# Argument selection for T703

The named quantity is Samart's real elliptic trilogarithm L32, with the logarithmic correction in section 1 and equation (2.10) of arXiv:1309.7730v2. The table follows the marking and parameter order (Delta,N,r,s) in family issue 311 and the live companion tables T700, T701 and T702. It does not include L31, the weight-two regulator components, or a Mahler measure as extra quantities.

Delta is an arithmetic_structure parameter: the initial selection is precisely the thirteen negative quadratic-order discriminants with class number one, hence rational singular modulus. They are -3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163. The list includes nonmaximal orders and is supported by Sage's cm_orders documentation and NumberDB T83. We use the principal marking tau=(-b+i sqrt(|Delta|))/2, with b=0 for Delta divisible by four and b=1 otherwise. We do not replace the negative q for odd Delta by its absolute value.

N is a discrete_index, the exact order in the elliptic curve's finite torsion group. The practical initial bound is 1<=N<=6. The origin and 2-,3-,4-torsion feature in the source's trilogarithm identities and K3 Mahler-measure relations. Order five adds the next prime torsion group, and six is the first mixed-prime order. This bounded arithmetic reference supports recognition and comparison of finite-group identities. It does not claim a documented external application for every selected torsion point. Six is a noncanonical cutoff independent of numeral notation.

The parameters r and s are arithmetic_structure coordinates constrained jointly by 0<=r,s<N and gcd(N,r,s)=1. We keep the lexicographically smaller pair of (r,s) and (-r mod N,-s mod N), including a self-inverse pair once. Both coordinates and all torsion directions are needed; this is not two independently sampled real meshes. The representative convention is noncanonical and notation-independent. L32 is even, so the omitted opposite point has the same value. No zero-locus exclusion from the odd weight-two functions is carried over. The origin is (1,0,0), and every 2-torsion point is retained.

Coverage was reviewed across lattice geometry and torsion structure before choosing depth: all thirteen orders include both rectangular and rhombic lattices, the extra-automorphism cases, and nonmaximal orders; all directions in the selected finite groups are included rather than just real-axis examples. 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 in total. The Cartesian-looking notation describes complete finite groups on an arithmetic CM slice, not a product of unrelated simple continuous inputs. Higher-class-number ideal classes and higher torsion orders remain outside the current extent. No general exploratory continuous-input group is included.

Sources checked: Samart, section 1, Corollary 2.8, Theorems 1.2 and 2.11; SageMath, Complex multiplication for elliptic curves, cm_orders and cm_j_invariants_and_orders; live T83, T700, T701 and T702 and family issue 311. The simplifying 3-torsion identity in Samart (2.13) is not imposed on rhombic markings. Instead the full finite-character distribution, including the constant correction, is used.

The skill evidence is ../20261110T015857Z-build-skills/manifest.json. Both archived versions have SHA-256 886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b. This builder justification is not an independent argument-selection acceptance; that belongs to the separate critique actor.