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# Argument selection for Values of the Hardy Z-function, T586
The sole free parameter is the real height t. Its mathematical domain is R;
the three selected groups consist of exactly defined intrinsic landmarks, not
rounded numerical inputs. Every row stores the same signed quantity Z(t).
No coupling, normalization selector, derivative order, or second quantity is
hidden in the parameter.
## Groups and finite extent
1. **Origin:** the single identity `0`, displayed as 0. The continuous
log-gamma phase has theta(0)=0, and Z(0)=zeta(1/2) fixes its sign
normalization. Hall, equation (1) and the discussion immediately following,
establishes this real, even convention. The corpus stores zeta(1/2) in
T14; the origin row links that exact entry.
2. **Phase crossings:** identities `gram:-1` through `gram:200`, displayed as
g_n, in increasing n. For every integer n >= -1, g_n is the unique solution
of theta(g_n)=n*pi in [7,infinity). FLINT's documentation, “Gram points and
counting zeros,” gives this branch restriction and role in zero counting.
These 202 consecutive landmarks support comparison of the signs and sizes
encountered in zeta zero searches. The prefix reaches the early failures
of Gram's sign rule at indices 126, 134 and 195; those signs are checked,
not assumed. The cutoff 200 is a practical, noncanonical reference bound,
consistent with the companion T583, rather than a real-height mesh.
3. **Stationary arguments:** identities `stationary:1` through
`stationary:100`, displayed as c_j, in increasing j. The positive distinct
roots of Z'(t) are intrinsically selected by the function. Their signed
values describe its local extrema; the two roots preceding the first zeta
zero are retained. Hall, page 125, distinguishes those exceptional extrema
from the later oscillations. A prefix of 100 covers these and a substantial
consecutive stretch of oscillations while bounding the cost of proving
that none were missed. The cutoff is computational and noncanonical,
consistent with T585; it does not attach special mathematical status to
the hundredth root. The generator covers [0,233] with certified intervals,
independently of a global interlacing theorem or the Riemann hypothesis.
These groups exhaust all 303 selected arguments and all have the category
`intrinsic`. A mixed classification is unnecessary because their roles have
the same category, but each group has been assessed separately. Their bounds
are independent of numeral notation. No reference_sample group, probability,
quantile, decimal mesh, or Cartesian product is used.
## Breadth and omissions
The breadth is initial normalization, phase-crossing values, and critical
values. It is not confined to the most famous single point of either sequence.
Negative heights are omitted because evenness gives the identical values;
zero heights of Z itself would give only the exact number zero. No arbitrary
integer or rational height grid has been added simply because the function
can be evaluated. Those choices do not rule out a later, separately justified
application or exploratory reference slice.
The enumeration is origin first, then Gram indices -1..200, then stationary
indices 1..100. The completeness note states these same sets and their
purposes. Parameters retain type R and the full real domain. Symbolic
identities denote exact roots, with labels showing the argument rather than
Z of that argument. `gram.py` and `stationary.py` recompute root enclosures
at greater precision than the output, and `generate.py` propagates their
widths into the signed values. No saved decimal root is a numerical input.
## Sources checked and policy
- R. R. Hall, *On the stationary points of Hardy's function Z(t)*,
Acta Arithmetica 111 (2004), 125–140, equation (1), page 125 and the
first page's discussion of the two low stationary points.
https://www.impan.pl/shop/en/publication/transaction/download/product/82433
- FLINT, acb_dirichlet documentation, “Hardy Z-functions” and
“Gram points and counting zeros.” The documented null-character convention
is the Riemann zeta case, and Hardy jets return Taylor coefficients.
https://flintlib.org/doc/acb_dirichlet/index.html
- mpmath documentation, `siegeltheta`, `siegelz`, and `grampoint`, provides
separate numerical implementations of the same phase and functions.
https://mpmath.org/doc/current/functions/zeta.html
- Live documents and source attachments of T583 and T585 were read from the
website, not inferred from local directories. Both use the same phase and
indexing conventions.
The archived policy manifest is
`agents/runs/20261028T165752Z-build-skills/manifest.json`, copied into the
attachments as `skill-manifest.json`. Both archived local and live skills
have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This document is the builder's justification, not an independent acceptance.
The separate critique must review all selection-controlling dependencies:
`generate.py`, `gram.py`, `stationary.py`, and `hardy_arb.pyx`; it should also
bind this justification. The builder does not write the critique sidecar.