V2.136: Renyi Entanglement Entropy Spectrum

Status: COMPLETE

Question

The framework derives Lambda from the Clausius relation delta_Q = T dS, which uses von Neumann entropy S_1 = -Tr(rho ln rho). But the Renyi entropies S_n = ln Tr(rho^n)/(1-n) are an equally valid one-parameter family of entanglement measures. Does the self-consistency ratio R_n = |delta_n|/(6 alpha_n) depend on the Renyi parameter n? If so, what makes von Neumann (n=1) special?

Method

1. Spherical angular momentum decomposition (V2.67 infrastructure) for a free massless scalar on a radial lattice (N=500, proportional cutoff C=8)
2. Compute symplectic eigenvalues ONCE per (l, n_sub), then evaluate ALL Renyi entropies from the same eigenvalues (efficient: no redundant diagonalizations)
3. Direct 3-parameter fit S_n(n_sub) = alpha_n * n_sub^2 + delta_n * ln(n_sub) + gamma to extract area coefficient alpha_n and log coefficient delta_n simultaneously
4. Compute R_n = |delta_n|/(6 alpha_n) for n in {0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 5.0, 10.0, inf}
5. 36 partition sizes n_sub in [10, 45]

Results

Renyi Entropy Spectrum

n	alpha_n	delta_n	R_n	R_n/Omega_Lambda
0.5	1.9167	17.604	1.531	2.235
0.75	0.6005	6.328	1.756	2.564
1.0	0.2847	3.134	1.835	2.678
1.5	0.1332	1.487	1.860	2.716
2.0	0.0956	1.068	1.862	2.718
3.0	0.0725	0.810	1.862	2.718
5.0	0.0604	0.675	1.862	2.718
10.0	0.0537	0.600	1.862	2.718
inf	0.0483	0.540	1.862	2.718

All fits have R^2 > 0.999999. Elapsed: 77s.

Key Observations

1. R_n is NOT constant across Renyi parameter. It rises from 1.531 (n=0.5) to a plateau of ~1.862 (n >= 1.5).

2. Von Neumann (n=1) does not minimize the single-field ratio. R_1 = 1.835, while R_0.5 = 1.531 is closer to Omega_Lambda = 0.685. However, this comparison is misleading — see interpretation below.

3. Remarkable plateau for n >= 1.5. R_n converges to 1.862 for large n, with R_n/Omega_Lambda -> 2.718 = e. This is likely a lattice artifact (the Euler number appearing in the ratio).

4. alpha_n decreases monotonically with n, consistent with the general property that Renyi entropy is monotonically non-increasing in n.

5. delta_n decreases monotonically with n, from 17.6 (n=0.5) down to 0.54 (n=inf).

6. Conformal predictions fail badly. The CFT prediction alpha_n = alpha_1 * (1 + 1/n) / 2 is wildly wrong on the lattice (ratios range from 0.34 to 4.49 vs expected 1.0). This indicates that the lattice UV regulator breaks conformal scaling for Renyi entropies more severely than for von Neumann.

Interpretation for the Lambda Prediction

Why this doesn’t invalidate the framework

The self-consistency argument for Lambda uses two separate ingredients:

1. alpha from the lattice (numerical area-law coefficient)
2. delta from QFT (exact trace anomaly: delta = -1/90 per scalar degree of freedom)

The trace anomaly coefficient delta = (c/6) * (universal) is SPECIFICALLY the coefficient of the logarithmic correction to von Neumann entanglement entropy. For Renyi entropy of order n != 1, the logarithmic coefficient is a different quantity — the Renyi trace anomaly — which depends on the Renyi parameter.

In the Clausius relation delta_Q = T dS, the entropy S MUST be the thermodynamic (von Neumann) entropy. This is not a choice — it is a consequence of:

• The zeroth law of thermodynamics (transitivity of equilibrium requires a unique temperature)
• The Clausius inequality (delta S >= delta Q / T, with equality for reversible processes)
• The KMS condition (thermal states are characterized by von Neumann entropy)

Renyi entropies S_n for n != 1 do not satisfy the Clausius relation and are not thermodynamic entropies.

What this experiment demonstrates

The R_n spectrum confirms that the self-consistency ratio depends on which entropy functional is used. The framework’s prediction of Lambda relies on the von Neumann entropy specifically because:

1. The Clausius relation singles out n=1 on thermodynamic grounds
2. The QFT trace anomaly delta = -1/90 is the von Neumann coefficient
3. The alpha_{SM} summation rule (118 effective scalars) assumes the von Neumann area law

If one were to repeat the Lambda calculation with Renyi-n entropy, one would need:

• The Renyi trace anomaly delta_n (different for each n, and NOT known exactly for most n)
• The Renyi area coefficient alpha_n (measured here)
• A modified thermodynamic relation (which doesn’t exist for n != 1)

The fact that R_n varies by ~20% across the Renyi spectrum (from 1.53 to 1.86) while the framework requires a SPECIFIC value matching Omega_Lambda means that the choice of entropy functional is physically consequential. Von Neumann is not just one option among many — it is the ONLY one with a consistent thermodynamic interpretation.

Implications for the Overall Science

• Positive for the framework: The Clausius relation’s requirement of von Neumann entropy is physically motivated, not ad hoc. This experiment shows that different entropy choices give different predictions, so the framework genuinely relies on thermodynamic principles to single out the correct one.
• Note: The single-scalar R_1 = 1.83 here is NOT the Lambda prediction. The full SM prediction uses the exact QFT delta and lattice alpha for 118 effective degrees of freedom, giving R = 0.657 (V2.101).
• Lattice artifact: The plateau R_n -> 1.862 for large n, with R_n/Omega_Lambda -> e, is suggestive but most likely a coincidence of the UV regulator.

Files

• run_experiment.py: Main experiment driver
• src/renyi_entropy.py: Renyi entropy from symplectic eigenvalues
• src/spherical_renyi.py: Spherical decomposition + direct 3-parameter fit
• tests/test_renyi.py: Validation tests (all pass)
• results/results.json: Full numerical data