QCM: A Quadratic Congruence Dynamics Revealing Divisor Hierarchies

QCM (Quadratic Capturing Method) is a quadratic iterative method defined on integers of the form
N = 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 form
U² ≡ 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.