V2.418 - Informed α Extraction — Fixing δ Does NOT Help
V2.418: Informed α Extraction — Fixing δ Does NOT Help
Objective
V2.417 showed δ_lattice varies 18,934% across schemes. Test whether using the known exact δ = -1/90 (trace anomaly) to constrain the extraction makes α scheme-independent.
Key Result: Zero Improvement
| Scheme | α_naive | α_informed | Difference |
|---|---|---|---|
| Srednicki | 0.02103 (−10.5%) | 0.02103 (−10.5%) | < 0.001% |
| Standard FD | 0.02103 (−10.5%) | 0.02103 (−10.5%) | < 0.001% |
| Numerov | 0.01609 (−31.6%) | 0.01609 (−31.6%) | < 0.001% |
| Volume-weighted | 0.00762 (−67.6%) | 0.00762 (−67.6%) | < 0.001% |
Spreads: Naive = 81.5%, Informed = 81.6%. Improvement: 1.0× (none).
Why It Doesn’t Help
The α extraction is dominated by the A term (constant in d²S = A + B/n²). The B term (which encodes δ) is divided by n², making it negligible at n ≥ 10. Fixing B has almost no effect because B/n² is already a tiny correction to A.
The scheme-dependence of α is in the area-law coefficient itself — how each discretization handles the UV spectral structure of the coupling matrix. This is NOT contamination from the log term. It’s intrinsic.
Phase 2: α_informed(n) Is Extraordinarily Stable
Within each scheme, α_informed varies by < 10⁻⁶ across n = 8..20:
- Srednicki: 0.021028 ± 0.000000
- Numerov: 0.016088 ± 0.000002
The n-dependence is negligible. All finite-size bias comes from the angular cutoff C, not the subsystem size n.
Phase 3-4: C-Convergence Unchanged
| C | Srednicki | Standard FD | Numerov | Vol-weighted | Spread |
|---|---|---|---|---|---|
| 2 | −31% | −31% | −37% | −71% | 71% |
| 3 | −17% | −17% | −32% | −69% | 78% |
| 4 | −11% | −11% | −32% | −68% | 82% |
| 5 | −7% | −7% | −29% | −67% | 83% |
C→∞ extrapolation: Srednicki → +9.8%, Numerov → −25.3%, Volume-weighted → −65.0%.
Implications
-
V2.417’s δ spread is a red herring. The 18,934% δ spread reflects poor lattice extraction of a subleading (1/n²) term, not a real physical effect. The framework correctly uses δ_exact from the trace anomaly.
-
α scheme-dependence is intrinsic. It arises from how each discretization samples the continuum mode structure, not from δ contamination. The UV spectral weight (V2.287: 96% UV-dominated) is captured differently by each scheme.
-
Only Srednicki/FD converge toward 1/(24√π). These two schemes (same universality class: φ = rR substitution + O(h²) FD) approach the target from below. Numerov and Volume-weighted converge to different values.
-
The framework’s prediction stands on the Srednicki scheme and the assumption that it correctly approaches the continuum. This is reasonable (it’s the natural Hamiltonian discretization) but unproven analytically.
What Would Resolve This
An analytical proof of α_s = 1/(24√π) from the continuum theory — without any lattice. Possible routes:
- Spectral zeta function of the Laplacian on S²
- Exact Toeplitz determinant asymptotics
- Conformal field theory methods in the continuum limit
Files
src/informed_extraction.py— 4 lattice schemes, naive/informed/3-param extractionrun_experiment.py— 5-phase analysistests/test_informed.py— 5/5 tests passing