Inside the Width: Trajectory Structure of Nonconservative Erasure

Suppose that some requirements or distinctions are withdrawn from a specification. When does their removal preserve exactly the same semantic content, and why can individually valid removal certificates fail when used together? I formulate the problem for an arbitrary antitone Galois connection. The central Endpoint Criterion states that erasing S ⊆ Q is conservative exactly when every erased item is certified by the closure generated by the final remainder: F(Q ∖ S) = F(Q) ⇔ S ⊆ G(F(Q ∖ S)). Conservative erasure is therefore determined by the remainder that actually survives, not by independent certificates issued against larger intermediate contexts. This yields a Certificate Transport law. The composite G ∘ F is a monotone closure operator, so certificates issued by a smaller remainder remain valid in larger remainders, while no corresponding transport principle holds generally in the reverse direction. Apparent failures of composition can therefore be understood as context-indexing failures: a certificate remains tied to the remainder against which it was established. The same structure distinguishes endpoint semantics from trajectory semantics. The final semantic value and the conservativity verdict depend only on the endpoint pair. If the endpoint is conservative, every intermediate value along every removal order collapses to the same value. Outside the conservative case, however, intermediate trajectories may differ; a minimal polarity provides a counterexample to unconditional history-independence. The result abstracts a retirement calculus for declared specifications while separating its generic order-theoretic component from domain-specific constraints such as forced obligation floors. No novelty claim is made for classical Galois or closure theory; the emphasis is on the semantics and typing of certificates under changing contexts.