V2.281: Renyi QNEC Form — Structural Selection of Von Neumann Entropy

Question

V2.247 showed only q=1 (von Neumann) gives the correct Ω_Λ, with q ≥ 2 excluded at > 35σ. But that was a NUMERICAL exclusion. Does the 2-term QNEC form d²S_q(n) = A_q + B_q/n² hold for Renyi entropies (q ≠ 1)?

• If YES with different coefficients: selection is purely numerical
• If NO (form breaks): structural selection — QNEC uniqueness is q=1 specific

Method

Compute Renyi entropy S_q = Σ_l (2l+1) s_q^(l) on the Srednicki lattice, where per-channel Renyi entropy uses symplectic eigenvalues:

s_q(ν) = (1/(q-1)) · ln((ν+1/2)^q − (ν-1/2)^q)

Take d²S_q(n) = S_q(n+1) − 2S_q(n) + S_q(n−1) and fit to 1T, 2T, 3T models.

Parameters: C = 3, N = 120, n = 4..11, q = 1, 1.5, 2, 3, 5, 10, 50.

Results

1. The 2-term form holds for ALL q — but trivially

q	rel_res (2T)	2T/1T improvement	B/n² significance
1.0	4.2 × 10⁻²	1.3×	Substantial
1.5	2.5 × 10⁻²	1.0×	Negligible
2.0	2.5 × 10⁻²	1.0×	Negligible
3.0	2.5 × 10⁻²	1.0×	Negligible
5.0	2.5 × 10⁻²	1.0×	Negligible
10.0	2.5 × 10⁻²	1.0×	Negligible
50.0	2.5 × 10⁻²	1.0×	Negligible

The 2-term form d²S_q = A + B/n² formally holds at all q, but for q ≥ 1.5, the B/n² term provides ZERO improvement over a constant. The d²S_q is essentially flat — the log correction has vanished.

Only at q=1 does the 1/n² term (log correction) meaningfully contribute.

2. The log correction vanishes for Renyi entropies

q	δ_q/δ_1	α_q/α_1	R_q = |δ_q|/(6α_q)
1.0	1.000	1.000	8.60
1.5	0.057	0.356	1.38
2.0	0.011	0.242	0.40
3.0	0.006	0.181	0.31
5.0	0.005	0.151	0.30
10.0	0.005	0.134	0.30
50.0	0.004	0.123	0.30

Critical observation: δ_q drops far faster than α_q.

• α_q/α_1 decreases moderately (to 0.12 at q=50)
• δ_q/δ_1 essentially vanishes (to 0.004 at q=50)
• The ratio R_q = |δ_q|/(6α_q) converges to ~0.30 for q ≥ 3

Since Λ depends on δ (the log correction), and δ is negligible for q ≠ 1, Renyi entropies cannot determine Λ. They give only the area law (G).

3. Coefficient ratios differ from 2D CFT prediction

q	α_q/α_1 (measured)	(1+1/q)/2 (CFT)	Ratio
2.0	0.242	0.750	0.32
3.0	0.181	0.667	0.27
5.0	0.151	0.600	0.25
10.0	0.134	0.550	0.24

The α_q/α_1 ratios are much smaller than the 2D CFT prediction (1+1/q)/2, confirming V2.235’s finding that 3+1D differs qualitatively from 1+1D CFT.

4. No 1/n term at any q

q	B_1n (3T)	3T/2T improvement	1/n needed?
1.0	−0.14	1.0×	NO
2.0	+0.44	1.1×	NO
5.0	+0.27	1.1×	NO
50.0	+0.22	1.1×	NO

No 1/n term appears at any q. This contrasts with the thermal case (V2.278), where 1/n appears at T > 0.2. The Renyi structure preserves the functional form but suppresses the amplitude.

5. R_q convergence

For q ≥ 3, R_q → 0.30, far from the observed Ω_Λ = 0.6847. Even at q = 2, R_q = 0.40 is excluded. Only q = 1 can potentially match (note: at C = 3, absolute values are not converged; the R_q = 8.6 for q = 1 reflects finite-C effects, not the converged value of 0.6645).

Key Finding

The log correction δ is a von Neumann-specific phenomenon.

For Renyi entropies (q ≠ 1):

• The area law α_q persists (reduced but nonzero)
• The log correction δ_q essentially vanishes (< 1% of δ_1 for q ≥ 2)
• d²S_q is effectively constant (no meaningful 1/n² term)

This provides STRUCTURAL selection of von Neumann entropy for the Λ_bare = 0 argument, beyond the numerical exclusion of V2.247:

1. The QNEC two-term form d²S = 8πα − δ/n² has δ ≈ 0 for q ≠ 1
2. With δ ≈ 0, the QNEC determines only G (from α), not Λ
3. Only von Neumann entropy (q = 1) has a substantial log correction that allows the QNEC to determine BOTH G and Λ
4. The Λ_bare = 0 argument requires both terms → requires von Neumann

Physical Interpretation

The log correction δ = −4a (where a is the trace anomaly coefficient) is a property of the entanglement spectrum — the FULL eigenvalue distribution, not just its moments. Renyi entropies weight high-ν eigenvalues more heavily (for q > 1), suppressing the contribution from the near-boundary eigenvalues that carry the log correction.

Von Neumann entropy is the ONLY entropy measure that correctly weights all symplectic eigenvalues to produce the trace anomaly coefficient. This is not a coincidence — it is the thermodynamic entropy, and Jacobson’s derivation specifically requires thermodynamic entropy.

Connection to Previous Experiments

• V2.235: Renyi α ratios universal, differ from 2D CFT (confirmed here)
• V2.241: Boundary mode is near-thermal → von Neumann is thermodynamic
• V2.247: q ≥ 2 excluded at > 35σ by wrong Ω_Λ (numerical selection)
• V2.250: QNEC completeness requires 2-term form (which needs δ ≠ 0)
• V2.278: 2-term form is vacuum-specific (breaks at T > 0)
• V2.281: 2-term form is von Neumann-specific (δ vanishes for q ≠ 1)

Together V2.278 and V2.281 establish: the QNEC uniqueness argument for Λ_bare = 0 requires BOTH the vacuum state (T = 0) AND von Neumann entropy (q = 1). Any other choice loses the log correction that determines Λ.