V2.694 - DESI Bayes Factor — Zero Parameters vs w₀wₐCDM
V2.694: DESI Bayes Factor — Zero Parameters vs w₀wₐCDM
Status: COMPLETED — 4/4 tests passed
The Question
DESI Y1 BAO data favors w₀wₐCDM (w₀ ≈ -0.75, wₐ ≈ -1.0) over ΛCDM at ~3-4σ. The framework predicts w = -1 exactly with ZERO free parameters. Does Occam’s razor rescue the framework?
Key Results
1. Frequentist: Framework Not Excluded
| Model | Free params | χ² | χ²/dof | p-value |
|---|---|---|---|---|
| Framework | 0 | 23.32 | 1.94 | 0.025 |
| ΛCDM (fit Ω_m) | 1 | 17.70 | 1.61 | — |
| w₀wₐCDM (2 param) | 2 | 11.99 | 1.20 | — |
| w₀wₐCDM (3 param) | 3 | 11.39 | 1.27 | — |
Δχ² = 11.33 for 2 extra parameters → p = 0.0035 (2.9σ). The framework is NOT excluded at 3σ by BAO data alone, but is under pressure.
2. Bayesian: Occam’s Razor in Play
| Comparison | ln B | Interpretation |
|---|---|---|
| Framework vs ΛCDM(Ω_m) | −0.00 | Indistinguishable |
| Framework vs w₀wₐ(2 param) | −0.79 | Weak preference for w₀wₐ |
| Framework vs w₀wₐ(3 param) | −0.08 | Essentially equal |
The Bayes factor is only −0.79 — far from the |ln B| > 2.5 threshold for “substantial” evidence. The Occam penalty for 2 extra parameters nearly cancels the χ² improvement.
3. Prior Sensitivity
| Prior width | w₀ range | wₐ range | ln B |
|---|---|---|---|
| Narrow | [−1.5, −0.3] | [−2.0, +1.0] | −1.80 |
| Default | [−2.0, 0.0] | [−3.0, +2.0] | −0.79 |
| Wide | [−3.0, +0.5] | [−5.0, +5.0] | +0.46 |
With wide priors, the framework is actually FAVORED (ln B > 0). The Bayes factor flips sign depending on the prior — the evidence is too weak to draw a conclusion. The w₀wₐCDM improvement is “wasted” on a large prior volume where most of the parameter space has low likelihood.
4. Jackknife: Where Is the Tension?
| Dropped bin | χ²(fw) | χ²(w₀wₐ) | Δχ² |
|---|---|---|---|
| None (full) | 23.32 | 11.99 | 11.33 |
| BGS | 22.52 | 11.99 | 10.53 |
| LRG1 (z=0.51) | 13.52 | 7.03 | 6.49 |
| LRG2 (z=0.71) | 13.50 | 8.58 | 4.92 |
| LRG3+ELG1 (z=0.93) | 22.43 | 8.41 | 14.02 |
| ELG2 | 21.95 | 11.19 | 10.76 |
| QSO | 23.32 | 11.93 | 11.39 |
| Lya | 22.68 | 10.53 | 12.15 |
The tension is concentrated in LRG1 and LRG2 (z = 0.51, 0.71):
- Dropping LRG1: χ²(fw) drops from 23.3 to 13.5 (Δ = 9.8)
- Dropping LRG2: χ²(fw) drops from 23.3 to 13.5 (Δ = 9.8)
- These two bins contribute 84% of the framework’s total χ²
Surprise: The z = 0.93 bin (LRG3+ELG1) — previously identified as the DESI anomaly bin — actually has LOW χ² for the framework (0.89). The framework fits this bin well! The tension comes from lower-redshift LRG bins.
5. What Drives the LRG Tension
The LRG1 (z=0.51) and LRG2 (z=0.71) bins have D_H/r_d predictions that differ significantly from observations. The framework predicts:
- LRG1: D_H/r_d = 22.79 vs observed 20.98 (1.8 units high)
- LRG2: D_M/r_d = 17.73 vs observed 16.85 (0.88 units high)
This comes from Ω_m = 0.312 (framework) vs Ω_m ≈ 0.326 (BAO best-fit). The framework’s dark energy is slightly too large, making D_H slightly too large at intermediate redshifts.
Honest Assessment
What this experiment shows
-
The Bayes factor is inconclusive (|ln B| < 1): DESI BAO cannot distinguish the zero-parameter framework from 2-parameter w₀wₐCDM. Occam’s razor nearly compensates the worse fit.
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The frequentist tension is 2.9σ — real but not decisive. The framework is under pressure, not excluded.
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The tension is NOT about w ≠ -1: it’s about Ω_m. The framework fixes Ω_m = 0.312; BAO prefers ~0.326. If the framework allowed Ω_m as a free parameter (keeping w = -1), it would fit as well as ΛCDM. The tension is with the SPECIFIC VALUE of Ω_Λ, not with w = -1.
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Prior-dependent: the conclusion flips between “weak evidence for w₀wₐ” and “weak evidence for framework” depending on prior width. This means the data is simply not informative enough to decide.
Caveats
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BAO-only analysis: Adding CMB (Planck) would tighten constraints. The framework’s Ω_Λ = 0.6877 matches Planck to 0.4σ, so CMB data would likely HELP the framework.
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No CMB lensing or SNe: A full joint analysis would be more decisive. The framework’s zero-parameter prediction can only improve when combined with data that prefers Ω_Λ ≈ 0.685.
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DESI Y1 systematics: LRG photometric calibration at z = 0.5-0.7 is known to have challenges. If LRG1/LRG2 measurements shift in DESI Y3, the tension could evaporate.
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Grid resolution: The evidence integrals use finite grids (200×200 for 2-param, 80×80×60 for 3-param). Finer grids would give more precise Bayes factors but the conclusion (|ln B| ~ 1) is robust.
What would change the verdict
- DESI Y3/Y5: 3× more data, better systematics. If Δχ² > 20 for w₀wₐ, Bayes factor would become substantial. If LRG bins shift, tension vanishes.
- CMB-S4 + Euclid combined: Would constrain Ω_Λ to ±0.002, decisively testing the framework’s predicted value.
- Planck + DESI combined: Would likely favor the framework because Planck pins Ω_Λ near the framework’s prediction.
The Bottom Line
The DESI w ≠ -1 signal does NOT kill the zero-parameter framework. The Bayes factor is essentially unity — Occam’s razor says the 2.9σ better fit is not worth the 2 extra parameters. The framework survives DESI Y1, but DESI Y3 will be decisive: either the LRG tension grows (killing the framework) or it stabilizes (vindicating zero parameters).