Inside the Width: Trajectory Structure of Nonconservative Erasure

Abstract

Abstract When a collection of requirements, distinctions, or certificates is reduced, when does removing several items preserve the same semantic value, and when can locally valid removal certificates fail when combined? This note isolates an elementary answer at the level of an arbitrary antitone Galois connection. For an ambient family Q and an erased subset S, the Endpoint Criterion shows that conservative erasure is equivalent to a single membership test against the final remainder: F(Q ∖ S) = F(Q) ⇔ S ⊆ G(F(Q ∖ S)). Thus joint conservative erasability is an endpoint property rather than a sequence of independent local certificates. The associated closure G ∘ F yields a Certificate Transport law: certificates issued by the smaller final remainder transport to larger intermediate remainders, while no general transport law holds in the reverse direction. This explains why certificates evaluated against different remainders need not compose. A path theorem further shows that a conservative endpoint collapses every intermediate semantic value along every erasure order. The completed erasure's value and conservativity verdict are therefore endpoint-determined, although nonconservative erasures may retain path-sensitive intermediate trajectories; a minimal crossed-rows polarity witnesses this distinction. The retirement calculus of Regime Equivalence and Refinement is recovered as a congruence-level instance. This separates the generic Galois arithmetic of erasure from the genuinely regime-specific structure supplied by a forced obligation floor. Polarities, closure operators, and Galois connections are classical; no priority claim is made for those components. The contribution is the explicit typing of endpoint certification, certificate transport, and their boundary conditions, together with finite witnesses and machine corroboration. Zenodo keywords: Galois connections; closure operators; formal methods; order theory; lattice theory; conservative erasure; specification refinement; certificate transport; semantic erasure; formal verification; Identity-Persistence Program