γ Regimes: Crossovers, Not Phase Transitions
Reparametrization — γ ≈ 1 clustering restates the Debye θ_D (research ledger: a known result in new notation)C(ρ) is a smooth curve: it has no sharp transition anywhere. What γ does is sort systems into three bands (few-body / uncorrelated, boundary, many-body collective), and the clustering of fitted γ near 1 restates the Debye temperature θD (1912) rather than adding new physics. This page describes those bands and why they are crossovers, not phase transitions.
Revision notes (history; not the current claim)
Until 2026-10-24 this page lived at/phase-transitions (old links redirect here) and opened “The coherence function predicts three distinct regimes, with transitions between them. The most interesting physics happens at the boundaries — especially at γ ≈ 1.” The slug, that sentence, the H1 and the badge said four different things (technical-writer simulated-visitor persona, same defect fixed on the visualizer 2026-10-23). The badge descriptor said “ledger Bucket 3”, a term the site did not define.Why the bands are no longer labelled “quantum” and “classical” (2026-10-26). The archive called the γ = 2 end “strongly quantum” (a single electron) and the γ → 0 end “classical” (macroscopic). That reading is backwards against statistical mechanics: quantum degeneracy needs many overlapping particles (nλdB³ ≳ 1), so the most quantum many-body states, condensates and superconductors, have Ncorr ≫ 1 and land at the far right, the “classical” end. That is the formula's sign inversion in one picture. The deeper reason is on parameter derivations: in every mean-field model a transition gets sharper as more units correlate (the Curie–Weiss rounding width shrinks as N−1/2), so a sharpness parameter should grow as √N, the reciprocal of γ = 2/√Ncorr. The band cutoffs (0.6 and 1.4 on the visualizer) are display conventions with no derivation behind them, and Ncorr itself has no operational definition (see the superconductor example below).
The γ ≈ 1 Boundary
Correction (2026-08-09): this section stated that at γ ≈ 1 (Ncorr ≈ 4) “the coherence function has maximum curvature.” It does not. With x = ρ/ρcrit, dC/dx∣x=0 = γ — strictly increasing in γ, no interior maximum anywhere; in log-density the peak sensitivity also rises monotonically with γ and merely saturates (≈0.32 at γ=1, ≈0.45 as γ→∞); and C is concave for every x ≥ 0, so it has no inflection point. Nothing in the shape of C distinguishes γ ≈ 1. The γ ≈ 1 clustering is an empirical fit result with no derivation behind it. Full retraction and the derivative on the γ boundary page. (Simulated-visitor Pass 3 found the /gamma-boundary instance; this second one was found by sweeping.)
What the chemistry sector actually has is a clustering of fitted γ near 1, and the list below is a description of where that clustering falls — not a consequence of the equation:
- Phase transitions cluster here (melting, boiling, sublimation)
- Catalysis operates at this boundary (enzymes, industrial catalysts)
- Superconductivity appears (Cooper pairs ≈ 2-body correlations)
- Biology emerges (the archive's gloss was “molecular recognition requires a quantum-classical interface”; see the band note above for why that labelling does not hold)
Which Ncorr for a superconductor? This list places superconductivity at γ ≈ 1 by counting the Cooper pair (Ncorr ≈ 2–4). The γ Calculator's BCS preset counts the electrons overlapping inside one coherence volume (Ncorr = 107, γ ≈ 6×10−4). Both counts are defensible, and the formula has no rule that picks one. The same system lands four orders of magnitude apart in γ depending on what is counted. That is the Ncorr-divergence problem in one example: until Ncorr has one operational definition, γ = 2/√Ncorr can place a system wherever it is needed.
1,703 chemistry sessions were logged (one “phenomenon type” per session, a counter rather than a census), and 89% gave γ values within the boundary region. The audit found that about 86% of that 89% restates the Debye model (θD, 1912), so the figure is a reparametrization, not a confirmed prediction. See the full chemistry analysis →
Transitions in Cosmology
The framework proposed the same picture at galaxy scale: a transition at ρ ≈ ρcrit from Newtonian (high-density inner region) to MOND-like (low-density outer region). This did not hold as a density law. Keyed on density, the threshold has the wrong velocity scaling (the data need ρcrit ∝ V−2), and the version that fits galaxy rotation is keyed on acceleration with γ fitted, where it becomes MOND. The tanh is also smooth, so the change is a crossover rather than a phase transition. See Galaxy Rotation.
What Doesn't Work
Real phase transitions have universality classes with specific critical exponents. C(ρ) uses a tanh motivated by mean-field theory, but has no self-consistency equation — ρ goes in, C comes out with no feedback loop. It belongs to the logarithmic compander family (μ-law / Hill / Naka–Rushton), not Landau theory, and has no critical point and no critical exponents.
What the Failed badge records. The research ledger lists “critical exponents: 2× off from observation” as a failure, citing the archive README. No exponent, no fit and no data set is recorded with it. What failed is the older framing that read this curve as a Landau-style phase transition and so implied it should have exponents. That reading is a category error: you cannot be 2× off on a quantity a function does not define. The badge stays Failed because the claim was made and does not hold; it is not a measured exponent of this tanh. The function captures where fitted γ values cluster, not how a transition unfolds.