V2.415 - Ratio Universality — Is R = |δ|/α Scheme-Independent?
V2.415: Ratio Universality — Is R = |δ|/α Scheme-Independent?
Motivation
V2.410 showed α_s varies 74% across 4 discretization schemes at finite lattice. This raised the question: does the framework’s prediction depend on the lattice scheme?
The physical prediction is Ω_Λ = |δ_total|/(6·α_s·N_eff). If α and δ shift together under scheme change (both being UV-sensitive), R = |δ|/α could be universal even when α and δ individually are not.
Results
The Answer: R_lattice is NOT Scheme-Universal
| Scheme | α | δ_lattice | R = |δ|/α | R² | |--------|---|-----------|-----------|-----| | Srednicki | 0.02102 | -0.0300 | 1.43 | 0.90 | | Standard FD | 0.02102 | -0.0419 | 1.99 | 0.95 | | Numerov | 0.01607 | -0.0407 | 2.53 | 0.94 | | Volume-weighted | 0.00762 | +0.0506 | 6.64 | 1.00 |
- α spread: 82% (confirms V2.410)
- δ spread: 596% (δ is FAR more scheme-sensitive than α!)
- R spread: 165% (worse than α alone, not better)
The ratio R is less universal than α alone, not more.
Why δ_lattice Is So Noisy
The true δ_scalar = -1/90 ≈ -0.011 is tiny compared to the area coefficient (A = 8πα ≈ 0.53). Extracting δ from d²S = A + B/n² requires resolving a signal that is ~2% of the dominant term. At the lattice sizes accessible here (n = 10–21), the extracted δ values are contaminated by:
- Higher-order corrections (1/n⁴, ln(n)/n² terms)
- Scheme-dependent finite-size effects that don’t cancel in the ratio
- Volume-weighted scheme gives δ > 0 (wrong sign!) — clearly not converged
C-Scaling: Srednicki Converges, Others Don’t
| Scheme | R(C=2) | R(C=5) | Change |
|---|---|---|---|
| Srednicki | 1.55 | 0.66 | -58% (converging toward 0.47) |
| Standard FD | 2.18 | 1.29 | -41% |
| Numerov | 2.40 | 2.05 | -15% |
| Volume-weighted | 7.61 | 6.68 | -12% |
Only Srednicki shows rapid convergence toward the target R = 0.4727.
N-Scaling Reveals the Problem
| Scheme | R(N=40) | R(N=100) | Trend |
|---|---|---|---|
| Srednicki | 0.49 | 0.96 | ↑ Moving away |
| Standard FD | 1.52 | 1.48 | ≈ Stable |
| Numerov | 1.38 | 1.82 | ↑ Diverging |
| Volume-weighted | 5.54 | 7.17 | ↑ Diverging |
The N-dependence of R shows that δ_lattice is not converged — the double limit (N→∞, C→∞) needs much larger lattices to stabilize δ than α.
Critical Realization: This Test Was Ill-Posed
The framework’s prediction uses δ_exact = -149/12 (from the trace anomaly), NOT δ_lattice. The trace anomaly is a topological invariant — it doesn’t need lattice extraction. It is computed exactly from QFT (Seeley-DeWitt coefficients).
This means:
- R_framework = |δ_exact|/(6·α_s·N_eff) depends only on α_s
- R_lattice = |δ_lattice|/α_lattice is a different quantity entirely
- The question “is R universal?” reduces to “is α_s universal?” (V2.410’s question)
The experiment is still valuable because it reveals:
- δ_lattice is scheme-dependent and unreliable at accessible sizes
- The framework is correct to use δ_exact (analytical) rather than lattice δ
- α is the ONLY lattice-dependent quantity, and it’s the sole remaining gap
Per-Channel Structure
A surprising finding: at n = 10–21, the per-channel α_l is negative for all l = 0–7, across all schemes. The total α > 0 comes from high-l channels (l >> 7) where (2l+1) weighting amplifies small positive contributions. This confirms V2.287’s finding that α is 96% UV-dominated.
Implications
What This Means for the Framework
-
δ is safe: The trace anomaly is exact and scheme-independent. Using δ_exact = -149/12 is not a choice — it’s the unique correct value.
-
α_s remains the gap: The framework’s only scheme-dependent input is α_s = 1/(24√π). V2.410 showed this isn’t confirmed across discretization classes. V2.412 showed the dimensional formula fails.
-
The framework is honest: It uses one analytical input (δ) and one empirical input (α_s). The prediction is only as good as α_s.
The Real Question
The experiment reframes the universality question:
It is not “does the ratio cancel scheme dependence?” (it doesn’t). It is “is the Srednicki lattice the correct discretization?”
If the Srednicki lattice captures the correct UV physics of entanglement entropy (and there are good reasons to think it does — it’s the natural discretization of the Hamiltonian on spherical shells), then α_s = 1/(24√π) follows.
The 0.011% match of α_s to 1/(24√π) within the Srednicki class (V2.288) is strong evidence. But proving this is the UNIQUE correct scheme requires either:
- An analytical derivation of α_s from the continuum theory
- Convergence of all schemes at n → ∞ (computationally prohibitive)
6/6 tests passing. Runtime: 20.5s.