V2.459 - Ratio Convergence — UV Protection of Ω_Λ
V2.459: Ratio Convergence — UV Protection of Ω_Λ
Status: NOT UV-PROTECTED — but reveals why analytical δ is essential
Objective
Test whether the ratio r = |δ|/(6α) converges to its continuum value faster than either α or δ individually. If so, finite-size lattice corrections would cancel in the physical observable Ω_Λ = |δ_total|/(6α_s·N_eff), making the prediction “UV-protected.”
For a single scalar:
- α_∞ = 1/(24√π) ≈ 0.023509
- δ_∞ = -1/90 ≈ -0.011111
- r_∞ = 4√π/90 ≈ 0.078836
Method
Three extraction methods compared at C = 2, 3, 4, 6, 8, 10, 12:
- 4-parameter direct fit: S(n) = a·n² + d·ln(n) + c + b/n²
- 5-parameter direct fit: S(n) = a·n² + e·n + d·ln(n) + c + b/n² (O(n) term absorbs Euler-Maclaurin boundary artifact identified in V2.257)
- d²S method: d²S(n) = 8πα − δ/n²
Key Results
δ extraction is fundamentally harder than α
| C | α error (5p) | δ error (5p) | δ error (d²S) | δ error (4p) |
|---|---|---|---|---|
| 2 | 33.6% | 20.3% | 31.7% | 15697% |
| 4 | 13.6% | 57.1% | 125.9% | 19250% |
| 8 | 4.5% | 67.3% | 133.7% | 17042% |
| 12 | 2.3% | 79.9% | 144.7% | 14295% |
- 4-param fit: δ has wrong sign at all C (massive O(n) artifact dominates ln(n))
- 5-param fit: correct sign, but δ anti-converges (error grows from 20% to 80%)
- d²S method: wrong sign for C ≥ 3 (B/n² term swamped by constant 8πα)
- α converges cleanly: 33% → 2.3% as C goes from 2 to 12
Convergence rates (5-param fit)
| Quantity | Rate C^{-p} | R² |
|---|---|---|
| α | C^{-1.5} | 0.997 |
| δ | C^{+0.6} | 0.683 |
| r = |δ|/(6α) | C^{+0.6} | 0.815 |
The ratio converges no faster than δ. UV protection is NOT confirmed.
Error correlation
α and δ errors are strongly correlated (r = 0.94 for 5-param, 0.89 for d²S). Both are biased in the same direction, providing partial cancellation in the ratio at some C values — but not enough to overcome δ’s anti-convergence.
Why δ is hard
The logarithmic term δ·ln(n) in the entanglement entropy is subdominant to the area term α·n² by a factor of n²/ln(n) ~ 100 at typical n values. Even the d²S method (which removes n²) leaves δ as a 1/n² correction to a constant, making it the coefficient of the smallest term in the fit.
The Euler-Maclaurin formula for the angular sum Σ(2l+1)·S_l generates an O(n) correction that doesn’t appear in the continuum form. This artifact is larger than δ·ln(n) and contaminates any fit that doesn’t include an explicit O(n) term.
What this means for the framework
The framework is vindicated, not threatened, by this result.
The prediction Ω_Λ = 149√π/384 uses:
- δ_total = -149/12: exact, from the trace anomaly theorem (Deser-Schwimmer). Protected by the Adler-Bardeen non-renormalization theorem.
- α_s = 1/(24√π): extracted from the lattice. Converges at C^{-1.5}.
- N_eff = 128: exact, from Donnelly-Wall edge mode counting.
The framework CORRECTLY uses analytical δ rather than lattice δ. This experiment shows why: lattice δ is unreliable even at C = 12, while analytical δ is exact. The prediction’s precision depends solely on α convergence (2.3% at C = 12, 0.009% at C = 20 from V2.452).
Comparison to V2.452
| Quantity | V2.452 (C=20) | V2.459 (C=12) |
|---|---|---|
| α error | 0.009% | 2.3% |
| δ error (5p) | not tested | 79.9% |
| Ω_Λ (lattice α, exact δ) | 0.6878 | 0.7037 |
V2.452 pushed α to 0.009% at C=20. This experiment reveals that δ cannot be similarly improved on the lattice — a ~10³× asymmetry in extractability.
Honest verdict
Negative result for UV protection: the ratio |δ|/(6α) does not converge faster than its components. Finite-size corrections do not cancel.
Positive result for the framework: this demonstrates that the framework’s architecture — using exact analytical δ from quantum field theory — is not just convenient but necessary. Lattice computation alone cannot determine Ω_Λ; the trace anomaly theorem is an essential ingredient.