Parameter Derivations
The coherence function has two kinds of parameters: the functional form (tanh, γ = 2/√Ncorr) which is derived from mean-field theory, and the scaling constants (A, a₀, Σ₀, R₀) which are derived from dimensional analysis with fundamental constants but show 3–12% errors against observations. Whether those errors reflect approximation limits or hidden calibration is an open question.
The Complete Chain
1. γ = 2/√Ncorr
ExactSessions #64-65
From 6D phase space (3 position + 3 momentum): effective DOF contracts to yield factor of 2. Ncorr enters via central limit theorem for correlated ensembles.
2. tanh form
ExactSession #66
Mean-field theory derivation. The bounded, monotonic, smooth sigmoid that arises from self-consistent ordering in many-body systems IS tanh. Not a choice; a consequence.
3. A = 4π/(α²GR₀²) ≈ 0.029
5% ErrorSession #66
Relates fine structure constant, gravitational constant, and characteristic length scale to the proportionality between critical density and rotation velocity. Empirical: 0.028. Derived: 0.029.
4. a₀ = cH₀/(2π) ≈ 1.08×10−10 m/s²
10% ErrorSessions #87-88
MOND's acceleration constant emerges from the coherence function. In MOND, a₀ is fundamental. In Synchronism, it's emergent from cosmological parameters c and H₀. Milgrom's observed: 1.20×10−10.
5. Σ₀ = cH₀/(4π²G) ≈ 110 M☉/pc²
12% ErrorSession #89
Freeman's surface density law. Observed: 124 M☉/pc². From first principles.
6. R₀ = V²/(3a₀)
3% ErrorSession #91
Characteristic radius from velocity and acceleration. 97% accuracy against observed values.
What's Notable
The derivation chain uses fundamental constants (c, G, α, H₀) plus one observable (Vflat). The functional form (tanh, γ) has zero free parameters. The scaling constants (A, a₀, Σ₀, R₀) show 3–12% agreement with observations.
Honest caveat: The 3–12% errors could reflect either (a) legitimate approximation limits in the dimensional analysis, or (b) implicit calibration through choice of Vflat as input. Distinguishing these requires independent derivation. The landing page describes the scaling constants as “fitted” — that may be the more conservative and accurate framing.
Prerequisites
Understanding these concepts first will help: