Galaxy Curve Plotter
- The dots are what telescopes measure: stars keep orbiting fast far from the centre.
- The dashed gray line is what the visible matter alone would give. It sags below the dots. That gap is the dark-matter puzzle. (Since 2026-11-10 the gray line is built from each galaxy's measured gas and stars, SPARC's own mass model, at a stellar mass-to-light ratio you can move with the slider below the galaxy buttons. Until then it was a stand-in built from the galaxy's speed.)
- Green (MOND: an existing theory, from 1983, that gravity gets slightly stronger than Newton predicts where it is very weak, as at the far edges of galaxies) follows the dots (with no per-galaxy tuning: one universal constant and the measured gas and stars, which is why it is the rival worth comparing against). Solid violet (this framework's equation as written) does not: it is the gray line scaled up, still falling where the dots stay flat. In plain words: the only version of this framework's equation that fits these dots is MOND's formula written differently. Fed acceleration instead of density, the equation at γ ≈ ½ is MOND's formula, so that version is the green curve; there is no separate line for it. The chart is the version that failed (violet) against the version that survives (green), and the one that survives is MOND. At these densities the violet line is a constant 1.78× stretch of the gray one at every radius (the floor binds; details in the physicists' block).
- “But First Encounter says the fitted γ ≈ 0.5 reproduces MOND?” It does, and that is not a win. The fit that gives γ ≈ 0.5 feeds the equation acceleration, which is MOND's own input, instead of the density the framework names; with that swap the equation becomes MOND's formula and you get the green line back. That is why the site calls it a reparametrization. The violet line is the equation as written, keyed on density, and it fails. Why matching MOND still counts as Failed: fitting the same curves as an existing theory with a renamed formula adds nothing new. A match would only be news if the equation also predicted something MOND does not and that prediction held. It made three such predictions (a cap on how much gravity can be boosted, an effect of a galaxy's surroundings, and a density knee that grows with rotation speed), and each failed when tested.
That is the whole lesson. You can ignore the other lines (an old hand-drawn stand-in and a second violet variant) and the “For physicists” block. Your next step is What Synchronism Is Not →
ⓘ What exactly the two badges cover (for physicists)
Failed: the published density law Cρ (γ = 2, ρcrit = 0.029·Vflat²) in the division wiring g = gbar/C, floored at Ωm (solid violet). At these densities Cρ ≈ 10−3sits under the floor everywhere, so what is plotted is G → G/Ωm: the floor (the boost ceiling) failing, not the shape of the density law. The quadrature wiring (dashed) is inert. Reparametrization: what matched MOND, the hand-drawn stand-in and the fitted γ ≈ 0.49 law (acceleration-keyed). Badges made plain 2026-11-05 after a beginner visitor met this spec as the first thing on step 3 of the Beginner Path.
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.
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 written (with the registered tests' Ωm floor), does not — wired by division (solid violet below) it is the visible-matter line scaled up by a constant, still falling where the dots stay flat; wired in quadrature (dashed light violet) it 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.
For physicists: the calibration, the two wirings, parsimony and the full audit trail (beginners can skip this)
Revision note: until 2026-09-15 the paragraph above said “all three fit the observations” and described the violet curve as refitting ρcrit; the chart shows neither. On 2026-10-20 the technical notes below were folded into this block after a casual simulated-visitor persona found Beginner Path step 3 showed “the expert's notebook”.
• γ = 2 — the pin refuted on the RAR at ΔBIC = +184 (Υ frozen; +46 with Υ free per law) — strictly, that fit used the acceleration-keyed Cg (keyed on gobs, no floor; not the floored Ca of TEST-09/10); the density-keyed Cρ drawn here lost its own head-to-head by ΔBIC +2843 even with γ free (Honest Assessment).
• ρcrit = 0.029·Vflat+2 (M☉/pc³ with V in km/s) — 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 at galaxy level (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; both drawn since 2026-10-08): the light-violet dashed 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. The solid violet curve now draws the division wiring with the Ωm floor the ledger's density-law tests used, which caps the over-boost at 3.17× in gravity (1.78× in speed). Without the floor it would be the ~30× curve. So the plot shows both failure modes: inert (quadrature), and a uniform boost with the wrong shape (floored division).
So the tool is best read not as “the theory's prediction” but as a working demonstration that the equation never switches on at its published calibration: C(ρ) stays below 0.01 on every shipped disc (0.0004 on UGC 128 up to 0.008 on the compact NGC 2403), so what you see is decided entirely by the wiring (no boost; a flat 3.17× in gravity, 1.78× in speed; or about 1000× in gravity, about 30× in speed, unfloored). The legend used to read “max C on this disk: 0.001 — the equation never switches on here: this galaxy is too spread out” under the quadrature curve, and “inert by construction” before 2026-09-25; “inert” described the display wiring, not the tested law (corrected 2026-10-08). 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 of Famaey & Binney (2005) — not the RAR fit function the ΔBIC table on Galaxy Rotation uses (McGaugh's ν); the two are different functions and are not interchangeable toward the Solar System (corrected 2026-10-10; this note previously called the simple function “the RAR-preferred choice”) — and also a function TEST-25's instrument finds Cassini excluding, at +15.3σ to +20.1σ (the +17.95σ quoted until 2026-11-05 is the framework's compander at the SPARC fit, not the simple function; see below). 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.
Try the twin pair: NGC 2403 and UGC 128 have the same flat speed (131 vs 129 km/s) but UGC 128's disc is 16× dimmer per unit area and 4× wider. This is the classic low-surface-brightness test (de Blok & McGaugh). MOND keys on acceleration, so it predicts the dimmer twin is more dark-matter dominated, which is what the dots show. A law keyed on local density makes its own prediction for the pair. Here both discs sit on the floor, so the violet line is just the gray line scaled up in both. (Observed points switched from hand-drawn approximations to real SPARC rows on 2026-10-26; the UGC 128 row had been a 55 km/s curve that is not this galaxy.)
Select a SPARC galaxy. The plot shows five things: what visible matter predicts (dashed gray), what we observe (dots), what Synchronism's equation as written, with the registered tests' Ωm floor, 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: two existing explanations both fit these curves: MOND (the green line) and dark-matter halos (the standard “NFW” halo shape; not plotted here); the framework's own equation, as written (with the registered tests' floor), 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 with points as independent, ≈ +11 with the galaxy as the unit — 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 violet curves on this plotter are two display wirings of Cρ (division and quadrature), not solutions of 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, Υ frozen) implies Ncorr≈17, still not a physical correlated-star count. See Galaxy Rotation for the full both-directions contradiction.
- Dashed gray — Newtonian prediction from the measured gas, stellar disc and bulge (SPARC's mass model at the slider's Υdisk; a BTFR toy disc until 2026-11-10). Drops off at the edges; this is the puzzle.
- Dots — observed rotation velocities from SPARC (up to 14 radii per galaxy, with their quoted errors as vertical bars; real data since 2026-10-26, hand-drawn approximations before). Flat at large radius; doesn't drop like Newtonian says it should.
- Violet solid (division reading) — Synchronism's real Cρ at γ=2 with the framework's asserted ρcrit = 0.029·V², wired as g = gbar/Cf with Cf = Ωm + (1−Ωm)·Cρ. The disk's density is thousands of times below ρcrit, so Cρ stays near zero, Cf sits on the floor, and the curve is the baryon line times √3.17 ≈ 1.78 at every radius: wrong shape. Light-violet dashed (quadrature reading) — the same Cρ added in quadrature, v² = vb² + (Vflat·Cρ)²; with Cρ ≈ 0.001 it 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 simple-ν function (Famaey & Binney 2005) on SPARC's measured baryons (gas, disc and bulge at the slider's Υdisk) with one global a₀ = 1.2×10−10 m/s² and zero per-galaxy knobs: a prediction, since 2026-11-10. Until then it ran on a BTFR-assigned mass Mb = 47·Vflat⁴, which set its flat level to 0.93·Vflat by construction (the same circularity this page counts against ρcrit = 0.029·Vflat²); a leading-edge reader pointed out on 2026-11-10 that the “visible matter” line was then MOND's own mass–velocity relation fed in as the baryons, on a gas-dominated default galaxy. Where the green line sits relative to the outer dots now depends on Υdisk and on the measured gas, not on the dots.
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 simulated-visitor persona (graduate physics) caught it. The numbers above come from a pre-registered run:maintainer/scripts/cassini_q2_mond_interpolating_functions.py.Toy mass model, now used only for the density ρ(r) that feeds Cρ and for the amber stand-in: M_b = 47·V_flat⁴ M☉ (empirical BTFR, McGaugh 2011), exponential disk with each galaxy's observed scale length R_d, h = 0.3 kpc. Until 2026-11-10 the gray line was this toy disc's Freeman (1970) curve, v_b²(r) = (2GM_disk/R_d)·y²[I₀(y)K₀(y) − I₁(y)K₁(y)], y = r/2R_d, with no gas component
Density-keyed coherence (both violet lines): Cρ(r) = tanh(2·ln(ρ(r)/ρcrit+1)), ρ(r) = Σ(r)/2h with Σ(r) = (M_b/2πR_d²)·e−r/R_d (M☉/pc², R_d in pc), ρcrit = 0.029·V_flat² (Σ(r) printed 2026-11-04 so a reader can reproduce ρ and Cρ; Cρ stays < 10−3 on every disc here, so the solid violet line is the Ωm floor, a constant 1/√Ωm = 1.78× stretch in v, and tests Ωm, not the tanh)
Violet SOLID, division reading (the wiring the ledger's tests use; failed): g = g_bar/Cf, i.e. v(r) = v_b/√Cf, Cf = Ωm + (1−Ωm)·Cρ, Ωm = 0.315 (boost ≤ 1/Ωm = 3.17 in g)
Light-violet DASHED, quadrature reading (display branch, not a tested law; inert at Cρ ≈ 0.001): v(r) = √(v_b² + [V_flat·Cρ]²)
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 simulated-visitor passes, 2026-08-08.)
Revision note
Until 2026-11-10 the gray curve was a toy exponential disc whose whole mass came from the BTFR, Mb = 47·Vflat⁴, with no gas component, so on the gas-dominated default galaxy the “visible matter” line was MOND's own mass–velocity relation fed in as the baryons (a leading-edge reader, 2026-11-10; the fix had been “queued” on this page). Before that, until 2026-10-29, the gray curve used spherical enclosed mass, √(G·M(<r)/r). That ran 14–16% low between 2 and 3 Rd (0.542 against 0.622 √(GM/Rd) at 2.2 Rd), so the visual gap near the peak was exaggerated; the two agree to under 3% beyond 10 Rd, where the discrepancy lives. An expert reader flagged it on 2026-07-17 and a graduate-physics reader again on 2026-10-29.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 M☉ pc−3 (km/s)−2 (A = 0.029 is in the same units). 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 dashed quadrature law, v² = vb² + [Vflat·Cρ]², takes the observed Vflat as an input, so even a match would not count as evidence. The solid division curve uses the site's tested wiring, g = gbar/C, but with the density-keyed Cρ; TEST-09/TEST-10 divide by the acceleration-keyed Ca (see the two-C's box inside the chart card). The failure verdicts rest on those refutations, not on this picture.
Detail for physicists (how Vflat enters each curve, units, the 47 vs 62.8)
It enters both violet curves through ρcrit = 0.029·Vflat² (here 64 M☉/pc³), through the toy disc density that feeds Cρ (Mb = 47·Vflat⁴), and the dashed quadrature curve a third time through its Vflat·C boost term. The gray baryon line no longer uses it (since 2026-11-10). A curve built from Vflat cannot be a prediction of Vflat. Units: the code compares ρcrit with a disk density in M☉/pc³ with Vflat in km/s, so the 0.029 carries units of M☉ pc−3 (km/s)−2; the archive ledger states units only for the V0.5 law that coefficient was originally derived under, not for the V² law drawn here. The green MOND curve no longer uses Vflat at all: since 2026-11-10 its baryons are SPARC's measured components, so its outer level is a prediction from one global a₀ = 1.2×10−10 m/s² and the measured gas and stars, with Υdisk the only knob, shared across galaxies. (Until then it ran on the BTFR mass 47·Vflat4, which set its flat level to 0.93·Vflat by construction: 47 corresponds to a₀ ≈ 1.6×10−10 while the curve uses 1.2×10−10, 1/(G·a₀) ≈ 63, and (47/63)1/4 ≈ 0.93. That circularity is gone.) See Honest Assessment.Which wiring each violet line draws (2026-09-15; rewritten 2026-10-12, it still described the pre-10-08 chart): the solid line is the division reading, g = gbar/Cf with Cf = Ωm + (1−Ωm)·Cρ, so Cρ → 0 means the maximal boost 1/Ωm = 3.17×. The dashed line is the quadrature reading, v² = vb² + [Vflat·Cρ]², so Cρ → 0 means no boost. The ledger's kills (TEST-09/TEST-10) use the division wiring too — but with a different C, the acceleration-keyed Ca (see the box below), not this chart's Cρ held at a floor. On this chart's Cρ the division boost is 1/C ≈ 1292× unfloored, or the flat 3.17× drawn, 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. Over the 2,856 measured radii of the same 123 discs, Ca itself runs 0.329 to 0.954 (per-radius 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 (one value per disc at its outermost radius, N = 123: median 0.585, s.d. 0.062 — a different sample from the per-radius Ca median above, which is why 1 − 0.515 ≠ 0.585; stated 2026-10-10 — 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.
Start here: compare the gray line (Newton, visible matter only) with the dots (observed), then the green (MOND) line.
How to read this chart, in three steps- Blue dots are what telescopes measure: how fast stars and gas orbit at each distance from the centre. The dashed gray line is what the visible matter alone should produce under ordinary (Newtonian) gravity.
- The gap between the gray line and the blue dots, growing toward the edge, is the missing gravity — the dark-matter puzzle. The green dashed line (MOND) closes it with no per-galaxy tuning.
- The solid violet line is Synchronism's density equation wired the way its tests wire it (gravity divided by C). The disk is far too thin to reach the equation's switch, so C sits on its floor and the boost in gravity is the same 3.17× at every radius, so in speed the curve is the gray line scaled up by √3.17 ≈ 1.78×. It overshoots the inner points and still falls at the edge, where the dots stay flat — that is the failure. This holds on every galaxy in the menu: with ρcrit = 0.029·Vflat², Cρ stays below 0.01 everywhere on every shipped disc (0.0004 on UGC 128 up to 0.008 on the compact NGC 2403; “about 0.001” before the 2026-10-26 switch to real SPARC rows), so the violet curve is the floor's constant boost, never the tanh's S-shape. In plain terms: on these discs the violet curve is the gray line times 1/√Ωm, and γ has no effect on it at all. What keeps the knee out of the discs is the calibration A = 0.029 in ρcrit = A·Vflat². That A is about 600× larger than its own Jeans formula gives (4.6×10⁻⁵; Parameter Derivations). With the formula's value, ρcrit ≈ 0.8 M☉/pc³ at 130 km/s, disc midplanes sit at about a tenth of the knee, and the curve's shape would start to matter (added 2026-11-02, graduate-physics simulated-visitor persona; not yet drawn here). The tool never exercises the curve's shape on a real galaxy, and that is the actual reason it fails. The light violet dashed line (off by default; checkbox below) is the same C wired the other way (added on top), which does nothing at all. The two wirings fail in opposite directions; without the floor, the division wiring would run up to 302× above the gray line, far off the chart.
Why should the gray line fall at all? Under ordinary gravity, stars far from a galaxy's centre should orbit more slowly, the way Neptune orbits the Sun more slowly than Earth, because almost all the visible mass is inside their orbit. A “rotation curve” is just orbital speed plotted against distance from the centre.
Advanced: two optional lines, and what ticking them shows
The framework never fixed how C enters gravity: divided into it (solid violet) or added on top (light violet). The second wiring is shown so you can check for yourself that it changes nothing, rather than take the page's word that the choice of wiring is not hiding a success. The amber stand-in is the hand-drawn curve this page used to show. What you learn by ticking them: amber shows the shape a working curve would need, so you can see how far violet is from it; light violet shows that the other wiring changes nothing, so the failure is not an artefact of how C was plugged in.
Legend in plain words (shortened 2026-10-14 after a beginner simulated-visitor pass could not tell which line was the theory, and again 2026-10-22). The violet line is Newton's shape ×1.78 in speed (= √3.17, the floored boost in gravity). The technical definition of each curve (division vs quadrature reading, the Ωm floor, max Cρ on this disk = 0.001, the simple-ν MOND function and its BTFR mass) is in the “Reading this plot” list and the formula box on this page.
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, with gravity divided by C as the tests do (in plain words: wherever C is small the theory multiplies gravity up, because dividing by a number below 1 makes it bigger; the floor caps that at 3.17×) — and it gets the shape wrong: the disk never gets dense enough for C to move off its floor (this galaxy's maximum C(ρ) is 0.001; the knee needs it to approach 1), so the boost is the same everywhere and the curve still falls where the dots stay flat. The light-violet dashed curve (behind a checkbox above the chart since 2026-10-22) wires the same C the other way and does nothing. The hand-tuned tanh(radius) stand-in this page displayed before 2026-07-08 (it fit because it was drawn to fit) is behind the checkbox above the chart. Full disclosure of the 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 fitted parameters, though the mass itself is read off each galaxy's plateau. The MOND curve has a circularity of its own (disclosed 2026-10-08 after a researcher visitor checked it): its baryonic mass comes from the empirical BTFR, Mb = 47·Vflat⁴, so part of its agreement with Vflat is built in. The 47 is also not exactly consistent with the a₀ used: 1/(G a₀) = 62.8 M☉/(km/s)⁴, so the green curve levels off at (47/62.8)¼ ≈ 0.93 Vflat, not at Vflat. The disk has no gas component, which matters most for DDO 154. A fair comparison uses SPARC mass models for both laws; that is what the ledger's tests do. 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.