QCM: A Quadratic Congruence Dynamics Revealing Divisor Hierarchies

QCM (Quadratic Capturing Method) is a quadratic iterative method defined on integers of the formN = 2k + c (with c odd). Starting from a fixed initial value, the method generates a modular quadratic trajectory whose goal is to capture N, meaning that the iteration reaches a quadratic congruence of the formU² ≡ k (mod N). The method does not rely on prior factorization.Instead, it reveals the divisorial structure of N dynamically through the capture process. Experimental results show that: capture is hereditary by division (divisor completeness), composite values appear only after their divisors, proper composites exhibit delayed capture due to global modular synchronization, captured values follow a strict quadratic geometric constraint. QCM provides a dynamic framework for studying quadratic congruences and divisor hierarchies, based on deterministic modular iteration.