V2.589 - Angular Anatomy of the Cosmological Constant
V2.589: Angular Anatomy of the Cosmological Constant
Question
Can the entanglement framework predict CMB large-scale anomalies — specifically the anomalously low quadrupole (C₂ ≈ 0.17 × ΛCDM)? More broadly: how is the cosmological constant assembled from entanglement at different angular momentum scales?
Method
Decompose the QNEC second derivative d²S into per-angular-momentum contributions on the Srednicki lattice (N=300, n=15, C=6.0, l_max=90). Each mode d²S_l is either:
- IR (saturated): d²S_l < 0 → contributes to δ → Λ (cosmological constant)
- UV (entering): d²S_l > 0 → contributes to α → G (Newton’s constant)
Weight by SM + graviton field content: 94 scalars (l≥0), 24 vectors (l≥1), 10 graviton modes (l≥2).
Key Results
1. The UV/IR transition in angular space
Every angular channel l contributes either to Λ (if saturated, l ≤ 16) or to G (if entering, l ≥ 17). The transition is sharp:
| Quantity | Angular range | Contribution |
|---|---|---|
| δ → Λ (cosmological constant) | l = 0..16 (IR) | 100% of IR budget |
| α → G (Newton’s constant) | l = 17..90 (UV) | 100% of UV budget |
Λ and G probe DIFFERENT angular scales of the entanglement spectrum. This is a structural result not available in any other approach.
2. Sector budget of Λ (IR contributions)
Of the IR (Λ-determining) entanglement budget:
| Sector | Components | l restriction | IR fraction | UV fraction |
|---|---|---|---|---|
| Scalars (Higgs + Weyl) | 94 | l ≥ 0 | 73.9% | 73.4% |
| Vectors (gauge bosons) | 24 | l ≥ 1 | 18.6% | 18.8% |
| Graviton (TT modes) | 10 | l ≥ 2 | 7.5% | 7.8% |
The graviton contributes 7.5% of the IR budget (Λ) and 7.8% of the UV budget (G), meaning the graviton is roughly equally important for both. The l=0,1 exclusion (spin-2 constraint) slightly reduces its IR share.
3. Angular convergence of δ
The IR budget (∝ δ) converges rapidly in angular momentum:
| l_max | % of total δ captured |
|---|---|
| 2 | 9.1% |
| 5 | 33.0% |
| 7 | 50% |
| 10 | 78.0% |
| 12 | 90% |
| 15 | 99.9% |
50% of the cosmological constant comes from modes with l ≤ 7. The UV budget (∝ α) converges much more slowly, requiring l > 60 for 50%.
4. CMB quadrupole prediction: NULL
The framework does NOT predict the low CMB quadrupole. The physical argument is definitive:
- The cosmological horizon has n_H ~ L_H/l_P ~ 10⁶¹ lattice sites
- CMB multipole l=2 corresponds to x = l/(C·n_H) ~ 10⁻⁶¹
- ALL CMB multipoles (l = 2 to 2500) are in the deep IR limit x → 0
- At x → 0, the per-l entanglement structure is featureless — no angular variation predicted
- The lattice l=2 at n=15 has x = 0.022, which is 10⁵⁸ times larger than the physical regime
- Therefore: lattice edge-mode physics at l=2 is a finite-size effect, not a CMB prediction
The observed C₂/C₂ᴸᶜᴰᴹ = 0.17 is cosmic variance (a ~5% probability in ΛCDM), not new physics from entanglement.
5. Graviton onset at l=2
The graviton enters the entanglement budget only at l ≥ 2 (spin-2 has no monopole or dipole modes). Changing the graviton’s angular range shifts R:
| Graviton placement | R (lattice C=6) | ΔR from l≥2 |
|---|---|---|
| l ≥ 0 (unphysical) | 6.500 | reference |
| l ≥ 2 (correct) | 5.716 | −0.783 |
| l ≥ 3 (wrong) | 5.209 | −1.291 |
At converged parameters (n→∞, C→∞), this becomes: R(l≥2) = 0.6877 vs R(l≥0) ≈ R_scalar × 128/128 = same α but different δ. The graviton l-restriction matters at the few-percent level for the absolute prediction.
6. Lattice R vs analytical
The lattice extraction at C=6.0 gives R ≈ 5.7 (SM+grav), far from the analytical R = 149√π/384 = 0.6877. This is the known convergence issue: the double limit C→∞, n→∞ with Richardson extrapolation (V2.184) is required for the absolute R value. The per-l anatomy (relative fractions) is robust at C=6.
What This Means
For CMB predictions
The framework makes zero predictions for CMB anomalies beyond standard ΛCDM:
- w = −1 exactly (same as ΛCDM)
- Ω_Λ = 0.6877 (ISW shift of 0.9% — negligible for the quadrupole)
- No per-l modification of the primordial spectrum
- The low quadrupole remains unexplained and unexplainable within this framework
This is an honest negative result that closes off a speculative avenue. Anyone claiming that entanglement-based dark energy explains CMB anomalies must explain how features at x ~ 10⁻⁶¹ could possibly appear in the entanglement spectrum.
For the framework’s unique predictions
The angular anatomy is novel physics even without CMB connections:
- Λ is assembled from low-l entanglement (l ≤ 12 captures 90% of δ)
- G is assembled from high-l entanglement (l ≥ 17 for UV modes)
- The graviton’s l ≥ 2 onset is a structural prediction tied to spin-2 statistics
- Sector budget: gauge bosons (vectors) contribute 18.6% of Λ despite being only 18.8% of modes — near-democratic
Implications for testability
The strongest unique predictions remain:
- Species-dependence: Λ changes calculably with field content (V2.555, V2.576)
- BH entropy log coefficient: γ_BH = −149/12 (V2.557)
- Graviton mode count: n_grav = 10 from Ω_Λ (V2.556, V2.328)
- w = −1 exactly: falsifiable by DESI/Euclid (V2.580, V2.588)
The CMB is not where this framework makes its mark. The cosmological constant itself is the prediction.
Parameters
- N = 300 (lattice size)
- n = 15 (subsystem, for anatomy); n = 10..21 (for R extraction)
- C = 6.0 (angular capacity ratio)
- l_max = 90 (angular momentum cutoff)
- Total runtime: 242s