Parameter Derivations

Sessions #64-65

The 1/√N_corr scaling follows central-limit-theorem statistics for correlated ensembles. The factor of 2 is motivated by phase-space dimensionality arguments (6D to 3 effective) but is not rigorously derived — integrating out momenta introduces temperature- and mass-dependent factors, not a clean factor of 2. Best understood as a physically motivated ansatz.

Session #66

tanh is a phenomenological choice from the Landau-universality family of sigmoid functions. Other sigmoids (logistic, erf) share the same qualitative properties and would produce indistinguishable physics near γ ≈ 1. The fractal coherence bridge failure (0/7 boundaries on 36 tests) is consistent with tanh being a generic sigmoid here, not a uniquely derived form.

Note on the Ising analogy: tanh arises in mean-field Ising models as m = tanh(βJzm) — but that tanh comes from the self-consistency equationm = tanh(βJz·m), where m feeds back into itself. C(ρ) has no such self-consistency loop: it is evaluated directly at the input ρ with no fixed-point iteration. The Ising tanh is derived; this tanh is chosen. These are structurally different justifications.

Sessions #53, #66

β_J = λ_Jeans / R_half is the dimensionless Jeans-length-to-galaxy-size ratio (Session 53). Empirically β_J ≈ 1.1 ± 0.2 across SPARC galaxies; β_J = 1.0 is the fiducial. From the Jeans criterion (ρ_crit = V² / (Gβ_J²R_half²)) with R₀ = 8 kpc and the 4π factor from spherical surface-area integration (Session 66), A = 0.0294 vs empirical 0.028 — 5% agreement.

Symbol note (2026-04-24 correction): β_J is the Jeans ratio λ_Jeans/R_half— an O(1) structural ratio, not the electromagnetic fine-structure constant α_em ≈ 1/137. The formula was previously written with α, which invited that misread. With α_em² ≈ 5×10⁻⁵, the formula yields A ≈ 550 (km/s)⁻² — 20,000× too large. The formula only closes at 5% with β_J = O(1). No electromagnetic coupling is implied.

Sessions #87-88

The MOND acceleration scale a₀ = cH₀/(2π) follows from Synchronism's coherence function. Milgrom's observed: 1.20×10⁻¹⁰ m/s² (10% error).

This dimensional relation a₀ ∼ cH₀ has been noted since Milgrom (1983) and independently derived by multiple frameworks (McCulloch 2007, Verlinde 2017, Smolin 2017) with the same geometric factor. The quantities c and H₀ are dimensionally sufficient to produce an acceleration — cH₀ is not a Synchronism-specific derivation. Classified as dimensional analysis / reparametrization on the honest assessment page.

Session #89

Freeman's surface density law (Freeman 1970): observed 124 M☉/pc², 12% error. The combination cH₀/G has dimensions of surface density, so this is dimensional bookkeeping — expressing Freeman's empirical value via cosmological constants, not deriving it from physics. Re-badged from “Validated” (2026-04-28): the 12% agreement is not sufficient to claim derivation of what is, in origin, an observational law.

Session #91

Characteristic radius from velocity and acceleration. 97% accuracy against observed values.