The MOND acceleration scale as a de Sitter curvature scale

One claim: the acceleration scale of the mass-discrepancy–acceleration relation is set by the dark-energy density.

Carl P. Zimmerman|Standing revision 4 — 30 July 2026|STANDING.md
a₀ = κ c √(G ρ_Λ) = c H_Λ / Z
κ = ½    Z = √(32π/3) = 5.78881    a₀ = 9.36 × 10⁻¹¹ m s⁻²

Realised as modified gravity — the modified-inertia arm was closed by lensing in August 2026 — with the exact relation g_obs² = g_bar² + a₀ g_bar and the Milgrom–Sanders (2008) kernel ν(y) = 1/(1 − e^(−√y)) as the operative interpolation.

Attribution — what is and is not original here

This law is not new, and neither is its derivation. Milgrom (1999, Phys. Lett. A 253, 273, Eqs. 6–9) derives this exact law from the de Sitter–Unruh balance and fixes its coefficient at a₀ = 2 c H_Λ. His Eq. (9) is identically the relation above — verified symbolically, difference exactly zero. The same law was independently re-derived entropically by Pikhitsa (2010) and Klinkhamer & Kopp (2011), both also landing on 2 c H_Λ.

So the de Sitter–Unruh argument does not leave the coefficient free — it predicts one, and that prediction is 11.58× the value used here. What this programme actually contributes is therefore narrow and should be stated as such: a re-normalisation of the coefficient to fit data (κ = ½ in place of Milgrom’s 2), plus the relativistic completion — the scale embedded in Aether–Scalar–Tensor theory — and its derived a₀(z). It is not a derivation of the law, and not a derivation of its scale.

What is earned

ResultStatus
Radial-acceleration relation on SPARC (175 galaxies), the framework’s own ν and a₀0.108 dex at Υ = 0.70 — beats regular MOND’s 0.122–0.140
The a₀-line, g_obs² − g_bar² = a₀ g_barExact identity, verified
The κ reduction a₀ = κ c √(G ρ_Λ)Every π, the 32 and the 3 cancel
Modified-inertia action (v1–v11); disformal lensing constructionPublished; constraint structure machine-verified, zero frame degrees of freedom; lensing Cassini-safe and Ostrogradsky-free
Seven structural theoremsPublished 30 July 2026 — DOI 10.5281/zenodo.21708842

What is postulated — not derived, and not presented otherwise

  • κ = ½. Its value is not derived. Ghost-freedom, unitarity and holography have each been shown insufficient to force it. This is a one-parameter effective theory, not a zero-parameter derivation. Z has the same status.
  • The law itself is not the Euler–Lagrange equation of the published action. It does arise variationally in a nonlocal, non-quadratic class — Milgrom’s own virial construction — but only on the two-parameter family of circular orbits. Infinitely many extensions share that slice and none is written down. Milgrom’s own status line still applies: “we do not have a modified-inertia theory for MOND at the level of satisfaction achieved for modified-gravity formulations.”
  • Two footings are carried on every dimensional number, always: canonical ρ_DE (a₀ = 9.36 × 10⁻¹¹) and alternative ρ_total (1.13 × 10⁻¹⁰).

Open liabilities, stated plainly

The exact law is in conflict with the inner-planet ephemerides

Held to all accelerations, the relation implies a constant sunward anomaly of a₀/2 = 4.68 × 10⁻¹¹ m s⁻² that never decays. Against the Earth 2σ limit derived from Sereno & Jetzer (2006) that is 1278× too large, and 119–189× too large even after the framework’s own external-field effect. Galaxy data does not require this: across SPARC the α = 1, α = 2 and α = ∞ tails fit within 0.0084 dex of one another. The honest reading is that the relation is an infrared statement — which costs nothing in galaxies but withdraws the claim that it is exact.

Galaxy clusters

η(R₅₀₀) = 2.33 median / 2.54 geometric mean on real eRASS1 (N = 9830) using the framework’s own kernel — significant at 4.1/2.7/2.0σ against a 0.10/0.15/0.20 dex systematic floor. The cluster acceleration scale is 21.6× the framework’s a₀. The framework’s lower coefficient makes this 13.2% worse than standard MOND. Real, soft, central, and shared with AeST by an in-corpus argument — not a published family-wide theorem.

A correction that runs the other way, reported at equal weight

The previously advertised “6–8σ” Lyman-α forest exclusion of the diffuse-baryon sector is withdrawn. Three defects compounded: the observed cutoff values were unsourceable, the error bar was invented, and the response kernel was evaluated at the Newtonian rather than the observed acceleration — inflating every significance by 1.9–5.6×. On the best estimator and the defensible error channel it is 0.4–0.9σ: a weak, convention-dominated tension, not an exclusion.

arXiv endorsement

Remains the blocker on everything that matters. Nothing here has been through peer review.

What would falsify it

The forward tests are pre-registered before the data, with targets and signs frozen and hash-stamped. A confirmation that lands in the wrong place is scored as a kill.

FrontPredictionClock
Wide binariesIn force (Amendment 10, Aug 2026): γ_v = 1.1614–1.1814 canonical / 1.1917–1.2267 alt footing, no-verdict edge 1.23 — hash-stamped, amended in the open before data. Earlier targets (1.09, 1.1582) superseded on the record.Gaia DR4, Dec 2026
Lorentz-violation dipole sTXSign frozen negative; margin 1.50× / 1.24× on the two footingsGaia DR4
a₀(z) evolutionBump-then-decline, not a monotonic rise. Dissolves if w → −1.DESI, ongoing

Where to look

Galaxy data

The radial acceleration relation and baryonic Tully–Fisher behaviour — the part that is empirically grounded.

Simulations

Rotation curves and the acceleration relation, computed from the law rather than illustrated.

STANDING.md

The single source of truth: earned, postulated, live, closed, retracted. Newer than this page if they disagree.

Retracted work

The cube×sphere numerology, the “19 derivations,” the T³/Z₂ topology, and the visualisations built on them. Kept as a record.