Task-Relative Continuity Through Changing Representations: Reversible Handoffs, Route Ambiguity, and the Limits of Stable Readout

A persisting component need not preserve a particular memory, and a recoverable memory need not be correctly used by its installed reader. We analyze these distinctions in finite models of representation transfer. A relational-memory example separates loss, invertible misencoding, and successful handoff between different storage carriers. For multiple bijective routes on a full source domain, one route-blind target decoder exists exactly when the target is invariant under every relative route transformation. The orbit quotient characterizes all exactly retained targets. Route metadata need distinguish only the distinct target-reader functions; we give its exact minimum alphabet. We also derive a sharp two-route error expression, a uniform-group Bayes law, and a finite-field signal-versus-nuisance recovery criterion. A three-position example preserves Hamming weight under unknown permutations while limiting recovery of the original first bit to three quarters and of the full ordered state to one half. Explicit counterexamples show why source support, route probabilities, decoder access, and hidden correlated variables must remain part of each claim. Conditional implications for a structural theory of experience are separated from the mathematics. The contribution is a task-sensitive theoretical synthesis with reproducible diagnostics, not a new foundation for invariant theory, a measured neural mechanism, or a criterion for consciousness. Preprint, not peer reviewed. Substantial ChatGPT assistance was used under human direction. Formal/finite counterexamples are not empirical validation of consciousness.