Escape-Broken Continuation, Brandt Partial Actions, and Order-Sensitive Survival Operations

Let π: F → Y be a noninjective local diffeomorphism. Continuation of a fiber point along a path is unique when it exists, but it may fail through nonproper escape. We develop a path-level theory of this partial continuation and determine exactly which mathematical objects retain or forget its history.

The groupoid of surviving-homotopy classes is canonically isomorphic to the fundamental groupoid of the source, Cont_hist(π) ≅ Π₁(F). It therefore retains only source homotopy and is blind to representative-dependent escape. Full escape information lives before homotopy quotienting, in a maximal partial-lift system on Moore paths that records lifting domains, endpoint maps, escape times, and maximal lifts. Its finite operational quotient is the continuation inverse monoid on a chosen fiber, while germs of loop-family transports retain only realized endpoint pairs.

For the Pinchuk presentation we compute the continuation inverse monoid exactly as the six-element Brandt monoid B₂¹, including an absorbing zero. Representing its elements by partial isometries produces completely positive, trace-nonincreasing continuation operations. Two itineraries containing identical continuation blocks in different orders can yield respectively a nonzero rank-one operation and zero, establishing order-sensitive survival.

The remaining structural question is whether a useful canonical quotient can retain the death certificates of escaping branches without retaining the entire raw path space.

Preprint, version 0.8. Independent expert verification remains outstanding. The lower-bound Pinchuk realizations carry the explicitly stated Campbell Figure 3 verification gate; the upper bound does not.