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# Argument selection for T645: Values of the Riesz function
This is the build author's justification, not the independent critique or its
acceptance sidecar. The critique must evaluate it independently.
## Scope and conventions
The sole free parameter is the continuous real argument x>=0. The value is
R(x)=x sum_{j>=0}(-x)^j/(j! zeta(2j+2)), with the ordinary Riemann zeta function.
All current arguments have exact integer identities; their type remains R.
Values retain signs. The header is R(x). There is no quantity selector.
Cisło–Wolf, arXiv:0807.2971, equation (2), defines this normalization. Equation
(10) relates it to the Báez-Duarte coefficients in T644. Equation (13) gives the
Möbius representation, with mu(1)=1. Paris, arXiv:2107.02800, §1, writes the
same function as -x S_{2,2}(x). These sources were read, not inferred from a
software function's name. T644 was read in full through the live API; its
coefficient signs, zeta convention and exact integer identities agree.
## Mixed selection, assessed group by group
1. Intrinsic group: ["0"]. The leading factor x in the entire defining series
proves R(0)=0 exactly. This is the initial value, not a rounded near-zero.
Source: Cisło–Wolf, equation (2).
2. Documented-application group: every string identity "1" through "100",
listed explicitly in the membership record. Cisło–Wolf, §2, Lemma 3 and
Theorem 1 compare R(k)/k and c_k at positive integers, quantify their
difference, and connect continuous and sequential RH criteria. This
selects an integer-argument family; it does not single out every integer
or prescribe our upper cutoff. The consecutive range preserves all early
comparisons, including the sign change between x=1 and x=2. The cutoff
100 is a practical, noncanonical choice balancing certified Taylor-series
cancellation and tail cost with an initial comparison reference. It is
independent of how these integers are written and of any numeral base.
Membership record (disjoint and exhaustive):
```json
{
"x": {
"category": "mixed",
"groups": [
{
"category": "intrinsic",
"values": [
"0"
]
},
{
"category": "documented_application",
"values": [
"1",
"2",
"3",
"4",
"5",
"6",
"7",
"8",
"9",
"10",
"11",
"12",
"13",
"14",
"15",
"16",
"17",
"18",
"19",
"20",
"21",
"22",
"23",
"24",
"25",
"26",
"27",
"28",
"29",
"30",
"31",
"32",
"33",
"34",
"35",
"36",
"37",
"38",
"39",
"40",
"41",
"42",
"43",
"44",
"45",
"46",
"47",
"48",
"49",
"50",
"51",
"52",
"53",
"54",
"55",
"56",
"57",
"58",
"59",
"60",
"61",
"62",
"63",
"64",
"65",
"66",
"67",
"68",
"69",
"70",
"71",
"72",
"73",
"74",
"75",
"76",
"77",
"78",
"79",
"80",
"81",
"82",
"83",
"84",
"85",
"86",
"87",
"88",
"89",
"90",
"91",
"92",
"93",
"94",
"95",
"96",
"97",
"98",
"99",
"100"
]
}
]
}
}
```
## Breadth and finite extent
The origin and the published integer-argument comparison serve different
purposes and are both retained. All intervening positive integers are included;
the selection is not pruned to recognizable isolated values. There is no
exploratory reference sample in this build, no borrowed rational mesh and no
Cartesian product. Additional rational inputs would require their own reference
purpose and bounds. Zeros and stationary arguments require independent isolation
and counting work; those are deferred in family #283 rather than approximated
by hand-picked near-roots. Negative and complex arguments lie outside the chosen
nonnegative-real table domain. The late oscillatory regime is not represented
by this finite initial selection, and these values do not test an asymptotic RH
claim. Paris §3 was read to distinguish those regimes.
At 100 significant digits, this selection has 101 entries, a longest written
entry of 103 characters and an approximately 13.7 KiB flat entries block. The
worst direct-series argument needs 505 terms. The independent Möbius method uses
16 direct terms and 100 Taylor terms with a proved remainder. These measurements
support the practical cutoff; the table was not expanded to reach a size target.
The default enumeration is exactly x=0,...,100 increasingly. The completeness
note describes the same two groups and finite bound. The generator's ability
to evaluate another nonnegative rational does not add it to this selection.
Every selection-controlling source is in generate.py; the other attachments
provide justification, checks and evidence rather than imported selection data.
## Sources and archived instructions
- https://arxiv.org/html/0807.2971 : equations (2), (10), (13), (33), §2
Lemma 3 and Theorem 1.
- https://arxiv.org/html/2107.02800 : §§1 and 3.
- https://dlmf.nist.gov/25.6 : equations 25.6.1–2 for the coefficient check.
- Live companion https://numberdb.org/T644 and family issue #283.
- Coordinated skill manifest:
agents/runs/20261101T075931Z-build-skills/manifest.json, attached as
skill-manifest.json. Both archived versions have SHA-256
a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.