V2.302: Locality Gap Spectrum — Connecting Double-Counting to Entanglement

Motivation

Previous experiments established:

• tr(P)/ρ = 1: vacuum energy is encoded in entanglement (V2.285, V2.295, V2.299, V2.301)
• S” = 2α − δ/n²: QNEC maps {α, δ} → {G, Λ} uniquely (V2.250)
• BW inconsistency: Λ_bare ≠ 0 violates Bisognano-Wichmann (V2.256)

Missing: the constructive link between the double-counting identity and entanglement entropy. HOW does the shared matrix K connect ρ_vac (total vacuum energy) to α (area law coefficient)?

This experiment identifies the locality gap ε_i = (K^{1/2}){ii,full} − (K^{1/2}{sub})_{ii} as the bridge. Entanglement entropy comes entirely from this boundary perturbation.

Method

For each angular channel l, compute:

1. Full K^{1/2} diagonal restricted to subsystem (sites 1..n_sub)
2. Interior block K_{sub}^{1/2} diagonal
3. Per-site gap ε_i = difference
4. Relate ε profile to entanglement entropy via perturbative expansion of symplectic eigenvalues

Key Results

1. Locality Gap Concentrates at Boundary

The gap |ε_i| is exponentially small in the interior and peaks sharply at the boundary site:

Site	ε (l=0, n=30)
1 (interior)	−3.8 × 10⁻⁷
25	−2.1 × 10⁻³
28	−8.8 × 10⁻³
29	−2.0 × 10⁻²
30 (boundary)	−8.5 × 10⁻²

93% of total |ε| is in the last 5 sites. The gap is ALWAYS negative (K^{1/2}_full < K^{1/2}_sub at boundary), meaning truncation overestimates frequencies at the cut.

2. Boundary Layer Width Is Fixed (~3 sites)

The boundary layer width (sites where |ε| > 10% of max) is:

• Constant at ~3 sites for all n_sub from 15 to 40 (n-independent!)
• Decreases with l: width = 3 (l=0) → 1 (l=40)
• This confirms entanglement is a UV/boundary phenomenon

3. Interior Has Zero Entanglement

Perturbative analysis: decompose covariances as P = P_int + δP, X = X_int + δX.

• Interior alone gives S_interior = 0 exactly (all ν_k = 1/2)
• Leading perturbation S_pert captures 90% of entropy on average
• Error decreases with l: 20% at l=0 → 0.5% at l=20

This proves: ALL entanglement comes from the boundary locality gap δP, δX.

4. Per-Channel log(S) vs log(||ε||) Correlation = 0.985

Across channels l=0..30, the per-channel entropy S_l and locality gap ||ε||₂ are tightly correlated in log-log space (ρ = 0.985). Larger gaps → more entropy, following a power law.

The S vs ε correlation across n_sub values is weaker (ρ = 0.86) because S grows linearly with n while ||ε||₂ saturates — consistent with the area law: each additional site adds the same boundary contribution.

5. Double-Counting Identity Verified at 0.02%

(2l+1)-weighted total: tr(P)/ρ_A = 1.00023 (deviation 2.3 × 10⁻⁴).

6. Perturbative Chain: δP/P ~ 2%

The locality gap perturbation is small: ||δP||/||P_int|| ~ 1-4%. Yet this 2% perturbation generates ALL the entanglement entropy. This is because the symplectic eigenvalues ν_k are exponentially sensitive to boundary coupling — a small perturbation lifts many modes from ν = 1/2 to ν > 1/2.

The Constructive Chain

ρ_vac = (1/2) tr(K^{1/2})     [vacuum energy, from K]
      ↓
K^{1/2} has exponential off-diagonal decay   [Benzi-Golub, V2.295]
      ↓
ε_i = (K^{1/2})_{ii} − (K^{1/2}_sub)_{ii}   [locality gap, this exp]
    → concentrates at boundary (~3 sites)
    → sum(ε) ≈ 0   [double-counting identity]
      ↓
δP = P_full − P_int   [boundary perturbation to momentum covariance]
    → ||δP||/||P|| ~ 2%   [small but non-zero]
      ↓
ν_k = √eig(X·P) shifts from 1/2   [symplectic eigenvalues perturbed]
    → captures 90% of S   [leading perturbation theory]
      ↓
S = α·n + δ·ln(n)   [area law + log correction]
      ↓
QNEC: S'' = 2α − δ/n²   [exactly 2 terms, V2.250]
      ↓
G = 1/(4α), Λ = |δ|/(2α·L_H²)   [uniquely determined]

Key insight: The locality gap ε bridges ρ_vac and S. Since:

• tr(ε) ≈ 0: total vacuum energy is fully encoded (no leakage)
• S comes entirely from boundary ε: all gravitational parameters from same K
• No additional source can contribute: K is the complete vacuum state

Adding Λ_bare would require an entropy source outside K — but K determines the complete vacuum. There is no room.

Honest Assessment

Achieved:

• Identified locality gap ε as the constructive bridge between ρ_vac and S
• Proved ε concentrates at boundary (93% in last 5 sites)
• Showed boundary width is n-independent (~3 sites) and l-decreasing
• Demonstrated leading perturbation captures 90% of entropy
• Confirmed all entanglement comes from boundary corrections (S_interior = 0 exactly)
• Per-channel correlation log(S) vs log(||ε||) = 0.985
• 13/13 unit tests pass, 5/6 experiment parts pass

Limitations:

• Perturbative expansion at l=0 captures only 80% (not 99%): higher-order corrections needed
• S vs ||ε||₂ correlation across n_sub is 0.86 (nonlinear relationship; area law means S grows linearly while ε saturates)
• Free scalar only — interacting theories not tested
• “No room for Λ_bare” argument remains qualitative, not a formal proof
• Does not derive α = F(K) in closed form

For the programme:

This completes the conceptual picture for Approach B: the double-counting identity is not merely a numerical coincidence but reflects the locality structure of K^{1/2}. The locality gap is the physical mechanism converting vacuum energy (full K^{1/2}) into entanglement (boundary δP from K^{1/2}). The formal proof would require showing the perturbative expansion converges and generates exactly the area law + log correction — a problem in random matrix theory for structured (banded) matrices.