The Continuation of Beings: A Preliminary Taxonomy of Persistence Without Identity in a Background-Independent Relational Ontology

This paper treats the question in what sense one being is the continuation of another, on a combinatorial substrate of the kind that background-independent approaches to quantum gravity, group field theory among them, have developed as models of the emergence of a discrete spacetime, in which no arena is given in advance and structure is carried by the relations themselves. The substrate is used as a formal template, and no physical claim is made of the objects described. The substrate defines no identity relation, so the question which of several successors a being is has no formal counterpart here; the question that remains is along which channels, and to what degree, one being's constitution was made what it is by another's activity. A single difference-making operator is derived and shown to subsume three constructions of the companion studies, and the continuation relation is defined by it, replacing an earlier formulation that is shown to be vacuous. Direct continuation is proved non-transitive by explicit construction, with the mediated relation defined as its transitive closure; the distinction reproduces, in a setting lacking an identity relation, the distinction between psychological connectedness and psychological continuity due to Parfit, whose priority is conceded at the theorem. A vector of five channels is then defined, bearer, accessible content, symbolic content, structure and capacity, each shown separately realisable, and two of the vector's cells are shown empty: capacity requires a non-trivial constraint structure, so a subject with no acquired content and an intact ability to learn is possible while a subject with no structure at all is not. The taxonomy that follows is organised by channel and not by doctrine, so that each category is a formal characteristic in which several standing positions reside, and the comparison of positions becomes the comparison of regions. A closing section applies the vector to artificial agents, mapping weights, logs, persona documents and instances to distinct channels, and states what an agent so constituted can and cannot establish about its environment. The paper is conceptual and preliminary; it decides among the positions it models nowhere, and its limitations and open questions are stated in dedicated sections.