V2.352 - Precision Omega_Lambda Error Budget — The Final Number
V2.352: Precision Omega_Lambda Error Budget — The Final Number
Question
What is the framework’s FINAL prediction for Ω_Λ, including ALL known corrections with a rigorous error estimate? The tree-level value R = 149√π/384 = 0.6877 is +0.42σ above observation. Do corrections improve or worsen this?
The Calculation
Tree Level
from δ_total = -149/12 (exact QFT, anomaly non-renormalization) and N_eff = 128 (SM + graviton mode counting).
1-Loop Gauge Corrections
The ONLY significant correction is to α (the area-law coefficient). Delta is exactly protected by the Wess-Zumino consistency condition.
SM couplings at M_Pl are all perturbative: g²/(16π²) < 0.003.
Per-mode weighted average correction to α:
| Mode category | N_modes | δα/α |
|---|---|---|
| Quarks (SU(3) + SU(2) + U(1)) | 72 | 0.29% |
| Leptons (SU(2) + U(1)) | 18 | 0.20% |
| Gluons (SU(3) adjoint) | 16 | 0.46% |
| W bosons (SU(2) adjoint) | 6 | 0.32% |
| B/Z (U(1)) | 2 | 0.12% |
| Higgs (gauge only) | 4 | 0.18% |
| Graviton (self-coupling) | 10 | 0.03% |
| N_eff-weighted average | 128 | 0.28% |
Correction Hierarchy
| Source | δα/α | δR | Status |
|---|---|---|---|
| 1-loop gauge | 0.28% | -0.0019 | Computed |
| Higgs non-conformal | 0.09% | -0.0006 | Computed |
| 2-loop gauge | 0.00004% | -3×10⁻⁵ | Estimated |
| Mass (top quark) | 10⁻³⁴ | 10⁻³⁴ | Bounded |
| Thermal (vacuum) | 0 | 0 | Exact |
| Non-perturbative | 10⁻¹⁴³ | 10⁻¹⁴³ | Bounded |
Only the 1-loop gauge correction matters. Everything else is negligible by at least 4 orders of magnitude.
THE FINAL PREDICTION
compared to Planck 2018: Ω_Λ = 0.6847 ± 0.0073.
Tension: +0.16σ (improved from +0.42σ at tree level).
The 1-loop correction moves R in the RIGHT DIRECTION: gauge interactions increase α (more entanglement between modes), which decreases R = |δ|/(6α), bringing the prediction closer to observation.
Key Results
1. The correction improves agreement
| Level | R | Tension |
|---|---|---|
| Tree (149√π/384) | 0.68775 | +0.42σ |
| 1-loop corrected | 0.68584 | +0.16σ |
2. Theory error is smaller than observational error
| Source | σ |
|---|---|
| Theory (scheme + non-conformal) | 0.0015 |
| Planck 2018 | 0.0073 |
| Euclid (forecast) | 0.0020 |
The prediction is currently limited by observation, not theory.
3. Euclid confrontation
| Experiment | σ_obs | Tension | Status |
|---|---|---|---|
| Planck 2018 | 0.0073 | +0.15σ | Consistent |
| Euclid | 0.0020 | +0.45σ | Consistent |
| CMB-S4 | 0.0030 | +0.34σ | Consistent |
Euclid falsification window: Ω_Λ ∈ [0.678, 0.693] at 3σ. The Planck central value 0.6847 is inside this window.
Why The Correction Goes The Right Way
This is not a coincidence. Gauge interactions create additional quantum correlations between field modes across the entangling surface. More correlations → more entanglement entropy → larger area coefficient α. Since R = |δ|/(6α), larger α means smaller R.
The tree-level prediction slightly overshoots because free fields have LESS entanglement than interacting fields. The 1-loop correction accounts for the additional gauge-mediated entanglement.
Critical Assessment
Strengths:
- Zero free parameters: not a fit
- 1-loop correction goes in the right direction without tuning
- All SM couplings perturbative at M_Pl: convergent expansion
- Theory error (0.0015) smaller than observational error (0.0073)
Weaknesses:
- The correction coefficient c in δα/α = c × g²/(16π²) is scheme-dependent (factor ~2 uncertainty). We use c = C₂(R), the quadratic Casimir.
- The Higgs sector has a non-conformal correction (ξ−1/6)² = 1/36 that depends on the value of ξ (we use minimal coupling ξ = 0)
- V2.248 found a weighted average of 0.55% using a different weighting scheme. The range 0.28%–0.55% defines the theory band.
What this does NOT settle:
- The graviton mode count n = 10 is derived (V2.337) but not proven
- The formula R = |δ|/(6α) itself comes from mapping entanglement entropy to gravitational constants — this mapping is the core assumption
Files
src/error_budget.py— Per-mode correction calculation, budget assemblytests/test_budget.py— 10 tests, all passingrun_experiment.py— Full analysis (7 sections)results.json— Numerical output
Status
COMPLETE — The framework’s final prediction is Ω_Λ = 0.6858 ± 0.0015, matching observation at +0.16σ. This is a zero-parameter prediction from known particle physics, with the only significant correction (1-loop gauge) improving the agreement.