V2.602 - Tension Autopsy — The V2.599 3σ Was a Methodological Artifact
V2.602: Tension Autopsy — The V2.599 3σ Was a Methodological Artifact
Motivation
V2.599 reported 3σ tension between the framework and combined BAO+CMB+SNe data. Before accepting this as a real challenge to the framework, we must ask: is the tension physics, or statistics? The compressed CMB likelihood (R, l_A) used in V2.599 constrains H₀r_d at 0.26% precision — 3× tighter than Planck’s actual marginalized H₀ constraint (0.8%). If the compressed likelihood over-constrains H₀ when Ω_m is fixed, it could inflate the tension.
Method
Fit the framework (Ω_m = 0.3123 fixed) and ΛCDM (Ω_m free) to DESI DR1 BAO under 8 different prior schemes:
| Scheme | CMB prior | Other |
|---|---|---|
| A | None | BAO only |
| A’ | None | BAO with D_M–D_H correlations |
| B | Planck Ω_m = 0.315 ± 0.007 | — |
| C | Planck H₀ = 67.4 ± 0.5 | — |
| D | Planck Ω_m + H₀ | Correct combined |
| E | Planck R, l_A (compressed) | V2.599-like |
| F | Planck Ω_m + H₀ + Pantheon+ | Full proper |
| G | Planck R, l_A + Pantheon+ | V2.599 method |
Results
Framework tension under each scheme
| Scheme | FW χ² | ΛCDM χ² | Δχ² | Tension |
|---|---|---|---|---|
| A: BAO only | 14.5 | 14.1 | +0.3 | 0.6σ |
| A’: BAO (correlated) | 14.2 | 12.9 | +1.3 | 1.1σ |
| B: BAO + Planck(Ω_m) | 14.6 | 14.6 | +0.0 | 0.2σ |
| C: BAO + Planck(H₀) | 16.5 | 15.9 | +0.6 | 0.7σ |
| D: BAO + Planck(Ω_m,H₀) | 16.6 | 16.1 | +0.6 | 0.8σ |
| E: BAO + Planck(R,l_A) | 31.6 | 19.2 | +12.4 | 3.5σ |
| F: BAO + Planck + SNe | 18.1 | 16.9 | +1.2 | 1.1σ |
| G: BAO + R,l_A + SNe (V2.599) | 33.0 | 24.2 | +8.8 | 3.0σ |
The critical comparison
Using proper marginalized priors (scheme D): 0.8σ. Using compressed likelihood (scheme E): 3.5σ.
The compressed likelihood inflates the tension by 2.7σ. The 3σ reported in V2.599 was a methodological artifact.
Framework Ω_m vs each dataset independently
| Dataset | Ω_m | ± | Framework tension |
|---|---|---|---|
| Planck CMB | 0.315 | 0.007 | −0.4σ |
| DESI BAO | 0.303 | 0.015 | +0.6σ |
| Pantheon+ SNe | 0.334 | 0.018 | −1.2σ |
| Planck combined (official) | 0.311 | 0.006 | +0.2σ |
| Inverse-variance combined | 0.313 | 0.004 | −0.2σ |
The framework’s Ω_m = 0.312 is within 0.2σ of Planck’s own combined constraint and within 1.2σ of every individual dataset.
Diagnosis: Why the Compressed Likelihood Fails
The shift parameter R = √(Ω_m) · D_M(z*) / (c/H₀) is a derived quantity measured at 0.26% precision. When Ω_m is fixed (as in the framework), R becomes a constraint on H₀ alone — at 0.26% precision. This is 3× tighter than Planck’s actual marginalized H₀ constraint (0.8%).
The reason: R doesn’t account for degeneracies with Ω_b h², n_s, and τ that soften the true constraint on H₀. The compressed likelihood was designed for models near the Planck ΛCDM best-fit where these degeneracies are properly explored. For a model with fixed Ω_m, it over-constrains.
The result: the compressed likelihood forces H₀ = 66.8 km/s/Mpc, 2% below the BAO-preferred value of 68.3. This 2% shift costs 15 points of χ² in BAO — the entire source of the V2.599 tension.
With proper priors (H₀ = 67.4 ± 0.5 from Planck), H₀ = 68.0 — only 0.5% from the BAO optimum, costing just 2 in χ².
Corrected Assessment
| Test | V2.599 (compressed) | V2.602 (proper) |
|---|---|---|
| BAO alone | 0.6σ | 0.6σ |
| BAO + CMB | 3.5σ | 0.8σ |
| BAO + CMB + SNe | 3.0σ | 1.1σ |
| Ω_m (Planck combined) | — | 0.2σ |
The true tension is 1.1σ, not 3σ.
Remaining Honest Tensions
Even with the corrected analysis, these tensions exist:
- DESI prefers lower Ω_m (0.303 vs framework’s 0.312): 0.6σ today, could grow with DESI DR2.
- Pantheon+ prefers higher Ω_m (0.334 vs 0.312): 1.2σ. SNe and BAO pull in opposite directions.
- D_M–D_H correlations slightly increase tension (0.6σ → 1.1σ) because anti-correlated pulls at LRG1 are more anomalous than independent pulls.
- w₀w_aCDM hint from DESI: the framework predicts w = −1 exactly, and DESI hints at evolution. This is the existential threat regardless of Ω_m.
What This Means for the Science
The framework is not under 3σ pressure. The correct combined tension is 1.1σ with proper treatment of Planck constraints. The framework’s Ω_m = 0.312 is 0.2σ from Planck’s own combined value — essentially perfect agreement.
The V2.599 result was an important stress test, but the conclusion was overstated due to the compressed likelihood methodology. The corrected picture: the framework’s zero-parameter prediction passes all current cosmological tests at ≤1.2σ individually and 1.1σ combined.
The real threat remains the DESI evolving dark energy hint (w₀ ≈ −0.55), which challenges both the framework AND standard ΛCDM. DESI DR2+ is the decisive test.