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# Argument selection for T643: Zagier traces of singular moduli
This is the build justification, not an independent critique or acceptance.
The governing archive is `agents/runs/20261101T063157Z-build-skills/manifest.json`.
Both `skill-local.md` and `skill-live.md` were read and have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
## Quantity and conventions
The single quantity is Zagier's weighted trace t_m(d) of J_m at all positive
definite integral binary quadratic forms of discriminant -d, modulo PSL_2(Z).
The sum includes imprimitive forms. The stabilizer orders are 3 for multiples
of [1,1,1], 2 for multiples of [1,0,1], and 1 otherwise. The CM root is
(-b+i sqrt(d))/(2a), with positive sqrt(d). J_m has principal part q^(-m)
and zero constant term, with q=exp(2 pi i tau); J_1=j-744. There is no division
by m. The m=0 class-number quantity is not included.
These conventions were checked in Don Zagier, *Traces of Singular Moduli*,
sections 1, 2 and 6, especially equation (16), equations (17)-(18), and Theorem 5:
https://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext.pdf .
The definition of the ordinary trace and its integrality are independently
stated by Bruinier and Funke, *Traces of CM values of modular functions*,
Introduction, equations (1.1)-(1.2):
https://www.maths.dur.ac.uk/users/jens.funke/cmtraces.pdf .
## m: discrete_index
The domain is the positive integers. It indexes the canonical sequence J_m,
equivalently the identity-class Hecke-type Faber polynomial P_(m,1A) evaluated
at J. It does not select unrelated functions. The stored extent is 1 <= m <= 4.
This includes the ordinary trace, the first two prime orders, and the first
composite order, where a nontrivial Hecke composition already occurs. The same
normalization is used by T635. The cutoff is a practical, noncanonical reference
choice. Larger orders grow exponentially in their integer lengths. Extending
the first order far deeper would give less breadth of trace identities than
covering these initial four orders together.
## d: arithmetic_structure
The domain is d>0, d congruent to 0 or 3 modulo 4: precisely the positive
magnitudes of imaginary quadratic order discriminants. It is not a mesh of
continuous CM arguments. The stored extent includes every such d <= 200,
including every nonfundamental discriminant in that interval. No class-number,
sign, numerical-size, squarefree or fame filter is applied. All thirteen
rational CM cases lie in this extent, along with larger CM orbits and repeated
imprimitive contributions (already present at d=12 and d=16).
These are the input discriminants of the trace formula and the indices of the
weight-3/2 Fourier coefficients, so the arithmetic family has a specific reason
independent of notation. The bound 200 is noncanonical: it gives a compact
initial reference spanning different class numbers and conductors, within the
already checked class-polynomial companion's range. Coverage is not restricted
to that companion's class-number-one special cases. Neither this bound nor the
order cutoff is claimed to be a mathematical endpoint.
## Joint coverage, ordering and measured cost
For every m=1,2,3,4, include all 100 admissible discriminants from 3 through 200,
400 entries total. Enumerate by increasing m, then increasing d. This rectangle
is an initial segment of two actual arithmetic/discrete indices of one named
trace construction; it is not a product of arbitrary simple continuous inputs.
Every pair defines a full weighted orbit trace. No additional reference-sample
or continuous argument group is hidden in either parameter.
The dry run found a maximum length of 78 characters at (m,d)=(4,200), with a
29.2 KB measured entries block. Thus the chosen extent remains readable, leaves
room for extension, and covers all proposed structural groups without expanding
only one direction. No count target was used to choose the cutoffs. The table's
completeness note and generator enumeration state exactly this coverage.
## Family compatibility and exclusions
T79 supplies the j expansion; T83 gives rational singular moduli; T129 contains
primitive-order Hilbert class polynomials; T635 contains the identity-class
Faber polynomials. Their live documents and actual slugs were read. A scalar
weighted sum is a different quantity from any of those inputs. T634's principal
rectangular samples for nonidentity classes are not full orbit sums, and T636's
polynomial roots are another operation. Neither is averaged or merged here.
There is no separate block of Hurwitz class numbers, no higher-level or
genus-character-twisted traces, no extra table of generating coefficients,
and no quantile or statistical critical-value construction.
The separate critic must independently assess these arguments and the numerical
evidence, then bind its assessment to the live selection and attachments. This
build writes no independent argument-review sidecar and performs no offer,
review or publication action.