Generative Analysis I

Abstract

This work opens the analytical block of the generative programme. If topology answers the question where an object exists, and algebra answers how it interacts, analysis answers the question how it becomes: how a trajectory in the generated space M(ε, δ) stabilizes, accumulates, and branches. We establish that a sequence is a trajectory γ : N → M0 with internal time; its primary elementary object is the increment ιnγ : an ⇝an+1 rather than a difference of states. An object-limit is defined as zero stabilization of a trajectory-stable kernel; alongside it, we introduce a mode-limit and a class-limit. A series is defined as iterative C-accumulation; for trajectories satisfying an explicit tree hypothesis, a space of potential continuations is constructed. Classical Cauchy-Weierstrass analysis is obtained as a special case under a specified metric specialization. The paper fixes definitions, conditional theorems, and a programme for further development. It does not yet introduce differentiation and integration; these constructions belong to Generative Analysis II. The work relies on the published OST [1] and continues the Generative Mathematics series after generative topology, geometry, and algebra [2, 3, 4]