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 (μ₁ = 3A/2B), matrix determinant (M < 1), and binary bit-consumption kinetics. Moving from the universal loop equation x^* = C/2B - 3A, it is algebraically demonstrated that the unique attractor in the set of positive integers is the 1 → 4 → 2 → 1 closed circuit, and all trajectories must collapse into this stable core due to phase space volume contraction.