# Argument selection for T585

The sole free parameter is the integer j >= 1, the increasing index of distinct
positive real roots of Z'(t). Its role is discrete_index. There is no free
coupling, phase normalization, or derivative-order parameter hidden beside it.
The phase uses log-Gamma analytic in the right half-plane and real at 1/4;
the derivative is with respect to real t. Zero is excluded. Multiplicity does
not change the index; the finite certification proves each selected root simple.

We select every j from 1 through 100, in increasing order. This consecutive
prefix covers the two exceptional stationary points before the first zeta zero
and the subsequent oscillations without singling out famous values. Hall,
"On the stationary points of Hardy's function Z(t)", Acta Arithmetica 111
(2004), pp. 125–126, defines this stationary family and discusses the initial
two roots. Our j=1,2 correspond to his u_{-1},u_0; later j correspond to
u_{j-2} in the certified simple prefix. The cutoff 100 is a noncanonical,
notation-independent reference choice that bounds the cost of proving no
roots were missed. It is not an assertion that the hundredth point is special.

Breadth was assessed before depth: the initial two roots must be included,
including their different low-height behavior. Negative roots are fixed by
evenness and would duplicate the same family under sign; zero is a symmetry
root deliberately excluded by the shared positive-root convention. Gram points,
Z values, and derivatives at zeta zeros are other named quantities belonging
to the family's separate tables, not additional quantities in this table.
There are no exploratory real-input samples or Cartesian products here.

The full prefix must be certified by an interval cover from zero through the
last root. Opposite endpoint signs and a nonzero Z'' enclosure certify each
simple root; exclusion bounds certify the complementary intervals. Near zero,
Z''>0 on a short interval makes the analytic extension of Z'(t)/t positive,
since Z'(t)/t = integral_0^1 Z''(ut) du. Root-finding alone is insufficient.
The generator, its default enumeration and the completeness note must all use
j=1,...,100. Independent argument-selection acceptance belongs to the later
critique actor, not this build.

Sources checked: Hall, equation (1) and introduction, pp. 125–126,
https://www.impan.pl/shop/en/publication/transaction/download/product/82433 ;
FLINT, acb_dirichlet_hardy_z, Hardy Z-functions section,
https://flintlib.org/doc/acb_dirichlet/index.html .

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