Galaxy Curve Plotter
The dark matter puzzle in one picture: Physics predicts that galaxies should rotate more slowly at their outer edges (like planets in the solar system — the further out, the slower). They don't. The outer stars rotate just as fast as the inner ones. Something invisible is adding gravity. Most physicists call it dark matter — a proposed invisible substance that has never been directly detected; we infer it only from its gravitational pull (whether it's real stuff or a placeholder for missing physics is exactly what's being debated). MOND (Modified Newtonian Dynamics) explains the same curves by changing the gravity law. Synchronism offers a third interpretation: the coherence function C(ρ) mimics the extra gravity via density-dependent coupling. All three fit the observations; none is confirmed over the others by rotation curve data alone — though the fits are not on equal footing: MOND uses one global constant (a₀) for every galaxy, while the violet Synchronism curve refits ρcrit per galaxy.
Select a SPARC galaxy. The plot shows five things: what visible matter predicts (dashed gray), what we observe (dots), what Synchronism's real equation gives (violet solid — it fails), the hand-tuned stand-in formerly shown (amber dotted — illustration only), and what MOND gives (green). Notice that the stand-in and MOND nearly overlap — the framework's own Honest Assessment labels this a reparametrization — plain words: the same curve wearing a different costume; fitting a known curve isn't discovering anything new. Plain verdict for casual readers: these curves look great but don't prove the idea — all three models (Synchronism, MOND, and NFW dark-matter halo) fit galaxy rotation curves about equally well. What matters is whether any makes a different, testable prediction, and the ensemble test (SPARC RAR, ΔBIC=+184) shows Synchronism collapses to MOND when γ is freed — curve-equivalence at fitted γ only, not theory-equivalence. (Corrected 2026-08-09: this read “the framework has no action, no Lagrangian, and no dynamics.” It has all three, postulated in Appendix D of the archive since 2025-12-01 — but the version stated there is eliminated a priori, and writing down the surviving one changes none of the refutations. See Honest Assessment.) The curve on this plotter is built from a quadrature stand-in, not from any field equation — see the formula box below. See what the tests actually say →
γ note: the violet curve pins γ=2, which via γ=2/√Ncorr implies Ncorr=1 — stars treated as uncorrelated. No galaxy satisfies that; the data-preferred fit (γ≈0.49, ΔBIC=+7 vs +184 for γ=2) implies Ncorr≈17, still not a physical correlated-star count. See Galaxy Rotation for the full both-directions contradiction.
- Dashed gray — Newtonian prediction using visible matter only (toy exponential disk). Drops off at the edges; this is the puzzle.
- Dots — observed rotation velocities. Flat at large radius; doesn't drop like Newtonian says it should.
- Violet solid — Synchronism's real C(ρ) at γ=2 with the framework's asserted ρcrit = 0.029·V². The disk's density is thousands of times below ρcrit, so C stays near zero and the curve sits on the baryon line. This is the 2026-07-02 audit result, rendered. Caveat (2026-08-05, revised same day): that “thousands of times below” is conditional on A = 0.029, since ρ/ρcrit ∝ 1/A — at A = 4.6×10−5 the ratio for NGC 3198 is 0.60 and the curve would lift off the baryon line. But the 635× between them is a law swap, not a scale choice: universal A with ρcrit ∝ V² versus per-galaxy A ∝ Rhalf−2 with ρcrit ∝ V0.5 — the two-law fork parameter derivations item 1 has documented since 2026-06-07. An earlier version of this caveat attributed the gap to an unstated coarse-graining length ℓ; that is withdrawn — a self-consistent ℓ smooths ρ and ρcrit alike and cancels, leaving ρ/ρcrit ≲ 0.019βJ² in every sector at every ℓ. This plot shows the framework under its own stated law.
- Amber dotted — the hand-tuned tanh(radius) stand-in previously displayed. It fits because it was drawn to fit; no ρ, ρcrit, or γ enters it. (Recolored from violet 2026-07-23: it is an illustration, not the theory.)
- Green dashed — MOND's real simple-ν function on a BTFR-assigned mass (one global a₀, zero per-galaxy knobs). It lands close to the dots — expect ~10% mismatches from the toy mass model, not from tuning.
Which MOND, and why it matters (added 2026-07-29): the green curve here uses the simple-ν family (ν = ½ + √(¼ + 1/y), power-law high-acceleration return) — the same family TEST-25 excludes at +17.7–18.0σ against Cassini planetary ephemerides. The Galaxy Rotation page's RAR/ΔBIC test uses McGaugh's different exponential-return ν = 1/(1−e−√x), which is Cassini-safe. Both are called “MOND” on this site; they are not the same function, and only one of them survives the solar system.
Gray “Newtonian”: v_b(r) = √(G·M(<r)/r), M(<r) = M_disk·[1 − e^(−r/R_d)(1+r/R_d)]
Violet SOLID “Synchronism (real)”: v(r) = √(v_b² + [V_flat·C(ρ(r))]²), C(ρ) = tanh(2·ln(ρ/ρcrit+1)), ρ(r) = Σ(r)/2h, ρcrit = 0.029·V_flat²
Amber DOTTED (old illustrative stand-in — what this page showed before 2026-07-08, kept for comparison): v(r) = √(v_toy² + [V_flat·tanh(0.4·r/2.5)]²), v_toy = 0.6·V_flat·√(1−e^(−r/2.5))
Green “MOND”: v(r) = v_b·√ν(y), ν(y) = ½ + √(¼ + 1/y), y = g_N/a₀, g_N = v_b²/r (real simple-ν)
For rotation curves specifically, though, the fork does not block the verdict — it doubles it. Take a gas-rich dwarf like DDO 154, which needs a boost of B ≈ 10 in acceleration to explain its outer curve (SPARC's most DM-dominated systems reach fDM = 0.927, i.e. B ≥ 13.7 — see TEST-10). At max C = 0.001 the quadrature branch delivers B ≈ 1 — a factor ~10 under. The division branch delivers B = 1/C ≈ 1000 — a factor ~100 over. Both branches are refuted by the same galaxy, from opposite sides. Registering a canonical coupling is still required for the EFE and cluster sectors; it is not required to conclude that the galaxy sector fails. (Raised independently by two expert visitor passes, 2026-08-08.)
Approximation note (added 2026-07-17, flagged by an expert reader): the gray Newtonian curve uses spherical enclosed mass for what is physically a thin exponential disk. The exact thin-disk result (Freeman 1970: v² = 4πGΣ₀Rd·y²[I₀K₀ − I₁K₁], modified Bessel functions) runs ~15–20% higher near the peak (r ≈ 2Rd), so the plotted Newtonian baseline understates the baryonic prediction there and the visual “dark matter gap” — this page's pedagogical centerpiece — is modestly exaggerated near the peak. The gap itself is real and much larger than this correction at large radius (where the discrepancy actually lives), so no conclusion flips; a Bessel-function implementation is queued for a future pass rather than rushed here.
What You're Seeing
The dashed line is what rotation curves should look like with only visible matter (stars + gas). The blue dots are what we actually observe. The gap is the “dark matter problem.”
The solid violet curve is Synchronism's real C(ρ) evaluated on the disk's density profile — and it fails to fill the gap: the disk never gets dense enough for the coherence boost to turn on (this galaxy's maximum C is 0.001; the knee needs C to approach 1). The dotted amber curve is the hand-tuned tanh(radius) stand-in this page displayed before 2026-07-08 — it fit because it was drawn to fit. Full disclosure of both formulas below the plot.
The green dashed curve is MOND (Modified Newtonian Dynamics) using its real simple-ν interpolating function and the acceleration scale a₀ ≈ 1.2×10⁻¹⁰ m/s², on a mass fixed by the baryonic Tully–Fisher relation — no per-galaxy tuning at all. The stand-in the framework needed to look like MOND is what the site labels a reparametrization; the real equation doesn't even manage the costume.
Note: Curves shown are simplified models for illustration. Actual SPARC fits use full surface brightness profiles and mass-to-light ratios. See the research data for precise fits.