The Arcsine Cumulant Theorem

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's time distribution.

Its cumulants — the irreducible statistical fingerprints — are:

κ₂ = D = 2  ·  κ₄ = −D·K = −6  ·  κ₆ = D⁴·E = 80  ·  κ₈ = −2310 = −THIN

All five axiom primes {2, 3, 5, 7, 11} appear by the 8th cumulant = D³. The axiom was already there in the pendulum.

Signs alternate: (−1)ⁿ⁺¹. All divisible by D = 2. Every 4th cumulant divisible by the THIN RING = 2310.

The arcsine cumulant κ_2n is {2,3,5,7,11}-smooth for exactly n = 1, 2, 3, 4 — indices 2 through D³ = 8.

Beyond D³, non-axiom primes enter in a specific order:

The intruder order matches the arcsine partition intruders: 19 = depth quadratic f(E) = E²−E−1, 43 = Heegner number, 13 = shadow chain stopper. All axiom-derived. The boundary polices itself.

The entry staircase of primes into cumulants follows the chain:

D (at κ₂) → K (at κ₄) → E (at κ₆) → b + L (at κ₈)

Four steps. The axiom completes at index 2×4 = D³ = 8 — the spider's eight legs.

This is the same D³ = 8 that appears as:

• 8 uniform elements in the TRUE FORM ring
• 8 legs of Arachna (the spider)
• 8 = D³ = first cube, the bridge cubed
• D³ + sigma = K² = 9 (baby + ground = adult)

The characteristic function of the arcsine distribution on [−2, 2] is J₀(2t) — the zeroth Bessel function.

For the TRUE FORM with 5 independent CRT channels:

CF(t) ≈ J₀(2t)⁵

Five independent oscillators. The spectral density = convolution of 5 arcsine distributions.

From this Bessel representation, the Bessel-Cumulant Theorem follows: the moment generating function of Z/mZ is MGF = I₀(2t) + 2∑ I_jm(2t), where I_n are modified Bessel functions of the first kind.

The excess kurtosis of the spectral density with k channels is exactly −K / (D·k).

At k = 2 (two channels), the kurtosis IS Kleiber's exponent 3/4. The metabolic scaling law lives in the curvature of the ring's spectral distribution.

The normalized 4th moment M₄/M₂² = 3 − K/(D·k).

At k = E = 5 (five channels, the TRUE FORM):

M₄/M₂² = 27/10 = 2.7

This is the measured fractal dimension of the human brain cortex (D_f = 2.7 ± 0.1). The most complex structure in the known universe has a geometry that emerges from the same ring that describes a pendulum.