Coherence Explorer
What 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.
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 controls the transition sharpness. High γ (> 1.4, small Ncorr) = single-particle / uncorrelated regime (ideal gases, free atoms); γ ≈ 1 = the boundary where chemistry and biology happen; 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 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. γ ≈ 1 is the regime boundary where chemistry and biology happen — not a steepness extremum. (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 the sharpness parameter, 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. For very negative u it returns ≈ 0; for large positive u it returns ≈ 1; near zero it rises steeply. Think of a dimmer switch that snaps rather than fading gradually.
- γ — controls how quickly the snap happens. Large γ (γ = 2, free atoms) = a sharp cliff. Small γ (γ ≈ 6×10⁻⁴, superconductors) = a long gentle ramp.
- 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.
Before you drag: the default γ = 0.49 is what galaxy data prefer when γ is left free. γ = 2 is what the framework originally said. The gap between them is part of why the framework failed its galaxy test.
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.
Collective (γ < 0.6)
Higher γ = sharper, more abrupt snap to coherent. Lower γ = gentler slope. Depends on Ncorr (correlated particle count): γ = 2/√Ncorr.
Opens at γ ≈ 0.49, the value SPARC galaxy data actually prefers when fit freely. 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), and at ~18σ in this site's unmarginalized run (TEST-25 on Tier 1); no γ satisfies both. And the S-shape you see is a property of the log axis: with the density-keyed knee where the galaxy fits put it, 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 actually occupy (explorer measurement 2026-09-03).
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 = transition starts at lower density (shifts curve left). Higher = transition occurs at higher density.
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 — it is the γ↔ρcrit degeneracy that makes SPARC's γ = 0.489 non-identifiable against a rescaled critical density, which is the crux of the whole galaxy sector, made visible. Read the abscissa as log10(ρ/ρcrit) — dimensionless — and the degeneracy is the only thing the slider can express. Raised by a 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