V2.349: CMB Quadrupole from Horizon Entanglement — A Smoking Gun Test

Question

If Λ arises from entanglement at the cosmological horizon, does the entanglement spectrum leave observable imprints in the CMB? Specifically, can it explain the anomalously low CMB quadrupole (C₂ ≈ 201 μK² vs ΛCDM prediction ~1200 μK²)?

Motivation

This would be a smoking gun: ΛCDM has no explanation for the low quadrupole. If the entanglement structure at the horizon preferentially suppresses low-l modes (whose wavelengths are comparable to the horizon), the framework would make a unique prediction connecting Λ to CMB anomalies.

Method

1. Compute the entanglement spectrum S_l on the Srednicki lattice for a scalar field with angular momentum barrier l(l+1)/r²
2. Extract the suppression of low-l modes (especially l=2) relative to high-l
3. Model the CMB power modification as C_l^{ent}/C_l^{BD} = 1 - tanh²(S_l) (entanglement with exterior reduces variance of interior modes)
4. Test size-dependence to check if the effect is physical or a lattice artifact

Results

1. Entanglement Spectrum

At n=60, n_in=30 (half-space partition):

l	S_l	S_l/S_0	1/(2l+1)
0	0.580	1.000	1.000
1	0.484	0.834	0.333
2	0.413	0.712	0.200
5	0.289	0.498	0.091
10	0.189	0.325	0.048

Key finding: S_l/S_0 is much larger than 1/(2l+1) — low-l modes are MORE entangled than the naive angular barrier would predict. The entanglement decays much more slowly with l than expected.

2. Per-l Scaling Coefficients

| l | α_l (area) | δ_l (log) | |δ_l/α_l| | |---|---|---|---| | 0 | 0.000208 | 0.158 | 758 | | 2 | 0.000449 | 0.149 | 332 | | 10 | 0.00222 | 0.080 | 36 |

|δ₂/α₂| is 9.2× larger than |δ₁₀/α₁₀| — log corrections dominate overwhelmingly at low l. The area law coefficient is negligible at l=0,1,2.

3. CMB Power Modification

Using C_l^{ent}/C_l^{BD} = 1 - tanh²(S_l):

l	Predicted C_l ratio	Observed (l=2)
0	0.727	—
2	0.847	~0.168
5	0.921	—
10	0.965	—

Predicted suppression at l=2: 15.3% Observed suppression at l=2: ~83% Ratio: factor of 5× too weak

4. Size Dependence

The entanglement fraction S_l/S_0 is NOT size-independent — it increases with n. This means it is not a universal, scale-free prediction. The lattice size enters as a physical scale, preventing a clean continuum prediction.

Conclusion: NEGATIVE RESULT

The entanglement spectrum at the horizon CANNOT explain the CMB quadrupole deficit.

The mechanism goes in the right direction — low-l modes are more entangled and therefore more suppressed — but the magnitude is 5× too small. Moreover:

1. The mapping S_l → C_l is an ansatz, not derived from first principles
2. The suppression is not universal — it depends on the lattice size
3. Cosmic variance at l=2 is 63%, so the “anomaly” may be a ~2σ fluctuation

What This Means for the Framework

The framework is NOT weakened by this negative result. The Λ prediction comes from the TOTAL entropy (summed over all l with (2l+1) weights), which is dominated by high-l modes. The per-l structure is interesting but does not feed back into the cosmological prediction.

The framework’s unique predictions remain:

1. Species-dependence curve (V2.346) — Λ shifts calculably with field content
2. BH log correction (V2.348) — γ = -149/12, disagrees with LQG by 8.3×
3. N_g = 3 selection — only 3 generations give the right Λ
4. w = -1 exactly — testable by DESI

The CMB quadrupole anomaly is outside the framework’s predictive reach.

Interesting Secondary Findings

1. Log dominance at low l: |δ_l/α_l| ranges from 758 (l=0) to 13 (l=15). The entanglement structure at low angular momenta is entirely controlled by the log correction, not the area law. This is consistent with V2.306’s finding that edge modes dominate at low l.

2. Slow l-decay: S_l/S_0 ≈ 1/(1 + 0.4l) rather than 1/(2l+1). The angular barrier is less effective at suppressing entanglement than naive scaling suggests.

Files

• src/cmb_entanglement.py — Srednicki lattice, entanglement spectrum, CMB ansatz
• tests/test_cmb.py — 7 tests, all passing
• run_experiment.py — Full analysis (7 sections)
• results.json — Numerical output

Status

COMPLETE — Negative result honestly reported. The CMB quadrupole is not a prediction of this framework. The unique testable predictions lie elsewhere.