V2.306 - Symplectic Spectral Flow — Per-Mode QNEC Structure
V2.306: Symplectic Spectral Flow — Per-Mode QNEC Structure
Motivation
The QNEC gives d²S/dn² = 8πα − δ/n² (2 terms, asymptotically). This is a sum over all symplectic modes: d²S = Σ_k d²s_k. Does each mode contribute a 2-term structure, or does the 2-term form emerge only from the sum?
If per-mode → Λ_bare=0 is a fundamental per-mode constraint. If emergent → Λ_bare=0 relies on collective cancellation across modes.
Method
Track symplectic eigenvalues ν_k(n) as subsystem size n varies from 8 to 27. For each mode k, compute per-mode entropy s_k(n) = (ν_k+½)ln(ν_k+½) − (ν_k−½)ln(ν_k−½), then fit d²s_k to:
- 2-term: A_k + B_k/n²
- 3-term: A_k + B_k/n² + C_k·ln(n)/n²
Modes are tracked by rank (top-1 = largest ν, top-2 = second largest, etc.).
Key Results
1. Two Classes of Modes
The symplectic eigenvalues split into two distinct populations:
Bulk modes (top-1 through top-5 at l=5): Well-separated from ν=0.5, with ν_max ≈ 0.517. These carry essentially all the entanglement entropy.
| Mode | R² (2-term) | R² (3-term) | C_k (log coeff) |
|---|---|---|---|
| top-1 | 0.9990 | 0.99998 | 8.7×10⁻³ |
| top-2 | 0.9979 | 1.00000 | 1.4×10⁻² |
| top-3 | 0.9921 | 0.99981 | 3.3×10⁻² |
| top-4 | 0.9821 | 0.99930 | 5.9×10⁻² |
| top-5 | 0.9561 | 0.99465 | 1.1×10⁻¹ |
Edge modes (top-6 through top-8): Near ν=0.5, appearing/disappearing as n changes.
| Mode | R² (2-term) | R² (3-term) |
|---|---|---|
| top-6 | 0.204 | 0.546 |
| top-7 | 0.224 | 0.252 |
| top-8 | 0.098 | 0.129 |
2. Per-Mode Structure Needs a Log Term
For bulk modes, the 2-term model (A + B/n²) achieves R² > 0.95, but adding a log correction (A + B/n² + C·ln(n)/n²) pushes R² to 0.999–1.000. The true per-mode structure is:
d²s_k = A_k + B_k/n² + C_k·ln(n)/n²
This 3-parameter structure holds across all channels tested:
| l | Mode | R² (2-term) | R² (3-term) |
|---|---|---|---|
| 0 | top-1 | 0.9994 | 1.000000 |
| 0 | top-2 | 0.9999 | 0.999999 |
| 2 | top-1 | 0.9985 | 1.000000 |
| 5 | top-1 | 0.9991 | 0.999998 |
| 10 | top-1 | 0.9997 | 0.999916 |
| 20 | top-1 | 0.9670 | 0.999905 |
The log term C_k grows for deeper modes (top-1 → top-5) and for higher l.
3. Edge Modes Are Chaotic
Modes near ν = 0.5 (the entanglement ground state) have erratic d²s_k because:
- New modes “activate” (cross from ν = 0.5 to ν > 0.5) as n increases
- This creates discontinuities in the per-mode tracking
- The edge modes have POSITIVE B_k (opposite sign to bulk modes)
At l=5, bulk modes have B_k ≈ −0.04 to −0.07, while edge modes have B_k ≈ +0.04 to +0.13. Partial cancellation between bulk and edge modes occurs in the total.
4. Sum Rule
For l=3, summing per-mode parameters and comparing to the total:
- Σ A_k vs A_total: 89% error
- Σ B_k vs B_total: 9.1% error
The A sum rule fails badly because the total 2-term fit itself is poor (R² = 0.78 for single-channel total at l=3). The per-mode decomposition is mathematically exact (d²S = Σ d²s_k), but fitting each piece independently introduces compounding errors from the edge modes.
5. Channel Dependence
| l | Total R² (2-term) | Top-1 R² (2-term) |
|---|---|---|
| 0 | 0.999 | 0.999 |
| 2 | 0.942 | 0.999 |
| 5 | 0.553 | 0.999 |
| 10 | 0.060 | 1.000 |
| 20 | 0.079 | 0.967 |
Low-l channels have clean total 2-term structure. High-l channels do not — the edge mode contamination dominates. The FULL QNEC (summed over all l with degeneracy 2l+1) must reconstruct the 2-term form through massive cancellation across channels.
6. Full QNEC at C=5
Summing over all channels with degeneracy (2l+1):
- 2-term: α = 0.0671, δ = −16.3, R² = 0.365
- 3-term (with log): R² = 0.518
The R² is low because n = 10–23 at C = 5 is below the asymptotic regime. The 2-term structure emerges only at larger n (known from V2.250, V2.296).
Physical Interpretation
The 2-Term QNEC Structure is EMERGENT
The per-mode structure is 3-parameter (A_k + B_k/n² + C_k·ln(n)/n²), not 2-parameter. The ln(n)/n² term is individually significant for each mode (improving R² from 0.99 to 0.99999).
The 2-term structure d²S = A + B/n² for the TOTAL entropy arises because:
- The ln(n)/n² contributions from bulk modes partially cancel
- Edge mode contributions (opposite sign B_k) partially cancel bulk modes
- The sum over channels with (2l+1) degeneracy provides further cancellation
This means Λ_bare = 0 (which follows from 2-term completeness) is a collective property of the entanglement spectrum, not a per-mode constraint. No individual mode “knows” that Λ_bare = 0 — it emerges from the conspiracy of all modes.
Connection to Trace Anomaly
The per-mode log term C_k·ln(n)/n² is reminiscent of the trace anomaly contribution to entanglement entropy (the δ·ln(A) term). At the per-mode level, the anomaly manifests as a ln(n)/n² correction to d²s_k. When summed:
- Σ B_k → δ (the log coefficient)
- Σ C_k·ln(n) → absorbed into the effective δ
The total δ is a sum over modal contributions that include both B_k and C_k·ln(n) terms.
Implications for Λ_bare = 0
This result slightly WEAKENS the Λ_bare = 0 argument from QNEC completeness. The 2-term structure (V2.250, V2.257) is exact in the total but emergent from a richer per-mode structure. This means:
- The “no room for Λ_bare” argument applies to the TOTAL entropy, not per-mode
- A subtle modification (adding Λ_bare) could potentially be absorbed into the per-mode log terms without changing the total structure
- The constraint is less rigid than if each mode independently demanded 2 terms
However, the argument remains valid at the level of the total entropy: the map {α, δ} → {G, Λ} is bijective, and the TOTAL d²S has exactly 2 macro-scale terms asymptotically.
Honest Assessment
Achieved
- First decomposition of QNEC into per-mode symplectic contributions
- Discovered per-mode needs 3 parameters (A + B/n² + C·ln(n)/n²), not 2
- Identified two mode populations: bulk (clean) and edge (chaotic)
- Established that 2-term QNEC structure is emergent, not fundamental
- The 3-term per-mode structure achieves R² = 0.99999+ for bulk modes at l=0
Not Achieved
- Cannot quantify how the log terms cancel in the full channel sum (would need larger n, C)
- Edge mode tracking is inherently noisy (modes appear/disappear)
- Sum rules have significant errors due to edge mode contamination
- Does not resolve whether a Λ_bare slot could hide in the per-mode log terms