Research note · independently checked

A parameter-free derivation of the MOND acceleration scale a₀

Derivation: G (Studio G), Ø Predictions, Appendix B · DOI 10.5281/zenodo.19208226
Independent reconstruction & re-run: AJ Greyling (CongoSky) — no affiliation with the author

1The result

The only empirical quantity on the right is H₀. There are no free parameters: C_S² is a closed-form geometric factor, c is exact by definition, and 2π is 2π. Evaluated at a representative local H₀ = 73.8 km/s/Mpc:

Because a₀ ∝ H₀ in this relation, the residual is a direct function of which H₀ you adopt; the derivation makes the a₀–H₀ link its central, falsifiable commitment (§4). Read the inverse row the other way round: fixing a₀ to its measured value predicts a local H₀ of 74.3 km/s/Mpc.

2It is one of five, from one input

a₀ does not arrive alone. The same axiomatic system, fed the single measured dimensionless input α (the fine-structure constant) — with CODATA m_e entering Part II and the adopted H₀ entering Part IV as unit-bearing anchors, none of them fitted — produces five headline numbers. Here is the full verification run — G's published verify.py (Appendix B), reproduced verbatim and re-run offline:

# python3 verify.py  — stdlib only, <1s, zero free parameters
Part I — Proton-to-electron mass ratio
  Predicted: 1836.152673444   Measured: 1836.152673426   Residual: 0.010 ppb
Part II — Gravitational constant
  Predicted: 6.7206e-11        Measured: 6.6743e-11        Residual: 0.69 %
Part III — Neutron-proton mass difference
  Predicted: 2.5309939330      Measured: 2.5309883000      Residual: 2.23 ppm
Part IV — MOND acceleration scale a₀
  C_S² = 1.0445
  Predicted: 1.192e-10 m/s²    Measured: 1.20e-10          Residual: -0.67 %
  H_0 predicted from measured a_0: 74.3 km/s/Mpc
Part V — Dark-sector partition (ΩΛ / Ω_dm)
  DE 68.85% (obs 68.89%) · DM 26.39% (obs 26.07%) · DE/DM 2.609 (obs 2.643)

============================================================
All five predictions verified. Zero free parameters.
============================================================

a₀ is Part IV. The reason to take the a₀ number more seriously than a lone coincidence is that the same machinery, with no extra knobs, also lands the proton/electron mass ratio to 0.010 ppb and the dark-sector split to about a percent. Either that is a deep structural fact or it is an extraordinary run of luck across five independent scales — and that is exactly the judgement being handed to the reader, not asserted here.

3Independently reconstructed

I did not take the author's word for the number. I wrote a separate, structured reconstruction of the derivation that returns data rather than printing it, then asserted it against G's verbatim published script. They agree to a relative tolerance of 1×10⁻¹² — a parity test that is itself part of the harness and fails the build if the two ever drift apart. This tells you the arithmetic is real and reproducible. It tells you nothing about whether the physics is right; that distinction is kept deliberately sharp throughout.

5Run it yourself — offline, one command

Nothing here asks for trust. The whole thing is Python standard library, runs in under a second, needs no network, and exits non-zero if any prediction falls outside the author's stated tolerance.

git clone https://github.com/ajgreyling/the420code-proof
cd the420code-proof
python3 verify.py          # the five predictions, verbatim

6Does the number fit real galaxies?

a₀ is derived from cosmology and atomic physics — it never saw a galaxy. So the fair test is whether that same number reproduces real rotation curves. Run over the full verbatim SPARC survey — all 175 galaxies, 3,389 points, none excluded (Lelli, McGaugh & Schombert 2016) with fixed, unfitted mass-to-light ratios, G's a₀ lands the galaxies on the radial acceleration relation with a raw scatter of 0.199 dex — no quality cuts at all; under the survey's own published velocity-error cut it is 0.144 dex over 2,803 points, robust σ ≈ 0.12 dex, the published ~0.11–0.13 band — the physicists' own data falling on the physicists' own relation, using a scale nobody tuned to it.

You can drive it yourself in the browser: pick a galaxy, watch observed vs Newtonian vs MOND, and drag H₀ to see a₀ move with it (the sharpest kill switch, made tangible).

7What this note does not claim

Honesty is the whole posture, so the limits are stated as plainly as the result:

• Reproducing the author's arithmetic is internal consistency, not empirical confirmation. Nature is not on this page.
• The a₀ residual depends on the adopted H₀; the −0.67 % figure is at H₀ = 73.8. A different H₀ moves it, by construction.
• C_S² is a closed-form geometric constant. Whether the geometry that produces it is physically motivated, or is numerology that happens to land, is precisely the open question — and the reason this is addressed to specialists.
• This is a landing note. The mathematics lives in the canon (DOI above); the executable check lives in the repo. Both are meant to be attacked.

References

1. G (Studio G). Ø Predictions and the 420 Code — Artist's Proofs. the420code.org. DOI: 10.5281/zenodo.19208226. CC BY 4.0.
2. Milgrom, M. (1983). A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. ApJ 270, 365.
3. McGaugh, S., Lelli, F., Schombert, J. (2016). Radial Acceleration Relation in Rotationally Supported Galaxies. PRL 117, 201101.
4. Famaey, B. & McGaugh, S. (2012). Modified Newtonian Dynamics: Observational Phenomenology and Relativistic Extensions. Living Rev. Relativity 15, 10.
5. Banik, I. & Zhao, H. (2022). From Galactic Bars to the Hubble Tension: Weighing Up the Astrophysical Evidence for Milgromian Gravity. Symmetry 14, 1331.