# Argument selection for T700

This is one named real quantity, Bloch's elliptic dilogarithm. Its real companion J and the elliptic trilogarithms are separate quantities, not additional values of a form parameter.

The marked torus and ordered integer parameters (Delta,N,r,s) follow family issue 311. All logarithms are natural. Set b=0 for Delta divisible by four and b=1 otherwise, tau=(-b+i sqrt(|Delta|))/2 and u=(r tau+s)/N; preserve the sign in q=exp(2*pi*i*tau). No modular replacement of the marking is made.

* Delta has role arithmetic_structure. The thirteen selected discriminants -3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163 are exactly the imaginary quadratic orders of class number one, equivalently rational singular modulus. Nonmaximal orders are included. This is a complete arithmetic slice, independently supported by T83 and Sage's cm_orders(1) documentation. Higher class numbers require ideal-class choices and are deferred, not silently collapsed onto principal representatives. Both rectangular and rhombic lattices and the extra-automorphism cases occur.
* N has role discrete_index. It is the exact finite order of a point, not a renamed continuous argument. The reference extent N<=6 is noncanonical. Orders 3,4,6 appear in Guillera–Rogers Theorem 4. Order 5 gives the next prime group; order 6 includes the first mixed-prime group. The purpose is recognition of torsion regulator values and finite-group comparisons. No individual application is asserted for every pair (Delta,N). The origin and order 2 give identically zero D_q and are omitted by proof, not a numeric threshold.
* r and s jointly have role arithmetic_structure, as coordinates in E[N]. They are not independently sampled real arguments: 0<=r,s<N and gcd(N,r,s)=1 enforce exact order. All directions of each selected group are covered modulo inversion. Keep the lexicographically smaller pair of (r,s) and (-r mod N,-s mod N), and omit 2s-br=0 modulo N, where x is real and every summand vanishes. The representatives are noncanonical but independent of numeral notation; omitted opposite points are recovered by oddness. No absolute value is taken.

The selection is a bounded arithmetic reference, not a Cartesian mesh of independent continuous inputs. The bound six limits group growth while covering prime and composite torsion; no entry-count target determined it. Breadth across all thirteen rational-j orders and all group directions is preferred to deeper torsion only on the square lattice. The expected 362 rows are a consequence to verify, not a quota. These roles do not mix in an unsupported exploratory continuous grid; the reference choice is the finite bound on a genuine discrete index. Later non-CM or independently chosen real inputs need their own reference policy.

Checked sources: Brunault, On Zagier's conjecture for base changes of elliptic curves, section 2 and Proposition 6 (https://perso.ens-lyon.fr/francois.brunault/recherche/LEF.pdf); Guillera and Rogers, Mahler measure and the WZ algorithm, section 3, Theorem 4 (https://arxiv.org/html/1006.1654v2); Sage CM documentation, cm_orders(1) (https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/cm.html); live T83. These sources establish the functions, arithmetic family, conventions and relationships, not a published use for every reference point.

The authoritative instruction evidence is ../20261110T001313Z-build-skills/manifest.json, copied as skill-manifest.json. Both archived skill versions have SHA-256 886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b. This builder's rationale is not the required independent argument-selection acceptance. The later critique must inspect the enumeration and interactions and bind its sidecar to the live selection and generate.py.