1997 | 91.1879 | 0.0020 | 91.02 | 2 | — → 4.7 | −0.184% | −84 |  | 91.1938(69): 4 digits, 3.0σ (0.8σ with its own error) SM |
| P32 | Weinberg angle sin²θ (on-shell) | 1986 | 0.22334 | 0.00015 | 0.22222 | 2 | 1.2 → 3.2 | −0.501% | −7.5 |  | 0.22347(12): 3 digits, 0.9σ (0.7σ with its own error) SM |
| P33 | his 'Higgs' level, GeV | 2000 | 125.13 | 0.11 | 114.9 | 1 | — → 3.1 | −8.18% | −93 |  | 112(19): 1 digit, −119σ (−0.7σ with its own error) SM |
| P34 | top quark, proton-based, GeV | 2001 | 172.60 | 0.27 | 174.97 | 2 | 1.5 → 2.8 | +1.37% | 8.8 |  | 173.6(15): 2 digits, 3.7σ (0.7σ with its own error) SM |
| P35 | top quark, graviton-based, GeV | 2001 | 172.60 | 0.27 | 174.78 | 2 | 1.5 → 2.8 | +1.26% | 8.1 |  | 173.6(15): 2 digits, 3.7σ (0.7σ with its own error) SM |

**Notes**

- **W1.** Printed 137.035915; his formula gives 137.0359148. No correction justified by his own rules has been found: closing the gap needs a +0.41 ppm change in the photon-unit relation (Part 1, U22).
- **W3.** The Rydberg energy follows from his α (hcR∞ = α²mₑc²/2), so it is the same result in another form. Planck's constant is likewise equivalent to α since the 2019 SI redefinition.
- **W4.** Printed 1836.15232. "Our correction" multiplies by (1 + α³/2), a numerical lead found by searching 60 candidate terms. Nothing in his theory motivates it, and it is not adopted (working paper section 6.5).
- **W5.** "Our correction" replaces his 1.875 cluster by exactly two electron masses (working paper section 7.5). It fails the out-of-sample test: applied to his muon lifetime (W10), the same replacement gives 3.69445 μs, 68% off.
- **W6.** "Our correction" gives the protons their own Thomson radius (working paper section 8).
- **W7, P26.** Today's values are worked out in electron masses and in atomic mass units, so the uncertainty of the conversion to MeV, which all the masses share, cancels instead of being counted once per mass.
- **W9.** Aspden did not publish a He⁺ value. This row applies his hydrogen formula scaled as Z⁴; the two forms of the scaling give 16,692 and 16,933 MHz, and the nearer is used. The measured value is the 2000 measurement by van Wijngaarden, Holuj and Drake, 14,041.13 ± 0.17 MHz. The working paper (v2.1) compared with their 1987 value, 14,041.9 ± 1.5 MHz; the result is +18.9% either way, and the working paper will be brought up to date in its next version.
- **W11, P06.** W11 is his 1983 cavity-model value with the α of his day. Both rows are now compared with the direct measurement of Fan et al. (2023), 0.00115965218059(13). Draft 0.1 used the CODATA 2022 value, 0.00115965218046(18). That value comes out of a least-squares fit that ties the electron anomaly to α through QED theory, so it is not a pure measurement, and it would favour QED in this comparison. Against the direct measurement P06 is 149σ off (109σ before), and W11 is −43,700σ (−31,500σ before).
- **U08.** He wrote that the cell energy makes it "tempting to suggest" a pair of virtual muons; this row compares it with two real muon masses.
- **P07.** He printed 1781 MeV (1986), 1781.062 MeV (1989d) and 3485.21 electron masses = 1780.94 MeV (1989b, 1996). P08 and P09 are his own revisions, made after the 1992 measurements.
- **P09.** He quotes the measurement as 1777.05 MeV "with an uncertainty of 0.02 MeV", which is probably a misquote; that is why "then" exceeds "now".
- **P13, P13b.** In 1989 he gave a range, 134.960 to 134.976 MeV, and called the lower end "the more representative".
- **P20.** He printed 1116 MeV; 1115.80 is his formula, 2^(1/4) × the proton mass.
- **P22, P23.** The PDG 2026 average is 878.3 ± 0.4 s, dominated by trapped-neutron measurements; neutron-beam measurements give 887.7 ± 2.2 s. Against the beam value his 898 s is 4.7σ off.
- **P24.** His −44/23; the table compares magnitudes.
- **P32.** His sin²θ = 2/9, compared with the on-shell definition, 1 − (W mass ÷ Z mass)², from the measured W and Z masses.
- **P27–P35.** The W mass is the PDG 2026 value without the CDF 2022 measurement, 80.3625(77) GeV; with that change P27 moves from 125σ (draft 0.2) to 217σ, and the measured sin²θ becomes 0.22334(15), which moves P32 from 3.8σ to 7.5σ.
- **P34, P35.** He called these "speculation" and "a quite curious coincidence".

**Notes on the QED / Standard Model column**

- **W1, W2.** QED needs α as an input. Turned the other way, QED and the measured electron anomaly give 1/α = 137.035999166(15) (Fan et al. 2023). That is the same test as rows W11 and P06, not a prediction of α.
- **W4.** Lattice QCD (Dürr et al. 2008) gives the nucleon mass, 936(25)(22) MeV, as the average of proton and neutron (938.92 MeV measured). Dividing by the electron mass, which is an input, gives 1832(65). It is scored against the proton/electron ratio.
- **W7.** Lattice QCD with QED (Borsanyi et al. 2015) gives mₙ − mₚ = 1.51(16)(23) MeV, here divided by the electron mass.
- **W8.** Horbatsch and Hessels (2016) tabulate the QED prediction for this Lamb shift against the proton radius. It is 1057.8322(3) MHz at 0.84087 fm, and so 1057.8322 MHz at the CODATA 2022 radius, 0.84075(64) fm; the radius adds ±0.0002 MHz. Yerokhin, Pachucki and Patkós (2019) give 1057.8341 MHz, because they include a 1.88 kHz hyperfine term that this measurement's convention leaves out. Our own check, from Bezginov's radius and the proton-size slope of 0.196 MHz per fm², gives 1057.8323 MHz (`scorecard.py`).
- **W9.** Yerokhin, Pachucki and Patkós (2019): 14041.517(9)(42) MHz. The Aspden value in this row is our extrapolation of his hydrogen formula, so the side-by-side comparison shows the row but does not count it.
- **W10.** The muon lifetime is how the Fermi constant G_F is measured, so the Standard Model cannot predict it independently.
- **W11, P06.** QED with α from rubidium (Morel et al. 2020): 0.00115965218023(9), as compiled by Jegerlehner (2026), including the 2024 revision of the five-loop term.
- **P04, P05.** Muon g-2 Theory Initiative (2025): 116 592 033(62) × 10⁻¹¹. The leading hadronic part comes from lattice QCD; the light-by-light part comes from a mix of methods.
- **P12, P13, P13b.** Lattice QCD with QED computes the difference between the charged and neutral pion masses, 4.534(42)(43) MeV (Feng, Jin and Riberdy 2022). With the charged-pion mass as input, the neutral-pion mass is 135.036(60) MeV.
- **P14.** Kampf and Moussallam (2009): a total lifetime of 8.04(11) × 10⁻¹⁷ s, from two-loop chiral perturbation theory built on the QED–QCD anomaly. Draft 0.2 quoted 8.14, which used the two-photon decay rate alone and left out the 1.2% Dalitz decay.
- **P15.** Lattice QCD gives the pion decay constant, f_π = 130.2(8) MeV (FLAG 2024), without using this lifetime. The PDG extracts 130.56 MeV from the measured lifetime, and the lifetime goes as 1/f_π², so the lattice value predicts 2.618(33) × 10⁻⁸ s.
- **P18, P19.** Recent lattice work tunes the charm quark mass to the J/ψ mass (HPQCD 2020). For the ψ(2S) no lattice value at the physical quark masses has been published; Fermilab/MILC (2019) could not determine the 2S states.
- **P20, P21.** Lattice QCD (Dürr et al. 2008): Λ 1114(15)(5) MeV, and Ω⁻ 1676(20)(15) MeV in the version of the calculation where the Ξ mass sets the scale.
- **P22, P23.** The relation τₙ = 4904.4 s ÷ (V_ud²(1 + 3g_A²)) is used. Its constant matches the radiative correction (2.454%) behind PDG 2024's V_ud = 0.97367(32). With the lattice g_A = 1.263(10) (FLAG 2024) it gives 894(12) s. Draft 0.2 used the 2006 constant, 4908.7 s, which goes with an older radiative correction. Using the measured g_A/g_V = 1.2754(13) instead gives about 880 s, but that takes the answer partly from neutron decay itself.
- **P24, P25.** Lattice QCD (Djukanovic et al. 2024): μₙ = −1.893(39)(58), μₚ = 2.739(63)(18).
- **P26, W6.** Nuclear-force models are fitted to the deuteron's binding energy and use its magnetic moment to fix a two-nucleon current. Lattice QCD cannot yet compute either at the physical quark masses.
- **P27–P35.** These come from the Gfitter global electroweak fit (Fischer et al. 2026). Each value is taken from a fit that leaves out the direct measurement of that quantity: W 80.3558(61), Z 91.1938(69), top 173.6(1.5), and Higgs 112 (+19/−17) GeV, where the upper error is used because the fit lies below the measurement. The on-shell sin²θ (P32) follows from the fit's W mass, so it is not a separate test of the Standard Model.

## What the table shows

- **Most digits right.** The proton magnetic moment (P25) and the electron anomaly (P06) have 8 digits right; 1/α (W1), the proton/electron mass ratio (W4), the muon/electron mass ratio (P01) and the neutron moment (P24) have 7. Of these, 1/α and the mass ratio come from his lattice with no step fitted to the target. P25, P06 and P01 each have a final step that chose among alternatives or added a correction, and their digits right match the digits measured when he wrote (7.6, 7.5 and 6.3 then). The neutron moment is the simple fraction −44/23; across the 36 particle numbers tested in Part 2, the chance of a fraction that simple landing that close by luck is about 1 in 30 (Part 2, section 6).
- **Ahead of the measurement of his day.** For 12 of the 35 numbers with a recorded measurement of the day, his value is closer to today's than that measurement was. The clearest cases are the proton/electron mass ratio of 1975 (his −0.19 ppm against the measurement's −0.63 ppm), the muon/electron mass ratio of 1983 (+0.12 against +0.33 ppm), the proton moment of 1988 (+8 against +15 ppb) and the tau mass of 1986 (+0.23% against +0.34%). The other eight need qualifying. For G (W5) and the Z mass in 1986 and 1989 (P29, P30) the margin is small. For the pion lifetimes and the weak mixing angle (P14, P15, P32) the measurement of his day was poor. The 1998 tau mass (P09) rests on a figure he probably misquoted, and the neutral-pion mass (P13b) is the upper end of a range whose lower end he preferred (P13 is not closer). For 21 of the 35 the measurement of his day was closer, including 1/α (the 1973 measurement was +0.30 ppm off today's value, his −0.61 ppm). For the remaining two (P06, P26) his value and the measurement of his day are the same number.
- **Digits and σ disagree most where measurements are best.** For 1/α, the proton/electron mass ratio and his cavity-model electron anomaly (W1, W4, W11), 5 to 7 digits right still leave 4,000 to 43,700 standard deviations, because those measurements now pin down 10 or 11 digits. His other electron anomaly (P06) has 8 digits right and is 149σ off. For the particle masses and ratios, 1 to 7 digits right against measurements of 4 to 8 digits gives anything from 1σ to a few thousand σ.
- **Consistent today.** Four numbers are within 2σ: the neutral-pion lifetime (P14, 1 digit right, 1.8σ), the upper end of his neutral-pion mass range (P13b, 6 digits, 1.6σ), the neutron moment (P24, 7 digits, 1.6σ) and his 1998 tau mass (P09, 4 digits, 1.0σ, after two revisions).
- **Our corrections.** The doubled electron improves G from 3 to 4 digits and from 14σ to 1.5σ, but fails out of sample. The α³/2 lead takes the proton/electron mass ratio from 7 to 10 digits, but was found by search and is not adopted. The deuteron refinement cuts the gap by more than half (0.79 to 0.33 ppm) but keeps the same 6 digits and is still 127σ off.

## QED and the Standard Model, scored the same way

Every mainstream value in the main table, with earlier or alternative mainstream values added as rows whose IDs end in "a". The columns are the same as for Aspden, with σ off given both ways.

| ID | Quantity | Kind | Year | Measured today | QED / Standard Model | Digits right | Digits measured now | Digits it claims | Off | σ off (as for Aspden) | σ off with its own error |
|---|---|---|---|---|---|---|---|---|---|---|---|
| W8 | hydrogen Lamb shift, MHz | QED | 2016 | 1057.8298(32) | 1057.8322(4) | 6 | 5.5 | 6.42 | +2.27 ppm | 0.8 | 0.7 |
| W8a | hydrogen Lamb shift, QED with the 2014 proton radius | QED | 2014 | 1057.8298(32) | 1057.8437(21) | 5 | 5.5 | 5.7 | +13.1 ppm | 4.3 | 3.6 |
| W9 | He⁺ Lamb shift, MHz | QED | 2019 | 14041.13(17) | 14041.517(43) | 5 | 4.9 | 5.51 | +27.6 ppm | 2.3 | 2.2 |
| W9a | He⁺ Lamb shift, earlier QED value | QED | 2004 | 14041.13(17) | 14041.474(42) | 5 | 4.9 | 5.52 | +24.5 ppm | 2.0 | 2.0 |
| W11 | electron anomaly (g − 2)/2 | QED | 2024 | 0.00115965218059(13) | 0.00115965218023(9) | 10 | 10.0 | 10.1 | −0.31 ppb | −2.8 | −2.3 |
| P06a | electron anomaly, QED with α from cesium | QED | 2018 | 0.00115965218059(13) | 0.00115965218158(23) | 9 | 10.0 | 9.7 | +0.85 ppb | 7.6 | 3.8 |
| P04 | muon anomaly (g − 2)/2 | SM | 2025 | 0.001165920715(145) | 0.00116592033(62) | 7 | 6.9 | 6.27 | −0.33 ppm | −2.7 | −0.6 |
| P04a | muon anomaly, Standard Model of 2020 | SM | 2020 | 0.001165920715(145) | 0.00116591810(43) | 6 | 6.9 | 6.43 | −2.24 ppm | −18 | −5.8 |
| P27 | W boson mass, GeV | SM | 2026 | 80.3625(77) | 80.3558(61) | 4 | 4.0 | 4.12 | −83.4 ppm | −0.9 | −0.7 |
| P32 | sin²θ on-shell (from the W mass) | SM | 2026 | 0.22334(15) | 0.22347(12) | 3 | 3.2 | 3.27 | +582 ppm | 0.9 | 0.7 |
| P28 | Z boson mass, GeV | SM | 2026 | 91.1879(20) | 91.1938(69) | 4 | 4.7 | 4.12 | +64.7 ppm | 3.0 | 0.8 |
| P33 | Higgs boson mass, GeV | SM | 2026 | 125.13(11) | 112(19) | 1 | 3.1 | 0.77 | −10.5% | −119 | −0.7 |
| P34 | top quark mass, GeV | SM | 2026 | 172.60(27) | 173.6(15) | 2 | 2.8 | 2.06 | +0.579% | 3.7 | 0.7 |
| P12 | neutral pion mass, MeV | lattice | 2022 | 134.9768(5) | 135.036(60) | 2 | 5.4 | 3.35 | +439 ppm | 118 | 1.0 |
| P14 | neutral-pion lifetime, 10⁻¹⁷ s | SM | 2009 | 8.43(13) | 8.04(11) | 1 | 1.8 | 1.86 | −4.63% | −3.0 | −2.3 |
| P14a | neutral-pion lifetime, leading anomaly term alone | SM | 1970s | 8.43(13) | 8.38(2) | 2 | 1.8 | 2.62 | −0.593% | −0.4 | −0.4 |
| P15 | charged-pion lifetime, 10⁻⁸ s | lattice | 2024 | 2.6033(5) | 2.618(33) | 2 | 3.7 | 1.9 | +0.565% | 29 | 0.4 |
| P20 | Λ mass, MeV | lattice | 2008 | 1115.683(6) | 1114(16) | 3 | 5.3 | 1.84 | −0.151% | −281 | −0.1 |
| P21 | Ω⁻ mass, MeV | lattice | 2008 | 1672.45(29) | 1676(25) | 3 | 3.8 | 1.83 | +0.212% | 12 | 0.1 |
| P22 | neutron mean life, s | SM | 2024 | 878.3(4) | 894(12) | 1 | 3.3 | 1.87 | +1.79% | 39 | 1.3 |
| P24 | neutron magnetic moment (size), μN | lattice | 2024 | 1.91304276(45) | 1.893(70) | 2 | 6.6 | 1.43 | −1.05% | −44,500 | −0.3 |
| P25 | proton magnetic moment, μN | lattice | 2024 | 2.79284734463(82) | 2.739(66) | 2 | 9.5 | 1.62 | −1.93% | −6.6 × 10⁷ | −0.8 |
| W4 | proton/electron mass ratio | lattice | 2008 | 1836.152673426(32) | 1832(65) | 3 | 11.0 | 1.45 | −0.226% | −1.3 × 10⁸ | −0.1 |
| W7 | (mₙ − mₚ)/mₑ | lattice | 2015 | 2.530988574(740) | 2.95(55) | 1 | 6.5 | 0.729 | +16.6% | 566,000 | 0.8 |

- **Scored as Aspden is, the Standard Model passes 2 of the 17 quantities it predicts.** The two within 2σ are the hydrogen Lamb shift (0.8σ) and the W mass (0.9σ). The rest miss:
  - the electron anomaly by 2.8σ;
  - the muon anomaly by 2.7σ;
  - the Z mass by 3.0σ, the top mass by 3.7σ and the Higgs mass by 119σ (the fit gives it only to ±19 GeV);
  - the neutral-pion lifetime by 3.0σ;
  - the neutron lifetime by 39σ;
  - lattice QCD's hadron values by 12σ to 10⁸σ.
- **Its earlier values missed too.** With the proton radius accepted before 2018 (W8a), QED misses the hydrogen Lamb shift by 4.3σ; that was the "proton radius puzzle". It was settled mainly in favour of the smaller radius, not by changing QED. The 2020 muon-anomaly prediction (P04a) is 18σ from today's measurement scored as for Aspden (5.8σ with its own error). It was 4.2σ from the measurement of its day and was replaced in 2025, when lattice QCD changed the hadronic part. Aspden's 1984 value, 0.001165918, sits next to it (116 591 800 against 116 591 810 × 10⁻¹¹). With α from cesium (P06a), QED misses the electron anomaly by 7.6σ.
- **With its own error bars, it passes nearly everything.** 15 of the 17 are within 2σ and all 17 within 3σ. The two outside 2σ are the electron anomaly (−2.3σ) and the neutral-pion lifetime (−2.3σ).
- **What separates the two tests is the precision the theory claims.** Where the Standard Model claims 4 or more digits (QED and the electroweak fit), it gets them. For the hadron quantities, where lattice QCD and chiral perturbation theory claim 1 to 3 digits, it gets 1 to 3.

## Side by side

These are the 19 quantities for which mainstream theory gives a value. For each, the table shows Aspden's best version: most digits right, then smallest σ. For the neutral-pion mass his best is P13b, the upper end of a range whose lower end he preferred (P13: 4 digits, −34σ). Two rows are shown but not counted. The He⁺ row compares our extrapolation of his formula, not a value he published. The mixing angle repeats the Standard Model's W prediction. That leaves 17 quantities counted.

| Quantity | His row | Measured today | Aspden | His digits right | His σ off | QED / Standard Model | Its digits right | Its σ off (as for Aspden) | Its σ off with its own error | Digits it claims | Kind |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Hydrogen Lamb shift | W8 | 1057.8298(32) | 1057.8479 | 5 | 5.7 | 1057.8322(4) | 6 | 0.8 | 0.7 | 6.42 | QED |
| He⁺ Lamb shift (his formula, our extrapolation) | W9 | 14041.13(17) | 16692.4 | 1 | 15,600 | 14041.517(43) | 5 | 2.3 | 2.2 | 5.51 | QED |
| Electron anomaly | P06 | 0.00115965218059(13) | 0.001159652200 | 8 | 149 | 0.00115965218023(9) | 10 | −2.8 | −2.3 | 10.1 | QED |
| Muon anomaly | P04 | 0.001165920715(145) | 0.001165918 | 6 | −19 | 0.00116592033(62) | 7 | −2.7 | −0.6 | 6.27 | SM |
| W boson mass | P27 | 80.3625(77) | 82.03 | 1 | 217 | 80.3558(61) | 4 | −0.9 | −0.7 | 4.12 | SM |
| sin²θ on-shell (Standard Model value follows from its W mass) | P32 | 0.22334(15) | 0.22222 | 2 | −7.5 | 0.22347(12) | 3 | 0.9 | 0.7 | 3.27 | SM |
| Z boson mass | P31 | 91.1879(20) | 91.02 | 2 | −84 | 91.1938(69) | 4 | 3.0 | 0.8 | 4.12 | SM |
| Higgs boson mass | P33 | 125.13(11) | 114.9 | 1 | −93 | 112(19) | 1 | −119 | −0.7 | 0.77 | SM |
| Top quark mass | P35 | 172.60(27) | 174.78 | 2 | 8.1 | 173.6(15) | 2 | 3.7 | 0.7 | 2.06 | SM |
| Neutral pion mass | P13b | 134.9768(5) | 134.976 | 6 | −1.6 | 135.036(60) | 2 | 118 | 1.0 | 3.35 | lattice |
| Neutral-pion lifetime | P14 | 8.43(13) | 8.2 | 1 | −1.8 | 8.04(11) | 1 | −3.0 | −2.3 | 1.86 | SM |
| Charged-pion lifetime | P15 | 2.6033(5) | 2.61 | 2 | 13 | 2.618(33) | 2 | 29 | 0.4 | 1.9 | lattice |
| Λ mass | P20 | 1115.683(6) | 1115.80 | 4 | 20 | 1114(16) | 3 | −281 | −0.1 | 1.84 | lattice |
| Ω⁻ mass | P21 | 1672.45(29) | 1674 | 3 | 5.3 | 1676(25) | 3 | 12 | 0.1 | 1.83 | lattice |
| Neutron mean life | P22 | 878.3(4) | 898 | 1 | 49 | 894(12) | 1 | 39 | 1.3 | 1.87 | SM |
| Neutron magnetic moment (size) | P24 | 1.91304276(45) | 1.913043478 | 7 | 1.6 | 1.893(70) | 2 | −44,500 | −0.3 | 1.43 | lattice |
| Proton magnetic moment | P25 | 2.79284734463(82) | 2.792847367 | 8 | 27 | 2.739(66) | 2 | −6.6 × 10⁷ | −0.8 | 1.62 | lattice |
| Proton/electron mass ratio | W4 | 1836.152673426(32) | 1836.152316 | 7 | −11,200 | 1832(65) | 3 | −1.3 × 10⁸ | −0.1 | 1.45 | lattice |
| (mₙ − mₚ)/mₑ | W7 | 2.530988574(740) | 2.5309419 | 5 | −63 | 2.95(55) | 1 | 566,000 | 0.8 | 0.729 | lattice |

- **Digits right.** The Standard Model has more digits right on 5 quantities and Aspden on 6; they tie on 6.
- **Nearer the measured value.** Aspden is nearer on 10 quantities, the Standard Model on 7.
- **Within 2σ, scored the same way.** Aspden is within 2σ on 3: the neutral-pion mass (upper end of his range, 1.6σ), the neutral-pion lifetime (1.8σ) and the neutron moment (1.6σ). The Standard Model is within 2σ on 2: the hydrogen Lamb shift and the W mass. With its own error bars it is within 2σ on 15.
- **The split follows the physics.** The Standard Model has more digits and is nearer for every quantity where QED or the electroweak theory does the work: the hydrogen Lamb shift, the two magnetic anomalies, and the W and Z masses. Aspden has more digits and is nearer where the strong force does the work and lattice QCD has to compute numerically: the proton/electron ratio, the neutron–proton difference, the neutral-pion mass, the Λ mass and the two nucleon moments. For the neutron moment and the neutral-pion mass he is also within 2σ. The other six quantities are ties on digits: the top and Higgs masses, the two pion lifetimes, the Ω⁻ and the neutron lifetime. Aspden is nearer on the Higgs, both pion lifetimes and the Ω⁻; the Standard Model is nearer on the top mass and the neutron lifetime.
- **What this means.** In hadron physics, mainstream precision is limited by computing power more than by theory; lattice QCD's error bars shrink every few years. Aspden's formulas give hadron numbers to 4 to 8 digits from a few lines of algebra, which lattice QCD cannot yet do. Against that:
  - only three of his values are within 2σ;
  - Part 2 shows that many of them were set to the measurement of their day;
  - his formulas come with no stated precision.

  The fair reading is that on the strong-force side his formulas hold numerical regularities worth testing further, for the number of free choices behind each and the odds of a chance match. On the QED and electroweak side the Standard Model is clearly better.

## Where the digits come from

This section asks both sides the same question: how much of each prediction's agreement comes from the physics, and how much from numbers put in or choices made with the answer in view? Draft 0.5 applies an independent check of draft 0.4 (Opus, extra effort). The check found that the two sides had not been labelled by the same rule, that the Standard Model's own choices had not been counted, and that several of Aspden's rows were mislabelled. All are corrected here.

**Method.** Each prediction is followed through the stages that built it. At every stage the agreement with today's value is measured in digits, D = −log₁₀(the size of value ÷ measured − 1); 6.0 means one part per million. The gain at each stage shows how many digits that stage supplied. One rule labels the stages on both sides:

- **Fixed structure**: computed from the theory's own parameters and rules. For the Standard Model these are its two dozen or so parameters, such as α and the particle masses, all of them measured. For Aspden they are his lattice integers and model rules, none of them measured.
- **Measured input**: a measured value of a quantity outside the theory's own parameters. Examples are the proton radius, collider data and the pion decay constant; for Aspden, the proton mass used as his unit and the measured electron anomaly.
- **Number encoding the target**: a number put in that already contains the answer, such as the integer 207 in his muon formula.
- **Choice**: a selection or revision made with the measurement in view. For the Standard Model this includes choosing between conflicting measurements, counted as the digits gained over the alternative.

Two cautions apply. The gains depend on the order of the stages, because the stage that closes the gap gets the credit. And the labels are judgements, each stated with its evidence in the ledgers at the end of this section and in `provenance.py`.

"Digits beyond chance" is −log₁₀ of the chance that a family of formulas this flexible would match this closely if the physics were wrong. It is given per formula. The sources are:
- the working paper's section 11 (α, the proton ratio, the deuteron, G);
- its section 9.3 (the cavity electron);
- Part 2's count of simple fractions (44/23);
- the density of Aspden's own particle-mass toolkit, averaged over 100 MeV to 12 GeV and taken within ±10% of each target;
- his threshold series for the W and Z.

Allowing for all the particle numbers tested in Part 2 would lower these figures; for 44/23 it lowers 3.0 to about 1.5.

### QED and the Standard Model

| Quantity | Digits of agreement | From fixed structure | From measured inputs | From choices between conflicting inputs | Note |
|---|---|---|---|---|---|
| electron anomaly | 9.5 | 8.3 | 0.77 | 0.44 | the choice of rubidium is not the most favourable: the weighted mean of the two α values would give D = 9.8; the 2024 revision of the five-loop coefficient moved the prediction away from the measurement (D 9.6 before, 9.5 after) |
| muon anomaly | 6.5 | 4.2 | 1.4 | 0.83 | the switch removed a discrepancy of 4.2σ (2021), 5.2σ (2023) and 5.8σ against the 2025 final measurement; the lattice result (2021) preceded Fermilab's final result but followed BNL's (2006); about 23 of the 223 × 10⁻¹¹ change came from the light-by-light term |
| hydrogen Lamb shift | 5.6 | 3.9 | 0.99 | 0.76 |  |
| W mass | 4.1 | 3.9 | 0.16 | — |  |
| neutral-pion lifetime | 1.3 | -0.89 | 2.2 | — | the higher-order result moved away from the measurement |
| neutron mean life | 1.7 | 1.7 | — | — |  |
| proton/electron mass ratio (lattice QCD) | 2.6 | 2.6 | — | — | computed after 4 constants are fixed once for all hadrons |
| (mₙ − mₚ)/mₑ (lattice QCD) | 0.78 | 0.78 | — | — | computed after 4 constants are fixed once for all hadrons |
| neutral pion mass (lattice QCD) | 3.4 | 1.9 | 1.5 | — | computed after 4 constants are fixed once for all hadrons |
| Λ mass (lattice QCD) | 2.8 | 2.8 | — | — | computed after 4 constants are fixed once for all hadrons |
| Ω⁻ mass (lattice QCD) | 2.7 | 2.7 | — | — | computed after 4 constants are fixed once for all hadrons |
| neutron moment (size) (lattice QCD) | 2.0 | 2.0 | — | — | computed after 4 constants are fixed once for all hadrons |
| proton moment (lattice QCD) | 1.7 | 1.7 | — | — | computed after 4 constants are fixed once for all hadrons |

- **Most of the digits come from computing the theory.** Across the six predictions traced in stages, 74% of the digits come from structure, 19% from measured inputs outside the theory's parameters and 7% from choices between conflicting measurements. With the seven lattice values added, the shares are 80%, 16% and 4.5%. None comes from a number that encodes the answer.
- **The structure rests on measured parameters.** QED's 8.3 structural digits for the electron rest on α, measured to 0.1 ppb. Aspden's 1/α and proton ratio use no measured number at all, which counts in his favour.
- **Its choices are real.** Each was made between two conflicting measurements or methods:
  - α from rubidium rather than cesium is worth 0.44 digits for the electron;
  - the smaller proton radius rather than the 2014 value is worth 0.76 for the Lamb shift;
  - the switch to lattice QCD for the muon's leading hadronic part is worth 0.83. It removed a discrepancy of 4.2σ (2021), 5.2σ (2023) and 5.8σ against the 2025 final measurement. The lattice result (2021) came before Fermilab's final measurement but after the BNL result of 2006.

  Each choice rests on independent evidence, and the rubidium choice is not even the most favourable one: the weighted mean of the two α values would agree better.
- **Its revisions go both ways.** The 2024 revision of the five-loop term moved the electron prediction slightly away from the measurement, and the two-loop pion lifetime is further from it than the leading term. Every revision of Aspden's in these ledgers moved toward the measurement of its day.
- **No constant is tuned to the quantity predicted.** Several predictions do need measured inputs of their own: the proton radius, collider data or the pion decay constant.

### Aspden

| ID | Quantity | Digits of agreement | From numbers encoding the target | From measured inputs | From fixed structure | From choices made with the measurement in view | Digits beyond chance (per formula) |
|---|---|---|---|---|---|---|---|
| W1 | 1/α | 6.2 | — | — | 6.2 | — | 2.9 to 3.9 |
| W4 | proton/electron mass ratio | 6.7 | — | — | 6.7 | — | 2.3 to 2.6 |
| W5 | G | 3.5 | — | — | 3.5 | — | 0.1 to 1.2 |
| W6 | deuteron magnetic moment | 6.1 | — | — | 2.6 | 3.5 | 0.16 to 2.1 |
| P01 | muon/electron mass ratio | 6.9 | 3.0 | — | 2.8 | 1.2 | not estimated |
| P04 | muon anomaly | 5.6 | — | 2.3 | 2.1 | 1.3 | not estimated |
| P06 | electron anomaly | 7.8 | — | — | 5.3 | 2.4 | 1.5 |
| P24 | neutron magnetic moment (size) | 6.4 | — | — | 1.3 | 5.1 | 3.0 |
| P25 | proton magnetic moment | 8.1 | — | — | 6.3 | 1.8 | not estimated |
| P10 | charged pion mass (in mₑ) | 4.7 | — | — | 4.2 | 0.49 | not estimated |
| P13 | neutral pion mass (in mₑ) | 3.9 | 3.8 | — | — | 0.11 | not estimated |
| P20 | Λ mass | 4.0 | — | — | — | 4.0 | 1.9 to 2.0 (local and global toolkit density) |
| P07 | tau mass | 4.3 | — | — | — | 4.3 | 0.49 to 0.65 (local and global toolkit density) |
| P21 | Ω⁻ mass | 3.0 | — | — | — | 3.0 | 0.88 to 1.0 (local and global toolkit density) |
| P18 | J/ψ mass | 3.2 | — | — | — | 3.2 | 0.84 to 1.2 (local and global toolkit density) |
| P27 | W mass | 1.7 | — | — | — | 1.7 | 0.59 |
| P31 | Z mass | 2.7 | — | — | — | 2.7 | 0.11 |
| P14 | neutral-pion lifetime | 1.6 | — | — | 1.6 | — | not estimated |
| P22 | neutron mean life | 1.6 | — | — | — | 1.6 | not estimated |

- **The overall split.** Across his 19 formulas:
  - 48% of the digits come from fixed structure;
  - 41% from choices made with the measurement in view;
  - 8% from numbers that already encode the answer: 207 for the muon, and the integer search behind the neutral pion's 2641;
  - 3% from measured inputs.
- **His strongest results are pure structure.** 1/α (6.2 digits) and the proton/electron ratio (6.7) use no measured number, no number that encodes the answer and no late choice. Allowing for the variants he could have tried, 2.3 to 3.9 digits remain beyond chance. Two cautions apply to α:
  - his mechanism neglects effects of 10 to 30% (working paper 11.4), so the agreement cannot reflect the calculation's precision;
  - his assumptions were put together in 1972, when α was known to about 1 ppm.
- **His most precise numbers get their last digits from late choices:**
  - **Proton moment:** the last 1.8 of 8.1 digits come from three revisions. The last adds his proton–electron charge difference, which was fitted to another residual and is excluded experimentally.
  - **Electron anomaly:** the cavity formula gives 5.3 digits, 1.5 of them beyond chance among 21 variants. The gravitational term adds 2.4 more and lands exactly on the 1981 measurement.
  - **Muon ratio:** 3.0 digits come from the integer 207, 2.8 from his structure and 1.2 from the resonance choice.
  - **Muon anomaly:** 2.3 digits come from taking the measured electron anomaly as input, 2.1 from his 1982 derivation and 1.3 from two revisions.
- **The deuteron's digits are mostly chosen.** 3.5 of its 6.1 digits come from the fraction 6/7, which is set by state weights that the chance test treats as free.
- **The neutron's 44/23 keeps 3.0 digits beyond chance as a single formula** (about 1.5 allowing for the 36 particle numbers tested), and it is still consistent today. But the 46 behind it comes from a four-state model with several free choices, which Aspden himself called tortuous, and the same model gives a neutron–proton mass difference 63σ off.
- **Most particle masses come from factors matched to particles already known.** For the Λ, tau, Ω⁻, J/ψ, W and Z, 0.1 to 2.0 digits of each are beyond chance; the Λ, at 1.9 to 2.0, is the highest. On 10 states he did not address, the same toolkit matched 2 within 0.3%, where chance predicts 2.8.

### Both together

- **The mainstream's digits are mostly computed, from measured parameters.** Inputs from other measurements supply between a sixth and a fifth, and choices between conflicting data 5 to 7%, each backed by independent evidence. Its weak points are those choices, and the fact that its parameters are themselves unexplained.
- **About half of Aspden's digits come from structure, and that structure uses no measured number.** But each structure was one of many he could have written down. Where his numbers are most precise, the last one to three digits come from steps chosen with the measurement in view, and about 40% of all his digits come from such choices.
- **What survives is short and specific.** 1/α and the proton/electron ratio are pure structure, with 2.3 to 3.9 digits beyond chance. The neutron's 44/23 has about 3 digits beyond chance as a single formula, but rests on a model that fails elsewhere. These are the results worth developing, and they are the targets for the correction programme and the pre-registered predictions recommended earlier.

### Stage-by-stage ledgers

**QED and the Standard Model**

| Quantity | Stage | Kind | Value | Digits of agreement | Digits added |
|---|---|---|---|---|---|
| electron anomaly | Schwinger term α/2π, with α measured by rubidium atom recoil | struct | 0.00116140973185 | 2.8 | +2.8 |
| electron anomaly | plus electron loops to fifth order (computed coefficients) | struct | 0.00115965217573 | 8.4 | +5.6 |
| electron anomaly | plus muon and tau loops (lepton masses are Standard Model parameters) | struct | 0.00115965217847 | 8.7 | +0.36 |
| electron anomaly | plus hadronic part (collider data) and electroweak part | input | 0.00115965218023 | 9.5 | +0.77 |
| electron anomaly | choice: α from rubidium rather than cesium (alternative gives D = 9.1) | choice | — | — | 0.44 moved from struct |
| muon anomaly | QED (α and lepton masses) | struct | 0.001165847188 | 4.2 | +4.2 |
| muon anomaly | plus electroweak part | struct | 0.001165848732 | 4.2 | +0.0092 |
| muon anomaly | plus hadronic part from collider data (2020) | input | 0.0011659181 | 5.6 | +1.4 |
| muon anomaly | leading hadronic part from lattice QCD instead (2025) | choice | 0.00116592033 | 6.5 | +0.83 |
| hydrogen Lamb shift | QED with a point proton | struct | 1057.6937967 | 3.9 | +3.9 |
| hydrogen Lamb shift | plus the proton-size term, radius 0.84075 fm (muonic hydrogen dominates it; the muonic-only radius gives the same value) | input | 1057.8322 | 5.6 | +1.8 |
| hydrogen Lamb shift | choice: the smaller (muonic) proton radius rather than the 2014 value (alternative gives D = 4.9) | choice | — | — | 0.76 moved from input |
| W mass | lowest order from α, the muon lifetime (G_F) and the Z mass | struct | 80.939228556 | 2.1 | +2.1 |
| W mass | with α run to the Z mass (uses collider data) | input | 79.9666165266 | 2.3 | +0.16 |
| W mass | plus loop corrections with the top and Higgs masses (Standard Model parameters; Gfitter 2026) | struct | 80.3558 | 4.1 | +1.8 |
| neutral-pion lifetime | the anomaly formula, with the measured pion decay constant | input | 8.38 | 2.2 | +2.2 |
| neutral-pion lifetime | plus two-loop corrections with estimated constants | struct | 8.04 | 1.3 | -0.89 |
| neutron mean life | Standard Model relation, with V_ud (a Standard Model parameter) and the lattice g_A | struct | 894.17 | 1.7 | +1.7 |
| proton/electron mass ratio | lattice QCD (inputs fixed once for all hadrons) | struct | 1832.0 | 2.6 | 2.6 |
| (mₙ − mₚ)/mₑ | lattice QCD (inputs fixed once for all hadrons) | struct | 2.95 | 0.78 | 0.78 |
| neutral pion mass | the measured π± mass (input to the lattice calculation) | input | 273.13244 | 1.5 | +1.5 |
| neutral pion mass | minus the lattice QCD + QED mass difference | struct | 264.25886 | 3.4 | +1.9 |
| Λ mass | lattice QCD (inputs fixed once for all hadrons) | struct | 1114.0 | 2.8 | 2.8 |
| Ω⁻ mass | lattice QCD (inputs fixed once for all hadrons) | struct | 1676.0 | 2.7 | 2.7 |
| neutron moment (size) | lattice QCD (inputs fixed once for all hadrons) | struct | 1.893 | 2.0 | 2.0 |
| proton moment | lattice QCD (inputs fixed once for all hadrons) | struct | 2.739 | 1.7 | 1.7 |

**Aspden**

| ID | Quantity | Stage | Kind | Value | Digits of agreement | Digits added |
|---|---|---|---|---|---|---|
| W1 | 1/α | 108π(8/N)^(1/6); N = 1843 is selected by his stated rule (odd, at or below the zero-potential value 1844.5); printed 137.035915 (1972) | struct | 137.0359148 | 6.2 | +6.2 |
| W4 | proton/electron mass ratio | P = Q/(√(3/2) − 1), Q = 944784π²/1843^(4/3); no new adjustable number (1975) | struct | 1836.152316 | 6.7 | +6.7 |
| W5 | G | his G from the lattice and the 1.875 cluster (struct, history unverified) | struct | 6.67216 | 3.5 | +3.5 |
| W6 | deuteron magnetic moment | the fraction 6/7 = (2 + 4)/7, set by the state weights 2:1:4 | choice | 0.8571428571 | 3.5 | +3.5 |
| W6 | deuteron magnetic moment | plus the mass-deficit term of his chain-state model (1986) | struct | 0.8574389111 | 6.1 | +2.6 |
| P01 | muon/electron mass ratio | the integer 207, the muon mass ratio rounded (1983 gives no derivation of it) | target | 207.0 | 3.0 | +3.0 |
| P01 | muon/electron mass ratio | three-charge formula 209 − (9/4)·207/(207 + δ), δ = √3 from his stated geometry (1983, P02) | struct | 206.7686704 | 5.7 | +2.8 |
| P01 | muon/electron mass ratio | δ = 1 + 2π/9, a 'wave resonance' (1983, P01) | choice | 206.7683078 | 6.9 | +1.2 |
| P04 | muon anomaly | the measured electron anomaly, which his formulas take as input | input | 0.001159652 | 2.3 | +2.3 |
| P04 | muon anomaly | 1982 derivation: resonance distance 2(R + r) | struct | 0.0011658682 | 4.3 | +2.1 |
| P04 | muon anomaly | 1983 revision (P05) | choice | 0.00116589 | 4.6 | +0.23 |
| P04 | muon anomaly | 1984 revision: average over two states (P04) | choice | 0.001165918 | 5.6 | +1.1 |
| P06 | electron anomaly | cavity formula 2/(1 − δ), δ = 1/(2π/α + 4/√3), α from N = 1843 (1981) | struct | 0.001159646913 | 5.3 | +5.3 |
| P06 | electron anomaly | plus a gravitational term of 5.281 × 10⁻¹⁰ | choice | 0.0011596522 | 7.8 | +2.4 |
| P24 | neutron magnetic moment (size) | his model's bare moment, 2 μN (the measured antiproton moment is 2.793 μN) | struct | 2.0 | 1.3 | +1.3 |
| P24 | neutron magnetic moment (size) | times (1 − 1/23): 44/23, from his four-state model (46 transitions; 23 is the largest prime factor of 207) | choice | 1.913043478 | 6.4 | +5.1 |
| P25 | proton magnetic moment | 1980: Physics Unified eq. (199), with his own α⁻¹ = 137.0359 and mμ = 206.7689 | struct | 2.792846 | 6.3 | +6.3 |
| P25 | proton magnetic moment | 1986: cavity resonance N/n = 9.373948 | choice | 2.7928467 | 6.6 | +0.32 |
| P25 | proton magnetic moment | 1988: 2:1 mixture of the 205 and 207 resonances | choice | 2.792847068 | 7.0 | +0.37 |
| P25 | proton magnetic moment | 1988: plus 1.07 × 10⁻⁷, his proton–electron charge difference, fitted to the proton/electron mass residual | choice | 2.792847367 | 8.1 | +1.1 |
| P10 | charged pion mass (in mₑ) | (π:P:π:π) system, π/P = 0.14874412, non-resonant (he notes it is 6σ off) | struct | 273.117 | 4.2 | +4.2 |
| P10 | charged pion mass (in mₑ) | resonant pion, bare half the time (1989) | choice | 273.127432 | 4.7 | +0.49 |
| P13 | neutral pion mass (in mₑ) | integer search: B/C = 2641/10 ('best C = 10'), the π⁰ mass to 4 digits | target | 264.1 | 3.8 | +3.8 |
| P13 | neutral pion mass (in mₑ) | plus the surplus at B/8A, the lower end of his range, which he preferred | choice | 264.11 | 3.9 | +0.11 |
| P20 | Λ mass | 2^(1/4) × the proton mass; the factor is empirical, matched to the known Λ (1986) | choice | 1115.799845 | 4.0 | +4.0 |
| P07 | tau mass | 3^(7/12) × the proton mass; empirical factor, matched to the known tau (printed 1781 MeV, 1986) | choice | 1781.0 | 2.6 | +2.6 |
| P07 | tau mass | revised 1996 after the 1992 measurements (P08) | choice | 1776.51 | 3.6 | +0.99 |
| P07 | tau mass | revised again 1998 (P09) | choice | 1777.02 | 4.3 | +0.67 |
| P21 | Ω⁻ mass | (τ + 3Q)/3^(1/3); empirical, matched to the known Ω⁻ (1986) | choice | 1674.0 | 3.0 | +3.0 |
| P18 | J/ψ mass | (√6 − 3/2)·2^(1/3)·g, with g = 2.587 GeV fitted in 1966 to the Σ, K and π; published 1978-79, after the J/ψ was found | choice | 3095.0 | 3.2 | +3.2 |
| P27 | W mass | 2(2N + 1)^(4/3) × proton mass with the integer N = 8, published after the W was found (1983) | choice | 82.03 | 1.7 | +1.7 |
| P31 | Z mass | 1984: same formula on the graviton, N = 4, after the Z was found | choice | 96.86 | 1.2 | +1.2 |
| P31 | Z mass | 1986: f(2) − f(1) | choice | 93.7935 | 1.5 | +0.34 |
| P31 | Z mass | 1989: (g/ω)³ g | choice | 92.59 | 1.8 | +0.27 |
| P31 | Z mass | 1997: minus 2 × 0.785 GeV | choice | 91.02 | 2.7 | +0.92 |
| P14 | neutral-pion lifetime | (2/1843) × his electron 'lifetime' 7.55 × 10⁻¹⁴ s (1987: 8.19 × 10⁻¹⁷ s; the 1989 paper gives 8.2 by another route) | struct | 8.2 | 1.6 | +1.6 |
| P22 | neutron mean life | 1981: simultaneous muon hits, condition (3), the one that matched 918 ± 14 s | choice | 921.8 | 1.3 | +1.3 |
| P22 | neutron mean life | 1986: dual-encounter formula, 'exactly the median of the recommended value' | choice | 898.0 | 1.6 | +0.34 |

## Sources of today's values

- CODATA 2022 recommended values (NIST): 1/α, the Rydberg energy, the mass ratios, G, the magnetic moments, the electron anomaly, and the particle masses in electron masses and atomic mass units.
- Particle Data Group, Review of Particle Physics 2026 (Int. J. Mod. Phys. A 41, 2630011): the W, Z, Higgs and top masses, the neutron mean life, and the lepton, pion, kaon, Λ and Ω⁻ values (pdg.lbl.gov/2026/tables). PDG 2024 for the J/ψ and ψ(2S) masses.
- Muon g-2 Collaboration, final result and world average, 2025: muon anomaly 116 592 0715(145) × 10⁻¹².
- N. Bezginov et al., *Science* 365, 1007 (2019): hydrogen Lamb shift, 1057.8298(32) MHz.
- A. van Wijngaarden, F. Holuj and G. W. F. Drake, *Phys. Rev. A* 63, 012505 (2000): He⁺ Lamb shift, 14041.13(17) MHz.
- X. Fan, T. G. Myers, B. A. D. Sukra and G. Gabrielse, *Phys. Rev. Lett.* 130, 071801 (2023): electron anomaly, 0.00115965218059(13), and 1/α = 137.035999166(15).
- The values of Aspden's day, and his own values, are those given in the working paper, Part 1 and Part 2, with their sources.

## Sources of the QED and Standard Model values

- Electron anomaly: F. Jegerlehner, "Lepton magnetic moments: what they tell us", arXiv:2601.08101 (2026): 0.00115965218023(9) with α from rubidium, including the 2024 five-loop revision. α from rubidium: L. Morel et al., *Nature* 588, 61 (2020); from cesium: R. H. Parker et al., *Science* 360, 191 (2018).
- Muon anomaly: R. Aliberti et al. (Muon g-2 Theory Initiative), arXiv:2505.21476 (2025): 116 592 033(62) × 10⁻¹¹. Earlier value: T. Aoyama et al., *Phys. Rep.* 887, 1 (2020): 116 591 810(43) × 10⁻¹¹.
- Hydrogen Lamb shift: M. Horbatsch and E. A. Hessels, *Phys. Rev. A* 93, 022513 (2016), arXiv:1601.01057; V. A. Yerokhin, K. Pachucki and V. Patkóš, *Ann. Phys. (Berlin)* 531, 1800324 (2019), arXiv:1809.00462.
- He⁺ Lamb shift: Yerokhin, Pachucki and Patkós (2019), as above; earlier value U. D. Jentschura and G. W. F. Drake, *Can. J. Phys.* 82, 103 (2004), arXiv:physics/0310152.
- W, Z, top and Higgs masses: Y. Fischer et al. (Gfitter Group), "The Higgs boson through the lens of electroweak precision data", arXiv:2607.09861 (2026). The PDG 2024 electroweak review gives 80.356(5) GeV for the W.
- Neutral pion: K. Kampf and B. Moussallam, *Phys. Rev. D* 79, 076005 (2009), arXiv:0901.4688 (lifetime); X. Feng, L. Jin and M. J. Riberdy, *Phys. Rev. Lett.* 128, 052003 (2022), arXiv:2108.05311 (mass difference); the leading-order width as quoted in M. Moinester, arXiv:2412.03669.
- Charged-pion lifetime: FLAG Review 2024, arXiv:2411.04268 (f_π); PDG 2024 review "Leptonic decays of charged pseudoscalar mesons" (f_π from the lifetime).
- Neutron mean life: PDG 2024 review of V_ud and V_us (the relation and V_ud); FLAG Review 2024 (g_A).
- Lattice masses: S. Dürr et al., *Science* 322, 1224 (2008), arXiv:0906.3599; Sz. Borsanyi et al., *Science* 347, 1452 (2015), arXiv:1406.4088.
- Nucleon magnetic moments: D. Djukanovic et al., *Phys. Rev. Lett.* 132, 211901 (2024), arXiv:2309.07491.
- Charmonium: D. Hatton et al. (HPQCD), *Phys. Rev. D* 102, 054511 (2020); A. Bazavov et al. (Fermilab/MILC), arXiv:1810.09983 (2019).