V2.419: Scaling Function for α_s — From Numerics to Analytics

Motivation

The area coefficient α_s = 1/(24√π) is verified numerically to 0.011% (V2.184) but has no analytical proof. V2.410 showed it’s Srednicki-scheme-specific. V2.414 showed δ extraction across schemes is too hard at accessible sizes.

New approach: Instead of trying to prove α_s directly, map out the per-channel scaling function F(x) = n²·d²s_l at x = l/n. If F(x) has a closed form, then α_s = (1/4π)∫ 2x·F(x)dx becomes analytically computable.

Method

For each angular channel l, compute the second finite difference: d²s_l = s_l(n+1) - 2s_l(n) + s_l(n-1)

The proper scaling variable is x = l/n. Since d²s_l ~ 1/n², define: F(x) = n²·d²s_l(n)

This converges as n → ∞ (unlike raw d²s_l which decays to zero).

Key Results

1. F(x) CONVERGES with a definite profile

x	n=15	n=20	n=30	n=40	Spread
0.0	-0.251	-0.269	-0.294	-0.311	19%
0.5	-0.076	-0.072	-0.074	-0.073	5%
1.0	-0.001	-0.001	-0.001	-0.001	14%
2.0	+0.032	+0.032	+0.033	+0.033	3.8%
3.0	+0.021	+0.021	+0.021	+0.021	1.8%

Convergence improves at larger x (UV channels). The x = 0 value has large finite-n corrections (boundary effects).

2. F(x) has a SIGN CHANGE at x ≈ 1.008

F(x) < 0  for x < 1   (IR channels: l < n)
F(x) = 0  at x ≈ 1    (l ≈ n)
F(x) > 0  for x > 1   (UV channels: l > n)

Physical meaning: The zero crossing at x = l/n ≈ 1 separates:

• IR channels (l < n): effective mass l/n < 1, correlation length > subsystem → these SUBTRACT from the area coefficient
• UV channels (l > n): effective mass l/n > 1, correlation length < subsystem → these ADD to the area coefficient

The area coefficient α_s is the net effect of cancellation between IR (negative) and UV (positive) contributions.

3. Best-fit model: F(x) = a·(x² - x₀²)·exp(-bx)

Model	R²	Parameters
(x² - x₀²)·exp(-bx)	0.9991	a=0.203, b=1.446, x₀=1.008
(x - x₀)·exp(-bx²)	0.976	a=0.166, b=0.344, x₀=1.047
sin(px)·exp(-bx)	0.803	a=-10, b=6.22, p=0.288

The (x² - x₀²)·exp(-bx) model fits at R² = 0.999 and is analytically integrable:

∫₀^∞ 2x·(x² - 1)·exp(-bx)dx = 2·[6/b⁴ - 1/b²] = 2(6 - b²)/b⁴

4. Two sources of α_s — a structural decomposition

Critical finding: The per-channel d²s_l accounts for only ~47% of α_s. The remaining ~53% comes from the angular spectrum growth — the fact that l_max = C·n scales with n, adding new channels as n increases.

Source	Contribution to α_s	Fraction
Per-channel d²s_l (fixed l)	0.0110	47%
Angular growth (new channels)	0.0125	53%
Total	0.0235	100%

This decomposition shows that the area law has two physically distinct origins:

1. Per-channel curvature: How each channel’s entropy curves with subsystem size
2. Angular multiplicity: How many channels contribute at each scale

5. Integral decomposition

The per-channel contribution further decomposes into:

• Negative (IR) part: α_neg = -0.0021 (16% of positive)
• Positive (UV) part: α_pos = +0.0131

The ratio neg/pos = 0.161 is remarkably stable across n values (varies < 0.5%).

Significance

What this reveals about α_s

1. α_s is NOT a simple spectral quantity — it emerges from cancellation between IR and UV contributions, with the UV part winning by a factor of ~6.

2. The zero crossing at x = 1 is robust — it’s a structural property of the Srednicki lattice, not a numerical coincidence.

3. The model F(x) = a(x² - 1)exp(-bx) suggests the per-channel contribution is related to the second derivative of the massive scalar entropy (mass = l/n), modified by an exponential UV cutoff.

Path to analytical proof

If the parameters a and b can be derived from the heat kernel of a massive scalar, then:

• α_per_channel = (a/2π)(6 - b²)/b⁴
• α_angular_growth can be computed from the boundary term of the Euler-Maclaurin expansion
• α_s = α_per_channel + α_angular_growth = 1/(24√π) would follow

The (x² - 1) factor suggests a connection to the Legendre function P₂(cos θ) or to the second Seeley-DeWitt coefficient.

Limitations

• F(x=0) converges slowly (19% spread at n=15..40)
• The model R² = 0.999 is good but not exact — sub-percent deviations remain
• The angular growth contribution has not been decomposed into a scaling function
• The connection to known heat kernel results is suggestive but not established

Files

• src/scaling.py — Per-channel d²s computation and scaling function
• run_experiment.py — 4-phase experiment
• tests/test_scaling.py — Convergence and sign-change tests