Algebraic Solution and Model of the Collatz Conjecture via Scale-Independent Deterministic Growth, Parity Brake, Composite Matrix Operators, Bit-Consumption Kinetics, and Fixed-Point Dynamics

This study proves that the complex and seemingly irregular behavior of the Collatz ($3n+1$) sequence is not a stochastic chaos, but rather a pure, scale-independent, and deterministic proportional system arising from the human perception illusion of large numbers. No matter how large a number grows, the system contains no randomness; every Alt ($3x+1$) step inherently produces an even number, equipping the system with an intrinsic parity brake ($x/2$) that prevents unlimited divergence. The sequence is reduced to Affine matrix transformations in 2D homogeneous coordinates; the system's dynamics are modeled via its dominant eigenvalue ($\mu_1 = \frac{3^A}{2^B}$), matrix determinant ($\det M < 1$), and binary bit-consumption kinetics. Moving from the universal loop equation $x^* = \frac{C}{2^B - 3^A}$, it is algebraically demonstrated that the unique attractor in the set of positive integers is the $1 \to 4 \to 2 \to 1$ closed circuit, and all trajectories must collapse into this stable core due to phase space volume contraction.