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 8 — 6 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) — and, as of 6 September 2026, proven underivable by the programme’s own covariant action: the MOND primitive enters the field equations only through its derivative and the cosmological background only through Λ + (2 − K_B) J(0)/2, a normalisation zero mode (DOI 10.5281/zenodo.22559892). The one coefficient-free alternative, the horizon form a₀ = c²/(2π L_dS) with κ = 0.461, is 8.5% away and exactly degenerate with the H₀ tension: κ = ½ on Planck’s H₀ and κ = 0.461 on SH0ES’s predict the same a₀ to 0.2%. 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 coefficient κ cannot be derived by the candidate action (6 September 2026)Zero-mode theorem; the Λ-free repair has the wrong sign and is 220× too small; a global constraint misses by 10⁵; a four-form promotion of a₀ fixes the sign and makes a₀ ∝ √(Gρ_Λ) structural but leaves κ as the free coupling ratio Z/β² = 8; the horizon coefficient 0.461 vs ½ is locked to H₀. Published — DOI 10.5281/zenodo.22559892
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, and since 6 September 2026 it is proven underivable by the candidate action itself: the additive zero of the MOND primitive is a zero mode of the field equations, the two repairs that could fix it fail on sign and by five orders of magnitude, and the four-form extension that fixes the sign turns the question into a free coupling ratio. It is an empirical boundary conditionthat only a stellar mass-to-light zero point, an absolute gas scale and a settled H₀ can pin. 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. On 6 September 2026 the last particle candidate, a thermal relic at the mass the N_eff bound allows (about 28 eV), closed as a pincer with no interior: N_eff needs at least 27.6 eV, the radial acceleration relation needs at most 11 eV, and the cluster profile worked only at 11.4 eV. Four condensate constructions are closed on the action’s own terms, and the last scale-selective mechanism, wave dark matter near 10⁻²⁴ eV, closed the same evening: a wave sea responds to a galaxy well exactly like a classical one, and the minimum size the uncertainty principle imposes on the captured mass pinches the Tully–Fisher and lensing bounds against cluster formation with no interior. No particle or condensate mechanism keeps the action’s dark fluid out of galaxies. 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 binariesTwo mutually exclusive arms, both registered before the data (Amendment 11, 6 Sep 2026, append-only and hash-stamped). Arm A, the frozen kernel as modified gravity: γ_v = 1.1614–1.1814 canonical / 1.1917–1.2267 alt, no-verdict edge 1.23. Arm B, the covariant candidate at its Cassini-minimal coherence length: γ_v ≤ 1.0450 / ≤ 1.0300, falling toward 1.000 as the length grows. Decision rule fixed in advance: A falsified below 1.056, B falsified at or above 1.129. Stated against interest: DR4 separates the arms at 4.2σ but cannot confirm Arm B over Newton beyond 1.6σ — a Newtonian result kills A and leaves B alive but unconfirmed, never a success. Newton = 1.000.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. The measurement itself — target gate, NIRSpec-then-ALMA CO(3–2) funnel, frozen statistic — is pre-registered for observers: DOI 10.5281/zenodo.22563139.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

Papers, September 2026 (Zenodo; AI-assisted drafts, not peer reviewed)

  • Does the MOND Acceleration Scale Evolve? A Pre-Registered Decisive Measurement: the Deep-MOND Tully–Fisher Zero Point of One Lensed Rotator at z ≃ 2.5, with a Two-Stage JWST/ALMA Funnel (6 Sep) — DOI 10.5281/zenodo.22563139
  • The Coefficient of the a₀–Λ Relation: a Zero-Mode Theorem for Local MOND Actions, Two Failed Repairs, a Four-Form Reframing, and an H₀ Degeneracy (6 Sep) — DOI 10.5281/zenodo.22559892
  • A Ceiling Dark Matter Cannot Impose: the Bounded-Boost Theorem for MOND-Class Kernels, and What It Says About Galaxies and Clusters (v4, 6 Sep) — DOI 10.5281/zenodo.22548669
  • The Filtered MOND Action: a Central Tidal Identity, Comparable-Mass Forces, a First Covariant Clock Action, and the Operator That Screens the Solar System (5 Sep) — DOI 10.5281/zenodo.22347632
  • Crispy Fried Chicken Matching Theorem (2 Sep) — DOI 10.5281/zenodo.22261001
  • The Retarded Nonlocal MOND Kernel Is Unstable on MOND Backgrounds: a Longitudinal Gradient Instability and a Deep-MOND Ghost Close the Nonlocal Door (v2, 2 Sep) — DOI 10.5281/zenodo.22255522
  • The Aether-Scalar-Tensor Dark Sector Is a γ = 2 Polytrope: the Cluster Helmholtz Phase Is Its Mass, It Pins Dynamically, and It Fills About a Quarter of the Cluster Gap (v2, 2 Sep; cosmology withdrawn) — DOI 10.5281/zenodo.22254075
  • An Obstruction Map for Relativistic MOND: the Conformal Lensing Barrier and the Cost of Its Repair (28 Aug) — DOI 10.5281/zenodo.22135510
  • A Conditionally Closed Constraint-Defined MOND Theory with Two Tensor Degrees of Freedom: Hamiltonian Certification, Kernel-Agnostic Chassis, and Solar-System Gates (v2, 27 Aug) — DOI 10.5281/zenodo.22133406
  • Carrier No-Go Theorems for Two-Degree-of-Freedom MOND: the F(A²) Class, the Auxiliary-Legendre Escape, and a Hamiltonian Audit of Causal Nonlocal MOND (27 Aug) — DOI 10.5281/zenodo.22132648

The full record, every deposit since June 2026 with its title and concept DOI, is the DOI index in the repository README.

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.