V2.456: Bayesian Evidence — Savage-Dickey Density Ratio

Status: VERY STRONG — BF = 70 (Planck+lensing+BAO), framework preferred over ΛCDM

Objective

Quantify the Bayesian evidence for the framework’s zero-parameter prediction Ω_Λ = 149√π/384 = 0.68775 against ΛCDM (which treats Ω_Λ as a free parameter).

Uses the Savage-Dickey density ratio — the standard method for comparing nested models where one fixes a parameter that the other leaves free.

Method

The Savage-Dickey density ratio:

BF_{framework/ΛCDM} = p(Ω_Λ = Ω_0 | data, ΛCDM) / π(Ω_Λ = Ω_0 | ΛCDM)

• Numerator: the ΛCDM posterior density evaluated at the framework’s prediction
• Denominator: the ΛCDM prior density at the same point
• BF > 1 means the data prefer the framework over ΛCDM

The posterior is approximated as Gaussian (standard for Planck constraints).

Results

1. Bayes Factors (flat prior [0,1])

Dataset	BF	ln(BF)	Tension	Jeffreys
Planck TT,TE,EE+lowE	50	3.91	0.42σ	Very strong
Planck + lensing	55	4.01	0.47σ	Very strong
Planck + lensing + BAO	70	4.24	0.21σ	Very strong
DESI DR2 + CMB (2025)	49	3.88	1.08σ	Very strong

All datasets give “Very strong” evidence for the framework on the Jeffreys scale.

The best result is Planck+lensing+BAO: BF = 70, with the framework only 0.21σ from the observed value.

2. Prior Sensitivity

Prior	Width	BF	Jeffreys
[0, 1]	1.00	50	Very strong
[0, 0.8]	0.80	40	Very strong
[0.3, 0.9]	0.60	30	Very strong
[0.5, 0.8]	0.30	15	Strong
[0.6, 0.75]	0.15	7.5	Substantial
[0.65, 0.72]	0.07	3.5	Substantial

Even with the tightest physically reasonable prior [0.65, 0.72], the framework is still preferred (BF = 3.5, “Substantial”). The BF scales linearly with prior width — this is exactly the Occam factor at work.

3. Information-Theoretic Metrics

Dataset	ΔAIC	ΔBIC	AIC prefers	BIC prefers
Planck TT,TE,EE+lowE	-1.83	-7.43	Framework	Framework
Planck + lensing	-1.78	-7.38	Framework	Framework
Planck + lensing + BAO	-1.96	-7.56	Framework	Framework
DESI DR2 + CMB (2025)	-0.84	-6.44	Framework	Framework

Both AIC and BIC prefer the framework across all datasets. BIC penalizes extra parameters more heavily (via ln(n_eff)), giving ΔBIC ≈ -7.

4. DESI Tension

DESI DR2 + CMB gives Ω_Λ = 0.6927 ± 0.0046, pulling higher than the framework’s 0.68775. The tension is 1.08σ — still well within acceptability, but the BF drops from 70 to 49 compared to Planck+BAO.

If DESI’s central value holds and errors shrink, this could become the first real challenge to the framework.

Honest Assessment

What we can say:

• The Bayes factor of 50–70 is “Very strong” evidence on the Jeffreys scale
• The framework is preferred by both AIC and BIC across all datasets
• The preference is robust to prior choice (BF > 3 even with very tight prior)
• Zero parameters beating one parameter by BF ~ 70 is a significant result

What we cannot say:

• This validates the framework’s derivation — only its prediction
• Any zero-parameter theory predicting Ω_Λ within ~0.5σ would score similarly
• The Gaussian approximation for the posterior is imperfect
• The Savage-Dickey ratio only applies to nested models
• The framework makes additional assumptions (α_s conjecture, field counting) that aren’t penalized in this comparison

The real significance:

The framework predicts a specific number (0.68775) from zero free parameters. That number happens to be within 0.21σ of the best-constrained observation. The Bayes factor quantifies what this means: the data are 70× more likely under the framework than under ΛCDM with a flat prior.

This is not a proof — but it is the kind of quantitative statement that makes the prediction worth taking seriously.

What Would Change the Picture

• DESI narrows to 1σ+: If Ω_Λ = 0.693 ± 0.003, BF drops to ~5
• DESI confirms Planck: If Ω_Λ = 0.689 ± 0.003, BF rises to ~130
• Euclid Stage IV: σ(Ω_Λ) → 0.002 would be decisive either way