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# Argument selection for T584
The reviewed policy is the coordinated canonical skill archived at
`agents/runs/20261028T151417Z-build-skills/manifest.json`. Both local and live
copies have SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The manifest is also attached as `skill-manifest.json`. This is the builder's
justification, not an independent critique or acceptance sidecar.
## Quantity and conventions
One named quantity is D(n,k) = Z^(k)(gamma_n). Both parameters are positive
integers; k is an argument (derivative order), not a selector between named
invariants. Derivatives are with respect to real height t, retain their signs,
and are actual derivatives rather than Taylor coefficients. The analytic
log-gamma branch real at 1/4 gives theta(0)=0 and Z(0)=zeta(1/2).
The gamma_n are positive critical-line zero heights, counted with multiplicity
from one. The selected zeros are certified simple and use the same numbering
as the nontrivial zeta zeros in T3. Rounded values from T3 are never inputs. The checks isolate 6710 distinct
critical-line roots up to height 7006 and certify that the total zero count,
including multiplicity, is 6710. Together these facts prove there are no
missing, off-line or multiple zeros in this finite range.
## n: mixed selection, with exhaustive disjoint groups
1. **discrete_index:** every integer 1 through 100. This consecutive prefix
covers the early oscillations and their derivative behavior without
selecting only exceptional cases. It includes the first Lehmer pair,
indices 34 and 35. A prefix of 100 is a noncanonical computational/reference
cutoff, not an assertion that index 100 is special or that each zero has
its own application. It bounds root isolation and independent checks while
covering both ordinary and close early zeros. Membership is exactly
`str(n)` for `n in range(1,101)`, expanded in numerical-evidence.json.
2. **documented_application:** exactly `6709` and `6710`. Simonič's Table 1
identifies these as Lehmer's historical pair at heights about 7005.06 and
7005.10. The accompanying first three derivatives are inputs to the
close-zero condition of Theorem 1 and Corollary 1.
These groups are disjoint and exhaustive. No other real arguments or hidden
coupling parameters are present. The source's third pair, indices exceeding
one billion, is not included: its very high height creates a substantially
more expensive certification regime. Breadth is supplied by consecutive low
zeros and the historically significant higher pair before extending either
prefix depth or height. That is a cost/coverage choice, not famous-only pruning.
## k: discrete_index
Exactly orders 1, 2 and 3 at every selected zero. Simonič's Theorem 1 uses
-Z'''/Z' + (3/4)(Z''/Z')^2, so these orders supply the complete derivative data
for that criterion. Order zero would just duplicate zero. Higher orders are
legitimate members of the underlying family but are not needed for this
application and are outside this finite build. The product of these three
orders with the selected zero indices supplies related derivatives together;
it is not a product of arbitrary continuous meshes.
## Checked sources and scope
- Aleksander Simonič, *Lehmer pairs and derivatives of Hardy's Z-function*,
arXiv:1612.08627v2, §1 (phase, Theorem 1), §3 (zeta derivative identities),
§5 Table 1 (indices, signs and derivative values), Table 2 (close-pair
diagnostics): https://arxiv.org/html/1612.08627v2 .
- FLINT/Arb documentation, *Hardy Z-functions* and *Riemann zeta function
zeros*: https://flintlib.org/doc/acb_dirichlet/index.html . The website
documentation was inaccessible to the browser; the upstream documentation
was read at https://raw.githubusercontent.com/flintlib/arb/master/doc/source/acb_dirichlet.rst .
It specifies Taylor coefficients, the NULL-character Riemann case,
numbered zero isolation and zero counting with multiplicity. Installed
Sage FLINT declarations were also inspected.
- T3 was read from the live API. Its first 100 ordinates supply an additional
independent corpus check, not the generator's root inputs. T228 and T387
concern zeta derivatives at rational arguments and their zeros; neither
duplicates the signed height derivatives here. T583 contains Gram points,
another phase landmark rather than these zero derivatives.
No reference_sample or arbitrary decimal grid is used. The completeness note,
program and generator enumerate exactly 102 zero indices times three orders,
306 entries. The noncanonical finite bounds are independent of numeral base.
The independent critic should assess n as mixed with explicit identities
1 through 100 in the first group, 6709 and 6710 in the second, and k as
discrete_index with identities 1, 2, 3. It must compute its own fingerprint.
Existing tags L-function, number theory and special values are used. The new
Riemann zeta function tag remains a proposal: T3, T14, T228, T387, T583 and
T584 would all fit it. No published table or tag was edited.