Coherence Explorer
Reparametrization — At γ = ½ this is a formula physics already had (MOND's, fed acceleration; ΛCDM in cosmology), not a new oneWhat this toy is for: seeing the shape of the S-curve and what γ does to it. It does not model any real material, and the density axis has no units. The curve drawn is the density version of the equation; the opening γ = 0.49 comes from fitting the acceleration version (a different function of a different variable, see the caption), so the default is illustrative here. One thing to try first: the page opens at γ = 0.49, so first drag γ up to 2 (the original guess), then drag it back down to 0.5 and watch the S flatten into a gentle ramp. That flattening is what happens when the curve is fitted to galaxies, and at γ = ½ the ramp has exactly the shape of MOND's formula (when it is fed acceleration instead of density), which is why the galaxy fit is not a new prediction.
What is coherence?
Coherence ≈ 0 is like a crowd of strangers in a plaza — everyone moving independently, no shared rhythm. Coherence ≈ 1 is like a marching band — everyone in lockstep. This curve shows how a system transitions between the two as density increases.
This tool plots the coherence function — the single equation at the heart of Synchronism. It takes a density (ρ) and returns a coherence value between 0 (sparse/independent) and 1 (dense/collective).
What this shows / what it gets wrong: the tool shows how the curve's shape responds to its three parameters — that part is faithful. What the formula gets wrong, and you should know before sliding: the γ = 2/√Ncorr sharpness rule runs backwards against real condensed-matter physics (the most correlated systems get the flattest curves; real BCS transitions are among nature's sharpest), and the “ρcrit” slider marks a saturation knee, not the curve's midpoint — at ρcrit and γ = 2 the curve already reads C ≈ 0.88, and the true C = 0.5 midpoint sits near 0.32·ρcrit. Both numbers are γ-specific: the live readout by the ρcrit slider recomputes them as you drag γ. Details in the caveats below.
Revision note
This note previously called site coherence “anti-correlated with quantum phase coherence” and, two sentences later, “orthogonal”; this note and the paragraph above it also labelled C ≈ 0 “quantum-like” and C ≈ 1 “classical-like.” Only “orthogonal” is defensible (condensates are both quantum-coherent and collective), so the anti-correlation wording and the quantum/classical labels were removed.γ = 2/√Ncorr was proposed as the transition sharpness. What it actually sets is the initial slope (C ≈ γx at low density) and how fast C approaches 1; for γ above ~1 it mostly moves the knee (toward 0.77 ρcrit/γ as γ grows, from 2u tanh u = 1; that asymptote is 15% low at γ = 1 and 8% low at γ = 2; the “knee” here is the inflection point of C on a log-density axis, and ρcrit itself is a reference scale, not the half-way point: C(ρcrit) = tanh(γ ln 2), 0.33 at the default γ) rather than sharpening it, so it trades off against ρcrit (derivation; added 2026-10-23). High γ (> 1.4, small Ncorr) = single-particle / uncorrelated regime (ideal gases, free atoms); γ ≈ 1 = where the archive's fitted chemistry γ values cluster — nothing in C(ρ) singles it out (γ Boundary, retracted 2026-08-09); low γ (< 0.6, large Ncorr) = many particles moving together (BEC, superconductors, superfluids) — which does not mean C near 1: low γ gives low C at any plottable density (see “What to notice” below). Note: these labels come only through γ = 2/√Ncorr, the relation this page's caveat below calls sign-inverted, so under that inversion they point the wrong way (flagged 2026-10-22, graduate-physics simulated-visitor persona). They describe the number of correlated degrees of freedom, not the standard quantum/classical distinction — BEC and BCS superconductors appear in the “collective” basin, which is correct (they have large Ncorr).
⚠ Sign-inversion caveat: γ = 2/√Ncorr is structurally inverted relative to real condensed-matter physics — more correlation (larger Ncorr) gives asmaller γ, i.e. a flatter transition, assigning the flattest curve to BCS superconductors (which have a sharp real Tc) and the sharpest curve to an ideal gas (which has no real phase transition). See the γ Calculator for the full sign-inversion accounting.
What to notice: Move γ from 2.0 downward and watch the curve flatten. Nothing special happens to the curve at γ ≈ 1: it is not a steepness extremum, and the old label placing chemistry and biology at a γ ≈ 1 boundary was retracted on γ Boundary (chemistry γ values were fitted there; no biology placement follows from C). Notice also the two presets γ = 0.49 (“galaxy fit”) and γ = ½ (“same curve as MOND”): they are almost the same number because the free galaxy fit lands on MOND's curve. That is why the fit is not a new prediction. (The slope is actually largest at ρ = 0 on a linear ρ axis; on a log axis, the peak slope grows as γ increases — e.g. peak dC/d(lnρ) ≈ 0.375 at γ=2 vs exactly 0.25 at γ=0.5, i.e. per decade on this chart's log₁₀ axis ≈ 0.86 vs ≈ 0.58 (multiply by ln 10 ≈ 2.303) — consistent with γ=2 being the sharpest curve. Corrected 2026-09-15: these were per-e-fold values labelled per decade.) The slider goes down to γ = 0.01: on the canonical ladder BEC sits near γ ≈ 2×10−3 (Ncorr = 106) and BCS superconductors near γ ≈ 6×10−4(Ncorr = 107) — both below the slider's floor (use the γ Calculator to reach those regimes). At very low γ the curve is nearly flat near C = 0 — a very long, gentle ramp. At γ = 0.01, C is still only 0.23 at 1010×ρcrit, and reaching C = 0.5 takes ~1024×. Turning γ down lowers C over any density range you can plot, even though the “many correlated particles” label above sounds like the opposite.
Why γ is decoupled from Ncorr here: This tool sets γ directly to explore how curve shape depends on γ, independently of any physical system. The relationship γ = 2/√Ncorr links γ to real systems — use the γ Calculator to enter a physical Ncorr and see where a real system lands.
- ln(x) — logarithm: it compresses huge ranges into small ones. A density 1,000× bigger becomes only ~7 units bigger inside the equation (natural log); on this chart's base-10 axis the same jump is 3 gridlines. The compression does not let one curve span quantum to cosmic scales: at any sharp γ the tanh saturates within ~1.6 decades of the knee, which is why the “80 orders of magnitude” unification was refuted (Session 633; corrected here 2026-09-14).
- tanh(u) — the “S-curve” shape. At u = 0 it returns 0; as u grows it rises, steeply at first, and levels off toward 1. Here u = γ ln(1 + ρ/ρcrit) is never negative, so C always runs from 0 (no density) to 1 (very dense). Think of a dimmer whose sharpness γ sets: gradual at γ ≈ ½, steeper at large γ.
- γ — small γ = a long gentle ramp; above γ ≈ 1, more γ slides the curve left without narrowing it. No γ makes the rise sudden: the 10–90% width stops near 1.17 dex (see the readout). Warning first: the framework's rule for which material gets which γ (γ = 2/√Ncorr: free atoms → 2, superconductors → ~6×10⁻⁴) runs backwards against real condensed-matter physics, where superconductors switch on abruptly. So treat this tool as a shape toy for learning the curve, not a map of real materials (details further down).
- ρcrit — the density where the dimensionless argument x = ρ/ρcrit equals 1; a reference point set by fitting, not a physical critical point and not the curve's midpoint: C(ρcrit) = tanh(γ ln 2) is γ-dependent (0.327 at γ = 0.49, 0.60 at γ = 1, 0.88 at γ = 2 — the three values printed on three pages of this site are all correct, at those three γ).
Adjust γ and ρcrit to see how the coherence function C(ρ) = tanh(γ · ln(ρ/ρcrit + 1)) responds. The dashed amber line marks C = 0.50; the dotted knee marker (ρ = ρcrit) is not where C = 0.5 — the curve's value there depends on γ (its label shows it).
Before you drag: the default γ = 0.49 is what galaxy data prefer when γ is left free in the acceleration version of this curve (the one that turns out to be MOND). This tool's x-axis is density, and the density version fitted freely to the same galaxies runs to γ = 0.046 — a slow, log-like rise (the preset sets γ only; the fit's knee is far off this axis). γ = 2 is what the framework originally said. The gaps between these numbers are part of why the framework failed its galaxy test.
Default γ = 0.49 is the galaxy-fit value, fitted with acceleration (g/a₀′) as the argument, not density; the same curve shape, a different variable. See the caveats below for why it fails in the Solar System (the presets quote that miss in σ, standard deviations: 8.7σ means the mismatch is 8.7 times the measurement uncertainty, far beyond chance). The tool opens on this value because it is the one galaxy data pick, not because the framework stands behind it; no γ here is one the framework predicts. All the γ values in use, with their variables: Which C? Which γ?
Presets are values of γ the site discusses, not physical systems: the density axis has no units, so no preset puts you “at” a galaxy, a metal, or a brain. For named systems via Ncorr, use the γ Calculator.
Many-body by γ = 2/√N_corr (γ < 0.6), yet C is lower here: the formula’s sign inversion (label from the audited-negative Ncorr map, whose sign is inverted; see the caveat above)
Higher γ = sharper, more abrupt snap to coherent. Lower γ = gentler slope. (The framework also wrote γ = 2/√Ncorr, which would make more correlated systems gentler. That runs backwards against real condensed-matter physics and is audited-negative; see the γ Calculator. Treat γ here as a shape dial only.)
Opens at γ ≈ 0.49, the value SPARC galaxy data prefer when the acceleration-keyed version is fit freely (0.489). The density-keyed version — the one this tool's ρ/ρcrit axis draws — fit freely to SPARC runs to γ = 0.046 instead: the fit keeps only a weak density response (boost ∝ ρ−0.10, with its knee fitted at 4.4×10−6 M☉/pc³, below every disc; it beats a constant boost by ΔBIC +945, so the fit's response is weak, not off; that response is gbar's, not density's: at fixed gbar, local density adds nothing (partial slope +0.003, 95% CI −0.022 to +0.027), and a model that never sees ρ reproduces the data's density slope to 0.2%, explorer 2026-11-04) (see Core Idea). γ = 2 was the framework's original galaxy guess — refuted (drag the slider up to see it). The default is dead on two fronts, not one (added 2026-09-05): γ ≈ 0.49 reproduces MOND on SPARC and the same γ fails the Cassini Solar-System bound — at 8.7σ as published by Desmond, Hees & Famaey 2024 (MNRAS 530, 1781; marginalized over a₀, M/L and RAR-fit uncertainty), (TEST-25 on Tier 1). This site's own unmarginalized run gives a larger number, but only by treating 2,807 rotation-curve points as independent; with the galaxy as the unit its retained γ range widens to 0.3–1.7, so the published 8.7σ is the figure to quote. At the γ galaxies prefer the bound fails; and once a0 is taken from SPARC at each γ, no γ up to 3 passes Cassini (an earlier scan with a0 held fixed had γ ≳ 1.5 passing; corrected 2026-11-06). And the S-shape you see is a property of the log axis: with the density-keyed knee at the published ρcrit = 0.029·V², SPARC samples x = ρ/ρcrit at a median of ~7×10−5, where C = γx to 1.8% at worst (0.22× the data's precision on C) — a straight line. Toggle to linear and zoom to x < 10−3 to see the regime the data occupy at that knee (explorer measurement 2026-09-03). The free density fit puts the knee elsewhere: at 4.4×10−6 M☉/pc³, below every disc, so its discs sit at x ~ 10³–10⁵ in the log regime, not the linear one (explorer 2026-11-02).
Two models, one curve (added 2026-09-06). “γ ≈ 0.49 fits SPARC and fails Cassini” is a fact about the acceleration-keyed branch, x = gobs/a0′ (the compander used as an implicit interpolating function and solved for gobs; see Galaxy Rotation). “SPARC sits at x ~ 7×10−5 where C = γx” is a fact about the density-keyed branch, x = ρ/ρcrit. The slider draws one curve; the site's galaxy numbers come from two different arguments of it, and no single model both fits the RAR and is linear over all of SPARC. The two are compared head-to-head on The Coherence Function (density-keyed loses at ΔBIC +2843).
Revision note
The caption above led with “fails the Cassini Solar-System bound at ~18σ”; it now leads with the published, marginalized 8.7σ, matching Honest Assessment. This box wrote the acceleration-keyed branch as x = gbar/a0; the SPARC fit that gives γ ≈ 0.49 keys on gobs implicitly (explicit gbar keying is the floored TEST-09/10 form).What the default view actually shows (added 2026-08-08): MOND. Set γ = 1/2 exactly and the compander collapses algebraically to C(x) = x/(x+2) = μsimple(x/2) — MOND's simple interpolating function, the factor of 2 absorbed by the fitted ρcrit. SPARC's 0.489 is 2.2% from that point, so the curve you are looking at on load is MOND's μ, not an alternative to it. Drag γ away from 0.5 and you are leaving MOND; the data pull it back.
Lower ρcrit moves the whole curve left, and higher moves it right. It sets the scale, not the shape, and it does not mark a transition.
This slider carries no shape information — and that is the point (added 2026-08-08). C depends on ρ and ρcrit only through the single ratio x = ρ/ρcrit, exactly, not just asymptotically. On the log axis this slider therefore performs a rigid horizontal translation by −log10ρcrit and changes nothing about the curve's shape: the tool is a one-parameter family plus a pan control, presented as two parameters. That is not a UI nit, but it is not a γ↔ρcrit degeneracy either (corrected 2026-09-29, graduate-physics simulated-visitor persona): a pan and a shape parameter are separately identifiable once data cross the knee, and SPARC does identify the shape (γ = 2 and γ = 0.489 differ by ΔBIC 184 with a₀′ profiled in both). What the galaxy fit cannot do is tell its shape from MOND's: the identified shape near γ = ½ is the simple μ, and in the deep regime only the combination γ/a₀′ is measured. Read the abscissa as log10(ρ/ρcrit) — dimensionless — and the pan is the only thing this slider can express. Raised by a simulated-visitor researcher pass.
Note: ρcrit is a saturation knee, not a critical density in the phase-transition sense. At ρ = ρcrit and the current γ = 0.49, C = tanh(0.49·ln2) ≈ 0.327 — the “+1” regulator inside the ln breaks sigmoid symmetry, so ρcrit is not the curve's midpoint. The C = 0.50 midpoint sits at ρ ≈ 2.07·ρcrit. This offset is γ-dependent and moves as you drag the slider (corrected 2026-07-27 — this note used to quote the γ=2 value 0.32 as if it were a general property, which goes wrong the moment the slider it accompanies is touched: at the SPARC-preferred γ = 0.49 the midpoint is at 2.07·ρcrit, i.e. above the knee).
Key Values
C(ρcrit)
0.3272
C(10ρcrit)
0.8259
C(100ρcrit)
0.9785