Galaxy Rotation Curves

Failed — density-keyed form ΔBIC +2843; γ = 2 compander disfavoured at about 2σ: ΔBIC ≈ +11 at galaxy level and +184 by point with Υ frozen, +2.7 and +46 with each law's own Υ Reparametrization — Acceleration-keyed fit is a MOND μ at every γ; MOND's simple μ exactly at γ = ½
γ ≈ ½ twice is not a cross-sector agreement. At γ = ½ the function satisfies 1/C − 1 = 2/x exactly. Keyed on acceleration (the galaxy fit's variable) that is MOND's simple interpolating function; keyed on the mean matter density (the cosmology's) it adds a constant to the Friedmann equation, which is Λ. So the galaxy fit (γ = 0.489) and the DESI fit (γ = 0.487) are each dataset choosing its own standard model, in different variables, at the one point where this family contains both. The match has no power to fail and is not evidence for the framework. Details: the cosmology side.

Sub-note: the registered ALFALFA test, TEST-03, was never run as registered (see correction below), so that one test is untested. Until 2026-09-19 that was this page's only badge, which undersold the page's own opening verdict; the two badges above are the ones the Galaxy Plotter already carried for the same content.

Galaxy 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.

Does Synchronism replace dark matter? No, not on current evidence. The idea was that gravity's strength depends on how densely matter is packed, which would make the extra pull without invisible matter. Written that way, keyed on density, the equation does not fit galaxies. In a real disc it barely changes gravity at all, which you can see on the Galaxy Curve Plotter. A version keyed on acceleration, with one dial fitted, does fit, but at that setting it is exactly MOND's formula. So it carries MOND's successes and MOND's problems, including the Solar System test below, and it never does better. Everything after this box is the technical detail behind those three sentences.
Read this first (promoted to the top 2026-07-23): the headline equation never reaches this page's fits. When this box was written no chain from C(ρ) to gobs was stated anywhere in the framework. Two have since been written down (division gobs = gbar/C, and a field equation that turned out to be Refracted Gravity's; see how C enters dynamics), and the density-keyed law has been fitted through them and lost. But none of the fits on this page use it — the galaxy fits shown below use McGaugh's standard RAR interpolating function (MOND phenomenology), with Synchronism contributing only a scatter hypothesis on top (since refuted by the 2026-07-14 environment run, filed on Tier 1 as TEST-03s — not the Freeman-law card TEST-08). The structural reason is in the next paragraph; the tested-and-killed compander variants are further down.How C entered the RAR fit, in one sentence: the compander was used as an implicit interpolating function keyed on gobs — the script solves gbar = gobs·tanh(γ ln(1 + gobs/a₀′)) for gobs with a₀′ profiled (simulations/sparc_tanhlog_profile.py, research repo, line 85). At γ = ½ that is Milgrom's simple μ with a₀ = 2a₀′. It was not evaluated explicitly on gbar (which would give an additive floor gbar + 2a₀′, not MOND), and not on ρ. So every quantitative galaxy fit on this page is a fit in MOND's own variable; the density-keyed law is tested on the Honest Assessment (head-to-head ΔBIC +2843, the Oort limit, and the globular-cluster fork).

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.

C-convention note (2026-08-04): the site ran three mutually exclusive readings of the substitution above: the Galaxy Curve Plotter originally coded v² = vb² + (Vflat·C)² (since its 2026-10-08 rewire that is the dashed curve; the solid curve is floored division, below), i.e. gobs = C·gbar shifted into an amplitude term; the fDM = 1 − C identity on Tier 1 implies gobs = gbar/C; and Relation to MOND's prose then said gobs = C·gbar directly — since corrected there to gbar/C, so two readings are live today: the plotter's amplitude term and gbar/C, whose field-equation form (and where the two part company in a disc) is written out on Coherence Function. On the site's own five plotter galaxies these three laws miss the observed curve in three different directions — roughly 10²–10³× high, converging on Newtonian, and 10²–10³× low — and no calibration reconciles them, because the disagreement is a functional-form mismatch (delivered boost falls exponentially with radius under an exponential disk; a flat curve needs it to rise roughly linearly), not an amplitude one. This plotter shows the amplitude reading only; it is not necessarily the framework's asserted one.
The published density law is handed Vflat twice, so it cannot predict a flat curve (stated in one place 2026-11-10). The calibration ρcrit = 0.029·Vflat² sets the knee from the galaxy's asymptotic speed, and the display wiring v² = vb² + (Vflat·C)² sets the boost's amplitude from the same speed. Vflat is the quantity a rotation-curve law exists to predict. A law whose knee and amplitude are both set by it is handed the answer, so a match would not have counted, and the mismatch it actually produces (C ≈ 10−5 at disc densities, a boost at the Sun of 3.5×104 under the division wiring) is a separate failure. The Milky Way knee at this calibration is 1.5×10³ M☉/pc³ and DDO 154's is 64, against disc midplane densities of 0.03–0.1, so every galaxy sits on the linear branch. This was said in pieces on Coherence Function, Parameter Derivations and the Galaxy Curve Plotter; a graduate-physics reader asked for it as one sentence. It is also an MRH violation on the framework's own terms: C(ρ(r), Mtotal) is not C(ρ(r)).

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
  • The coherence function fits within the scatter only at free γ ≈ 0.49, where it is MOND's simple μ (inherited, not a Synchronism result). The framework's pinned γ = 2 is disfavoured: ΔBIC ≈ +11 with the galaxy as the unit (1.6σ by galaxy-level cross-validation with Υ, D, i frozen; 1.9–2.3σ with them refitted per galaxy), at the threshold, not decisive. Read as points treated independently, it is +184.
  • Environment dependence: null. The 2026-07-14 environment run (Tier 1 TEST-03s; the archive numbers it S177 / TEST-08) gives r² = 0.0001 (N = 141, p = 0.89) against a registered prediction of > 20% of the scatter and a kill bar of r² < 0.09, so the kill fired (see Tier 1). These two bullets read “fits within observational scatter” and “environment effects visible” until 2026-09-27.

ALFALFA-SDSS Dataset

14,585 galaxies from the ALFALFA HI survey cross-matched with SDSS photometry — that is the registered cross-match; the definitive session actually analysed 14,435 after a quality cut. Unresolved rotation curves but massive statistical power. [ALFALFA data] [SDSS DR17]

  • No environment result exists for this sample: the registered TEST-03 environment-density test on it was never run. The p = 5 × 10−6, R² = 0.14 figure this bullet used to show is a different measurement on a different sample (SPARC, N ≈ 130–175) of a different predictor: Hubble type, not environment (see the Honest Caveat below).
  • σint = 0.086 ± 0.003 dex is not a BTFR scatter: it is independent of V (shuffling V across galaxies gives 0.0870 vs 0.0864; explorer 2026-10-23), the residual of an in-sample photometric mass model, so it has no CDM or MOND verdict (corrected 2026-10-24; the like-for-like inverse BTFR on the same 677 galaxies is 0.073 dex in log V, against abundance matching's 0.064). Before that it read: equal to the source session's own CDM figure, not to published ΛCDM predictions (Desmond 2017 abundance matching gives ~0.25 dex; a verdict suspended on 2026-10-09 and withdrawn 2026-10-24; 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 scatter: no executed comparison on this sample.

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). That ansatz was tested on SPARC with an external density catalogue and killed (r² = 0.0001, N = 141; see the Honest Caveat below). What remains is the McGaugh-2016 function.

gobs = gbar / (1 − e−√(gbar/a₀))
RAR interpolating function (McGaugh et al. 2016)

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%. Against this framework's own SPARC fit the comparison changes but gets no sharper: the scale the fit identifies is a₀′/γ (the compander's deep-regime limit; 2a₀′ at γ = ½). It stays at 1.07–1.09×10⁻¹⁰ across the whole Υdisk = 0.4–0.6 band while a₀′ itself moves 3.6×, but only because γ floats along a valley with Υ (0.28 → 0.96 across that band). Pin γ = ½, the reading this page calls surviving, and a₀ moves 1.44 → 1.10 → 0.87×10⁻¹⁰, so closing cH₀/2π needs H₀ ≈ 93 / 71 / 57 at Υ = 0.4 / 0.5 / 0.6 (maintainer/scripts/gamma2_global_upsilon_per_law.py). The scale is identified only jointly with γ. The free-γ value reads 1.16–1.18×10⁻¹⁰ under the velocity-χ² fits. cH₀/2π sits 3–12% below it at H₀ = 67.4. With γ free, closing the gap needs H₀ ≈ 69–70.5 for the Υ-sweep values (inside the Planck–SH0ES range) and ≈ 75–76 for the velocity-χ² values (above SH0ES's 73), so only the low end is absorbed by the H₀ tension. Still Milgrom's 1983 coincidence, still a reparametrization (maintainer/scripts/a0_identified_scale_across_upsilon.py). 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

0.086
σint (dex), ALFALFA–SDSS: not a BTFR scatter: it is independent of V (shuffling V across galaxies gives 0.0870 vs 0.0864; explorer 2026-10-23), the residual of an in-sample photometric mass model; not the RAR scatter either (SPARC RAR ≈ 0.057); which relation(labelled 2026-10-23 as a BTFR scatter; relabelled 2026-10-24, explorer: it is not one)
r² = 0.0001
environment effect on RAR scatter (null; kill fired)
SPARC × Cosmicflows-4, N = 141, p = 0.89 (TEST-03s). This tile showed “5 × 10−6: p-value for environment effect” until 2026-10-21; that p belongs to Hubble type, not environment
14,610
galaxies run (175 SPARC + 14,435 ALFALFA–SDSS (quality cut))
mechanism tested on 175 SPARC curves; the TFR scatter test never ran as registered

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: a reparametrization of MOND. The environmental scatter term proposed on top of it was killed on SPARC (Honest Caveat below).

Honest Caveat

Is there an environment effect in the RAR scatter? On the one run that measured environment, no. SPARC galaxies placed in the Cosmicflows-4 density field give r² = 0.0001 (N = 141, p = 0.89), against a registered prediction of more than 20% of the scatter, so the kill fired. The p = 5×10−6, R² = 0.14 result that this paragraph called “real and statistically significant” environment dependence until 2026-10-21 measures something else: RAR scatter against Hubble type on SPARC (Sessions 376–378). Session 381 then tested whether type stands in for environment. It does not: within each morphological type an isolation proxy predicts nothing (r = +0.003, p = 0.97), and rotation-curve roughness accounts for the type signal (type becomes non-significant once roughness is controlled, p = 0.61). So the significant number is a real morphology effect with a structural cause (late types have rougher curves), and it carries no environment content. Standard MOND plus mass-to-light corrections already explains essentially all of the RAR variance. MOND's External Field Effect predicts an environment dependence of its own; Chae et al. (2020/2021) report it at ~4σ, but Paranjape & Sheth (2022) and Freundlich et al. (2022) show that ΛCDM produces similar correlations, so that detection is not yet a clean MOND signature.

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: No ΔBIC for the environmental scatter term has been computed on the ALFALFA-SDSS dataset; on SPARC the term has no signal to score (r² = 0.0001). 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:

ModelRMS (dex)ΔBIC vs McGaugh (point level, N = 2,807)
McGaugh ν (standard MOND)0.1437— (reference)
Compander μ, γ=2 pinned0.1485+184
Same, each law with its own global Υ (γ=2 at 0.61, McGaugh at 0.54; added 2026-11-09)0.1443 vs 0.1431+46
Compander μ, γ free (best fit: γ=0.49)0.1437+7.1

Kill criterion triggered — under point independence and frozen Υ. Both numbers hold Υ_disk = 0.5 for both laws. With one global Υ per law (γ = 2 best at 0.61, McGaugh at 0.54; same parameter count), ΔBIC is +46 by point (galaxy-block bootstrap 95%: −26 to +132) and +2.7 by galaxy. A noise-free twin puts a McGaugh world at +44 and a γ = 2 world at −43, so the data sit about 2σ from a γ = 2 world, in line with the 1.9–2.3σ from per-galaxy nuisances. Read this row as disfavoured at about 2σ. (maintainer/scripts/gamma2_global_upsilon_per_law.py, PREREG 5a95614; twin post-hoc.) ΔBIC = +184 ≫ 10 refutes the γ=2 compander as the galaxy mechanism if the 2,807 points are independent. They are not: they come from 166 rotation curves, and the mass-to-light, distance and inclination errors move a whole curve together. With the galaxy as the replication unit (run 2026-09-29, pre-registered): γ = 2 is worse than the free fit in 94 of 166 galaxies (57%; exact sign test p = 0.051 one-sided, 0.10 two-sided; the one-sided test asks whether γ = 2 is worse, which is the registered direction); a 10-fold galaxy-level cross-validation prefers free γ by 1.6σ; a galaxy-block bootstrap puts the full-N ΔBIC at 179 with 95% interval [23, 351], a spread that implies an effective N of about 150; and ΔBIC recomputed with N = 166 is +10.9, at the threshold. How +184 becomes +10.9, both at frozen Υ (formula added 2026-10-24, after two simulated-visitor personas could not reconstruct it): the comparison is γ = 2 against McGaugh's ν, each with one fitted acceleration scale, so the parameter counts are equal and the BIC has no penalty term. With Gaussian residuals and a fitted variance, ΔBIC = N ln(RSSγ=2/RSSMcGaugh) = 2807 × ln(0.1485²/0.1437²) = +184. The galaxy-level figure keeps the per-point likelihood ratio and replaces N with Ngal = 166: 166 × 0.0656 = +10.9 (= 184 × 166/2807). A persona reconstruction that treated it as pinned-versus-free γ (Δk = 1) and re-applied a ln N penalty got +6.2; that applies to a comparison this number is not. The bootstrap [23, 351] is the sampling spread of the full-N statistic, not of the galaxy-level one; scaled by the same 166/2807 it is [1.4, 20.8], which straddles 10. That is why both read “at the threshold”. (On the Kass–Raftery scale +11 would read “very strong”; that scale assumes the BIC approximation holds, and the bootstrap spread of ±82 on the full-N value is why this page reads it as 1.6–2.2σ instead.) Seven galaxies carry 98% of the net excess, five pull the other way. So the pin is disfavoured at the threshold, not decisively refuted; the previously quoted “effective N ≈ 500–1000, ΔBIC ≈ 33, still decisive” was a guessed discount (retired 2026-09-29), and the “~8σ per bin” also assumed independent points. Script: maintainer/scripts/gamma2_pin_galaxy_level.py (+ PREREG, output). Refit with the published method (explorer 2026-09-29, pre-registered): with Υ, D and i profiled per galaxy under Li+2018 priors and a velocity χ² (Desmond, Hees & Famaey 2024's treatment), the pin sits at z = 1.9 galaxy-level, 2.2–2.3 with their intrinsic scatter, worse than free γ in 52–56% of galaxies: disfavoured, not refuted. Marginalizing the nuisances does not narrow the galaxy-level interval (its width stays 3–8× the point-level width), and the free fit moves to γ̂ ≈ 0.6 [0.43, 0.95] (a 95% galaxy-block bootstrap interval on the best-fit γ, with Υ, D and i profiled per galaxy). Script: explorer/findings/scripts/gamma2_pin_nuisance_refit.py. (This sentence read “Not done: refitting per-galaxy nuisances” until 2026-10-04.)

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 (with Υ frozen at 0.5 and points unweighted; with per-galaxy nuisances refitted the free fit moves to γ̂ ≈ 0.6 [0.43, 0.95]). ΔBIC=+7 is entirely the BIC penalty for the extra parameter — the fit improvement is negligible (Δχ² ≈ −0.84, derived below). Made quantitative: this is the table's third row, the free-γ compander against McGaugh's ν, scored on points. It is not the galaxy-level +10.9 above, which compares γ = 2 with McGaugh's ν (one fitted scale each, so no penalty term), with N = 166 in place of 2,807. 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: a₀ is floated in both arms, so Δk = 1 — McGaugh's ν is fitted with k = 1 (a₀), the free-γ compander with k = 2 (a₀, γ); see explorer/scripts/rar_transition_shape_real_sparc.py, where the BIC is computed as bic(ssr_F,2) − bic(ssr_M,1).) Note the contrast with the γ=2 row: ΔBIC = +184 there (points as independent; ≈ +11 with the galaxy as the unit) is pure likelihood (Δk = 0). The two numbers do different work and are not one ladder. Every γ is a MOND interpolating function: C ≈ γx in the deep regime, so a₀ = a₀′/γ, and γ only picks the transition shape and the return exponent q = 2γ. So there is no γ at which the compander is distinct from MOND in kind. The γ = 2 result selects among MOND's functions (disfavoured at about 2σ once each law has its own Υ), and free γ at Υ = 0.5 lands near the simple μ.

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. Within a one-parameter family, the SPARC transition shape maps onto q ≈ 1 (a free Hill fit on the same data gives n = 0.975); the asymptotic return itself is not constrained by SPARC — the data reach only x ≲ 10, and McGaugh's exponential-return function fits the same 2,807 points with the same RMS. q = 1 is 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 γ = 2 disfavouring restated mechanistically.

Stated exactly, not asymptotically. The paragraph above argues MOND-equivalence through a tail exponent. It holds identically, and the exact version is both shorter and more damning. Because eγln(1+x) = (1+x)γ,tanh(γ ln(1+x)) = [(1+x)2γ − 1] / [(1+x)2γ + 1]with no approximation. At γ = ½ the exponent cancels and this collapses to a Möbius transform of its argument x (in this passage an acceleration ratio, not a density):C(x) = x/(x+2) = μsimple(x/2)— MOND's simple interpolating function for all x, not just in the tail, with a₀ → 2a₀. (“γ = ½” is an L2 statement, the algebraic or plain field-equation wiring. Under the action-derived L3 wiring the effective μ is C + ½xC′, the tanh-log family maps onto itself, and the same fitted curve sits at γ ≈ 0.44 with identical RMS; explorer 2026-10-13.) x is defined once, and it matters which: here x = gobs/a₀′, and the fit solves gbar = gobs·C(gobs/a₀′) implicitly for gobs, the μ-form. The identity does not survive keying on gbar: with gobs = gbar/C(gbar/a₀′) at γ = ½ one gets gobs = gbar + 2a₀′, a constant acceleration offset (v² rising linearly with r), not a flat curve and not MOND: at gbar/a₀′ = 0.01 the μ-form gives gobs/a₀′ = 0.147 (deep-MOND √(2x) = 0.141) while the gbar-keyed division gives 2.01. The “gobs = gbar/C” written elsewhere on this page is the floored TEST-09/10 form, a different function. The free-γ SPARC fit lands at 0.489, which is 2% from the unique γ at which this model does not resemble MOND but is MOND. The same cancellation runs the cosmology sector: at γ = ½, (1−C)/C = 2/x, so the substituted dark-energy density ρDE = ρm(1−C)/C = 2ρcrit — ρm cancels identically and a cosmological constant falls out. One special point, two variables: exactly MOND-simple when x is an acceleration (the galaxy fit's variable), exactly Λ when x is the mean matter density (the cosmology's). The galaxy fit (γ = 0.489) and DESI DR2 (γ = 0.487) both land there, but they are fits of the same function in different variables, so this is each sector's incumbent fitting its own data, not one parameter measured twice. The density-keyed galaxy law, which shares cosmology's variable, fits SPARC best at γ → 0.046 and is refuted. See Coherence Function and Dark Energy.
The fitted γ has a consequence beyond MOND-equivalence: the Solar System (TEST-25). A QUMOND interpolating function used for galaxies also acts in the Solar System. There the Galactic field induces a small quadrupole in the Sun's potential, and Cassini bounds it. That quadrupole is set near the Sun's MOND radius, √(GM☉/a₀) ≈ 7,000 AU, where the Sun's field is about a₀, so it tests the interpolating function in its transition. TEST-25 status (2026-11-08). Published, for MOND's own slowly-transitioning functions: 1.9–15σ depending on mass model and sample (Desmond, Hees & Famaey 2024: 8.7σ fiducial, 1.9σ with M/L freedom and bulge galaxies removed; Park et al. 2026: 3–15σ). In-house, for this compander with a₀ refit to SPARC: 14.5–19.3σ for every γ from 0.3 to 3 with M/L pinned at Υ = 0.5/0.7 (13–18σ without bulges; the γ = 0.3 row is solver-limited to about ±6σ). Those σ values track the M/L prior (at γ = 3, about 10σ with Υ free per galaxy at 0.1 dex and about 7σ at 0.2 dex), so the number that travels is the M/L each family needs in order to pass. MOND's sharpest member tested (δ = 6, no bulges) passes at Υ_d ≈ 0.54, near stellar-population values, at about 1.8σ on SPARC. The compander needs Υ_d ≈ 1.2–1.4, more than twice that, which makes 36–42% of discs over-maximal and costs 2.9σ on SPARC at γ = 3; at γ ≤ 1 it does not pass for any Υ_d up to 1.5, and neither does McGaugh's RAR function. So the root is shared with MOND at δ = 1 / γ ≈ ½, and the family-wide failure is the compander's own. Earlier "2σ edge" and "narrow, not empty" statements compared against a Cassini pass that held a₀ fixed; both were superseded on 2026-11-06. The measured bound is Q₂ = (3 ± 3)×10⁻²⁷ s⁻² (Hees, Folkner, Jacobson & Park 2014, PRD 89, 102002); an updated DE440 analysis gives (1.6 ± 1.8)×10⁻²⁷ s⁻² and puts the galaxy tension at 3–15σ depending on mass modelling (Park et al. 2026, PRD 114, 024066). The in-house numbers below already use the 2026 value; the 3–15σ range is theirs. Our own pre-registered squeeze used the point-count SPARC interval (ΔBIC ≤ 10 ⇒ γ = 0.425–0.600) and found an empty intersection there: +17.95σ at the SPARC fit, M/L pinned. With the galaxy as the unit, the retained interval widens to roughly γ ≈ 0.3–1.7 (the registered statistic: the ΔBIC ≤ 10 band against the best fit, N = 166, nuisances frozen as in the original run; not the same object as the 95% interval [0.43, 0.95] on the best-fit γ above, which profiles the nuisances), and the intersection is still empty. With SPARC's own a₀ refit at each γ and M/L pinned, Cassini fails at 14.5–19σ for every γ from 0.3 to 3, and at 13–18σ with the 32 bulge-dominated galaxies dropped (2026-11-06 and 11-07; scripts maintainer/scripts/cassini_q2_at_sparc_profiled_a0_high_gamma.py and compander_cassini_nobulge_profiled_a0.py). So TEST-25's in-house kill does not depend on Neff, and it sits in the count's rule-independent floor on /honest-assessment. This holds for a modified-Poisson (QUMOND) reading of the law; an algebraic g = gbar/C law has no external-field quadrupole, and its Solar-System test is the direct tail below.
History of this paragraphFrom 2026-10-24 to 2026-11-06 this paragraph said the galaxy-level intersection was “narrow, not empty”, with a steep-return survivor near γ ≈ 1.5–1.7, so the in-house kill fired only with points as independent. That relied on a Cassini pass at γ ≳ 1.5 computed with a₀ held fixed at 1.03×10⁻¹⁰ for every γ; with SPARC's a₀ no γ up to 3 passes. The conclusion was left behind for a day after the pass was withdrawn (found by a graduate simulated-visitor persona, 2026-11-07).


This failure is shared with MOND's own functions. On the same instrument, McGaugh's RAR function ν = 1/(1−e−√y), the one fitted on this page, fails by +15.9σ to +20.9σ. Milgrom's simple function fails by +15.3σ to +20.1σ. The grid covers a₀ ∈ {1.128, 1.20}×10⁻¹⁰ and TEST-25's three external-field values. Desmond, Hees & Famaey (2024, MNRAS 530, 1781) report 8.7σ after marginalizing. In the δ-family, which contains McGaugh's function at δ = 1, nothing enters the Cassini 95% interval until δ = 4 with a₀ held at McGaugh's value. With a₀ refit to SPARC at each δ, δ = 4 fails at +9.2σ and no δ up to 6 passes; with bulge-dominated galaxies dropped, δ = 6 reaches +3.0σ at a SPARC cost of ΔBIC +164 (explorer, 2026-11-06). That matches Desmond, Hees & Famaey, whose 8.7σ falls to 1.9σ with extra M/L freedom and no bulges. The compander gains much less from the same cut (13–18σ): above γ ≈ 1 its γ mainly moves the knee rather than sharpening the transition, so it has no sharp member to reach for. So TEST-25 closes one scale-universal tanh-log QUMOND realization, and it closes the RAR-preferred MOND functions just as hard. See TEST-25 for the scope statement (it does not reach modified inertia, dark-matter models, or the umbrella ontology). Script and pre-registration: maintainer/scripts/cassini_q2_mond_interpolating_functions.py.

A separate, larger effect: the direct tail. The compander returns to Newton as a power law, 1−C ≈ 2(1+x)−2γ, while McGaugh's function returns exponentially. At Saturn (x ≈ 5×10⁵) the fractional anomaly is ≈ 5×10⁻⁶ against e−707. Small as a fraction, it is not small as an acceleration: with g ≈ 6.5×10⁻⁵ m/s² it is δg ≈ 3×10⁻¹⁰ m/s², an a₀-sized extra pull (for any simple-μ-class return, ν−1 ≈ a₀/g, so δg ≈ a₀). The Cassini-bound external-field quadrupole gives δg ~ Q₂·r ≈ 3×10⁻²⁷ s⁻² × 1.43×10¹² m ≈ 4×10⁻¹⁵ m/s² at the same place — about 10⁵ times smaller. Nor can the tail hide in GM☉: a near-constant δg makes δg/g grow as ~r² from planet to planet. The literature finds this branch (simple μ, slow power-law return) already disfavoured by planetary ephemerides, independently of the external-field effect (Hees et al. 2016; Blanchet & Novak 2011). Computed for the compander (explorer 2026-10-15): on EPM2011 supplementary perihelion precessions (Pitjeva & Pitjev 2013; mas/cy) the direct tail excludes γ < 0.87 at 3σ (Saturn binds; INPOP15a's conservative full-refit interval gives 0.78), and the SPARC-preferred γ = 0.489 misses Saturn by about 2×10⁴σ (−9,576 vs −0.32 ± 0.47 mas/cy). That empties the point-level SPARC intersection without the quadrupole (ΔBIC +59 to reach 0.87). At galaxy level it does not: the published-nuisance interval reaches 0.95, so a reading with no external-field effect survives at γ ≈ 0.87–0.95 for about 1.2σ of SPARC cost, less than the ~2σ the Q₂-passing γ ≳ 1.5 was thought to need (with SPARC's a₀ at each γ no γ up to 3 passes Q₂, 2026-11-06). For readings with no external field, the binding Solar-System number is 0.87, not 1.5. Globular clusters disfavour that corner at about 3σ; they do not close it (explorer 2026-10-30, pre-registered): refit per law with a photometric tidal-radius prior, the compander at γ = 0.9 with no external field costs Δχ² = +54 ± 19 against Newton (cluster bootstrap; +93 ± 32 with tangential-or-isotropic orbits), conditional on the mass model. The statistic sits in the deep regime, where only a₀′/γ enters, so this is MOND-without-EFE's disfavouring, inherited. (GC table). It is also inherited, not framework-specific — at γ = ½ the compander is simple μ. Two Solar-System bounds, two observables. The compander returns to Newton as a power law, 1 − C ≈ 2x−2γ, and that tail is what the perihelion bound (γ < 0.87 excluded) probes. McGaugh's ν returns exponentially (ν − 1 ≈ e−√y), so it is immune to this direct tail; it fails only through the external-field quadrupole Q₂ that TEST-25 uses, where the compander was thought to pass at γ ≳ 1.5; with a0 taken from SPARC at each γ it passes at no γ up to 3 (2026-11-06). When McGaugh's ν and simple μ are listed together as failing TEST-25, that is the quadrupole channel only. (finding; supplementary precessions fitted without the tail in the dynamical model, the standard Iorio-style test.)
Revision noteUntil 2026-10-16 this box said the literature result “has not been re-executed on this site, and the size of the exclusion for the compander specifically is not yet computed.” The explorer computed it on 2026-10-15. This box used to explain the empty intersection as a tail-shape mismatch, with the exponential tail “indistinguishable from zero at any precision.” It concluded that the compander sits in the simple-μ branch the Solar System had closed while McGaugh's branch survives. That compared the direct anomaly at Saturn, but Cassini bounds the external-field quadrupole set near 7,000 AU, and there McGaugh's function fails too. A simulated-visitor persona (graduate physics) raised it, and a pre-registered run on TEST-25's instrument confirmed it. An earlier correction on this box had already fixed the naming (simple μ returns as a power law; the exponential return belongs to McGaugh's RAR ν). Until 2026-09-22 the box then called the direct tail “a separate, smaller effect” that did not bear on the Solar System verdict. That compared fractions, not accelerations; in acceleration it is about 10⁵ times the Cassini Q₂ effect at Saturn (visitor graduate-physics and researcher personas, 2026-09-22). The Hill-fit sentence above also previously called the n = 0.975 fit an independent cross-validation; it uses the same SPARC data.

Galaxy tests that selected Synchronism over MOND: 0, by execution (this RAR-shape test cannot separate them at free γ). 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 environment test refuted its registered amplitude, 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.

MOND+EFE / wide-binary avenue: CLOSED (2026-06-03) MOND with the External Field Effect (EFE, Bekenstein-Milgrom 1984) also predicts environment-dependent wide-binary dynamics. The quantitative divergence was computed on 2026-06-03: a bounded C(a) is the only form whose EFE prediction differs from MOND (in Tidal Dwarf Galaxies; the velocity-dispersion intervals once quoted here are retracted on MOND Unification), but that same boundedness caps the acceleration boost at ≈3.17 while ~42% of SPARC RAR data points require boosts above that (up to ~34×), giving RMS 0.224 vs McGaugh's 0.146. The boost ceiling and RAR fit trade off monotonically with opposite sign — no ceiling simultaneously fits the RAR and stays distinct from MOND+EFE. This is the same fork as the RAR transition-shape test: the one non-degenerate form is refuted by the data. Wide-binary density stratification cannot discriminate Synchronism from MOND+EFE. See TEST-02 and Honest Assessment for the full picture.

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.

Open gap: This statement has not been formally derived from the framework's equations. The coherence field couples to density (ρ), not directly to the metric, and no covariant (relativistic) action containing C(ρ) has been written down, so the relationship between C(ρ) and GW propagation is undefined. (Scope, added 2026-11-10 after a graduate-physics reader found this box contradicting the “action-derived L3” wiring above: L3's μeff = C + ½xC′ comes from varying a non-relativistic Lagrangian in Φ, with C(ρ)|∇Φ|² as its kinetic term and a striction force as its by-product (explorer 2026-10-13). That is a Newtonian action. It says nothing about photons, wave speed or polarization, which is why GW170817, cT and lensing are not well-posed questions here; see dynamics per branch.) Until a covariant action exists, “GW170817 doesn't apply” is a reasonable expectation but not a proven constraint.
Cosmological extension — disfavored 2.4σ on σ₈ amplitude (reframed 2026-07-02): Session 107's prediction of suppressed growth (fσ₈ ≈ 0.418) was compared against DESI DR1. Full-shape result (arXiv:2411.12021): combined σ₈ = 0.841 ± 0.034 vs predicted 0.76 → 2.4σ tension on the σ₈ amplitude, a GR-conditioned statistic. The 0.76 was calibrated to the weak-lensing S8 value, so this 2.4σ restates the known lensing-vs-clustering S8 difference; a calibrated number cannot test the framework (note added 2026-10-04). The registered kill criterion (fσ₈(z = 0.51) > 0.46 at >3σ) was not met: ~1.5σ, underpowered as registered. The LRG1 fσ₈/(fσ₈)_fid = 1.09 (+0.12/−0.14) “enhancement” is a single ~0.6σ bin, not load-bearing. Verdict: post-hoc retrodiction, disfavored 2.4σ on amplitude. (Prior 2026-05-25 “correction” was itself an error — misattributed z≈0.07 PV value.) Honest Assessment: TEST-04a →
Next on Intermediate Path: The γ ≈ 1 Boundary →Next on Physics Track: Relation to MOND →CDM Discrimination →Galaxy Curve Plotter →

Prerequisites

Understanding these concepts first will help:

Dark Matter ReframedPatterns interacting indifferently: gravity only, no EMReference Density (ρ_crit)ρ_crit = A V_flat²: the saturation knee of C(ρ), a reference scale, not a transition point

Related Concepts

RAR ScatterEnvironment-dependent RAR scatter — registered >20% amplitude, refuted as registered (r² = 0.0001)CDM Discriminationσ_int = 0.086 dex is not a BTFR scatter (independent of V), so no CDM verdictGalaxy Curve PlotterPlot SPARC rotation curves with Synchronism overlays
Revision notes (history; not the current claim). Dated corrections moved here from the running text on 2026-11-08, verbatim, in page order.