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 6 — 2 September 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 Milgrom–Sanders (2008) kernel ν(y) = 1/(1 − e^(−√y)) as the operative interpolation. κ = ½ is fitted, not derived: measured 0.551 ± 0.043 (distance-free) and 0.465 ± 0.076 (Tully–Fisher). Because a₀ tracks the dark-energy density it cannot track the expansion rate: a₀(z)/a₀(0) = √(ρ_DE(z)/ρ_DE(0)) exactly, which is flat if Λ is constant, 13% lower by z = 2 on DESI’s w₀wₐ fit, and within ±20% of today’s value out to z = 3 for any dark energy the data allow — against ×3 for a₀ ∝ H(z) and ×1.8 for ΛCDM’s emergent scale. That is the one statement here that ΛCDM does not make.

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
The condensate polytrope (September 2026)The dust of the Aether–Scalar–Tensor completion is a γ = 2 polytrope whose sound speed in a well is the well depth, c_s² = |Ψ| c²; the static Helmholtz equation is its hydrostatics. Algebra published — DOI 10.5281/zenodo.22242701 (the cluster cosmology built on it is withdrawn, see below)
Nonlocal MOND kernels (Deffayet–Woodard class) are unstableIn-in linear analysis: longitudinal gradient instability and a deep-MOND ghost; an independent tensor-speed kill. Published — DOI 10.5281/zenodo.22253953
Modified-inertia action (v1–v11); disformal lensing constructionPublished; the arm is closed as physics (lensing, 21σ); kept as mathematics
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.
  • Dark matter exists, at full Ω_dm, and it is cold. The CMB needs a pressureless component and the relativistic completion (v9, on Skordis & Złośnik’s Aether–Scalar–Tensor chassis) supplied it with a condensate. On 2 September 2026 that condensate’s own equation of state closed the door: read on the cosmic background it fixes c_s²(z) = 4πG ρ_dm(z)/μ², which pins the theory’s free amplitude 18–300× above its own power-spectrum ceiling, and for any such field a galaxy well today is the background at z ≲ 16 with the same sound speed — so a field cold enough for the Lyman-α forest falls into galaxies. The slogan “no dark matter in galaxies” has no kinetic mechanism left.
  • A two-field metric MOND that light and matter both see is forced to carry a third field. At quadratic order around de Sitter, any elliptic auxiliary that enters the lapse equation frees one dust-like scalar; couplings to the spatial curvature free none but split lensing from dynamics. That is why TeVeS, AeST and the superfluid all carry a genuine extra field, and it is why the hoped-for constraint-only theory does not exist.
  • 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

Dark matter falls into galaxies, and MOND is still there

The completion needs an Ω_dm-worth of cold energy for the acoustic peaks and the 100-Mpc clustering. Cold means it clusters into galaxies, where the MOND boost is also acting: by the repository’s own numbers that double count overshoots rotation curves by 2.7–4.4×. Every kinetic escape has now been run and closed — pressure support, a superfluid phase, a rising a₀, a Hubble-scaled filter — each one a committed script with checks that can fail. This is the programme’s blocking problem, and it is the same one MOND-plus-halos always had.

Galaxy clusters

The framework’s own kernel leaves clusters short by η(R₅₀₀) ≈ 1.9–2.1. The condensate polytrope pins a core worth 23–33% of the missing mass (published, DOI 10.5281/zenodo.22242701, v2 22254075), but the cosmology behind that mechanism is the one excluded on 2 September 2026: the static algebra stands, the yield is withdrawn as a live number. Recorded in RETRACTIONS.md.

The inner-planet ephemerides — discharged

The earlier “exact” α = 1 law implied a constant sunward anomaly 1278× over the Earth bound. The exponential kernel now in force (Route A, 2 August 2026) makes the departure from Newton exponentially small in the Solar System, 2.7 × 10⁻²² at the Sun. The word “exact” stays withdrawn.

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
Deep-MOND Tully–Fisher zero-point at z ≈ 2.5Framework: 0.00 dex (−0.09 with DESI dark energy). ΛCDM’s emergent halo scale: +0.33 dex. One clean low-acceleration rotator measured to ±0.13 dex decides at 20:1. On present data the two are undecided and prior-dominated; the apparent rise seen by MUSE and JWST is also what ΛCDM’s halo structure produces. A robust measured rise kills the framework’s law either way.JWST / ALMA, a handful of objects
a₀(z) evolutiona₀(z)/a₀(0) = √(ρ_DE(z)/ρ_DE(0)), footing-independent: constant to <1% for z ≤ 5 if Λ is constant (the v9 completion switches it off only above z ≈ 20), a ~13% decline by z = 2 on DESI’s w₀wₐ fit, and never more than ±20% out to z = 3 for any allowed w. ΛCDM’s emergent scale rises ×1.8 by z = 2; to mimic a flat a₀ its haloes would have to be diluted to 0.61 (z = 2) and 0.40 (z = 3) of their N-body concentrations.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.