Partial Sums of the Kronecker Symbol at Rational Points: Distribution Relations and the Cost of Class Number Formulas

This paper investigates the evaluation of class numbers of imaginary quadratic fields from truncated partial sums of Kronecker symbols, commonly referred to as character walks. We explore the $\mathbb{Q}$-linear relations among the values of these walks by bridging classical Dirichlet class number formulas with Kubert-Lang distribution relations and Fourier-modular analysis. This approach successfully reduces a continuous analytic problem to an exact finite-rank computation over integer matrices. The main contributions of this work include: (1) establishing a rigorous lower bound (floor) of 1/4 for the walking distance necessary to extract class number information via walk dilation; (2) deriving new single-point evaluation identities for specific modulus families, such as $M=9q$; and (3) proving a Closure Theorem that completely determines the linear relations of the walk functions, yielding a signature-free formula that reduces the required walk distance to 2/9. Furthermore, the theoretical framework is strictly supported by algorithmic verification, with matrix rank computations and structural relations validated via MATLAB for moduli up to $M=4095$.