Phase Boundary Visualizer

Despite the name: C(ρ) is analytic everywhere — a smooth crossover, not a thermodynamic phase transition. “Boundary” here is a visual convenience (full naming note below).

This tool animates a refuted relation — on purpose. The γ = 2/√N_corr map you are about to drag is audited-negative: it predicts the opposite of real condensed-matter behavior. The lesson is the inversion itself. As you move the slider, each regime card carries a reality check line — your task is to catch the formula being wrong at every stop, not to learn its placements. (Reframed 2026-07-23: an earlier version presented the map first and the inversion as a caveat; reviewers kept reporting that the animation out-taught the caveat.)

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 N_corr 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/√N_corr; N_corr 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 γ actually classifies: γ = 2/√N_corr encodes the counting convention used for N_corr, not a physical invariant of the system. Two unrelated systems with the same estimated N_corr(e.g., ideal gases and galaxies, both assigned N_corr = 1) share a γ value for methodological reasons, not physical ones. This visualizer shows how systems cluster under the current counting convention; it does not imply they have the same microphysics. See the caveat box below for details.

Sharpness direction is inverted (open problem): this tool positions systems by transition sharpness, and γ = 2/√N_corr assigns the most strongly correlated systems the flattest C(ρ) curves. In real physics the opposite holds: BCS superconductors and BECs — the highest-N_corr systems shown here — undergo transitions that are among the sharpest in nature. So a condensed-matter reading of this map is backwards: more correlation should not mean a smoother transition. This is the same audited-negative sign inversion documented on the γ Calculator (1/√N is a width, not a rate); it is stated here because this is the page where systems are visually placed. Also note the axis itself: the quoted BCS value (γ ≈ 6×10⁻⁴) sits three orders of magnitude off the left edge of the displayed 0–4 range — the marker below is a direction indicator, not a position.

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.

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.

Liquid waterEnzymesPhase transitionsChemical bondsNeural firingConsciousness threshold

⚠ N_corr-method caveat (important): Ideal gases and galaxies both map to γ = 2.0 (N_corr = 1), despite having completely different microphysics. This is not a universality result — it is an artifact of how N_corr 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 N_corr recipe is established, every cross-scale γ comparison is method-dependent. BCS superconductor placement uses N_corr = 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 (N_corr = 1) was rejected on the SPARC RAR ensemble at ΔBIC = +184; the free fit gives γ ≈ 0.49 → N_corr ≈ 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

Summary index only — move the slider above to read each regime's full physics in the highlight card (duplicated text removed 2026-07-17).

γ < 0.6 — High-N_corr (strongly correlated)

SuperconductorsBECSuperfluidsCooper pairsQuantum computers

γ ≈ 1 — Boundary

Liquid waterEnzymesPhase transitionsChemical bondsNeural firingConsciousness threshold

γ > 1.4 — Single-particle

Ideal gasesSingle atomsFew-particle systemsUncorrelated matter

Related Concepts

Phase Transitionsγ < 1, γ ≈ 1, γ > 1 regimes The γ ≈ 1 Boundary1,703 phenomena at the quantum-classical edge