Relation to MOND
Reparametrization — Dimensional Analysis — 6–13% Below Milgrom's a₀, H₀-DependentThe MOND acceleration scale a₀ is close to cH₀, and this is not unique to Synchronism. What is shared is the dimensional origin, not the number: authors who write the relation put different prefactors in front of it, and none is forced. This is dimensional analysis, not a derivation from first principles.
| Source (read 2026-11-08) | What it writes | × cH₀ |
|---|---|---|
| Milgrom 2008 (arXiv:0801.3133 §VI), of his 1983 paper | “observed that 2πa₀≈cH₀”; no derivation | 0.16 |
| Sanders & McGaugh 2002 (astro-ph/0204521) | a₀ ~ cH₀/6; Milgrom noticed a₀ ≈ cH₀ “to within a factor of 5 or 6” | 0.17 |
| Famaey & McGaugh 2012 (arXiv:1112.3960 §4.1) | “a₀≈cH₀/2π”; no derivation | 0.16 |
| McCulloch 2007 (astro-ph/0612599, Eq. 12) | 0.99×cH, “about six times larger” than MOND's value; a Pioneer-anomaly paper | 0.99 |
| Verlinde 2016 (arXiv:1611.02269, Eq. 1.7) | aM = a₀/6 in d = 4, with his a₀ defined as cH₀; a factor from d | 0.17 |
| Smolin 2017 (arXiv:1704.00780, Eq. 3) | a₀ ≈ aΛ/8.3, aΛ = c²√Λ; an empirical ratio | ~0.17 (at ΩΛ ≈ 0.69) |
| This framework | cH₀/2π; the 2π has no derivation here, and the named Hubble-sphere calculation gives cH₀/2 | 0.16 |
Milgrom's 1983 paper is not on arXiv and was not read directly, so the first row is his own later description of it. cH₀/6 and cH₀/2π differ by about 5%, which a0 alone cannot separate.
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, with several different prefactors (table above). 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, revised 2026-08-04 (this page not updated until 2026-09-05): positing a₀ = cH/2π (the relation is posited, not derived from C) is not cost-free — taken at every epoch it is a commitment to a₀(z) = cH(z)/2π. That differs from a constant a₀, but it is prior art (cosmologically coupled MOND), not this framework's own, and it does not discriminate on the RC100 sample. It has been tested: Ciocan et al. 2026 (MUSE-DARK III, A&A 709, L16; arXiv:2604.22613) fit the RAR directly in 79 galaxies at 0.33 < z < 1.44 and find a₀ growing with redshift. How fast depends on the estimator: the paper's global line gives a₀ ∝ E(z)1.15, faster than H(z); the site's fit to the paper's four quantile bins gives n = 0.71 ± 0.13, slower than H(z) (bins read off the paper's figure, whose bars state no confidence level; read here as 1σ). Details below. This sentence read “disfavored by 2σ–6σ” for a month after Parameter Derivations had already revised the verdict: the level deviation depends on the anchor (four published a₀(0) values disagree by 69%; against the McGaugh+2016 anchor the site uses elsewhere, branch (A) is consistent at 0.5σ; revised 2026-10-12: the anchor does not enter the slope, the prediction's own statistic, below), and ΛCDM+baryons simulations predict the same growth (Mayer et al. 2023, MNRAS 518, 257). Verdict: non-discriminating, not disfavored, and not counted. The coincidence has a second face the site long omitted — a₀ ∝ c√Λ, which predicts zero evolution and fares worse against Ciocan than the H(z) branch; if the paper's global n = 1.15 holds, a₀ running faster than H(z) embarrasses both readings of the 1983 coincidence, not this framework specifically (on the bin fit, 0.71, it runs slower instead). Full four-anchor table on Parameter Derivations, row 4. The three numbers, like for like: measured a₀(z ≈ 1) = 2.38 (+0.12/−0.10, a 95% CI; 1σ ≈ 0.06) (Ciocan); constant a₀ = 1.20 (McGaugh+2016 anchor): ~21σ off on the measurement error alone (the paper's own figure is 19σ), 4.4σ once the anchor's own ±0.26 is carried; cH(z)/2π on the same anchor: 2.15, which is ~4σ and 0.5σ under the same two conventions. (Corrected 2026-10-12: Ciocan's errors are 95% CIs; earlier σ's divided them as 1σ, so the measurement-only figures shipped 2026-10-10 were about half size. The anchor-carried figures barely move.) Against a constant a₀ the forced commitment comes out closer, under either convention. The level comparison is not the prediction's own statistic, though. Branch (A) is a ratio, a₀(z)/a₀(0) = E(z); its statistic is the slope n in a₀ ∝ E(z)n with the level free, and the anchor does not enter the slope (adding the SPARC anchor moves n by 0.05; explorer 2026-10-10, pre-registered). On the slope: Ciocan's four quantile bins give n = 0.71 ± 0.13, which excludes constant a₀ (n = 0) with the normalisation free at 5σ (bars read as 1σ; this is Ciocan's own apparent-a₀ result, degenerate with a drift in Υ or gas fraction) and n = 1 at 2.1σ; the paper's own global line gives n = 1.15 and its MOND-track refit 0.99. n = 1 lies between the paper's two estimators, which disagree with each other at 3σ: the row is estimator-dominated; the anchor does not enter the slope. That is not evidence for the framework. ΛCDM+baryons sits at n ≈ 1 too: Mayer, Teklu, Dolag & Remus 2023 (MNRAS 518, 257; arXiv:2206.04333) find the RAR-fitted a₀ grows by a factor ≈3 from z = 0 to 2 in Magneticum ΛCDM simulations, which is n ≈ 0.99. And at high z, “a₀(z) rises” and “the stellar M/L (Υ), gas fraction or pressure-support correction drifts with z” predict the same shift in the RAR intercept, so the level and the slope both inherit that degeneracy. The other intermediate-z sample, RC100, gives n = 0.0 ± 0.5 (galaxy-bootstrap 95% interval about [−1, +1]; explorer 2026-10-06), 1.4σ from Ciocan's bins: RC100's slope error (±0.5) is too wide to test either. (Revised 2026-10-12: a sentence here, shipped 2026-10-10, said the two samples conflict more than either conflicts with any model; at 1.4σ that was not supported, and it was withdrawn.) Full slope table on Parameter Derivations, row 4. Why it is not counted either way: the site's counting rule (glossary, “Refutation (executed)”) admits a test only when a failure condition was stated before the comparison. TEST-25's was pre-registered, so its failure counts although MOND shares it; a₀(z) had no failure condition registered before Ciocan published, so neither its miss on three anchors, nor its hit on one, nor the bins' 2.1σ slope miss enters the tally. One rule, applied both ways. Why not use high-z Tully–Fisher zero points instead? They cannot decide this: Milgrom (2017, §4) already notes that Vmax zero points do not constrain a₀(z), and modelled, the lever is 0.12–0.24 dex against a 0.25–0.35 dex spread between two surveys measuring the same relation at z ≈ 0.9 (explorer 2026-09-19). The level-free test, the slope of a₀(z) against E(z), has now been run on both direct-RAR samples (RC100 2026-10-06, Ciocan's bins 2026-10-10; results above).
The Dimensional-Analysis Chain (not a derivation)
Step 1: Critical Density of the Universe
Standard cosmology. The density at which the universe is flat. This is measured, not assumed.
Step 2: The Literal Computation Gives cH₀/2, Not cH₀/2π
This step used to say the threshold is “the gravitational acceleration from ρcrit over a Hubble-scale volume” and that “the 2π factor arises from the spherical geometry of the causal horizon.” Do that calculation: a sphere of radius R = c/H₀ filled at ρcrit has M = (4π/3)ρcritR³, so
That is 3.27 × 10−10 m/s² — a factor π above the value in Step 3 and 2.7× Milgrom's. Spherical geometry supplies 4π/3 against 8π/3, which cancels to ½; no 2π appears anywhere in it. Step 2 does not produce Step 3. The 2π is a choice that lands near 1.2 × 10−10; no argument on this site or in the archive derives it. What is left is the dimensional statement a₀ ~ cH₀ × (a number of order 0.1–1), which is Milgrom's 1983 coincidence (see the prefactor table) and is badged accordingly on Parameter Derivations. The open question this leaves: is there any construction in the framework that fixes the prefactor before looking at Milgrom's number? None is known.
Revision notes
2026-09-19: a graduate-student reader carried out the computation the old Step 2 described and got cH₀/2. Checked by hand and confirmed. The three cards had been formatted as a derivation since the page was written, while the page badge and Parameter Derivations already said “dimensional analysis”.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%.
Comparison
MOND (Milgrom 1983)
- a₀ is a fitted acceleration scale; whether it is constant is open within MOND
- Value determined empirically from galaxy fits
- The near-equality a₀ ≈ cH₀ has been discussed in the MOND literature for decades, with no accepted derivation
- Extremely successful at fitting rotation curves
Synchronism
- a₀ ≈ cH₀/2π is posited, not derived from C
- Value from dimensional analysis of H₀ and c (shared with other frameworks)
- Two interpolating functions have been used, in different fits: early RAR comparisons used the standard McGaugh et al. (2016) ν; the framework's own tanh-log compander, when its γ is left free, lands on γ ≈ ½, which is Milgrom's simple μ exactly (C = x/(x+2)). These are different functions (ΔBIC between them is quoted on Coherence Function); Galaxy Rotation says which fit used which
- Predicts EFE = 0 structurally for the strict C(ρlocal) reading — the load-bearing premise is that C is independent of Φ, not locality per se (archive 2026-08-24: a fully non-local, Φ-independent C still gives EFE = 0 to 10−13; a ∇Φ-keyed C gives 0.046). At the fitted γ = ½ the compander that actually reproduces rotation curves is keyed on acceleration and its EFE is MOND's identically — so the EFE = 0 prediction has no live carrier that fits galaxies. If one reads the acceleration-keyed compander with EFE = 0 anyway, its Solar-System floor is γ ≳ 0.87 from planetary precessions (the direct tail, explorer 2026-10-15), not Cassini's quadrupole, and its next adjudicator is the galaxy-side EFE detection on SPARC (Chae et al. 2020, ApJ 904, 51: >4σ), which has not been confronted in-house: untested, not refuted. See correction below and the reconciled caveat on Tier 1
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 — which is Refracted Gravity's field equation, Matsakos & Diaferio 2016, published a decade before this sector rediscovered it, with a permittivity identical to the floored C in closed form — reducing to g = gbar/C only in spherical symmetry) 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 satisfies the Strong Equivalence Principle by construction and predicts EFE = 0 exactly. That holds for the algebraic reading only. The field-equation completion refracts an external field, as the next box shows. It is a sharper structural claim than “0.3–0.4× MOND.” Three qualifications an expert reader supplies immediately, stated here so they don't have to: (1) EFE = 0 is the Strong Equivalence Principle, which is also ΛCDM's prediction for internal galaxy dynamics — so this channel is refutation-only: a confirmed EFE detection wounds this framework and ΛCDM alike while selecting MOND, and no EFE outcome can ever select this framework over ΛCDM. (2) The reported ~4σ detection (Chae, Lelli, Desmond, McGaugh, Li & Schombert 2020, ApJ 904, 51) is contested in the literature — a ΛCDM-mimicry argument (Paranjape & Sheth 2022) and a no-EFE result in Coma ultra-diffuse galaxies (Freundlich et al. 2022) — a debate cited here in both directions, noting it currently cuts in this framework's favor. (3) Per the ledger's 2026-08-05 execution, EFE = 0 is not-evaluable against Chae's data anyway: at Chae's own measurement radii the framework's density law misses the rotation curves by 3–4 dex — 38–92× the entire EFE signal — so a 10–17% velocity effect cannot refute (or support) a model whose baseline is off by 10³–10⁴ on the same points.
• On its own it does not conserve momentum. Integrating the force density −ρ∇Φ over an isolated system leaves a net self-force F = −(1/8πG)∫|∇Φ|²∇C d³x. That vanishes for reflection-symmetric C profiles (hence spherical symmetry), not in general: a lopsided system would push itself. The variational form, with action term −∫C(ρ)|∇Φ|²/8πG, adds a striction force density −ρ∇(C′(ρ)|∇Φ|²/8πG), and that cancels the self-force exactly. Momentum conservation is the striction term. The site's Refracted Gravity disc runs (For Researchers) and the globular-cluster window solve the field equation without it.
• Striction is not small wherever the knee is inside the object. At this framework's A·V² knee it is ≤2×10⁻⁵ of gravity, because that knee never enters a disc. With the knee inside a disc it reaches 164× the vertical restoring force. At globular-cluster knee shells it is 1.7–13× gravity and points outward (explorer, 2026-09-16).
• The external field is refracted, not ignored. Linearity in Φ gives superposition, but an external field is solved with the system's own C(x), so wherever ∇C ≠ 0 it bends, like a field in a dielectric. Without striction (“L2”) that is a linear external-field effect of order gext·ΔC/C: about 0.3–0.37 gext beyond the knee in a halo globular cluster. With striction (“L3”) the internal residuals are 1–30 gext and there is a quadratic term, a nonlinear EFE. EFE = 0 holds exactly only for the algebraic reading, or when C is uniform across the system. This linear refraction EFE is distinct from MOND's and has not been confronted with data. For Refracted Gravity it is order unity for low-density satellites, about (Δε/ε) × (gext/gint), and dipolar (m = 1, aligned with the host), unlike Newtonian tides, which are quadrupolar. Separately, RG's spherical ceiling 1/ε0 (exact by Gauss's law) is below the boost every classical dwarf spheroidal needs under RG's disc calibration (≥ 4σ, including Leo I at 9–15σ, which RG keeps in equilibrium; explorer 2026-10-21). The effect belongs to the field-equation class, so it is RG's prediction (prior art). This framework's own permittivity is already excluded on SPARC and by lunar laser ranging, so testing it here could add a second root to that refutation but could not move the confirmed count (what “excluded” covers).
• Floored vs unfloored C. “The knee never enters a disc, so none of this touches the framework's parameters” holds for the floored form, where C ≈ Ωm below the knee. For the unfloored form C = γρ/ρc below the knee, ∇ln C = ∇ln ρ exactly, whatever the knee and γ. The equation then becomes ∇²Φ + ∇ln ρ·∇Φ = 4πGρc/γ, and refraction is order R/h in any thin disc.
The vacuum problem remains in both forms: empty space has C → 0 (or the floor), and that controls the exterior field of an isolated mass. Full derivations:
explorer/findings/efe-zero-survives-momentum-objection-but-the-substitution-was-never-evaluated.md, explorer/findings/l2-is-not-l3-for-a-disc-and-the-action-adds-a-force-the-tests-omit.md, explorer/findings/under-the-action-gc-knee-shells-are-striction-dominated-and-the-gc-window-is-an-l2-object.md.Revision note
From 2026-08-04 this box was titled “EFE = 0 survives the obvious momentum-conservation objection”. It said the completion “conserves momentum” and that, being linear in Φ, “EFE = 0 is preserved exactly.” The first is true only in spherical symmetry or with striction included. The second confuses superposition with the absence of refraction. The research archive recorded the momentum point on 2026-08-26 and the refraction point on 2026-09-16; a researcher simulated-visitor persona found the box still asserting both on 2026-09-17.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 is a second distinct prediction, separate from the refraction above, and like it, untested and on the excluded density-keyed branch. A tidal argument (archive, 2026-08-21) estimates it is systematics-limited at signal/systematics ≈ 4×10−3.
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 the 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. Scope: that run refuted the registered amplitude (> 20% of the scatter from Mpc-scale density). It did not hold gext fixed or measure gas density at a satellite, so it did not run the two-satellite discriminator above. Two independent expert simulated-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 — Nested inside MOND's interval — 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 (with the observed a₀ = 1.2×10⁻¹⁰; the framework's cH₀/2π gives 9.1 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
The dimensional relation a₀ ≈ cH₀/(2π) was proposed in Sessions #87–88 of the autonomous research program. Session #91 reached it again via Freeman's Law, but that is the same relation read backwards: Σ₀ = a₀/(2πG) is an exact rewriting of a₀, so the two routes are one number, not two derivations, and their agreement is not a consistency check. As Step 2 says, no calculation on this site or in the archive yields the 2π (the literal Hubble-sphere computation gives cH₀/2). The match is also weak evidence on its own: a grammar of 43 simple prefactors on cH₀ hits 89% of targets in the range 0.1–0.3 cH₀ to within ±10%, and the archive has used five such routes. The framework fixes no relation between the dark-energy knee and a₀′. A link through the constant knee would force a constant a₀ (n = 0 in a₀ ∝ E(z)ⁿ), which Ciocan's redshift bins disfavour at about 5σ, conditional on a redshift-flat gas systematic. Different z = 0 matches imply n anywhere from −1 to +1.7, so the slope, not the level, is the test (explorer finding shared-scale-rho-lambda-a0-the-level-is-free-and-the-slope-is-the-bet.md).
Revision note
Until 2026-10-15 this paragraph read “Derived in Sessions #87–88 … independently stress-tested in Session #91 … Both derivations agree, providing internal consistency.” A simulated-visitor graduate-physics persona pointed out that it contradicted Step 2 and /parameter-derivations.Related Concepts
Revision notes (history; not the current claim). Dated corrections moved here from the running text on 2026-11-08, verbatim, in page order.
- Revised 2026-10-12: this sentence read “has not been run” after both runs.
- Corrected 2026-07-22: the 1.08 / “~10%” previously shown here belongs to H₀ = 70, not the 67.4 stated in this very step.
- the direction the site's own fDM = 1−C identity implies — corrected 2026-08-04, this previously read “C(ρ)·g,” the opposite direction
- rewritten 2026-08-12; this sentence previously ended “already in tension with Chae et al. 2020”, a claim the research ledger withdrew on 2026-08-05
- Until 2026-10-21 this sentence called it “the one genuinely distinct external-field prediction of this family” with no scope.