Classical and Free Cumulants of the ω-Gaussian Central Limit Law

We study the classical and free cumulants of the ω-Gaussian central limit laws arising from braidable quantum coin tosses. Taking the known central limit theorem and oriented-pairing moment formula as input, we carry out a cumulant-level analysis of the resulting laws. At the imaginary parameter, we show that the first obstruction to semicircularity occurs at the eighth free cumulant, thereby identifying precisely where the limiting law first departs from semicircular behavior. Near the Gaussian endpoint, we prove that all higher even classical cumulants of order at least six vanish to at least second order. We further compute the classical cumulants through even order 16 and find a stronger order-dependent vanishing pattern, indicating increasing cancellation at the Gaussian endpoint as the cumulant order grows. These results reveal higher-order effects of the underlying quantum braiding that are not apparent from the first few moments alone and lead to an all-order divisibility conjecture together with several related open problems for the oriented-pairing expansion.