Physics

DME Against MOND on Real Data: Where the Two Laws Agree, Where They Don’t, and Why the Substrate Density Isn’t a Fitted Number

For fifty years, galaxies that rotate faster than their visible mass allows have had two serious rival explanations: add an undetected particle species (dark matter), or change the law of gravity itself at low acceleration (MOND). BFUT Paper 52 tests a third option directly against MOND, on the same public data, under the same locked conditions, with no gravity-side knob given to either law: DME, the Dark Matter Effects equation. The result is a close numerical contest with a sharp physical distinction underneath it: and a companion clarification in the current version of Paper 18 that fixes a subtlety worth stating plainly, since it changes how DME’s own domain equation should be described.

First, a Precision Fix: DDR’s Rotational Term Is Not What Produces DME’s Results

Paper 18 derives two related but genuinely separate quantities from the same substrate density, ρ_s. The first is the DDR equation, R_d = (3M/8πρ_s)^(1/3), the finite domain radius within which a mass’s gravitational influence is organised. If that same mass is given an effective inertial boost from organised rotation, M_eff = M(1 + v_rot²/c²), and this is inserted back into the DDR formula, the algebra produces an exact identity: R_eff = R_d(1 + v_rot²/c²)^(1/3). It’s a clean, exact result: the rotational correction factor comes straight out of the cube root with no approximation needed.

But the current paper is explicit about what this result is and isn’t. For ordinary galactic rotation speeds, v_rot/c is of order 10⁻³, which makes the entire correction (1 + v_rot²/c²)^(1/3) − 1 work out to a few parts in 10⁶: a shift in the domain boundary that is real but, for any galaxy, utterly negligible. R_eff is, in the paper’s own words, “a domain-boundary quantity… not the mechanism used in the SPARC or KiDS tests,” recorded for completeness as a consistent consequence of combining DDR with rotational mass enhancement, but explicitly “not fitted to rotation curves or lensing stacks” and “not required for the results” reported anywhere in the paper’s validation appendices.

This matters because it’s exactly the kind of thing that’s easy to conflate if you’re not reading carefully: two formulas that both involve rotation and both derive from ρ_s, sitting in the same section, describing what sound like related effects. They aren’t. R_eff’s rotational correction to the domain boundary is a six-decimal-place footnote. The actual mechanism that produces flat rotation curves and the observed lensing convergence, the one tested against 175 SPARC galaxies and four KiDS-1000 stacks, is DME, a distinct equation for extra organised mass, standing on its own.

The DME Equation Itself

DME’s construction is direct. Newton’s law is left completely unmodified. What changes is the mass distribution: a rotating disk organises the surrounding Spaticle substrate, and that organised, compressed medium contributes real additional mass, which then produces additional gravitational support through ordinary Newtonian gravity. The resulting circular-speed law is

v²(R) = v_b²(R) [1 + a_s R / v_b²(R)]^(1/2)

where v_b is the Newtonian circular speed from the observed baryons alone, and a_s = c(Gρ_s/3)^(1/2) = 1.09 × 10⁻¹⁰ m/s² is computed directly from the substrate density, G, and c: not fitted to any galaxy. The same density has an exact energy-equivalence to the cosmological constant, ρ_s c² = ρ_Λ = 5.30 × 10⁻¹⁰ J/m³, established independently of any rotation-curve or lensing data. When the baryonic acceleration is much larger than a_s, the extra term vanishes and Newton is recovered exactly. When it’s much smaller, the equation reduces to v⁴ = GM_b a_s: the observed baryonic Tully-Fisher relation, arrived at as extra enclosed mass rather than a change to the force law.

Running It Against MOND, Fairly

The comparison in Paper 52 is deliberately structured so neither law gets an advantage the other doesn’t. Both are tested on all 175 SPARC galaxies with the mass-to-light ratio locked at the standard survey value, Υ = 0.5, a photometric convention rather than a gravity fit, and MOND is given no per-galaxy tuning either, using the simple interpolating function with the standard a₀ = 1.20 × 10⁻¹⁰ m/s². This matters because published MOND results often let stellar mass-to-light ratio, distance, and inclination float per galaxy, which measurably improves MOND’s scores; that’s a different experiment from the one run here.

On SPARC (175 galaxies, 3,391 velocity points): shape agreement is identical for both laws, 92.0% (161/175), with 98.8% of flat rotation curves correctly identified by each. DME comes out slightly tighter on the metrics that matter most for the outer, dark-matter-dominated region: median outer relative residual is 0.100 for DME against 0.116 for MOND, and median galaxy χ²/N is 11.1 (DME) against 11.4 (MOND) across all radii, widening to 7.3 against 8.3 on the outer half specifically, which is the regime the whole comparison is actually about. MOND edges ahead on the global χ²/N summed across all 3,391 points (56.6 against DME’s 84.9): but that figure is dominated by a small number of very tight inner-galaxy error bars where bulge decomposition and beam smearing, not the outer extra-gravity law, are doing most of the work. The median-galaxy figures are the fairer comparison, and DME is marginally ahead on those.

On KiDS-1000 weak lensing (four isolated-lens stellar-mass bins, the same point-mass baryonic input fed into the identical DME equation): both laws produce a nearly flat equivalent-speed floor and the same v ∝ M_gal^(1/4) scaling. DME’s floors come out at 119, 152, 172, and 190 km/s against MOND’s 122, 156, 176, and 195 km/s. On χ²/N across all 60 points: 10.6 for DME, 9.5 for MOND, 32.8 for unmodified Newton, both substantially outperforming Newton alone, with MOND a few percent ahead of DME.

Why the Scores Are Close, and Why That Doesn’t Make the Theories Equivalent

The numerical proximity has a specific, identifiable cause: a_s and a_0 are numerically close, with a_s/a_0 ≈ 0.91. That’s a coincidence of magnitude, not a sign the two theories are saying the same thing. MOND changes the dynamical law itself once acceleration drops below a₀, with a₀ standing as an empirical constant introduced specifically to make rotation curves come out flat. DME changes nothing about the force law: it computes a_s from a substrate density that was already fixed by evidence with nothing to do with galaxy dynamics, and the value it produces simply happens to land close to MOND’s posted constant. One is a fitted acceleration scale; the other is a computed consequence of an independently measured physical quantity. Getting a similar number out of two different starting points is not the same as the two starting points being physically identical.

The Sensitivity Analysis: How Much Room Does DME Actually Have?

This is the part of Paper 52 that does the most work in distinguishing “close fit” from “coincidentally close, but fragile.” The paper takes ρ_s and shifts it by 25% up, 25% down, and by a full order of magnitude in each direction, holding Υ fixed, and reruns the entire SPARC comparison at each step:

Shift

ρ_s (kg/m³)

a_s (m/s²)

Median outer residual

Shape agreement

Standing

7.3 × 10⁻²⁷

1.09 × 10⁻¹⁰

0.100

92.0%

+25%

7.4 × 10⁻²⁷

1.36 × 10⁻¹⁰

0.101

90.9%

−25%

4.4 × 10⁻²⁷

8.15 × 10⁻¹¹

0.121

91.4%

×10

7.3 × 10⁻²⁶

1.09 × 10⁻⁹

0.677

9.7%

÷10

7.3 × 10⁻²⁸

1.09 × 10⁻¹¹

0.359

46.3%

The result is exactly what a genuine physical constraint should look like and exactly what a loosely fitted parameter would not: modest 25% shifts degrade the fit only slightly, while order-of-magnitude shifts in either direction are catastrophic, collapsing shape agreement from 92% down to below 10% or up to under half. In the KiDS deep regime, where the predicted speed scales as a_s^(1/4), a factor-of-ten shift in a_s produces an equally large mismatch on the lensing stacks. This is the signature of a real constraint operating near its correct value, not a parameter with generous room to be adjusted after the fact: the value of ρ_s used here isn’t being tuned to fit these galaxies; it’s the same value independently fixed elsewhere in the programme, and the fit holds up precisely because that value is close to correct.

What Actually Separates the Two Theories

Set the residual tables aside and the distinction is entirely physical, not statistical. MOND answers the question “why do outer rotation curves stay flat?” by saying the force law itself changes below a posted threshold. DME answers the same question by saying Newton never changes, and the extra gravitational support comes from real, organised, compressed substrate: the same medium independently required to explain the cosmological constant’s energy density, and the same medium whose finite propagation capacity gives the speed of light its physical meaning elsewhere in the BFUT programme. The residual tables show the two theories currently make almost the same numerical predictions on the data tested so far. They do not show the two theories are asking the same question.

Data, methodology, and sensitivity analysis from BFUT Paper 52, “Dark Matter Effects without a Dark-Matter Particle: The DME Equation Compared with MOND on SPARC and KiDS-1000.” DDR/DME relationship clarified in BFUT Paper 18, “Beyond General Relativity: A Unified Gravitation Equation.”

Download BFUT papers, simulation code, and companion materials: vijayshankarsharma.com/downloads/