Galaxy Rotation Curves
Untested — TEST-03 Never Run As Registered — see correction belowGalaxy rotation curves are the most important empirical test for any theory of gravity at galactic scales. Stars and gas in the outer regions of disk galaxies orbit faster than Newtonian gravity predicts from visible mass alone. The Radial Acceleration Relation (RAR) captures this: observed acceleration correlates tightly with the acceleration predicted from baryonic mass, but systematically exceeds it below a critical scale.
One structural reason to expect this fit to reparametrize rather than derive MOND: C(ρ) is a function of local density, while the RAR is organized by gbar, a non-localenclosed-mass acceleration. A pointwise map cannot in general reproduce an acceleration-space relation across systems with different mass geometries except by per-system calibration — see the local-vs-non-local structural no-go. The Galaxy Curve Plotter's hand-tuned radius stand-in is that no-go showing through in a single demo.
The literature already contains both this obstruction and its known escapes (noted 2026-07-10; corrected 2026-07-27): density-dependent transitions are a mature field (symmetron/chameleon screening). This paragraph previously claimed that “the ρ-vs-gbar variable mismatch is the standard reason screened scalars don't reproduce MOND.” That attribution was never sourced and is wrong: the screening literature does not make it, and Burrage, Copeland & Millington, PRD 95, 064050 (2017) reproduce the radial acceleration relation for 153 SPARC galaxies with a scalar keyed on local volumetric ρ(r) and universal Lagrangian parameters. Their extra force is the gradient of a function of ρ, not a multiplier on g — which is what lets a density-keyed field track an acceleration-keyed relation, and is precisely what C(ρ) does not do. See the counterexample paragraph on For Researchers. The other constructive escape is superfluid dark matter (Berezhiani & Khoury 2015): a local density criterion switches on a condensate, but the MOND-like force is carried by the condensate's phonons — the force scale enters independently of the switch. C(ρ) conflates switch and force into one local function; that conflation is what the data kill. See the escape taxonomy on For Researchers.
Dataset Results
SPARC Dataset
175 galaxies with high-quality photometry and resolved rotation curves (Lelli, McGaugh & Schombert 2016). The gold standard for RAR studies. [Public data]
- Tight RAR confirmed (σint ≈ 0.057 dex) — McGaugh et al. 2016 measured value, restated
- Coherence function fits within observational scatter
- Environment-dependent effects visible but sample too small for strong statistics
ALFALFA-SDSS Dataset
14,585 galaxies from the ALFALFA HI survey cross-matched with SDSS photometry. Unresolved rotation curves but massive statistical power. [ALFALFA data] [SDSS DR17]
- Environment-dependent RAR scatter detected at p = 5 × 10−6, R² = 0.14 — but this statistic is SPARC-scale (N ≈ 130–175), not a valid measurement of this 14,585-galaxy sample (see correction below). The registered TEST-03 environment-density test on this sample was never run.
- σint = 0.086 ± 0.003 dex — CDM-consistent (z = +0.5 in the source session's definitive run), not below CDM. An earlier −6.2σ “below CDM” reading from an intermediate session was retracted in-archive as premature once distance-noise modeling was added; the verdict is also modeling-choice-dependent (z ranges +0.5 to +64). See CDM Discrimination for the full history.
- Cluster vs. field galaxies show different scatter — the effect is real but sub-threshold, not a confirmed prediction
The Interpolating Function
Attribution: The equation below is the standard RAR interpolating function from McGaugh, Lelli & Schombert (2016) — already in the literature for a decade and widely used to fit SPARC. Synchronism's specific contribution is not the function itself, but the environmental scatter ansatz on top of it (the claim that σint depends on local density). All fit-quality claims refer to that ansatz and to the McGaugh-2016 baseline together.
In Synchronism, the acceleration scale a₀ ≈ cH₀/(2π) is a 40-year-old dimensional coincidence (Milgrom 1983), discussed by Famaey & McGaugh (2012), McCulloch, Verlinde, and Smolin under different theoretical umbrellas. The 2π prefactor is chosen to match the observed value; cH₀/(2π) ≈ 1.04×10⁻¹⁰ m/s² at the site-standard H₀ = 67.4 undershoots the observed 1.2×10⁻¹⁰ by ∼13%. Synchronism's contribution is the coherence-function mechanism that provides areason for this scale — not the identification itself. See Honest Assessment and Parameter Derivations for prior-art acknowledgment and the canonical reparametrization framing.
Key Results Summary
Structural Failure: Dark Matter Mechanism (March 2026)
Synchronism's CFD viscosity interpretation mapped low coherence (dark matter) to high viscosity — predicting dark matter should be stickier than baryons. The Bullet Cluster (1E 0657-558) shows the opposite: dark matter halos pass through each other with negligible self-interaction (σ/m < 0.47 cm²/g, Harvey et al. 2015). That specific ansatz has the wrong sign — see the full failure analysis.
Scope note (added 2026-07-09): this is a documented failure of the specific CFD viscosity ansatz, not of Claim 3's headline form (“dark matter is incomplete decoherence”), which asserts a modified gravitational response, not a substance with a viscosity to get the sign of. The Bullet Cluster still refutes any local-density-sourced-gravity reading of Claim 3, via a different and stronger argument: the lensing-mass peaks track the collisionless galaxies while the X-ray gas — most of the baryonic mass — is stripped and lags behind (Clowe et al. 2006, ApJ 648, L109). Any theory where gravity responds to localbaryon density must put the extra gravity where the baryons are; it isn't there. This is the same local-vs-non-local no-go documented above, applied at cluster scale — not a second, independent failure. (MOND needs additional collisionless mass at cluster scale for the same reason: Angus, Famaey & Zhao 2006.)
The galaxy rotation fit results on this page are independent of the CFD interpretation and stand as-is (reparametrization of MOND with an environmental scatter term).
Honest Caveat
The environment-dependent scatter is real and statistically significant (p = 5×10−6), but the R² = 0.14 / p = 5×10−6 pair is only mutually consistent at SPARC scale (N ≈ 130–175) — see correction below. Furthermore, standard MOND plus mass-to-light ratio corrections already explains essentially all of the RAR variance. Synchronism adds at most a small, detectable effect on top of what MOND already provides — it does not replace MOND's success, and MOND's own External Field Effect predicts the same qualitative environment dependence (Chae et al. 2020/2021), so the effect does not discriminate the two.
Correction (2026-07-09) — TEST-03/TEST-05 provenance: This page previously said R² = 0.14 was TEST-03's result on this 14,585-galaxy sample and that it tripped TEST-03's kill criterion. It doesn't: at N = 14,585, R² = 0.14 implies p of order 10−500, not 5×10−6 — the pair is self-consistent only at SPARC scale, TEST-05's sample. TEST-03's actual environment-density result on this sample was never computed as registered. TEST-05, which does own this statistic on its registered terms, met its own criterion (rejecting independence) — briefly reclassified MOND-shared (2026-07-09), a tie dissolved on adjudication 2026-07-15: the two frameworks' environment levers differ by ~50×–5,000× (local density vs external acceleration), so the axis discriminates; honest status is “never run as registered.” Full trace on Tier 1: TEST-03 and TEST-05.
ΔBIC note: The incremental value of the environmental scatter term above baseline MOND has not been quantified via ΔBIC on the ALFALFA-SDSS dataset. A separate ΔBIC analysis was run for the RAR transition-shape question (see below) — which is the more fundamental discriminating question.
RAR Transition Shape: The Decisive Test — CLOSED (2026-05-21)
The environment-dependent scatter tests (TEST-01, TEST-05) probe the size of deviations from MOND's RAR. A more fundamental question is whether the compander's functional form (μSyn = tanh(γ ln(1+x)), γ=2) differs from McGaugh's interpolating function ν(x) = 1/(1−e−√x) in the shape of the transition — the only test that could discriminate the two frameworks without assuming any environmental coupling.
This test was run on 2807 real SPARC points (Lelli-McGaugh-Schombert 2016, 10% velocity-error cut) with a₀ free and standard M/L priors. Result:
| Model | RMS (dex) | ΔBIC vs McGaugh |
|---|---|---|
| McGaugh ν (standard MOND) | 0.1437 | — (reference) |
| Compander μ, γ=2 pinned | 0.1485 | +184 |
| Compander μ, γ free (best fit: γ=0.49) | 0.1437 | +7.1 |
Kill criterion triggered. ΔBIC = +184 ≫ 10 refutes the γ=2 compander as the galaxy mechanism. Conservative correction for intra-galaxy point correlation (effective N≈500–1000): ΔBIC ≈ 33 — still decisive. The residual is a coherent S-shaped ≈0.05–0.10 dex signature at the transition, significant at ~8σ per bin.
Note on γ=2: γ=2 follows from Ncorr=1 (individual stars taken as uncorrelated), but Ncorr is asserted, not independently counted. The SPARC-preferred γ≈0.49 back-implies Ncorr≈17, contradicting the independent-stars premise. γ=2 is the most charitablepin for galaxies given the framework's own logic — not a prediction derived from independent evidence.
The fork that closes the question: free-γ converges to γ≈0.49 with RMS identical to McGaugh to four digits. ΔBIC=+7 is entirely the BIC penalty for the extra parameter — the fit improvement is zero. Made quantitative (2026-07-27, external-reviewer point): with N = 2807, ln N = 7.94, so one added free parameter costs +7.94 in BIC before any likelihood is considered. The observed +7.1 therefore implies Δχ² ≈ −0.84 — the free-γ compander fits the data marginally better than McGaugh's ν and loses purely on the complexity charge for a knob MOND does not need. That is a sharper statement than “collapses to MOND equivalence”: the compander is a strictly worse-parameterized reparametrization of the RAR, buying Δχ² < 1 for one degree of freedom. (Bookkeeping the fit record should state: whether a₀ was floated in both arms, i.e. Δk = 1 or 2. At Δk = 2 the penalty is 15.9 and the compander fits better by Δχ² ≈ 8.8 — different number, same conclusion.) Note the contrast with the γ=2 row: ΔBIC = +184 there is pure likelihood (Δk = 0). The two numbers do different work and are not one ladder. There is no γ for which the compander is both (a) distinct from MOND and (b) consistent with SPARC. Pin γ=2 → refuted. Fit γ → MOND.
Why 0.49? Mechanism identified (2026-07-22 form-selection run): γ controls the Newtonian-return exponent — algebraically, tanh(γ·ln(1+y)) approaches 1 like (1+y)−2γ, so the return exponent is q = 2γ. The fit pins q ≈ 0.98, cross-validated by an independent free-Hill fit landing n = 0.975 — two parametrizations converging on q ≈ 1, the value hard-coded in MOND's “simple” μ-function. γ = 0.49 was never a constant awaiting derivation; it is the tanh family's encoding of MOND's simple μ. The framework's asserted γ=2 means q=4 — re-Newtonianizing far too abruptly — which is the +184 kill restated mechanistically.
The fitted γ has a consequence beyond MOND-equivalence: the Solar System (TEST-25, added 2026-07-28). γ ≈ 0.49 landing on MOND's simple-μ family (q ≈ 1, above) is not merely a reparametrization finding — it exposes the compander to a constraint that has nothing to do with SPARC. Simple-μ's slow high-acceleration return is disfavored by planetary ephemerides (Blanchet & Novak 2011; Hees et al. 2016), because a slow return produces an anomalous perihelion precession under the same Galactic external field that shapes the RAR. A pre-registered joint squeeze (Cassini quadrupole bound vs the full SPARC-retained γ interval, ΔBIC ≤ 10 ⇒ γ = 0.425–0.600) found a robust empty intersection: every retained γ value fails Cassini by +17.7σ to +18.0σ. No γ satisfies both galaxy rotation and the Solar System under one scale-universal tanh-log QUMOND realization — this needs no new γ preference from SPARC, only the one already fitted. See TEST-25 for the full scope statement (it closes the realization, not modified inertia, dark-matter models, or the umbrella ontology).
Why the intersection is empty (added 2026-07-30, visitor persona pass): a tail-shape mismatch, not a tuning failure. The compander saturates as a power law, 1−C = 2(1+x)−2γ; McGaugh's simple-μ approaches Newtonian as an exponential, ν−1 ~ e−√x. At Saturn, x = gSaturn/a₀ ≈ 6.5×10⁻⁵/1.2×10⁻¹⁰ ≈ 5×10⁵. At the SPARC-preferred γ ≈ 0.489 the compander's fractional anomaly is ≈2·(5×10⁵)−0.978 ≈ 5×10⁻⁶; the exponential form gives e−√(5×10⁵) = e−707, indistinguishable from zero at any precision. The two tails are numerically identical over the SPARC range (10⁻² < x < 10²) — which is why “curve-equivalent to MOND” holds there — but a power-law tail decays far slower than an exponential one, so any sufficiently strong field separates them, and the Solar System is far enough into that regime that the gap is five-plus orders of magnitude rather than a fitting margin. This is the framework's one genuinely non-MOND-degenerate piece of physics, and it is a direct consequence of choosing a power-law compander for the deep-field limit — the same choice that gives the bounded-C formulation its finite boost ceiling (Bmax = 1/Ωm, see Parameter Derivations). Full derivation back-annotated to the research archive alongside TEST-25's registered result.
Net discriminating galaxy tests vs MOND: 0, by execution. Script + full analysis: explorer/scripts/rar_transition_shape_real_sparc.py, finding: explorer/findings/rar-transition-shape-real-sparc-result.md.
Wide Binaries
The RAR environmental scatter (TEST-03) tripped its kill criterion, and the RAR transition-shape test (above) has now closed the main galaxy-scale question. The wide binary test (TEST-02) was previously identified as a discriminator, but with the compander collapsing to MOND at its best-fit γ, the wide-binary question is now whether MOND+EFE and the compander (at fitted γ≈0.49) diverge in a density-stratified wide-binary sample. Wide stellar binaries (separations > 0.1 pc) probe the sub-a₀ acceleration regime in a fundamentally different mass and density environment from galaxy rotation curves — no dark matter halos, no baryonic feedback, just two stars in a nearly Keplerian orbit.
The key papers are Pittordis & Sutherland (2023) and Hernandez et al. (2024), using Gaia DR3. They disagree on the strength of the MOND signal. Synchronism predicts a density-dependent wide binary anomaly (stronger anomaly in denser stellar fields). Gaia DR3 is public; stratifying the sample by local stellar density is a tractable one-week analysis. See TEST-02 for the kill criterion.
GW170817 and the Speed of Gravity
The binary neutron star merger GW170817 constrained |cGW − c|/c < 10−15, killing TeVeS, Bekenstein-Sanders, and large swaths of Horndeski parameter space (Sakstein & Jain 2017, Ezquiaga & Zumalacárregui 2017). Synchronism is marketed in part as a modified-gravity alternative, so this constraint is relevant.
Synchronism's posture: The coherence function C(ρ) is not a tensor-vector-scalar theory and does not introduce new propagating tensor degrees of freedom. It does not modify the graviton kinetic term or introduce derivative couplings of a scalar to the metric that would change gravitational wave speed. Therefore the framework's prediction is cGW = c by construction, and GW170817 does not apply.