# Argument selection for T530

The only free parameter is the positive integer moment order n. Its category
is `discrete_index`: each row is the raw moment of one fixed random variable,
not a real-valued input sampled on a numerical grid. The first four moments
specify the mean and the raw ingredients of variance, skewness and kurtosis.
Higher integer moments are coefficients in the Laplace-transform expansion;
Perman and Wellner, *On the Distribution of Brownian Areas* (1996), section 6,
uses moments in Laguerre expansions of these distributions. Janson (2007),
section 25, equations (262), (265)–(269), defines this area and its moments.

The model is standard Brownian motion B on [0,1], started at zero, with zero
mean, covariance min(s,t), and generator (1/2)d²/dx². The positive part is
taken before integration: A = integral_0^1 max(B(t),0) dt. Duration one and
variance rate one are the standard scaling normalization in those sources.
There is no conditioning on the endpoint, drift, barrier, or hidden coupling.
For duration T, Brownian scaling gives A_T equal in law to T^(3/2) A;
therefore varying duration would only duplicate a specified scaling.

The default enumeration is n = 1,2,...,12 in ascending order. Twelve is an
editorial truncation of an infinite discrete sequence, not a distinguished
mathematical boundary. This supplies a short reference of expansion
coefficients including the low moments; no target entry count or decimal grid
selects it. The zeroth moment is identically one and is a normalization
control rather than a stored row. The parameter domain remains all n >= 1;
`complete: no` and `complete-note` give the finite stored range and its reason.
Even moments are exact rationals. Odd moments are rational multiples of
sqrt(2/pi), stored with certified decimal enclosures of 100 significant digits.
No second quantity, recurrence coefficient, central moment, or cumulant is
included as a row.

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