Parameter Derivations
Honest framing: Despite the page title, this is not a derivations page in the mathematical sense. γ = 2/√Ncorr is a motivated ansatz (the factor 2 is not rigorously derived; CLT is invoked for correlated DOF where CLT doesn't apply). ρcrit = A·Vflat² with A ≈ 0.029 is calibrated to the Jeans criterion — Vflat is the input. The scaling constants are dimensional analyses with 3–12% errors that may reflect implicit calibration rather than predictive accuracy. The functional form tanh is motivated by analogy with the sigmoid/compander family (μ-law, Hill, logistic), not uniquely derived. Update (2026-06-07): A-from-Jeans — the only surviving first-principles candidate — is now audited-negative: the derivation that produces 0.029 uses a different scaling law (ρcrit ∝ V0.5) than the framework's stated ρcrit ∝ V², and the stated formula gives A ≈ 4.6×10⁻⁵ (600× off). Zero parameters have independent first-principles derivations. A more accurate title: Parameter Calibration & Honest Ansätze. Read this page before concluding the equation is derived.
The coherence function has two kinds of parameters: the functional form (tanh, γ = 2/√Ncorr) which is motivated by the sigmoid/compander family (μ-law, Hill, logistic), and the scaling constants (A, a₀, Σ₀, R₀) which are calibrated to observational anchors with 3–12% errors. Whether those errors reflect approximation limits or implicit calibration is an open question.
Badge labels on this page: Motivated Ansatz and Motivated Choice are sub-types of Speculative — physically motivated but not uniquely derived. Freeman's Law Re-expressed and Dimensional Analysis are sub-types of Reparametrization — reproducing known observational laws in different notation. 3% Error below uses the deprecated Validated label (pre-2026-05 convention — do not interpret as passing any current audit criterion; the underlying R₀ derivation is a dimensional analysis, not an independent first-principles result). See badge taxonomy for the current two-family system.
The Complete Chain
1. γ = 2/√Ncorr
Motivated AnsatzSessions #64-65
The 1/√Ncorr scaling borrows from central-limit-theorem (CLT) statistics. 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.
Internal inconsistency (2026-05-20): The CLT's 1/√N scaling governs the standard error of the mean for iid (independent, identically distributed) variables. But Ncorr is by construction the count of correlated degrees of freedom — exactly the regime where the iid hypothesis fails and 1/√N does not apply. Invoking CLT for correlated Ncorr is self-contradictory. The scaling is borrowed by analogy, not derived from the CLT. It is an ansatz with a fitted prefactor.
2. tanh form
Motivated ChoiceSession #66
tanh is a phenomenological choice from the sigmoid/compander family (μ-law, Hill, logistic, erf). Other sigmoids 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. Not Landau: a saturating compander with argument ≥ 0 has no critical point, no diverging correlation length, and no critical exponents — nothing to put it in a universality class. The Landau framing was retired at the landing page (compander family); this page now matches.
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.
3. A = 4π/(βJ²GR₀²) ≈ 0.029
Audited-Negative — Chain-of-Custody FailureSessions #53, #66 — decisive test run 2026-06-07
βJ = λJeans / Rhalf is the dimensionless Jeans-length-to-galaxy-size ratio (Session 53). Empirically βJ ≈ 1.1 ± 0.2 across SPARC galaxies.
- Wrong scaling law: The only computation that yields A ≈ 0.0294 uses ρcrit ∝ V0.5 (Session 65: exponent B=0.5) with a fitted R₀ = 0.07 kpc/(km/s)^0.75 — not R₀ = 8 kpc and not the framework's ρcrit ∝ V² used everywhere else (equations.ts). The derivation that hits 5% underpins a law the framework does not use.
- Stated formula gives 600× off: A = 4π/(βJ²·G·R₀²) with βJ=1, R₀=8 kpc gives A ≈ 4.6×10⁻⁵ — not 0.029. The Session 66 markdown bridges them with an unexplained 644× “unit conversion.”
- Number detached from computation: 0.0294 propagated ~600 sessions without anyone re-running the stated formula. Same failure mode as the 2026-05-25 DESI epistemic regression.
a-from-jeans-chain-of-custody-failure.md; back-annotation:a_from_jeans_chain_of_custody_closure.md (Synchronism Research repo, 2026-06-07).Symbol note (2026-04-24 correction): βJ is the Jeans ratio λJeans/Rhalf— an O(1) structural ratio, not the electromagnetic fine-structure constant αem ≈ 1/137. The formula was previously written with α, which invited that misread. With αem² ≈ 5×10−5, the formula yields A ≈ 550 (km/s)−2 — 20,000× too large. The formula only closes at 5% with βJ = O(1). No electromagnetic coupling is implied.
4. a₀ = cH₀/(2π) ≈ 1.08×10−10 m/s²
Dimensional AnalysisSessions #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.
5. Σ₀ = cH₀/(4π²G) ≈ 110 M☉/pc²
Freeman's Law Re-expressedSession #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.
6. R₀ = V²/(3a₀)
Dimensional Analysis — 3% Error (⚠ deprecated Validated label)Session #91
Characteristic radius from velocity and acceleration. 97% accuracy against observed values.
What's Notable
The derivation chain uses fundamental constants (c, G, H₀) plus one structural ratio (βJ ≈ 1 from the Jeans criterion) and 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. a₀ and Σ₀ reproduce known observational relations (Milgrom 1983; Freeman 1970) using dimensional bookkeeping — classified as Reparametrization, not derivation. The effective novel parameter in this chain is A (Jeans criterion, 5% agreement), which has a derivation path independent of the observational coincidences.
Prerequisites
Understanding these concepts first will help: