Phase Transitions
Reparametrization — γ ≈ 1 clustering restates the Debye θ_D (ledger Bucket 3)The coherence function predicts three distinct regimes, with transitions between them. The most interesting physics happens at the boundaries — especially at γ ≈ 1.
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. (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 (molecular recognition requires quantum-classical interface)
1,703 chemical phenomena were fitted, 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. The critical exponents are off by ~2× not because of a calibration miss, but because a compander has no critical point to generate exponents from. The function captures where transitions happen but not how they unfold.