V2.612: CMB Quadrupole from Horizon Entanglement Spectrum

Status: HONEST NULL RESULT with interesting secondary findings

Hypothesis

If the cosmological constant arises from entanglement entropy at the cosmological horizon, the per-angular-momentum structure of this entropy is a parameter-free prediction. This per-l “entanglement spectrum” could potentially connect to CMB power spectrum anomalies, particularly the anomalously low quadrupole (l=2). If confirmed, this would be a smoking-gun prediction unique to entanglement-based cosmology.

What We Computed

1. Per-l entanglement entropy s_l(n) on the Srednicki lattice (N=200)
2. Per-channel area coefficient alpha_l and anomaly coefficient delta_l
3. SM-weighted spectrum including graviton onset at l=2
4. Suppression ratios at low l relative to smooth high-l envelope
5. Entanglement temperature T_l per angular momentum channel
6. CMB comparison: entanglement suppression vs observed C_l/C_l^LCDM

Key Results

1. Per-l Entropy Structure (CLEAN)

The per-channel entanglement entropy s_l is well-computed and shows clear monotonic decrease with angular momentum (barrier suppression):

l	s_l	(2l+1) s_l	fraction
0	0.691	0.691	0.48%
1	0.544	1.633	1.14%
2	0.463	2.314	1.61%
3	0.408	2.857	1.99%
5	0.335	3.681	2.56%
10	0.232	4.866	3.39%

The weighted spectrum (2l+1) s_l RISES with l (degeneracy growth dominates over barrier suppression), peaking at l ~ n/2 before declining. Low-l modes carry less total entropy than high-l modes.

2. Per-Channel Alpha Extraction (UNRELIABLE)

Per-channel area coefficients alpha_l are O(10^-6) — the area law is a COLLECTIVE property of the sum over all l, not an individual-channel feature. The fit S_l = 4pialpha_ln^2 + delta_lln(n) + gamma_l gives alpha_l ~ 0 because individual channels are dominated by the constant gamma_l term.

Lattice convergence of alpha_l is terrible (CV ~ 10^23 across N=100,150,200), confirming that per-channel alpha extraction is not a useful observable at these lattice sizes.

3. Graviton Feature at l=2 (SMALL, WRONG DIRECTION)

The graviton onset at l=2 adds 10 components (128 vs 118 at l=1):

• Graviton fraction at l=2: 7.8% of SM-weighted power
• Step ratio (P_sm vs interpolated no-graviton): -7.1%
• Component ratio l=2/l=1: 128/118 = 1.085

This creates a tiny feature in the WRONG direction for explaining the low quadrupole. The graviton ADDS power at l=2, while the CMB anomaly is a DEFICIT. The feature is also too small (7.8%) to explain a factor-of-6 suppression.

4. Low-l Suppression Analysis (INCONCLUSIVE)

Fitting alpha_l to a power law for l >= 8 and extrapolating to low l:

• l=2 ratio (actual/envelope): 0.563 (44% suppression)
• l=3 ratio: 0.631 (37% suppression)
• l=4 ratio: 0.701 (30% suppression)

But these numbers come from the unreliable per-channel alpha_l, so they should not be trusted.

5. CMB Comparison (QUALITATIVE MATCH, QUANTITATIVE FAILURE)

l	CMB C_l^obs/C_l^LCDM	Entanglement ratio	Match?
2	0.139	0.563	No (off by 4x)
3	0.822	0.631	Qualitative
4	0.650	0.701	Close
5	1.443	0.768	No

• Correlation: r = 0.893 across l=2,3,4,5
• But: only 4 data points, and the entanglement ratios come from unreliable alpha_l extraction. This correlation is NOT meaningful.

The quadrupole prediction FAILS: The framework predicts alpha_2/envelope = 0.563, while the CMB shows C_2^obs/C_2^LCDM = 0.139. The entanglement suppression is in the right qualitative direction (both < 1) but off by a factor of 4 in magnitude.

6. Entanglement Temperature Per Mode (GENUINELY INTERESTING)

Each angular channel has a different effective “entanglement temperature”:

l	T_l	_l
0	0.634	0.538
2	0.492	0.525
5	0.413	0.516
10	0.348	0.512

• T_l decreases monotonically with l (CV = 21.9%)
• This VIOLATES Bisognano-Wichmann universality (which predicts T = const)
• BW is exact for planar entangling surfaces; spherical surfaces introduce l-dependent corrections
• Ratio T_0/T_10 = 1.82: low-l modes are 82% “hotter” than high-l modes

This is consistent with V2.268 (CHM wins over BW for spherical surfaces). The l-dependent temperature is a real effect of the sphere geometry, not a lattice artifact.

7. Delta Per-l Decomposition (CONFIRMS V2.312)

Per-channel delta_l is positive for all computed l (0 through 20):

• delta_0 = 0.163 (analytical prediction: 1/3 from partial sum difference)
• All delta_l slowly decrease with l
• Cumulative sum 15.35 at l=10 (still positive, has not converged to -1/90)

The convergence to the total anomaly requires l_max >> n, which requires much larger lattices. This is consistent with V2.312 findings.

Honest Assessment

What Worked

• The per-l entropy decomposition is clean and robust
• The SM field content weighting is straightforward
• The entanglement temperature variation is a genuine new finding
• The computational infrastructure works correctly

What Failed

• The central hypothesis: The entanglement per-l spectrum does NOT predict the CMB low quadrupole at quantitative level
• Per-channel alpha extraction: Completely unreliable (O(10^-6) with terrible convergence)
• The graviton feature: Too small and in the wrong direction
• The correlation: Likely spurious (4 points, unreliable data)

Why the Connection Probably Doesn’t Exist (As Naively Formulated)

The fundamental issue: entanglement entropy per-l and CMB C_l are different observables. The per-l entropy s_l measures the quantum entanglement of field modes at angular momentum l across a spatial surface. The CMB C_l measures the variance of temperature fluctuations at angular scale l on the last scattering surface. These are related to different aspects of the quantum state:

• s_l depends on the ENTANGLEMENT structure (off-diagonal correlations between inside/outside the horizon)
• C_l depends on the FLUCTUATION amplitude (diagonal elements of the density matrix)

A connection would require that the entanglement structure constrains the fluctuation spectrum, which is true in general (entanglement implies correlations) but the mapping is not direct. The entanglement entropy is dominated by UV modes near the surface, while the CMB is sensitive to super-horizon correlations — different physics.

What’s Salvageable

1. The entanglement temperature spectrum T_l is a genuine prediction. If T_l varies with l as computed, this means different angular scales of the horizon “see” different effective temperatures. This is a testable prediction in the context of de Sitter QFT.

2. The graviton onset at l=2 is real physics. While it doesn’t explain the CMB quadrupole, it DOES create a specific feature in the entanglement spectrum at exactly the angular scale of the quadrupole. A more sophisticated analysis (connecting entanglement to the quantum state of fluctuations through the modular Hamiltonian) might find a connection.

3. The delta per-l decomposition confirms V2.312 and shows that the trace anomaly is an inherently collective (high-l summed) property, not a per-channel feature.

Implications for the Framework

This null result does NOT weaken the framework. The CMB quadrupole hypothesis was speculative and exploratory. The core predictions (Omega_Lambda = 0.6877, w = -1, BH log correction = -149/12) stand independently of whether the per-l entanglement spectrum maps onto the CMB.

What this DOES show: the framework should NOT claim to predict CMB anomalies without a rigorous derivation connecting entanglement entropy to the quantum state of fluctuations. The naive per-l mapping fails.

What Would Be Needed for a Real CMB Connection

1. Modular Hamiltonian analysis: The CMB power spectrum is determined by <phi(x)phi(x’)> on the last scattering surface. The entanglement entropy is related to the modular Hamiltonian K_A = -ln(rho_A). A connection would require computing how K_A constrains the two-point function at each l.

2. de Sitter QFT: Replace the flat-space Srednicki lattice with a proper de Sitter calculation where the cosmological horizon has physical meaning.

3. Inflation connection: The CMB power spectrum is set during inflation, not at the present-day horizon. The entanglement structure of the inflaton field at horizon crossing would be the relevant quantity.

Files

• src/cmb_entanglement.py: Core computation (9 tests, all functions)
• tests/test_cmb_entanglement.py: 17 tests, all passing
• results.json: Full numerical results