# Argument selection for T667

This is the build's justification, not the independent critique's decision.
The governing instructions are archived in
`agents/runs/20261102T031339Z-build-skills/manifest.json`. Both archived local
and live skills have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The matching manifest and its files were read before building.

## Quantity and sole free parameter

The stored integer is c(D) in the full K3 elliptic genus
Z_K3(tau,z) = sum c(4n-ell^2) q^n y^ell, with q=exp(2*pi*i*tau),
y=exp(2*pi*i*z), and constant Laurent polynomial 2/y+20+2*y.
There is one free parameter, D, of type Z. Its infinite domain is D >= -1,
D congruent to 0 or 3 modulo 4. This is a genuine discrete Fourier index,
classified as `discrete_index`, not a real input encoded as an integer.
The elliptic transformation law makes c depend on the discriminant alone.
For index one its residue also determines ell modulo 2, so no independent
charge parameter is needed. The quotient removes repeated elliptic translates.
D is not restricted to fundamental discriminants or squarefree integers.

The convention is checked in Kawai, hep-th/9710016, equations (4.28)-(4.33).
Eguchi, Ooguri and Tachikawa, 1004.0956, equation (1.2), independently fixes
the full normalization. The generator uses Kawai's two theta components;
the checker expands EOT's bivariate quotient without importing the generator's
arithmetic. Kawai uses the opposite sign for the odd theta function in (4.13)-(4.14).
We explicitly use theta1=-i*sum (-1)^r q^((r+1/2)^2/2)y^(r+1/2),
which is compatible with EOT (1.4); this affects the written mu convention,
not the squared genus quotient or Kawai component formula. The standard theta nome is q^(1/2), as in the CM companions
T504, T505 and T507. It is not the Fourier variable q used here.

## Selection and extent

Enumerate every admissible D in -1 <= D <= 1000, in increasing order:
-1, 0, 3, 4, ..., 999, 1000. This gives 502 integers. Equivalently, include
c(4n-1) and c(4n) for every n=0,...,250. The initial polar and constant
coefficients are included; both positive-discriminant residue classes are
complete to the same Fourier grade. Signs and any structural zeros are
retained. No dimension coincidences are used to select entries.

The purpose is a coherent initial reference for the weak Jacobi form's Fourier
multiplicities. These same coefficients are the exponents in the product
formula in Kawai (5.13), which supplies context beyond evaluating a missing
function. It does not require an application citing each individual grade.
The cutoff 250 is a practical, noncanonical choice. It extends well beyond
the ten printed normalization controls while retaining compact exact integers
and allowing a complete independent two-variable expansion. The largest
written integer occupies 42 characters and the measured entries block is
about 31 KiB (the dry-run's synthetic keys make it 32 KiB). There is ample
room for extension; storage capacity was not used as a target.

Breadth was assessed before depth: both theta components and the polar term
are present, with no restriction to famous coefficients and no repeated
charge rectangle. All possible intrinsic index classes have been represented.
Twining classes, symmetric-product indices and mock-modular character
coefficients would change the named quantity. They are outside this table,
not missing free parameters. There is no continuous-input reference sample
and no mixed category requiring a reference_policy. The selected finite
initial segment is independent of decimal notation.

## Companion boundary and review

Family issue #295 was read first. The live family-coverage tables were read,
and text and sample-value searches found no existing K3 elliptic genus table.
T504, T505 and T507 are linked with their actual URLs as theta-constant
ingredients; T652 is linked as an analogous Monster series. The proposed
Mathieu/umbral companion is unbuilt, so no identifier or slug is invented.
The N=4 relation is stated in EOT's full normalization, and the checker tests
its normalized H series at two independent elliptic arguments through grade
250, including all nine positive coefficients printed in EOT (1.12).
That verification attachment does not create a second table.

A separate critique actor must assess D, the complete enumeration, this
justification and the current live attachments. Its fingerprint must bind
at least generate.py; include checks.py as well to bind the numerical claims.
The build does not write or accept an independent argument-review sidecar.