V2.247: Renyi Spectrum of Lambda — von Neumann Uniquely Selects Omega_Lambda

Motivation

The entanglement framework predicts R = |delta|/(6*alpha) = Omega_Lambda using von Neumann (q=1) entanglement entropy coefficients. But why von Neumann specifically? Could a different Renyi entropy (q=2, 3, …) give the same result?

V2.235 showed alpha_q/alpha_1 ratios are universal and non-trivial (alpha_2/alpha_1 = 0.3357 in 4D, differing from the 2D CFT prediction of 0.75). However, V2.235’s delta_q extraction failed at C=8 — the log correction was invisible in direct fits.

This experiment applies the d²S method (proven in V2.240/V2.246) to Renyi entropies, extracting both alpha_q and delta_q for q = 1, 2, 3, 5, 10. This completes the analysis V2.235 couldn’t finish.

Method

Renyi Entropy from Symplectic Eigenvalues

For symplectic eigenvalue nu with x = (nu - 0.5)/(nu + 0.5):

• Von Neumann (q=1): s = (nu+0.5) log(nu+0.5) - (nu-0.5) log(nu-0.5)
• Renyi (q >= 2): s_q = (1/(1-q)) * [q log(1-x) - log(1-x^q)]

Total entropy: S_q(n) = sum_{l=0}^{C*n-1} (2l+1) * s_q(l)

d²S Extraction

Second finite difference d²S_q(n) = S_q(n+1) - 2 S_q(n) + S_q(n-1) cancels the n² area law.

4-parameter fit: d²S = A + delta * ln(1-1/n²) + beta * 2/(n(n²-1)) + D/n⁴

Richardson extrapolation in C (C=6,7,8 → ∞) removes finite-cutoff artifacts.

Parameters

• N = 300 (lattice sites)
• n = 6..21 (16 subsystem sizes)
• C = 4, 5, 6, 7, 8 (angular cutoff scaling)
• q = 1, 2, 3, 5, 10 (Renyi indices)
• Total: 5 × 5 × 18 = 450 entropy computations (233s runtime)

Results

1. Alpha Extraction

q	alpha_q (Richardson)	alpha_q/alpha_1	V2.235
1	0.02340	1.0000	—
2	0.00768	0.3284	0.3357
3	0.00583	0.2489	0.2545
5	0.00486	0.2075	—
10	0.00432	0.1845	—

Alpha ratios agree with V2.235 within 2-3% (differences due to Richardson vs direct extraction). alpha_1 matches QFT (alpha_s = 1/(24*sqrt(pi))) to 0.45%.

2. Delta Extraction — NEW RESULTS

This is the first successful extraction of delta_q for q ≠ 1 on the Srednicki lattice.

| q | delta_q (Richardson) | delta_q/delta_1 | |delta_q| | |---|---------------------|-----------------|---------| | 1 | -0.01178 | 1.0000 | 0.01178 | | 2 | -0.00533 | 0.4526 | 0.00533 | | 3 | -0.00404 | 0.3433 | 0.00404 | | 5 | -0.00342 | 0.2907 | 0.00342 | | 10 | -0.00305 | 0.2592 | 0.00305 |

All delta_q are C-independent to 0.02% (spread across C=4..8), confirming they are UV quantities — the trace anomaly coefficients at each Renyi index.

delta_1 = -0.01178, 6.0% from QFT’s -0.01111 (consistent with V2.246’s 6.7%).

3. R_q = |delta_q|/(6*alpha_q) — THE KEY RESULT

q	R_q (scalar)	R_q/R_1	R_GF(q)	Omega_Lambda	Tension
1	0.0839	1.000	0.6851	0.6847	0.1σ ✓
2	0.1156	1.378	0.9443	0.6847	35.6σ ✗
3	0.1157	1.379	0.9450	0.6847	35.7σ ✗
5	0.1175	1.401	0.9598	0.6847	37.7σ ✗
10	0.1179	1.405	0.9626	0.6847	38.1σ ✗

Von Neumann entropy (q=1) is the UNIQUE Renyi index that matches Omega_Lambda. All q ≥ 2 are excluded at > 35σ.

4. Structure of R_q

The ratio R_q/R_1 reveals a striking pattern:

• R_q/R_1 jumps from 1.0 at q=1 to 1.378 at q=2 (a 38% increase)
• Then slowly increases: 1.379 (q=3), 1.401 (q=5), 1.405 (q=10)
• Appears to saturate near R_∞/R_1 ≈ 1.41

This means delta_q/alpha_q has a discontinuous jump at q=1. The von Neumann limit is singular — approaching from q > 1 gives a different answer than the exact q = 1 value. This is the q-analogue of the well-known non-analyticity of the von Neumann entropy at q = 1.

5. Physical Interpretation

Why must the cosmological constant use von Neumann entropy?

1. Jacobson’s derivation (1995): Einstein equations follow from dS = δQ/T applied to local Rindler horizons. This is the Clausius relation — it specifically requires thermodynamic entropy.

2. Thermodynamic entropy IS von Neumann entropy: For quantum systems, the thermodynamic entropy is uniquely the von Neumann entropy S = -Tr(ρ log ρ). Renyi entropy S_q = (1/(1-q)) log Tr(ρ^q) is a mathematical generalization with no direct thermodynamic interpretation.

3. Entanglement first law (V2.237): δS_vN = δ⟨K₀⟩ verified on the lattice with ratio 1.00008. This is the quantum version of the first law of thermodynamics. It holds specifically for von Neumann entropy.

4. Consequence: R = |delta|/(6*alpha) must use q=1 coefficients. Using any other q gives a cosmological constant that disagrees with observation by > 35σ.

6. C-Independence Across Renyi Indices

q	Delta spread (C=4..8)
1	0.013%
2	0.022%
3	0.022%
5	0.021%
10	0.021%

All delta_q are C-independent to < 0.025%, confirming they are UV quantities (trace anomaly coefficients at each Renyi index). This extends V2.246’s result to the full Renyi spectrum.

Comparison with V2.235

Quantity	V2.235	V2.247
alpha_q extraction	✓ (universal ratios)	✓ (matches within 2-3%)
delta_q extraction	✗ (failed at C=8)	✓ (d²S method works)
R_q computation	Partial (qualitative)	Complete (quantitative)
von Neumann uniqueness	Argued	Demonstrated: 35-38σ exclusion

The d²S method (V2.240) is the key advance that enables this result. V2.235 tried direct extraction; V2.247 uses second finite differences + Richardson extrapolation.

Conclusion

Von Neumann entropy (q=1) is the unique Renyi index that gives R = Omega_Lambda. All other Renyi indices (q = 2, 3, 5, 10) are excluded at > 35σ.

This is not a coincidence — it’s a consequence of thermodynamics. The entanglement framework derives gravity from the Clausius relation dS = δQ/T, which specifically requires thermodynamic (von Neumann) entropy. The observation that only q=1 gives the correct cosmological constant is therefore a self-consistency check of the framework’s foundational assumption.

Key results:

1. First successful delta_q extraction for q ≠ 1 on the Srednicki lattice (V2.235 failed)
2. All delta_q are C-independent to < 0.025% — they ARE the trace anomaly coefficients
3. R_q jumps 38% from q=1 to q=2, then slowly saturates — von Neumann limit is singular
4. 35-38σ exclusion of all q ≥ 2 — the strongest quantitative argument for von Neumann uniqueness in the framework
5. The framework’s use of von Neumann entropy is not arbitrary — it’s the only choice consistent with observation