Arcsine Cumulants

k2 = 2, k4 = -6, k6 = 80, k8 = -2310

A pendulum spends most of its time near the turning points, least at the center. This is the arcsine distribution -- the universal shape of any oscillatory system. Its cumulants (irreducible statistical fingerprints) factor into the primes {2, 3, 5, 7, 11}. All five primes appear by the 8th cumulant, which equals -2310 = the primorial of 11. Beyond order 8, the pattern breaks.

The First Eight Cumulants

The arcsine distribution on [-2, 2] is symmetric, so all odd cumulants are zero. The even cumulants:

n	k_{2n}	Factorization	11-smooth?
1	+2	2	Yes
2	-6	-2 * 3	Yes
3	+80	2^4 * 5	Yes
4	-2310	-2 * 3 * 5 * 7 * 11	Yes (= primorial(11))
5	...	contains 19	No
6	...	contains 43	No
7	...	contains 13	No

Signs alternate: (-1)^(n+1). All are divisible by 2. The 4th cumulant k_8 = -2310 = -primorial(11), the product of all five primes. At n = 5, the prime 19 enters and the smooth run ends permanently.

The primes enter one at a time: 2 at n=1, then 3 at n=2, then 5 at n=3, then 7 and 11 together at n=4. Four steps to complete the set. The smooth zone ends at order 2*4 = 8.

Bessel Connection

The characteristic function of the arcsine distribution on [-2, 2] is J_0(2t), the zeroth Bessel function. For a ring Z/N with k independent CRT channels, the spectral density is the convolution of k arcsine distributions:

Bessel-Cumulant Theorem (PROVED)

The characteristic function of the spectral density is J_0(2t)^k. For Z/12,612,600 with 6 channels: CF(t) = J_0(2t)^6. The moment-generating function for Z/mZ involves modified Bessel functions I_n(2t).

Excess Kurtosis: -3/(2k)

The excess kurtosis of the spectral density with k independent channels is exactly -3/(2k):

Ring	k (channels)	Excess kurtosis
Z/210	4	-3/8
Z/2,310	5	-3/10
Z/970,200	5	-3/10
Z/12,612,600	6	-1/4
(any ring, k=2)	2	-3/4

The formula -3/(2k) is purely a consequence of the arcsine distribution: kurtosis = k_4 / k_2^2 = -6/4 = -3/2 for one channel, which scales as -3/(2k) for k independent channels. At k = 2, this gives -3/4.

Contrast

Arcsine distributionDescribes pendulum oscillation time. Classical probabilityIts cumulants factor into {2,3,5,7,11} for the first four even orders. The 4th cumulant -2310 = primorial(11).Smooth run to order 8The first few cumulants happen to have small prime factorsExactly 4 smooth cumulants (orders 2,4,6,8). The smooth zone ends at order 8 = rank(E8).Kurtosis -3/(2k)A formula for convolutions of arcsine distributions= -3/(2k) for k CRT channels. At k = 2 gives -3/4. At k = 6 (Z/12,612,600) gives -1/4.

Explore: Cumulant Calculator

Enter n (1-4) to see the even cumulant k_{2n} of the arcsine distribution. Beyond n = 4 (order 8), non-smooth primes enter.

Cumulant index n:

Try: 1 (k_2 = 2), 2 (k_4 = -6), 3 (k_6 = 80), 4 (k_8 = -2310 = primorial(11)).

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