The Algebraic Genesis of Fibonacci Numbers: A Diagonal Projection Theory from Pascal's Combinatorial Space

This dissertation establishes a comprehensive algebraic framework demonstrating that the Fibonacci sequence emerges as the result of a diagonal projection operator TTT applied to Pascal’s triangle. We introduce the diagonal projection operator as a linear transformation that maps the two-dimensional binomial coefficient space onto one-dimensional recursive sequences, providing the first rigorous operator-theoretic characterization of this classical relationship.