Phase Boundary Visualizer
The three regimes of γ map qualitatively different correlation structures. Drag the slider to explore where different systems are estimated to fall. Labeled positions are approximate (γ = 2/√Ncorr; Ncorr values are estimated, not precisely measured for most systems).
Naming note: despite the historical name, C(ρ) is a smooth compander (μ-law/Hill/logistic family) with no critical point — “boundary” on this page means a regime boundary along the γ axis, not a phase transition.
γ ≈ 1 — Boundary
The regime where collective and independent behavior balance. Systems whose estimated γ falls here include liquid water, enzymes, and neural dynamics. Note: C(ρ) itself is a smooth compander with no critical point — "boundary" here means a regime boundary in γ, not a mathematical phase boundary. The consciousness threshold conjecture (C ≈ 0.50) maps to this regime, though D and S remain undefined — see hard-problem page.
⚠ Ncorr-method caveat (important): Ideal gases and galaxies both map to γ = 2.0 (Ncorr = 1), despite having completely different microphysics. This is not a universality result — it is an artifact of how Ncorr is counted: both systems happen to be assigned “1 correlated particle” under the current counting convention. When two physically unrelated systems produce the same γ, γ is classifying the counting method, not the system. Until a scale-invariant Ncorr recipe is established, every cross-scale γ comparison is method-dependent. BCS superconductor placement uses Ncorr = 107 (γ = 2/√107 ≈ 6×10−4, matching the γ Calculator preset); physical Cooper-pair volumes contain ~106–109 pairs depending on material.Galaxy placement (2026-06-11): the asserted γ = 2 (Ncorr = 1) was rejected on the SPARC RAR ensemble at ΔBIC = +184; the free fit gives γ ≈ 0.49 → Ncorr ≈ 17, contradicting the independent-stars premise. Both markers are shown so the refutation is visible, not hidden. The γ values shown here are illustrative, not measured.
The Three Regimes
γ < 0.6 — High-N_corr (strongly correlated)
Many correlated particles (large N_corr, small γ) — the strongly correlated regime. BEC, BCS superconductors, and superfluids sit here because they have enormous N_corr, giving them very small γ (BCS: N_corr = 10⁷, the γ Calculator preset, giving γ = 2/√10⁷ ≈ 6×10⁻⁴). Important: "collective" in the C-axis sense (C→1) is separate from "large N_corr" here — small γ gives a nearly flat tanh curve, so C stays near 0 at physically accessible densities despite large N_corr. See γ Calculator caveats for this documented inversion.
γ ≈ 1 — Boundary
The regime where collective and independent behavior balance. Systems whose estimated γ falls here include liquid water, enzymes, and neural dynamics. Note: C(ρ) itself is a smooth compander with no critical point — "boundary" here means a regime boundary in γ, not a mathematical phase boundary. The consciousness threshold conjecture (C ≈ 0.50) maps to this regime, though D and S remain undefined — see hard-problem page.
γ > 1.4 — Single-particle
Few correlated particles (small N_corr, large γ) — the single-particle / uncorrelated regime. Individual behavior dominates; rapid decoherence. γ = 2 (N_corr = 1) is the value the framework asserts for galaxies — but the SPARC RAR ensemble test rejected γ = 2 at ΔBIC = +184; the free fit gives γ ≈ 0.49, which back-implies N_corr ≈ 17 and contradicts the N_corr = 1 premise. The galaxy marker below is shown at the asserted value with that refutation flagged.