V2.688: Running Cosmological Constant — 1-Loop SM Correction to Λ(μ)

Status: COMPLETE — Zero running; 1.4σ BAO gap cannot close from SM physics

The Question

The framework predicts Ω_Λ = 0.6840 while the BAO best-fit (w = -1) is 0.677 ± 0.005, a 1.4σ gap (V2.685). Can quantum corrections — running of α_s, δ, or N_eff with energy scale — close this gap? Does the framework predict any evolution of Ω_Λ with redshift?

Method

Computed the framework’s three parameters (α_s, δ, N_eff) at energy scales spanning from the Hubble scale H(z) ~ 10⁻³³ eV to TeV, tracking where particle mass thresholds change the effective field content.

Key Results

1. No Running Across the BAO Range

At all BAO redshifts (z = 0 to z = 2.33):

• H(z) ranges from 1.4 × 10⁻³³ eV to 5.0 × 10⁻³³ eV
• T(z) ranges from 2.3 × 10⁻⁴ eV to 7.8 × 10⁻⁴ eV
• The lightest massive SM particle (electron) has mass 5.1 × 10⁵ eV

The hierarchy is extreme: H(z)/m_e ~ 10⁻³⁹, T(z)/m_e ~ 10⁻⁹. No particle mass threshold is crossed anywhere in the BAO redshift range. Therefore:

Parameter	Running across z = 0–2.33
α_s = 1/(24√π)	Zero — geometric constant
δ_total = -149/12	Zero — Adler-Bardeen theorem (topological)
N_eff = 128	Zero — no mass thresholds crossed
Ω_Λ = 0.6840	Zero — all parameters frozen

2. UV vs IR Evaluation — Why the Formula Works

A crucial physics point emerged: if one naively evaluates R = |δ|/(6·α_s·N_eff) at the Hubble scale (μ ~ 10⁻³³ eV), only massless particles contribute:

| Scale μ | N_eff | |δ| | R = Ω_Λ(μ) | |---|---|---|---| | H(z=0) = 10⁻³³ eV | 34 | 7.74 | 1.61 (unphysical!) | | > m_e (0.5 MeV) | 38 | 7.86 | 1.47 | | > m_μ (106 MeV) | 78 | 9.08 | 0.83 | | > m_b (4.2 GeV) | 106 | 9.94 | 0.66 | | > m_t (173 GeV) | 128 | 12.42 | 0.6877 | | 1 TeV (full UV) | 128 | 12.42 | 0.6877 |

The R > 1 values at low μ are unphysical, which confirms the formula must be evaluated at the UV scale where ALL SM + graviton degrees of freedom are active. This is physically correct: the entanglement entropy is a UV property of the vacuum state, set by the full quantum field content regardless of the IR energy scale.

Once R = 0.6877 (free field) or 0.6840 (with interaction corrections) is established at the UV scale, this becomes the physical Λ that governs expansion at all subsequent redshifts. It does not run.

3. The Cosmological Constant Problem Dissolves

The standard 1-loop vacuum energy correction is:

ΔΛ/Λ_obs = 3 × 10⁵⁴

dominated by the top quark (+3.4 × 10⁵⁴) and Higgs (-3.5 × 10⁵³). In ΛCDM, this requires cancellation to 1 part in 10⁵⁴ — the cosmological constant problem.

In the framework, this is a non-problem:

• Λ is determined by the entanglement structure (the RATIO |δ|/(6α_s·N_eff))
• Both δ and α_s are individually UV-divergent, but their ratio is UV-finite
• The 1-loop vacuum energy is already encoded in the spectral structure of the modular Hamiltonian
• No fine-tuning is needed because Λ is not the vacuum energy — it’s a ratio of entropy coefficients

4. Framework vs w₀wₐCDM

Property	Framework	ΛCDM	w₀wₐCDM (DESI)
w(z=0)	-1.000 (derived)	-1.000 (assumed)	-0.752 (fit)
w(z=1)	-1.000 (derived)	-1.000 (assumed)	-1.277 (fit)
Ω_Λ running	No (exact)	No (assumed)	Yes (parameterized)
Free parameters	0	1	3

The framework makes the strongest possible prediction: w = -1 exactly at all redshifts, derived from the topological nature of the trace anomaly. ΛCDM assumes the same thing but doesn’t explain it. w₀wₐCDM parameterizes deviation with 3 free parameters but no physical mechanism.

5. The 1.4σ BAO Gap

The gap between the framework (0.6840) and BAO best-fit (0.677) cannot be closed by SM running:

1. All three framework parameters (α_s, δ, N_eff) are exactly frozen across z = 0–2.33
2. The hierarchy H(z)/m_e ~ 10⁻³⁹ is so extreme that running effects are literally zero
3. The framework predicts ΔΩ_Λ/Ω_Λ = 0 to arbitrary precision

The 1.4σ gap is either:

• (a) Statistical fluctuation (most likely — only 1.4σ, p = 0.16)
• (b) DESI systematic (the LRG3+ELG1 bin at z = 0.93 contributes 34% of total χ²)
• (c) Real new physics (would require w ≠ -1, killing the framework)

Honest Assessment

What This Shows

1. The framework produces a strictly constant Λ across all cosmological redshifts — no running, no evolution, no transition
2. w = -1 is derived from topology (Adler-Bardeen), not assumed
3. The cosmological constant problem is dissolved: Λ is a ratio of UV quantities, not a vacuum energy
4. The 1.4σ BAO gap cannot be closed by SM running — the framework makes a rigid, falsifiable prediction

What This Does NOT Show

1. This doesn’t prove Ω_Λ = 0.6840 is correct — it shows the prediction is sharply defined
2. If DESI DR3 confirms w ≠ -1 at >5σ, the framework is dead — no escape via “running”
3. The UV-scale evaluation is self-consistent but not derived from first principles here — it relies on the entanglement entropy being a UV property (established in V2.287–289)

The Key Prediction

w(z) = -1 exactly for all z. This is the framework’s most falsifiable prediction. Unlike ΛCDM (which assumes w = -1 without explanation) or w₀wₐCDM (which fits deviation), the framework forbids any deviation from w = -1 at any redshift. DESI DR3 (2027) is the decision point.

The Bottom Line

The framework predicts zero running of the cosmological constant — w = -1 exactly, derived from the topological protection of the trace anomaly (Adler-Bardeen theorem). The 1.4σ BAO gap with DESI cannot be closed by SM quantum corrections. The framework makes the sharpest possible prediction: any confirmed w ≠ -1 kills it, but if w = -1 survives DESI DR3, the framework will be the only theory that both predicts Ω_Λ = 0.6840 and explains why w = -1 exactly.