V2.299 - 2+1D Double-Counting Identity — Dimensional Universality
V2.299: 2+1D Double-Counting Identity — Dimensional Universality
Motivation
V2.295 proved the Benzi-Golub mechanism for tr(P_sub)/ρ_A = 1 in 3+1D: K^{1/2} has exponential off-diagonal decay (ξ ≈ 14), so interior diagonal elements match whether computed from the full or truncated matrix. V2.297 showed that 2+1D has surprising differences: δ_2D ≠ 0 (log correction exists without trace anomaly) and S”(n) lacks the constant term (perimeter law → d²(αn)/dn² = 0).
Key question: Is the double-counting identity tr(P)/ρ_A = 1 dimension-universal (algebraic, from matrix locality) or D=4-specific? This determines whether Approach B is a general mathematical identity or an accidental 3+1D coincidence.
Method
Compute tr(P_sub) and ρ_A per angular channel in 2+1D using the radial lattice:
- 2+1D chain: K_{jj} = 2 + m²/j², K_{j,j+1} = -(j+0.5)/√(j(j+1)), degeneracy g₀=1, g_m=2
- 3+1D chain: K_{jj} = ((j-0.5)² + (j+0.5)² + l(l+1))/j², K_{j,j+1} = -(j+0.5)²/(j(j+1)), degeneracy 2l+1
- Same site-level approximation in both: ν = √(X_nn·P_nn)
- Compare K^{1/2} off-diagonal decay, per-channel ratios, boundary scaling
Key Results
1. Identity Holds Per-Channel in Both Dimensions
| Channel | 2+1D ratio | 3+1D ratio |
|---|---|---|
| 0 | 1.00676 | 1.00643 |
| 5 | 1.00281 | 1.00255 |
| 10 | 1.00116 | 1.00107 |
| 20 | 1.00028 | 1.00026 |
All channels have ratio within 0.7% of unity. The 2+1D deviations are systematically ~6-10% larger than 3+1D at the same channel, but follow the identical pattern: both converge to 1 with increasing channel number.
2. K^{1/2} Localization Is Nearly Identical
| Channel | ξ (2+1D) | ξ (3+1D) |
|---|---|---|
| 0 | 14.04 | 13.60 |
| 5 | 9.73 | 9.43 |
| 10 | 7.42 | 7.25 |
| 20 | 5.05 | 4.97 |
ξ_2D ≈ 1.03 × ξ_3D — virtually the same localization length. The Benzi-Golub mechanism operates identically in both dimensions because both chains are tridiagonal (bandwidth 1).
3. Same Boundary Scaling Law
| Dimension | Power law | R² |
|---|---|---|
| 2+1D | dev ~ n^{-0.92} | 0.9994 |
| 3+1D | dev ~ n^{-0.87} | 0.9985 |
Both dimensions show power-law convergence to ratio = 1 with comparable exponents. The 2+1D exponent is slightly steeper (−0.92 vs −0.87).
4. N-Independence Confirmed in 2+1D
Per-channel ratio is independent of chain length N to CV < 0.0001%, identical to the 3+1D behavior found in V2.289. The identity is a property of the boundary structure, not the IR.
5. Weighted Total with Degeneracies
Including degeneracies (g_m = 1, 2, 2, … for 2+1D vs 2l+1 for 3+1D):
- 2+1D total ratio: 1.00022 at n_sub=15 (0.02% deviation)
- 3+1D total ratio: 1.00006 at n_sub=15 (0.006% deviation)
Both converge to 1 with increasing n_sub. The 3+1D total converges faster because the (2l+1) degeneracy weights high-l channels more heavily, where the identity is more precise.
6. Same Sign Pattern: 3D/2D Ratio ≈ 0.92
The deviation ratio (3D/2D) is remarkably stable at ~0.91–0.95 across all channels. This suggests the two dimensions share the same mathematical structure with a single O(1) prefactor difference tied to the shell weighting (circumference vs area).
Interpretation: Universality + D=4 Selection
The double-counting identity tr(P)/ρ = 1 is dimension-universal. This is expected from the Benzi-Golub theorem: for any banded matrix K, the function f(K) = K^{1/2} has exponential off-diagonal decay, making diagonal elements insensitive to truncation. Since both 2+1D and 3+1D chains are tridiagonal, the mechanism is identical.
For the Λ_bare = 0 derivation, this separates two ingredients:
- Universal: tr(P)/ρ = 1 (vacuum energy encoded in entanglement) — holds in any D
- D=4 specific: S” has two independent terms → QNEC maps to {G, Λ} uniquely
In 2+1D, the identity holds but S”(n) = −δ/n² (no constant term, V2.297), so there is only one gravitational parameter to determine. The Λ_bare = 0 conclusion requires BOTH the universal identity AND the D=4 area-law structure.
Honest Assessment
Achieved:
- Confirmed tr(P)/ρ_A = 1 per-channel in 2+1D (max dev 0.68%, same order as 3+1D)
- Showed ξ_2D ≈ ξ_3D to 3% (identical Benzi-Golub mechanism)
- Demonstrated same power-law boundary scaling (n^{-0.92} vs n^{-0.87})
- N-independence to CV < 0.0001% in 2+1D
- 15/15 unit tests pass, 6/6 experiment parts pass
Limitations:
- Site-level approximation (V2.287: overestimates α by ~57%)
- Only tested free scalar (not interacting theories)
- Pure exponential fit R² ≈ 0.86–0.99 (decay has power-law × exponential structure)
- Small n_sub range (5–30)
For the overall programme:
Strengthens Approach B by proving the double-counting identity is algebraic/universal, not an accident of D=4. Combined with V2.297 (S” structure differs in 2+1D) and V2.295 (Benzi-Golub mechanism), this completes the picture: vacuum energy is ALWAYS encoded in entanglement via matrix locality, but only in D=4 does the QNEC structure allow extracting Λ_bare = 0 from it.