Galaxy Curve Plotter
Tool type: Model Explainer (the Interactive Tools index grouping — shows how the equation works, not a verdict) · Claim status: the two badges above. The label says what kind of tool this is; the badges say how its claims stand.
• γ = 2 — the pin refuted on the RAR at ΔBIC = +184 (Honest Assessment).
• ρcrit = 0.029·Vflat+2— an a₀-tracking knee would need ρcrit ∝ V−2, and the knee measured on SPARC has no velocity dependence at all (V−0.16 ± 0.19, median 0.161 M☉/pc³), so V+2 is a wrong exponent, excluded at ~11σ (Parameter Derivations). The “240–300,000×” magnitude this line used to quote was withdrawn 2026-08-28: it applied the V² law to a coefficient derived for V0.5.
• Per-system normalization — keying the knee to Vflat makes it a function of the very observable being predicted; the site names that move disqualifying, and MOND by contrast uses one global a₀ for all galaxies.
Which wiring (added 2026-09-24): the violet curve adds the coherence term in quadrature, v² = vb² + (Vflat·C)². That display wiring takes the observed Vflat as an input, and with C ≈ 0.001 it does nothing. It is also circular by construction: if C ever approached 1, the “prediction” would simply hand back the observed Vflat (extended 2026-09-25 after a graduate-physics visitor flagged it). The ledger's tests use the division wiring, g = gbar/C, which at the same C fails the other way: about 10³× too much gravity, roughly 30× in velocity, far off the top of this plot. So the tool shows one failure mode (inertness), and the tested law has the opposite one (over-boost).
So the tool is best read not as “the theory's prediction” but as a working demonstration that the equation is inert at its published calibration — which is exactly what the DDO 154 annotation on this page says in one line (max C on this disk: 0.001 — the equation never switches on here: this galaxy is too spread out; this read “inert by construction” until 2026-09-25). At Vflat = 47 km/s that calibration puts the knee at 64 M☉/pc³ against disc densities of order 10−2: no galaxy in the dropdown can lift off the Newtonian baseline, for any of them.
One more cross-link the plot owes you: the green MOND reference uses the simple-ν function — the RAR-preferred choice, and also precisely the function TEST-25 reports Cassini excluding at +17.95σ. The benchmark drawn here is a function this site refutes in the Solar System; that is a real tension in the comparison and not a drafting slip.
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 proposed a third interpretation: the coherence function C(ρ) would mimic the extra gravity via density-dependent coupling. Only the first two fit the observations: dark-matter halos and MOND both reproduce these curves (MOND with one global constant, a₀, for every galaxy). The framework's own equation, as published, does not — the violet curve below stays on the visible-matter line. What matched MOND was either the hand-drawn amber stand-in or, in the archive's SPARC work, a fit that re-tunes ρcrit for each galaxy (one extra free parameter per galaxy). The violet curve is not a fit of any kind: nothing in it is adjusted to the dots. (Corrected 2026-09-15: this paragraph said “all three fit the observations” and described the violet curve as refitting ρcrit; the chart beside it shows neither.)
Select a SPARC galaxy. The plot shows five things: what visible matter predicts (dashed gray), what we observe (dots), what Synchronism's equation as published gives (violet solid — it fails, at a calibration refuted three ways; see the red box above), 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: MOND and dark-matter (NFW) halos (not plotted here) both fit these curves; the framework's own equation, as published, does not (solid violet); only the hand-tuned stand-in (dotted amber) does, and it was drawn to. (Corrected 2026-09-15: this verdict used to say all three models fit about equally well, which the chart beside it contradicts.) Fitting alone would not settle much anyway — what matters is whether a model makes a different, testable prediction, and the ensemble test (SPARC RAR, ΔBIC=+184 — a fit in acceleration, MOND's own variable, not in the density this plotter uses; see Honest Assessment for which test kills which version) 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 →
explorer/findings/the-argument-of-C-three-functions-each-killed-by-its-own-distinguishing-feature.md, Result 3. Vflat is taken from existing SPARC/MOND fits, not independently predicted. See parameter derivations for full accounting.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 — shown only to illustrate what a working boost would look like; not a prediction of any theory. It is the hand-tuned tanh(radius) stand-in previously displayed, and 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.
Revision note
This note used to call McGaugh's exponential-return function “Cassini-safe” and say only the simple function fails. That reasoning used the high-acceleration tail at Saturn, but Cassini constrains the quadrupole set in the transition region. A visitor persona (graduate physics) caught it. The numbers above come from a pre-registered run:maintainer/scripts/cassini_q2_mond_interpolating_functions.py.Gray “Newtonian”: v_b(r) = √(G·M(<r)/r), M(<r) = M_disk·[1 − e^(−r/R_d)(1+r/R_d)]
Violet SOLID “C(ρ), quadrature wiring (a display branch, not the ledger's gbar/C law)”: 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.
Two assumptions behind that picture:
- “The equation never switches on here” is a property of the calibration A = 0.029, not of the equation's form. Parameter Derivations records that A = 0.029 does not follow from its own stated formula, which gives A ≈ 4.6×10−5. At that value the knee sits inside dwarf-disc densities: a visitor's estimate puts C ≈ 0.5 near the DDO 154 midplane (an estimate, not a computation this page performs). The chart draws only A = 0.029.
- The violet law, v² = vb² + [Vflat·C]², takes the observed Vflat as an input, so even a match would not count as evidence. It is also a different law from the g = gN/C used in the site's refutations (see the wiring note inside the chart card). The failure verdicts rest on those refutations, not on this picture.
Which wiring this chart draws (2026-09-15): the violet curve couples C in quadrature, v² = vb² + [Vflat·C]², so C → 0 means no boost. The ledger's kills (TEST-09/TEST-10) use the division wiring gobs = gbar/C — but with a different C (see the box below), not this one clipped at a floor. Applying a floor to this chart's C would give a boost of 1/C ≈ 1275× unfloored, or a flat 3.17× floored. For DDO 154 (needs ≈ 10×) that is quadrature ≈ 1× (under), floored division ≈ 3× (under), unfloored division ≈ 1000× (over) — every wiring fails; the fork box above has the detail. Radius is in kpc (kiloparsecs): 1 kpc = 1,000 parsecs ≈ 3,260 light-years.
• Cρ = tanh(γ·ln(1+ρ/ρcrit)), keyed on density — what this chart draws, and what “max C = 0.001” reports. It carries no floor. The 0.001 figure does generalise off the toy: over 123 SPARC discs the per-disc maximum has median 1.2×10−3 at γ = 2 (1.6×10−4to 4.9×10−2), and 0 of 123 reach Ωm anywhere.
• Ca = Ωm + (1−Ωm)·x/(1+x), x = (gbar/a₀)1/φ, keyed on acceleration — what the TEST-09 and TEST-10 scripts evaluate. Its floor is part of the functional form, not a clip. On the same 123 discs it runs 0.329 to 0.954 (median 0.515, IQR 0.235), and 0.00% of the 2,856 radii sit within 1% of the floor. The applied boost spans 1.05×–3.04×, and never reaches the 3.17 ceiling.
So the two corollaries the persona drew are refuted on the framework's own data: the predicted fDM is not a delta function at 0.685 (median 0.585, s.d. 0.062, and no galaxy within 0.01 of the cap), and TEST-09's slope is not ceiling-independent — it moves 0.62 across Bmax ∈ [2, 100]. What the sweep did find, and it is adverse: TEST-09's kill is convention-dependent, firing only under the site's own undefended ceiling. Pre-registered at site commit
89e0467; script maintainer/scripts/which_C_carries_the_floor.py (+ _output.txt), identity control reproduces TEST-09's published n = 3.35 ± 0.07 exactly.The misreading is this page's defect, not the reader's. Both functions were written “C” and the sentence above this box put one function's output range next to the other function's floor. There are now three live objects behind the one symbol — Cρ, Ca, and the quadrature-vs-division wiring fork — and only the third was labelled.
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.