Crossover Regime Visualizer
Tool type: Model Explainer · Artifact Lesson (the Interactive Tools index Kind and Status tags — Artifact Lesson is an operational state, not a level: a tool kept to show a failure, not a verdict) · Claim status: Audited-Negative, the badge above.
Formerly “Phase Boundary Visualizer” (the URL keeps that name). C(ρ) is analytic everywhere — a smooth crossover, not a thermodynamic phase transition — so the regimes below are ranges of γ, not phases (full naming note below).
What this tool is for: drag the slider to see which physical systems the γ formula groups together — and why that grouping is now known to be misleading (the placements carry the formula's documented sign inversion: real BCS/BEC transitions are among the sharpest in nature, yet the formula files them at the flat end).
One more thing to know before reading the map: the Ncorr values driving every placement (10⁷ for BCS, 17 for galaxies, …) are asserted, not counted — no independent counting procedure exists on any of the 17 scales (the derivations page documents this). Treat the positions as the formula's claims, not as data.
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.
What to notice as you drag: BCS superconductors and BECs land at the far flat (left) end — yet their real transitions are among the sharpest in nature — while an ideal gas, which has no phase transition at all, lands at the sharp end (γ = 2). That inversion is the lesson.
Band edges in Ncorr: inverting γ = 2/√Ncorr gives Ncorr = 4/γ², so γ = 1.4 ⇔ Ncorr ≈ 2.04 and γ = 0.6 ⇔ Ncorr ≈ 11.1. With whole-number counts, the single-particle band holds only Ncorr = 1 and 2, the γ ≈ 1 band holds 3–11, and everything with 12 or more correlated units lands in the bottom band. That includes the γ Calculator's enzyme preset (Ncorr = 30, γ ≈ 0.37) and BCS (107). This page's enzyme placement near γ ≈ 0.85 implies Ncorr ≈ 5.5, a different estimate.
γ ≈ 1 (0.6–1.4) — Boundary · N_corr = 3–11
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) was historically placed in this band; C is a different axis from γ, and D and S remain undefined, so that placement has no calculation behind it — see hard-problem page.
Reality check: no shared transition physics is known to connect the systems grouped here — water, enzymes, and neural firing land together because of how their N_corr was estimated, not because of any measured common behavior at γ ≈ 1. The grouping is the counting convention, visualized.
⚠ 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. Which law (added 2026-09-24): the +184 and the 0.49 both come from the formula keyed on acceleration (MOND's variable), while this axis is the density-keyed law's γ = 2/√Ncorr. Fitted in density, the galaxy law goes to γ → 0.046 (Ncorr ≈ 1.9×10³) and is refuted at ΔBIC +2843. So the 0.49 marker is placed on an axis it was not measured on. The γ values shown here are illustrative, not measured.
The Three Regimes
Summary index only — move the slider above to read each regime's full physics in the highlight card (duplicated text removed 2026-07-17).