# Argument selection for T647

The only free parameter is the integer total number of Bessel factors n >= 3.
Its category is `discrete_index`. For m=floor((n-1)/2), the determinant uses
all rows a=1,...,m and all columns r=2b-1 with b=1,...,m. These are defining
matrix indices, not independently chosen arguments. Every Bessel function
has order zero and argument t with common unit scale. There is no external
mass, continuous coupling, variable Bessel order, or additional quantity key.

Zhou, arXiv:1711.01829v2, equations (1.3)-(1.6) and Section 4, defines this
arbitrary-size family and proves the two parity product formulas. Broadhurst,
arXiv:1604.03057, Section 7.5, equations (147)-(149), and Section 7.9,
equations (179)-(180), gives the same matrices and normalization. The
moments agree with T521's IKM(a,b;k) by b=n-a and k=r. The notation D_n in
this table indexes the total factor count; it is not Broadhurst's D_m in
his equation (180), which describes only the even case with m=n/2.

The selected finite prefix is every n from 3 through 22, in increasing order.
It contains matrix orders 1 through 10 in both parity classes. Reviewing
breadth first retains both parity classes and the scalar controls n=3,4,
the familiar two-by-two matrices n=5,6, and higher orders without a
famous-points-only restriction. The order-ten cutoff is a practical initial
reference choice, not a canonical mathematical boundary or a storage target.
It includes larger determinants while keeping the matrices small enough for
independent numerical determinant checks. Formula evaluation is cheap; that
alone is not a reason to extend to hundreds of increasingly large matrices.
The family itself comes from nonlinear relations among equal-mass Bessel
periods, not simply from an unevaluated function in a library.

No exploratory continuous sample is used, and there is no hidden Cartesian
product. The vacuum-containing determinants and distinguished minors have
different defining matrices; the family map assigns those their own scope.
They are not included here. Default enumeration, the parameter constraint,
the program snippet, and the completeness note state this same choice.

The stored values are signed determinants, not their magnitudes or logarithms.
The product formulas make them positive in this row and column order. Values
for n=3 and n=4 agree with scalar moments through the one-by-one determinant
definition; they are retained as the base members of the indexed family.

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