Delta Extraction with Global Cutoff
Experiment V2.73: Delta Extraction with Global Cutoff
Motivation
V2.72 showed proportional angular cutoff (l_max = C×n) inflates alpha by 18% for scalars and 586% for Dirac. The question: does the cutoff convention also affect the trace anomaly coefficient delta, extracted via third finite differences (d3S)?
Method
Compute scalar S(n) at N=1000, n=25..100, C=10 with both proportional and global cutoff. Apply three extraction methods to each:
- d3S: Third differences → delta = A/2 from fit d3S = A/n³ + B/n⁴
- d2S: Second differences → alpha from d2S ≈ 8πα, delta from d2S ≈ 8πα - δ/n²
- 5-param fit: S = α·4πn² + β·n + δ·ln(n) + γ + ε/n
Also sweep C = 5, 8, 10, 13, 15 at N=400 to study convergence.
Results
Phase 2: d3S Delta — THE CRITICAL FINDING
| Method | Proportional | Global |
|---|---|---|
| d3S 2-param | -0.01099 (1.07% err) | -137.14 (garbage) |
| d3S 1-param | -0.00703 (36.8% err) | -15.87 (garbage) |
| d3S large-n | -0.01191 (7.2% err) | -781.83 (garbage) |
Global cutoff completely destroys the d3S method. The extracted delta is off by a factor of 12,000.
Why Global Cutoff Breaks d3S
With proportional cutoff (l ≤ 10n), each S(n) includes channels 0..10n. The truncation correction is:
f_trunc(n) = Σ_{l=10n+1}^{∞} (2l+1) S_l(n)
Because l_max = 10n scales linearly with n, and S_l(n) decays as l^{-3.37}, this correction is approximately polynomial in n. Third differences kill polynomials up to degree 2, so d3S(f_trunc) ≈ 0 — the truncation correction cancels.
With global cutoff (all l ≤ 1020 for every n), S_global(n) includes extra channels l = 10n+1..1020. The number of extra channels DECREASES with n:
- n=25: 770 extra channels (l=251..1020)
- n=100: 20 extra channels (l=1001..1020)
This creates a correction g(n) that decays rapidly with n and is NOT polynomial. Third differences amplify this non-polynomial component rather than canceling it, producing a d3S signal orders of magnitude larger than the true delta.
Phase 3-4: Alpha and 5-Param Fit
| Quantity | Proportional | Global |
|---|---|---|
| alpha (d2S) | 0.02278 | 0.02089 |
| alpha (5-param) | 0.02278 | 0.01942 |
| delta (5-param) | -0.01150 (3.5% err) | -391.07 (garbage) |
| R² (5-param) | 1.000000 | 0.999999970 |
The 5-param fit with global cutoff also gives garbage delta because the same non-polynomial correction corrupts all coefficients (beta, delta, gamma, epsilon all blow up).
The global alpha from 5-param fit (0.01942) matches V2.72’s value (0.01926 at N=400) to 0.8%, confirming N-independence.
Phase 5: Convergence with C
Alpha (5-param fit):
| C | alpha_prop | alpha_global | Diff |
|---|---|---|---|
| 5 | 0.02123 | 0.01084 | +96% |
| 8 | 0.02244 | 0.01706 | +32% |
| 10 | 0.02278 | 0.01897 | +20% |
| 13 | 0.02304 | 0.02055 | +12% |
| 15 | 0.02315 | 0.02118 | +9.3% |
Both converge toward the same value as C → ∞ (the “true” alpha), but from opposite sides: proportional from above, global from below. At C=10, the gap is 20%. Neither is exact at finite C.
Delta (d3S):
- Proportional: stable at -0.01578 for all C (42% error at N=400)
- Global: garbage at all C values
The d3S delta at N=400 is -0.01578 (42% error) vs N=1000: -0.01099 (1.07% error). Large N is essential.
Phase 6: Implications for R
Single scalar R = |delta|/(12·alpha):
| Convention | delta | alpha | R |
|---|---|---|---|
| Proportional (V2.67) | -0.01099 | 0.02278 | 0.0402 ≈ 1/(8π) |
| Global alpha, prop delta | -0.01099 | 0.01942 | 0.0472 |
| Theory delta, global alpha | -0.01111 | 0.01942 | 0.0477 |
The neat V2.67 result R ≈ 1/(8π) = 0.0398 used proportional alpha. With global alpha, R increases to 0.048 — 20% above 1/(8π). The 1/(8π) agreement was partly an artifact of the proportional convention.
Full SM R_SM = 0.36 (unchanged — uses V2.72 ratios with global alpha).
Key Conclusions
-
The proportional and global cutoffs serve complementary roles:
- Proportional: makes truncation correction polynomial → d3S works → correct delta
- Global: eliminates n-dependent bias → correct alpha
- No single convention works for both simultaneously
-
Best practice: use different conventions for different quantities:
- delta: proportional d3S at N ≥ 1000 → delta = -0.01099 ≈ -1/90
- alpha: global fit at any N ≥ 300 → alpha = 0.0193
-
The single-scalar R ≈ 1/(8π) from V2.67 was scheme-dependent. With global alpha, R = 0.048 (20% higher). This means R is not universal — it depends on the regularization scheme.
-
Both alphas converge as C → ∞, but slowly. At C=15, the gap is still 9%. Getting to <1% would require C > 50 at enormous computational cost.
-
R_SM = 0.36 is robust and doesn’t depend on which scalar alpha we use (the ratio alpha_Dirac/alpha_scalar is convention-independent for species with the same decay rate, and Dirac is always 4.6× scalar).
Open Questions
- Can we define a “renormalized” alpha that is convention-independent? The C → ∞ limit should give the physical alpha, but convergence is slow.
- Is the self-consistency condition R = |delta|/(12·alpha) itself scheme-dependent? If alpha depends on the regulator, the Friedmann equation derived from entanglement entropy inherits this dependence.
- Should the self-consistency formula use a different normalization?
Runtime
246 seconds total (N=1000 computations: 120s, Phase 5 convergence sweep: 100s).