Abstract
Problem Statement. Standard algebraic semantics identifies mathematical objects solely by their extensional value, discarding construction history—a loss that obscures intrinsic computational differences between structurally distinct but value-identical terms. Framework Introduction. This monograph introduces Genesis Mathematics, a formal framework in which objects carry their construction history as a first-class mathematical structure. We define genesis objects Γ = (V, G, H, C), equipped with an Observational Collapse Axiom and a multi-layer metric measuring structural, semantic, and computa- tional distance. Formal System. We instantiate the framework over the signature {Z, S, Add}, defining a Genesis Term Rewriting System (GTRS) and introducing K-bounded rewrite systems as models of bounded observational reasoning. Main Theorem. Our Genetic Separation Theorem proves that for every bound K, there exist genesis objects Γn and ∆n with identical value n, yet any K-bounded rewrite system requires Ω(n) steps to identify their histories. Implication. This proves that intensional structure is computationally irreducible un- der bounded rewrite systems, establishing that construction history is not an epistemic artifact but a mathematically real invariant property.