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3049 bytes, as of the version from 2026-11-02 04:21 (current). Recorded here, not run.
# Argument selection for T668
This is the build's rationale, not the independent critique acceptance.
The controlling skill copies are recorded in
`agents/runs/20261102T034954Z-build-skills/manifest.json`.
Both read copies have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
Every argument is a discrete index of the same named coefficient C_X(r,s).
There is no continuous input, exploratory rational grid or quantity selector.
* **X (discrete_index).** A Niemeier root system identifies an untwined umbral
module. Initial coverage comprises A1^24, A2^12, A3^8, A4^6, A6^4 and A12^2.
These are exactly the six pure A cases with m-1 dividing 12 and a nonzero
extremal weight-zero Jacobi form (CDH 1204.2779, Theorem 2.2 and §2.5;
CDH 1307.5793, §4.3). This is a mathematically specified subfamily of the
23 nonempty Niemeier root systems, not a selection of group names chosen
by convenient coefficient coincidences. Mathieu is included once as A1^24.
The other 17 cases need additional constructions; they are deferred.
* **r (discrete_index).** Every independent component 1<=r<m_X is included
for each selected X. The restriction follows DGO 1503.01472, equations
(1.3)–(1.4) and the definition of I^X. The remaining residues modulo 2m
are zero or determined by antisymmetry. No component is discarded because
its coefficients are small or zero. The six systems have 28 components.
* **s (discrete_index).** Positive integer numerator of the actual Fourier
exponent, with s=-r^2 mod 4m_X, in the full module normalization. Include
every admissible 0<s<=120*m_X, equivalently every positive exponent <=30.
This is 30 consecutive positive grades in each component, not the same
integer n=1,...,30 when r^2/(4m)>1. The practical cutoff is noncanonical
and independent of decimal spelling. It provides a common initial segment
across all components, extends beyond familiar low Mathieu dimensions,
stays within the published multi-component coefficient controls, and keeps
exact values short. Breadth across the six cases is preferred to a much
deeper Mathieu-only list. The polar coefficient is defined in prose and
excluded from this positive-grade quantity. Zero coefficients are retained.
Coverage is 840 entries. The default enumeration, document and completeness
note must agree with this set; the dry run measures the actual written length.
The acting conjugacy class is fixed to 1A, not a free parameter. The Coxeter
number is derived from X. All six groups and their quotient interpretation
are metadata, not extra values. Twined traces and K3 elliptic-genus Fourier
coefficients are separate quantities and are not added to this table.
Checked supporting sources: CDH 1204.2779v3, §2.3, Theorem 2.2, equations
(2.49)–(2.68), Appendix C; CDH 1307.5793v2, Table 3, §4.3 and Appendix A.4;
DGO 1503.01472v2, equation (1.4) and Theorem 1.1; EOT 1004.0956v2,
equations (1.7)–(1.12). The initial cutoff is our reference choice, not a
claim made by those papers.