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# Argument selection for Chowla–Selberg period constants (T669)
The only free parameter is the negative fundamental field discriminant D.
Its role is `arithmetic_structure`: each D identifies an imaginary quadratic
field, its primitive quadratic character and its maximal ideal class group.
The character-weighted Gamma product is exactly b_K of Li, Long and Tu,
SIGMA 14 (2018), 090, §2.1, equation (2.1), DOI 10.3842/SIGMA.2018.090.
The character sum does not allow independent choices of Gamma arguments:
a/|D| and every exponent are forced by D. Class number h_D and unit count
w_D are derived data, not additional parameters. There are no freely sampled
continuous inputs and no reference grids.
The default enumeration includes every such field with |D| <= 1000, ordered
by increasing |D| from -3 and -4. The cutoff is a practical, noncanonical
choice. This modest initial discriminant interval allows complete arithmetic
coverage across odd and even fundamental discriminants, class number one
and higher class numbers, prime and composite discriminants, and real and
nonrectangular reduced representatives. Restricting to the nine class-number-one
fields would omit the class-product structure which distinguishes this table
from individual elliptic values. Extending only those nine cases by varying
orders would repeat the same field constant. The chosen breadth preserves
one named quantity and one field index.
Products have fewer than 1000 possible Gamma factors and entries remain
100 significant digits. The bound permits full independent eta-class-product
checks at higher precision, rather than spending the initial budget on a
deeper partial list. The 305-member count is a consequence, not a size target.
checks.py compares this set against an independently constructed squarefree
radicand enumeration and the negative-discriminant entries of T230. It also
measures the written lengths and block size. Enumeration, completeness metadata
and attached sources must be reviewed together by the independent critic.
The normalization is a specific positive representative of the CM period
class. It does not identify the actual period of an unspecified curve or
differential. Different orders in the same field do not define extra rows.
The separate proposed order-indexed stable-height family is not absorbed here.
Barquero-Sanchez et al., Research in Number Theory 3 (2017), 13, Theorem 5.1,
DOI 10.1007/s40993-017-0077-7, supplies the independent full ideal-class eta
product, specialized to conductor one. These sources support the function,
its conventions and its arithmetic organization; no claim is made that each
field in the chosen finite interval has a separate named application.
The governing archived skill manifest is
`agents/runs/20261102T045202Z-build-skills/manifest.json`.
The local and live copies were both read and independently hashed; both SHA-256
values are `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
This justification is a build artifact, not independent critique acceptance.
The selection is controlled by generate.py; checks.py and this note explain
and validate it and should also be bound in the later critique fingerprint.