# Argument selection for T702

Policy read: `agents/runs/20261110T013256Z-build-skills/manifest.json`.
Both archived versions, `skill-local.md` and `skill-live.md`, have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
This is the builder's justification, not an independent critique acceptance.

The named quantity is Samart's elliptic trilogarithm L^E_{3,1}, the q-average
of the single-valued real trilogarithm. L^E_{3,2} is a different quantity and
is not included. This construction occurs in weight-three elliptic regulator
and K3 Mahler-measure relations (Samart, arXiv:1309.7730v2, sections 1–2,
Theorem 1.2 and Corollary 2.8). The classical polylogarithm and Epstein-zeta
tables do not store this family. T700 and T701 supply the weight-two
companions and fix the same marking and ordered parameter keys.

* Delta (`arithmetic_structure`): exactly the thirteen imaginary quadratic
  order discriminants of class number one: -3,-4,-7,-8,-11,-12,-16,-19,-27,
  -28,-43,-67,-163. The classification is independently listed in Sage's
  cm_orders(1) documentation and T83. It is the complete rational-singular-
  modulus slice, including nonmaximal orders, not a modulus mesh. This
  breadth covers both rectangular and rhombic lattices before extending
  torsion depth on a single curve.
* N (`discrete_index`): exact torsion order, with the origin at N=1 and
  1<=N<=6. Orders 2,3,4 enter Samart's torsion and Mahler relations. Order 5
  adds the next prime torsion group and 6 the first mixed-prime group. These
  are bounded arithmetic reference cases, not a claim of a published
  application at every individual point. Six is a noncanonical practical
  cutoff independent of numeral notation. Larger orders increase the
  number of directions quadratically and are deferred.
* r and s (`arithmetic_structure`, jointly): coordinates of every point of
  exact order N in E[N]=(Z/NZ)^2. Require 0<=r,s<N and gcd(N,r,s)=1; keep the
  lexicographically smaller of (r,s) and (-r,-s) modulo N. They are constrained
  coordinates of one finite group, not two free continuous sampled inputs.
  Inversion is an actual even symmetry of L31. Each order includes all its
  directions, not just real-axis points. No symmetry-forced-zero exclusions
  from the odd weight-two companions are transferred to this even function.

The lattice marking is tau=(-b+i sqrt(abs(Delta)))/2, with b=0 for 4|Delta
and b=1 otherwise, the positive square root, and q=exp(2 pi i tau). It is
never replaced by a modularly equivalent tau, nor is negative q replaced by
its magnitude. The exact point is u=(r tau+s)/N. These conventions determine
all input arguments without rounded reference values.

There are 1,3,4,6,12,12 representatives at exact orders 1,2,3,4,5,6,
respectively, hence 38 per discriminant and 494 overall. Origin and 2-torsion
are important independent distribution controls. Signs and coincidences are
retained. All discriminants are ordered by increasing absolute value.
The family domain permits other negative quadratic-order discriminants and
larger exact orders; the finite bounds belong only to the completeness note.

Sources checked: Samart, arXiv:1309.7730v2, section 1 definition, Corollary
2.8 equation (2.9), Theorem 2.11; SageMath CM reference, cm_orders and
cm_j_invariants_and_orders; live T83, T700, T701, T145, T175, T176 and T402.
A separate critique must inspect the joint gcd and inversion conditions and
all four parameters, verify numbers independently, and bind its own sidecar
to generate.py, checks.py and this attachment. No review verdict is made here.