γ Calculator
Formula Audited-Negative — Sign Inverted for All Collective Systemsγ = 2/√Ncorr assigns the sharpest coherence transition to the least-correlated system (ideal gas) and the flattest to the most-correlated (BCS superconductor) — the opposite of real condensed-matter transitions. This is a structural inversion in the formula (Caveat 2 below), not a calibration issue. This tool is preserved for exploration; treat its outputs as “what the inverted formula predicts” rather than as physical estimates.
Three caveats before using this tool:
- γ = 2/√Ncorr is motivated, not rigorously derived. The 1/√N scaling is a dimensional ansatz inspired by fluctuation theory — not a consequence of the CLT (which governs sample-mean fluctuation, not transition sharpness). The factor of 2 is not derived from first principles. See Parameter Derivations for what is and isn't derived →
- The direction of the Ncorr→sharpness mapping is inverted relative to the stated analogy (2026-06-06). In fluctuation theory, 1/√N is a width — more correlation → smaller width → sharper transition. But in γ = 2/√Ncorr, more correlation → larger Ncorr → smaller γ → flatter tanh. This assigns the sharpest transition (γ=2) to the least-correlated system (ideal gas, no real phase transition) and the flattest (γ≈6×10−4) to the most-correlated (BCS superconductor, which has a real sharp Tc). The sign of the analogy is inverted — a structural issue independent of the prefactor. See Parameter Derivations and research proposal
gamma_ncorr_sign_inversion_sharpness.md. - Preset Ncorr values are back-fits, not measurements. For BCS superconductors, the physical Cooper-pair coherence volume contains 106–109 pairs; the preset uses Ncorr = 107 (mid-range of physical estimates) — fitted to produce a plausible γ (6.32×10−4), not derived from the Hamiltonian. No protocol exists for converting a system's Hamiltonian into Ncorr without first fitting γ to observed behavior. Every γ “prediction” is therefore a consistency check on a back-fitted parameter, not a first-principles result.
What this tool does: Ncorr is the count of particles that move as a correlated unit — dimensionless, no units. For a single atom, Ncorr = 1. For a crystal oscillating in phase, Ncorr can reach millions. This tool maps that count to γ = 2/√Ncorr and shows which physical regime results.
γ ≈ 1 marks the boundary where chemistry and phase transitions happen. The presets below cover common systems: BCS = Bardeen-Cooper-Schrieffer superconductors (electrons paired by phonons, conventional: Al, Nb, Pb); BEC = Bose-Einstein Condensate (ultra-cold atoms collapsed into a single quantum state). Both are quantum systems that appear in the “collective” regime because they have large Ncorr.
Input Ncorr (number of correlated particles) and see the resulting γ = 2/√Ncorr and what physical regime it falls in.
Ncorr = 4
γ = 1.0000
Boundary (γ ≈ 1)
Transition zone. Phase transitions, chemistry, consciousness threshold sit near this boundary.
Presets
⚠ Preset caveat: Ncorr values in the presets are approximate estimates, not measured physical pair counts. The BCS superconductor preset uses Ncorr = 107 (mid-range of physical Cooper-pair coherence volumes); physical estimates span 106–109 pairs (Al vs. Nb vs. Pb differ significantly). The Phase Boundary Visualizer uses different Ncorr estimates for some systems. A scale-invariant counting recipe for operational Ncorr is an open research question — see ncorr-operational-definition-recipe in the explorer topic queue.
Quick Reference
| Ncorr | γ | Regime | Example |
|---|---|---|---|
| 1 | 2.000 | Weakly Correlated (γ-sharp) | Ideal gas |
| 4 | 1.000 | Boundary (γ ≈ 1) | Liquid water |
| 30 | 0.365 | Strongly Correlated (γ-flat) | Enzyme site |
| 100 | 0.200 | Collective Regime (γ-flattest) | Ferromagnet |
| 10,000,000 | 6.32e-4 | Collective Regime (γ-flattest) | BCS superconductor (10⁷ — mid of physical 10⁶–10⁹) |
| 1,000,000 | 2.00e-3 | Collective Regime (γ-flattest) | BEC |