γ Calculator
Formula Audited-Negative — Sign Inverted for All Collective SystemsThis tool lets you explore how a failed formula fails. It computes γ = 2/√Ncorr, the framework's original guess for γ, which an internal audit found gets the direction backwards for collective systems. It is kept so you can watch that happen, not as a way to estimate γ.
Tool type: Model Explainer · Artifact Lesson (the Interactive Tools index Kind and Status tags — Artifact Lesson is an operational state, not a level: a tool kept to show a failure, not a verdict) · Claim status: Audited-Negative, the badge above. (Added 2026-09-15: the index and this page showed different words with no key between them.)
In 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.
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.
Run the framework's own galaxy parameters through this tool and it refutes them. The galaxy sector is the one place 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. Flipping the sign does not rescue it either. An earlier finding (2026-06-06) was that flipping to γ = 2√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 of 17, from 2/√17 ≈ 0.485 to 2√17 ≈ 8.25. Run the other way, the flipped map needs Ncorr = (0.489/2)² ≈ 0.06 to give the SPARC γ = 0.489. That is less than one correlated unit, which no count of correlated units can reach.
Revision note
This block was added 2026-07-27 as a fourth numbered caveat, headed “Four caveats”, and began “Until today this page never applied its map…”. It is a finding about the framework's parameters rather than a caveat on using the tool, so it now stands on its own. Its last sentence said the flip “moves γ by a factor ~8”. The factor is Ncorr = 17.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
Crossover zone (N_corr = 3–11). C(ρ) has no phase transition here: tanh is smooth, so this is a crossover in γ, not a critical point. Fitted chemistry γ values cluster near this band, and the consciousness-threshold conjecture was placed here historically; neither placement follows from the equation.
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). Within this one-parameter family the SPARC transition shape maps onto q ≈ 0.98 (a free-Hill fit to the same data gives n = 0.975); the asymptotic return itself is not constrained by SPARC, which reaches only moderate accelerations (an exponential return fits equally well). 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.
Before you type a number: no protocol exists for getting Ncorr from a system's physics without first fitting γ to its observed behavior (Caveat 3), so a value entered here is a guess you supply, not a measurement the tool can check. The presets are back-fits too.
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 Crossover Regime Visualizer uses different Ncorr estimates for some systems. How to count Ncorr for a real system is an open question: no operational, scale-invariant recipe for Ncorr exists yet.
Quick Reference
| Ncorr | γ | Regime | Example |
|---|---|---|---|
| 1 | 2.000 | Weakly Correlated (γ-sharp) | Ideal gas (N_corr = 1, γ = 2; also the framework’s stated galaxy value) |
| 4 | 1.000 | Boundary (γ ≈ 1) | Liquid water |
| 17 | 0.485 | Strongly Correlated (0.2 ≤ γ ≤ 0.6) | Galaxy (back-fitted from SPARC γ ≈ 0.49, N_corr ≈ 17) |
| 30 | 0.365 | Strongly Correlated (0.2 ≤ γ ≤ 0.6) | Enzyme site |
| 100 | 0.200 | Strongly Correlated (0.2 ≤ γ ≤ 0.6) | Ferromagnet |
| 1,000,000 | 2.00e-3 | Collective (γ < 0.2, flattest) | BEC |
| 10,000,000 | 6.32e-4 | Collective (γ < 0.2, flattest) | BCS superconductor (10⁷ — mid of physical 10⁶–10⁹) |