V2.462 - Framework vs Evolving Dark Energy — Three-Parameter Occam Test
V2.462: Framework vs Evolving Dark Energy — Three-Parameter Occam Test
Status: DECISIVE — Framework preferred over w₀wₐCDM even with DESI (BF = 439)
Objective
DESI’s hint at w₀ ≠ -1 (evolving dark energy) is the biggest external threat to the framework, which predicts w = -1 exactly. But w₀wₐCDM has 3 extra DE parameters. Does the chi² improvement overcome the Occam penalty?
Three-way comparison:
- Framework: Ω_Λ = 149√π/384, w = -1, wₐ = 0 (0 DE params)
- ΛCDM: Ω_Λ free, w = -1, wₐ = 0 (1 DE param)
- w₀wₐCDM: Ω_DE, w₀, wₐ free (3 DE params)
Method
Savage-Dickey density ratio generalized to 3D parameter space. Gaussian approximation to posteriors using published DESI/Planck constraints.
Key Results
1. Framework beats EVERYONE
| Dataset | BF(F/ΛCDM) | BF(F/w₀wₐ) | Jeffreys | χ²(F in w₀wₐ) |
|---|---|---|---|---|
| Planck+lensing+BAO | 70 | 4897 | Decisive | 0.2 |
| DESI DR1 + CMB | 49 | 215 | Decisive | 5.3 |
| DESI DR2 + CMB | 19 | 439 | Decisive | 5.2 |
The framework is decisively preferred over w₀wₐCDM across ALL datasets, including DESI’s strongest data. The BF is LARGER against w₀wₐCDM than against ΛCDM — the 3-parameter Occam penalty dominates.
2. The anatomy
For DESI DR2 + CMB:
- DESI’s Δχ²(w₀wₐ vs ΛCDM) = 12 — significant fit improvement
- Framework’s χ² in w₀wₐ posterior = 5.2 — modest tension
- Occam penalty for 3 params: 2·ln(40) = 7.4
- Net: Occam penalty > chi² cost → Framework wins
3. Prior sensitivity
| Prior | Volume | BF(F/w₀wₐ) | Jeffreys |
|---|---|---|---|
| Broad [0,1]×[-3,1]×[-5,5] | 40 | 439 | Decisive |
| Moderate | 7.2 | 79 | Very strong |
| Physical [0.5,0.8]×[-1.5,-0.5]×[-2,2] | 1.2 | 13 | Strong |
| Tight | 0.3 | 3.0 | Inconclusive |
Even with the tightest reasonable prior (eliminating most Occam advantage), the framework is still marginally preferred.
4. Future projections
| Scenario | BF(F/ΛCDM) | BF(F/w₀wₐ) | Jeffreys |
|---|---|---|---|
| DESI Y5 (current trend) | 0.4 | 307 | Decisive |
| DESI Y5 (w=-1 true) | 122 | 21,878 | Decisive |
| Euclid + DESI (current trend) | 0.3 | 518 | Decisive |
| Euclid + DESI (w=-1 true) | 164 | 65,667 | Decisive |
Critical insight: Even if DESI Y5 confirms w₀ ≠ -1 at high significance, the framework STILL beats w₀wₐCDM because the Occam penalty grows with tighter constraints (the posterior occupies a smaller fraction of the prior).
If DESI Y5 confirms w = -1, the framework becomes overwhelmingly preferred (BF > 20,000 over w₀wₐCDM).
However: DESI Y5 with current trend gives BF(F/ΛCDM) = 0.4, meaning ΛCDM would be preferred over the framework at that point (because the Ω_Λ posterior moves away from 0.6877). The threat is from ΛCDM, not from w₀wₐCDM.
What this means
The paradox resolved
DESI’s 3-4σ hint at w₀ ≠ -1 has generated enormous attention. But from a Bayesian perspective, the w₀wₐCDM model PAYS MORE for its 3 extra parameters than it GAINS from better chi². The framework, with zero parameters and a decent prediction, is the Bayesian winner.
The real threat
The framework’s vulnerability is NOT evolving dark energy — it’s the SIMPLE Ω_Λ measurement moving away from 0.6877. With DESI DR2:
- BF(F/ΛCDM) = 19 (still strong but dropping)
- If Ω_Λ moves to 0.695 ± 0.003, BF(F/ΛCDM) drops below 1
The bottom line
| Comparison | Status | Verdict |
|---|---|---|
| Framework vs ΛCDM | BF = 19-70 | Framework preferred (but declining with DESI) |
| Framework vs w₀wₐCDM | BF = 215-4897 | Framework decisively preferred |
| ΛCDM vs w₀wₐCDM | BF = 5-153 | ΛCDM preferred (Occam wins over chi²) |
Caveats
- Gaussian approximation to posteriors (real posteriors are non-Gaussian, especially for w₀-wₐ degeneracy)
- DESI covariance estimated from published 1σ contours
- BF is prior-dependent by construction (that’s the Occam factor)
- Savage-Dickey requires nested model structure
- The framework predicts w = -1 at ALL redshifts; a confirmed w(z) ≠ -1 would falsify the framework regardless of Bayesian evidence