Perspective Algebra: A Non-Well-Founded Banach Algebra for Relational Physics

This paper presents the formal mathematical foundations of Perspective Algebra, the algebraic structure underlying the Zero-Present Cosmology (ZPC). Perspective Algebra is a non-well-founded Banach algebra defined on the space of all perspectives(P), equipped with a commutative entanglement sum (⊕) and a non-commutative Lie bracket derived from phase-winding of the Time dimension. The algebra is shown to be complete (Banach), with a well-defined dual space of observables and a C*-algebra representation that recovers standard quantum mechanics as a special case. The noncommutative bracket is used to derive spin and gauge interactions, while the topological closure of the universal present (Ψ) is used to prove the identity 0 = 1 as a topological boundary condition. This paper is intended as a standalone mathematical foundation for the physical applications presented in the companion paper, ”The Zero-Present Cosmology (ZPC): A Perspective Framework for Gravity, Quantum Mechanics, and Cosmic Dynamics.”