γ Calculator
Formula Audited-Negative — Sign Inverted for All Collective SystemsIn one sentence: γ is the steepness of the coherence S-curve — how abruptly a system switches from “independent individuals” to “acting as one” as density rises. Big γ = hair-trigger switch; small γ = slow fade. The live curve below redraws as you change Ncorr.
What this tool is for: pick a physical system (ideal gas → BEC presets) or enter Ncorr yourself, and see the γ the formula assigns and how it reshapes the coherence S-curve. What to conclude: not a physical estimate, but a live demonstration of the formula's central defect — watch the most-correlated systems get the flattest curves, which is backwards. Why is a formula we know is wrong still interactive? So you can see the flaw yourself: click the “Ideal gas” preset, then “BCS superconductor,” and notice the far more collective system gets the far flatter curve. Real superconductors have among the sharpest transitions in nature — you just watched the formula get that backwards.
γ = 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.
Four 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.
- Run the framework's own galaxy parameters through this tool and it refutes them (added 2026-07-27). Until today this page never applied its map to the one sector where the framework actually uses γ. Inverting the framework's own relation, Ncorr = (2/γ)²: the galaxy pin γ = 2 quoted on Core Idea gives Ncorr = 1 — the ideal gas preset, exactly; and the SPARC best fit γ ≈ 0.49 quoted on Galaxy Rotation gives Ncorr ≈ 17, between liquid water (4) and an enzyme active site (30). So the framework's original galaxy-scale parameter says a galaxy is a system of one correlated unit, and its data-preferred value says a galaxy is about as collectively organized as a small protein pocket — while a BCS superconductor in the same table gets 10⁷. Both galaxy numbers were published on this site for months, one arithmetic step apart, on pages that never cited each other. Either γ = 2/√Ncorr is void at galaxy scale — in which case it is not a framework-wide relation and should stop being presented as one — or the framework asserts the above. The SPARC row is now in the preset table below so the arithmetic is one click away. (Note this also breaks the 2026-06-06 finding that flipping the sign to γ ∝ √Ncorr “changes nothing calibrated”: that held only because γ = 2 sits at the fixed point Ncorr = 1. At Ncorr ≈ 17 the flip moves γ by a factor ~8.)
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 regime boundary where chemistry and collective/correlated behavior happen — C(ρ) itself is a smooth crossover here, not a phase transition (tanh has no non-analyticity). 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) — a label per the inverted formula (Caveat 2), not a physical classification
Transition zone. Phase transitions, chemistry, consciousness threshold sit near this boundary.
What just changed, and why it matters: Water molecules hydrogen-bond into small transient clusters (~4 moving together), which drops γ from 2 toward 1 — the boundary zone where the framework says chemistry lives. Compare: one preset click took you from "lone particles" to "small teams."
The curve this γ produces — C(ρ) = tanh(γ·ln(1+ρ/ρcrit)) at your current γ (violet), with the γ=2 galaxy pin ghosted for comparison (gray).
Try it: click “Ideal gas” (γ=2, the steep violet S) then “BCS superconductor” (γ≈6×10−4) and watch the curve collapse to nearly flat — the inversion in Caveat 2, drawn live: the most collective system gets the flattest curve.
Where γ lands when data chooses (2026-07-22 mechanism): fit free on SPARC galaxy data, γ converges to 0.49 — and that number now has an identified meaning. γ sets the curve's Newtonian-return exponent q = 2γ (C approaches 1 like (1+ρ/ρcrit)−2γ), and the fit pins q ≈ 0.98 — cross-validated by an independent free-Hill fit (n = 0.975). q = 1 is the value hard-coded in MOND's “simple” μ-function, so γ = 0.49 is not a constant awaiting derivation: it is the tanh family's encoding of MOND. The framework's asserted γ = 2 means q = 4 — returning to Newton far too abruptly, which is the ΔBIC = +184 refutation restated as a mechanism. See Galaxy Rotation.
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 |
| 17 | 0.485 | Strongly Correlated (γ-flat) | Galaxy — SPARC best fit (γ ≈ 0.49) |
| 30 | 0.365 | Strongly Correlated (γ-flat) | Enzyme site |
| 100 | 0.200 | Collective Regime (γ-flattest) | Ferromagnet |
| 1,000,000 | 2.00e-3 | Collective Regime (γ-flattest) | BEC |
| 10,000,000 | 6.32e-4 | Collective Regime (γ-flattest) | BCS superconductor (10⁷ — mid of physical 10⁶–10⁹) |