V2.597: Precision Error Budget for the Ω_Λ Prediction

Status: COMPLETE — 28/28 tests passing

The Question

How precise IS the framework’s prediction, really? What are all the sources of theoretical uncertainty? Is the prediction limited by theory or by data? What single measurement would most improve it?

Central Prediction

Ω_Λ = 149√π / 384 = 0.68775

Tension with Planck: +0.42σ. Zero free parameters.

The Error Budget

Source	δΩ_Λ	% of variance	Status	Reducible?
Unknown BSM particles	0.010	74.3%	bounded	Yes (CMB-S4)
n_grav (graviton modes)	0.004	10.5%	predicted + constrained	Yes (DESI/Euclid)
Interaction corrections	0.003	8.8%	estimated	Yes (interacting lattice)
α_s (area coefficient)	0.003	6.4%	measured + predicted	Yes (larger lattice)
δ (trace anomaly)	0 (exact)	0%	theorem	N/A
Curvature corrections	10⁻¹²⁴	0%	negligible	N/A
Thermal corrections	10⁻³³	0%	negligible	N/A
TOTAL (quadrature)	±0.012
Observational (Planck)	±0.007

Theory/observation ratio: 1.59 — the prediction is currently theory-limited, not data-limited. Even with current Planck data, improving the theory matters more than improving the observations.

The Surprise: BSM Particles Dominate

The dominant uncertainty is NOT α_s, NOT n_grav — it’s whether unknown light particles exist. If even 2 undiscovered real scalars are lighter than the Hubble scale, the prediction shifts by 0.010 (1.4σ with Planck, 5σ with Euclid).

This transforms the error budget into an argument:

1. If the framework is correct, CMB-S4’s ΔN_eff measurement is not just “nice to have” — it’s the single most important measurement for the prediction’s precision.
2. If CMB-S4 finds ΔN_eff = 0 (no BSM), the theoretical error drops from ±0.012 to ±0.006 — suddenly comparable to Euclid’s projected ±0.002.
3. The framework’s precision is gated on particle physics, not cosmology.

Graviton Mode Count: Still Critical

Despite being only 10.5% of the variance, n_grav determines the central value:

n_grav	Ω_Λ	Tension	Status
0	0.746	+8.4σ	EXCLUDED
2 (TT only)	0.734	+6.7σ	EXCLUDED
8	0.699	+1.9σ	Allowed
9 (data-preferred)	0.693	+1.2σ	Allowed
10 (predicted)	0.688	+0.4σ	Best fit
11	0.682	−0.3σ	Allowed

Classical graviton (n=0) excluded at 8.4σ. TT-only graviton (n=2) excluded at 6.7σ. The allowed window is n_grav = 8–12, with n=10 predicted and n=9 data-preferred.

Precision Crossing: Theory vs. Data

Experiment	Year	σ_obs	σ_theo	Limited by
Planck 2018	2018	0.0073	0.0116	Theory
Planck + DESI DR1	2024	0.0050	0.0116	Theory
DESI DR3	2026	0.0035	0.0116	Theory
Euclid DR1	2028	0.0030	0.0116	Theory
CMB-S4 + Euclid	2030	0.0020	0.0116	Theory

The prediction is theory-limited at ALL current and projected data precisions. Even CMB-S4 + Euclid (σ = 0.002) won’t be limited by theory — but only if CMB-S4 first eliminates the BSM uncertainty. If ΔN_eff = 0, the theoretical error drops to ~0.006, and by 2030 the test becomes truly balanced.

What Is Protected By Theorem

The trace anomaly coefficients δ_i are exactly zero uncertainty:

• δ_scalar = −1/90, δ_Weyl = −11/180, δ_vector = −31/45, δ_graviton = −61/45
• Protected by the Adler-Bardeen non-renormalization theorem
• No loop corrections, no RG running, no mass dependence
• This is the framework’s most robust feature

Two numbers are negligible by construction:

• Curvature corrections: O(l_P²/L_H²) ~ 10⁻¹²³
• Thermal corrections: O(kT_CMB/E_Planck) ~ 10⁻³²

The Path Forward

Priority 1: CMB-S4 measurement of ΔN_eff (eliminates 74% of variance) Priority 2: DESI DR3 + Euclid constraint on n_grav (eliminates 10%) Priority 3: Interacting lattice QFT computation (eliminates 9%) Priority 4: Larger lattice for α_s (eliminates 6%)

After priorities 1–2 are achieved (~2030), the theoretical error should be: σ_theo ≈ √(0.003² + 0.003²) ≈ 0.004

This makes the framework’s prediction Ω_Λ = 0.688 ± 0.004 (theory) ± 0.002 (data) — a 0.9% theoretical prediction testable at 0.3% observational precision.

Honest Limitations

1. The BSM uncertainty is conservative. We assumed up to 2 undetected light scalars could exist. If the SM is complete (no BSM), this drops to zero. If the dark sector is richer, it could be larger.

2. The interaction correction is estimated, not computed. We used 0.5% based on V2.248 (QCD: 0.55%) as a conservative estimate. A proper interacting lattice computation could settle this.

3. n_grav = 10 is a theoretical prediction, not a measurement. The framework predicts 10; the data prefers 9.0 ± 0.7. This 1.4σ tension is the most concrete internal stress in the prediction.

4. The theory-limited finding is sobering. It means even perfect data cannot confirm the prediction to better than ±1.7% without first settling the particle physics (BSM content) and graviton mode count. The framework’s fate is tied to particle physics experiments, not just cosmological surveys.

Conclusion

The framework predicts Ω_Λ = 149√π/384 = 0.68775 ± 0.012 (theoretical). The dominant uncertainty (74%) is whether unknown light BSM particles exist — making CMB-S4’s ΔN_eff measurement the single most important experiment for the framework’s precision. The trace anomaly coefficients are exact by theorem. The prediction is currently theory-limited, not data-limited: improving the theory matters more than improving the observations. After CMB-S4 + Euclid, the theoretical precision should reach ±0.004 (0.6%), testable against ±0.002 (0.3%) observational precision.