Composable Future: An Algebraic Theory of Paradigmatic Transitions

We propose Composable Future, a formal theory of paradigmatic transitions as composable algebraic structures. A future is a 4-tuple F = (S0, τ, S1, Φ) where S0 and S1 are paradigmatic states, τ is a trajectory of change, and Φ is an affordance set of accessible futures from S1. We define four primitive operators (sequential bind, parallel tensor, fork, merge) and prove fundamental laws: identity (substantive in the affordance component, ADR-0005), closure, well-formedness preservation, and associativity. The trajectory carries an explicit path of intermediate states; sequential bind concatenates paths, so associativity is substantive - it holds by List. appendₐssoc on the concatenated paths, not by definitional collapse of trajectories - and is further structured via an indexed monad construction for path-dependent trajectories. We resolve the central open problem (associativity under path-dependence) by carrying path-dependence in the trajectory itself, complemented by Orchard et al. 's indexed monad framework, and establish the correct equivalence relation between futures (OP3): path-trace isomorphism implemented as FutureIso + PathIso + TrajectoryEquiv in Lean 4, giving ComposableFuture the structure of a setoid. The theory occupies the intersection of category theory, paradigm change studies, process algebra, affordance theory, and futures formalization - a structure not currently named or investigated in any of these domains