γ Calculator — a γ formula kept to show how it fails
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 sets how early the coherence S-curve starts to rise as density rises. Small γ = a slow fade stretched over many powers of ten. Big γ = the rise comes earlier, but it does not become a switch: however big γ gets, going from 10% to 90% still takes about a factor of 15 in density. 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” (a metal whose electrons pair up and move in lockstep, so it conducts with zero resistance below a critical temperature: the most collective system here), and notice the far more collective system gets the far flatter curve. Real superconductors have among the sharpest transitions in nature (sharp in temperature; this curve's axis is density, so the comparison is loose) — you just watched the formula get that backwards. Does that contradict “superconductors score low C” on other pages? No, it is the same fact. The flat curve is the low C. “Backwards” compares that output with the real material. And C(ρ) has no temperature in it, so it cannot describe a superconductor's transition at all: the only input is an Ncorr assigned by hand, which has no measurement protocol.
γ = 2/√Ncorr assigns the narrowest coherence curve 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. The deeper defect is the family, not the sign: flipping the sign would not produce a sharp transition, because this curve has none at any γ. One vivid case: γ = 2, the framework's original galaxy value, maps to Ncorr = 1, the ideal-gas row, so on its own map the galaxy guess says a galaxy is “uncorrelated”. 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. And γ = 8.25 is not sharp either: its 10–90% width is 1.2 dex, against 1.17 dex at any larger γ.
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, the illustrative value used on γ Parameter is Ncorr = 10²⁴ (every atom). No protocol exists for measuring Ncorr, so any value is illustrative. This tool maps that count to γ = 2/√Ncorr and shows which physical regime results.
What to do with it: press the BEC and BCS presets. The formula gives them the flattestcurves on the site; in reality they switch on sharply at a critical temperature. That mismatch is the audited failure this tool exists to show. (An earlier version of this line placed a chemistry regime boundary at γ ≈ 1; that was retracted on γ Boundary: chemistry γ values were fitted near 1, and nothing in C(ρ) singles it out.) 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 γ = 2 ghosted for comparison (gray).
Two different “galaxy” γ's, on purpose. The gray γ = 2 is the framework's original guess for galaxies (it is also the ideal-gas preset, Ncorr = 1); SPARC rotation curves disfavour it. The “Galaxy” preset's γ ≈ 0.49 is the best fit to those curves, read back into Ncorr ≈ 17; that fit is MOND's formula and fails a Solar System test. It is the fit with the equation fed acceleration; fed density, as the framework states it, the best fit is γ = 0.046 and it loses to the acceleration version by ΔBIC +2843 (see the Coherence Explorer), so a third “galaxy γ” exists and it is the refuted one. Neither is a value the framework predicts. The full list of γ values in use is in the Glossary: Which C? Which γ?
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. A superconductor is one of the most collective states known, so it should get the sharpest switch. The formula gives it the flattest.
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 (Υ frozen; +46 with Υ free per law) result restated as a mechanism. See Galaxy Rotation.
Ncorr = how many particles move together as one correlated unit (1 for a single atom; millions for a crystal oscillating in phase). Dimensionless.
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. (Moved above the input 2026-10-10.)
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 > 1.4: earliest rise) | Ideal gas (N_corr = 1, γ = 2; also the framework’s original galaxy guess, now disfavoured) |
| 4 | 1.000 | Boundary (γ ≈ 1) | Liquid water |
| 17 | 0.485 | Strongly Correlated (0.2 ≤ γ ≤ 0.6: later rise) | Galaxy, fitted to data (γ ≈ 0.49 from SPARC, so N_corr = (2/0.49)² ≈ 17, not a count of anything; the pre-fit guess was γ = 2) |
| 30 | 0.365 | Strongly Correlated (0.2 ≤ γ ≤ 0.6: later rise) | Enzyme site |
| 100 | 0.200 | Strongly Correlated (0.2 ≤ γ ≤ 0.6: later rise) | 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) |