V2.231: Analytic Scaling Function — Identifying f(x) to prove alpha_s = 1/(24*sqrt(pi))

Executive Summary

V2.230 discovered a universal scaling function f(x) = S_l(xn) governing per-mode entanglement entropy, with the integral identity alpha_s = (1/(2pi)) * integral f(x)*x dx. If f(x) could be expressed in closed form, this would convert the Lambda prediction from numerical to analytic.

This experiment systematically fits f(x) to 12 candidate analytic forms and analyzes its structure. Result: no simple closed-form function matches f(x). The best fit is a stretched exponential with exponent c = 0.63, but it misses the integral by 5%. However, two important structural insights emerge:

1. f(0) diverges logarithmically: f(0, n) = (1/(2*pi)) * ln(n) + const, connecting the s-wave entropy to the logarithmic correction delta.
2. The tail is slower than exponential: d(ln f)/dx decreases with x, ruling out simple exponential decay. The best tail description is a stretched exponential with c ~ 0.3-0.6.

The Identity to Prove

alpha_s = (1/(2pi)) * integral_0^inf f(x) * x dx = 1/(24sqrt(pi))

Equivalently: integral_0^inf f(x) * x dx = sqrt(pi)/12 = 0.14770

Part 1: Dense f(x) Computation

Computed f(x) at n=20 and n=30 with l_max = 10*n:

x	f(x), n=20	f(x), n=30	Converged?
0.0	0.586	0.651	NO (grows as ln n)
0.5	0.132	0.133	~1%
1.0	0.0559	0.0559	YES (0.02%)
2.0	0.0140	0.0139	~1%
3.0	0.0047	0.0047	~1%
5.0	0.00095	0.00093	~2%

The spectral integral at n=15 converges to within 0.3% of the target at x_max=15:

x_max	integral f*x dx	Error from sqrt(pi)/12
5	0.1340	-9.3%
10	0.1447	-2.1%
15	0.1472	-0.3%
20	0.1482	+0.4%

The integral converges slowly because f(x) has a long tail extending to x ~ 20-30.

Part 2: Logarithmic Divergence of f(0)

Key finding: f(0, n) = S_0(n) grows logarithmically with n.

n	f(0, n)	(1/(2*pi))*ln(n) + 0.104
8	0.4387	0.4372
12	0.5038	0.5014
16	0.5502	0.5474
20	0.5863	0.5834
25	0.6223	0.6184
30	0.6515	0.6477

The coefficient B = 0.1611 matches 1/(2*pi) = 0.1592 to 1.2%.

Physical interpretation: The s-wave (l=0) mode is a 1D radial chain with no centrifugal barrier. Its entanglement entropy grows logarithmically, S_0(n) ~ (c_eff/3)ln(n), where c_eff ~ 3/(2pi) ~ 0.477. This logarithmic growth is the origin of the log correction delta in the total entropy — the same physics that determines the trace anomaly coefficient.

The scaling function f(x) is well-defined for x > 0 (convergence < 2%) but diverges at x = 0. This means f(x) cannot be any normalizable function — it must have a logarithmic singularity at the origin.

Part 3: Tail Behavior

The tail of f(x) was analyzed for x > 1, 2, and 3:

Tail form	R² (x > 1)	R² (x > 2)	R² (x > 3)
exp(-b*x)	0.664	0.829	0.921
exp(-b*x^2)	0.254	0.518	0.732
x^(-p)	0.586	0.967	0.995
exp(-b*sqrt(x))	0.960	0.971	0.985
exp(-b*x^c)	0.999	0.998	0.998

The stretched exponential wins everywhere. The best-fit exponent c decreases from ~0.6 at x > 1 to ~0.3 at x > 3, indicating the tail becomes increasingly power-law-like at large x.

The log-derivative d(ln f)/dx is diagnostic:

x	d(ln f)/dx	Implication
0.5	-1.95	steep decay
1.0	-1.58
2.0	-1.23
3.0	-0.98
5.0	-0.67
7.0	-0.50	slow decay

A pure exponential would give constant d(ln f)/dx. The monotonic decrease rules out simple exponential decay and points to a function that transitions from exponential-like at small x to power-law-like at large x.

Part 4: Moment Analysis of x*f(x)

The x-weighted distribution g(x) = x*f(x)/integral determines alpha_s:

Moment	Value	Gaussian	Exponential
Mean	1.88	—	—
Std dev	1.78	—	—
Skewness	1.85	0	2
Excess kurtosis	3.69	0	6

The skewness (1.85) is close to but less than the exponential value (2), and the excess kurtosis (3.69) is between Gaussian (0) and exponential (6). This is consistent with a distribution that is “softer” than exponential but strongly right-skewed.

Part 5: Fitting 12 Candidate Functions

Rank	Function	R²	Integral error	Parameters
1	Stretched exp	0.99987	-5.0%	a=0.654, b=2.47, c=0.626
2	Double exp	0.99889	-24.6%	a1=0.32, b1=12.6, a2=0.32, b2=1.71
3	Rational exp	0.99776	+9.4%	a=0.62, b=0.55, c=5.55
4	Exp + Gaussian	0.99287	-28.2%	—
5	exp(-b*sqrt(x))	0.98938	+67.2%	—
6	Exponential	0.97112	-52.0%	—
7	Heat kernel	0.96148	-58.4%	—
8	Power law	0.95658	+1183%	—
9	erfc	0.93828	-63.3%	—
10	sech^2	0.90887	-59.8%	—
11	Gaussian	0.88811	-69.6%	—

The stretched exponential f(x) = 0.654 * exp(-2.47 * x^0.626) dominates with R² = 0.99987.

The stretching exponent c = 0.626

Nearby simple fractions:

• 5/8 = 0.625 (diff = 0.001, closest)
• 2/3 = 0.667 (diff = 0.041)
• 3/5 = 0.600 (diff = 0.026)

The exponent is tantalizingly close to 5/8, but this could be coincidental. The convergence study shows c decreasing with n (0.74 at n=8 → 0.63 at n=30), so the n→∞ value might be different.

Why no candidate captures the integral

The stretched exp integral is (a/c) * Gamma(2/c) / b^(2/c). At c=0.626, this gives 0.140, which is 5% below the target 0.148. The mismatch comes from two sources:

1. The logarithmic divergence at x=0 (not captured by any smooth function)
2. The very extended tail x > 10 (not well-sampled in the fitting data)

Part 6: Convergence of Fit Parameters with n

n	c (stretched exp)	R²
8	0.744	0.99988
12	0.705	0.99990
16	0.679	0.99991
20	0.659	0.99990
25	0.640	0.99989
30	0.626	0.99987

The exponent c is still decreasing at n=30. Extrapolating (roughly): c(n→∞) ~ 0.55-0.60, possibly approaching a value like 1/2 or 3/5. This remains uncertain without larger-n data.

What This Means

Negative result: no closed form found

None of the 12 candidate functions simultaneously:

1. Fits f(x) with R² > 0.999
2. Gives integral = sqrt(pi)/12 within 1%

The analytic form of f(x) is not a simple special function. It likely involves:

• A logarithmic singularity at x=0 (from the s-wave)
• A stretched-exponential body (from the centrifugal barrier)
• A slow power-law tail (from ultraviolet modes)

Positive insights

1. f(0) ~ (1/(2*pi))*ln(n): The s-wave entropy connects to the log correction. The coefficient 1/(2*pi) is exact to 1.2%.

2. The numerical integral confirms alpha_s = 1/(24*sqrt(pi)) to 0.3% at x_max=15, consistent with V2.184’s 0.011% determination. The spectral integral route works but converges more slowly than the second-difference method.

3. The stretched exponential exponent c ~ 0.63 is a new structural characterization. This tells us f(x) decays as exp(-const * x^{0.63}), which is intermediate between Gaussian (c=2) and power law (c→0).

4. The tail extends to x ~ 20-30: 90% of the area law comes from x < 6.3 (V2.230), but the remaining 10% comes from a very extended tail up to x ~ 30.

Path forward

The analytic form of f(x) may require:

1. Exact diagonalization of the modular Hamiltonian in the angular momentum basis
2. Continuum limit of the radial Hamiltonian (Euler-Maclaurin corrections)
3. Connection to heat kernel on the hemisphere, including all subleading terms
4. Separate treatment of the x=0 logarithmic singularity (1D CFT contribution) and the x>0 body (area-law contribution)

The fact that f(x) is NOT a simple function suggests that alpha_s = 1/(24*sqrt(pi)) may arise from a non-trivial integral identity rather than from f(x) having a recognizable form.

Limitations

1. Maximum lattice size n=30 with x_max=10 in the main fits. Larger n and x_max would improve the tail characterization.
2. The stretching exponent c is still evolving at n=30; the n→∞ value is uncertain.
3. Only 12 candidate functions were tested. More exotic forms (Meijer G-functions, Fox H-functions) were not attempted.
4. The logarithmic divergence at x=0 means f(x) is not a conventional function — it’s a distribution that requires regularization at the origin.