MOND Unification
Reparametrization — Dimensional Analysis — 13% ErrorThe MOND acceleration scale a₀ is related to cosmological parameters. However, this relationship is not unique to Synchronism: Milgrom noted the a₀ ~ cH₀ coincidence in his original 1983 paper. McCulloch (2007) derived a₀ = cH₀/(2π) from quantized inertia. Verlinde (2017) obtained a similar relation from emergent gravity. The 2π factor is the standard geometric factor arising from any argument involving a spherical causal horizon. This is best understood as dimensional analysis with a geometric prior, not a unique derivation from first principles.
The Significance
In Modified Newtonian Dynamics (MOND), a₀ is the acceleration below which gravity deviates from Newton's law. Milgrom observed it empirically:
For 40 years, the coincidence that a₀ ≈ cH₀ has been noted by many researchers. Multiple frameworks produce the same relation with the same geometric factor. In Synchronism, the coherence function provides a physical narrative for why this relationship holds, but the result itself is shared with other approaches.
Update 2026-08-01: treating a₀ as emergent rather than fundamental is not cost-free — it is a forced commitment to a₀(z) = cH(z)/2π, which is now measurable and currently disfavored by 2σ–6σ against real high-z RAR data (Ciocan et al. 2026). See the epoch-fork row on Parameter Derivations for the full account — disfavored, not refuted, and it does not move the site's refutation count.
The Derivation Chain
Step 1: Critical Density of the Universe
Standard cosmology. The density at which the universe is flat. This is measured, not assumed.
Step 2: Coherence Transition
At the coherence transition (C ≈ 0.5), the gravitational acceleration from ρcrit over a Hubble-scale volume defines the threshold where dynamics change. The 2π factor arises from the spherical geometry of the causal horizon.
Step 3: The Result
Plugging in H₀ = 67.4 km/s/Mpc and c = 3 × 108 m/s gives 1.04 × 10−10 m/s². Milgrom's observed value: 1.20 × 10−10. Error: ~13%. (Corrected 2026-07-22: the 1.08 / “~10%” previously shown here belongs to H₀ = 70, not the 67.4 stated in this very step.)
Comparison
MOND (Milgrom 1983)
- a₀ is a fundamental constant
- Value determined empirically from galaxy fits
- No explanation for why a₀ ≈ cH₀
- Extremely successful at fitting rotation curves
Synchronism
- a₀ is an emergent scale
- Value from dimensional analysis of H₀ and c (shared with other frameworks)
- Uses the standard McGaugh et al. (2016) RAR interpolating function
- Predicts EFE = 0 structurally (C depends only on local ρ); real environmental coupling is an untested ambient-density effect, not an EFE — see correction below
The External Field Effect — Correction
Correction (2026-08-02), itself corrected 2026-08-09: the paragraphs below previously attributed the External Field Effect to “the nonlinear Poisson equation that implements the coherence function,” and the 0.3–0.4× figure was never actually derived from one. That much stands. But the 2026-08-02 correction justified itself with a claim that was false — it said the object “does not exist in this framework” and cited Honest Assessment as stating “correctly and repeatedly” that there is no field equation anywhere in the galaxy sector. The archive has had one since 2025-12-01: Appendix D §D.2 states ∇²Φ = 4πGρ/C, §D.3 effective Einstein equations, §D.5 a worldline action. The reason the 0.3–0.4× figure is not derivable is not that no field equation exists; it is that the one the archive states (L1) is eliminated a priori by a vacuum source floor, and the one the site's tests actually use (∇·[C∇Φ] = 4πGρ, equivalently g = gbar/C) gives EFE = 0, not 0.3–0.4×. The conclusion below is unchanged; its stated reason is.
Applying the framework's actual structure instead: C is a function of local matter density ρ alone. A uniform external gravitational field does not change ρ. So an algebraic gobs = gbar/C(ρ) modification (the direction the site's own fDM = 1−C identity implies — corrected 2026-08-04, this previously read “C(ρ)·g,” the opposite direction) satisfies the Strong Equivalence Principle by construction and predicts EFE = 0 exactly — a sharper and more discriminating structural claim than “0.3–0.4× MOND,” and one already in tension with Chae, Lelli, Desmond, McGaugh, Li & Schombert (2020, ApJ 904, 51), who report a ~4σ detection of MOND's EFE in SPARC rotation curves.
EFE = 0 survives the obvious momentum-conservation objection (2026-08-04): an algebraic g = gbar/C(ρ) modification looks like it violates momentum conservation, since it isn't obviously sourced by a field equation. It survives: the one-line completion ∇·[C(ρ)∇Φ] = 4πGρ conserves momentum, reproduces g = gN/C exactly in spherical symmetry, and its extra polarization force is ≤2×10⁻⁵ of gravity — negligible. Because this completion is linear in Φ, EFE = 0 is preserved exactly, and the same linearity is why the exterior field of an isolated mass diverges as C → 0 in vacuum: “a uniform external field does not change ρ” (⇒ EFE = 0) and “empty space has C = 0 however strong the field” (⇒ divergent exterior field) are the same statement, not two separate properties. Full derivation: explorer/findings/efe-zero-survives-momentum-objection-but-the-substitution-was-never-evaluated.md.
There is a real environmental effect in this framework — it is just not the EFE. Ambient medium density adds to local ρ, raising C and suppressing the boost: an ambient-density effect, keyed on ρambient rather than MOND's gext ∝ M/r². Two satellites at the same external acceleration but different host gas content would behave identically under MOND and differently here — that variable difference, not a scalar EFE ratio, is the genuine discriminator.
Correction (2026-08-03): the paragraph above previously called this “an open, unclaimed test” that had not been checked against SPARC or Chae et al. (2020). That was stale — it already has an ID and a verdict. TEST-05 computes exactly this lever comparison (MOND+EFE's external-acceleration coupling vs. this framework's ambient-density coupling), cites Chae et al. 2020/2021's ~4σ EFE detection, and the registered run (SPARC RAR offsets vs. Cosmicflows-4 ambient density, N = 141) gives r² = 0.0001 — the kill bar fires. Two independent expert visitor passes (2026-08-03) flagged this page and For Researchers as stating incompatible things (“0 tests that could select it” vs. an uncatalogued live discriminator); the resolution is that it was never uncatalogued, the cross-reference from this page just never pointed to it.
Tidal Dwarf Galaxy Test
For a 107 M☉ TDG at gext = 1.0 a₀, Synchronism predicts σ ~ 10.5–14.5 km/s while MOND predicts σ ~ 10.9–40.9 km/s. Observable with the NGC 5291 system (Bournaud et al. 2007, Lelli et al. 2015).
Untested — Can only tie or refute — see caveatCorrection (2026-08-01): this is not an independent novel prediction. Three problems, on the site's own numbers elsewhere: (1) the intervals are nested — [10.5, 14.5] sits almost entirely inside MOND's [10.9, 40.9]. Only σ > 14.5 discriminates, and that outcome falsifies Synchronism while leaving MOND untouched — this is the site's own nested-submodel argument showing up as a concrete interval. (2) The lever generating it is already dead: the weaker EFE is a consequence of the bounded boost B ≤ 1/Ωm ≈ 3.17, and that boost ceiling is exactly what TEST-09 and TEST-10 fire on in Tier 1. (3) The 0.3–0.4× factor is read off the same RAR fit that converges to MOND's simple-μ function (see Galaxy Rotation), so it is not independent of that fit. This card is kept for the record; it should not be read as a standing discriminating prediction, consistent with For Researchers and Tier 1 both stating zero discriminating tests remain.
Further correction (2026-08-02): the mechanism this card's numbers were built on (the “nonlinear Poisson equation”) has been retracted above — the structural prediction is EFE = 0, not a weakened MOND EFE, so the quoted 10.5–14.5 km/s interval does not follow from anything currently on the site. Independently, an isolated-deep-MOND check for a 107 M☉ system gives σ = (4GMa₀/81)1/4 ≈ 9.4 km/s, radius-independent — below both quoted intervals, which the EFE (in MOND) can only lower further, not raise. No radius is stated for either interval and MOND's 40.9 km/s upper bound does not reconstruct from the stated mass. The nested-interval, non-discriminating conclusion above still stands independent of this arithmetic; the specific numbers in this card do not and should not be cited.
Sessions and History
Derived in Sessions #87–88 of the autonomous research program. The derivation was independently stress-tested in Session #91, where the same result was obtained from a different starting point (via Freeman's Law). Both derivations agree, providing internal consistency.