The Reference-Boundary Projection-Divergence Theory (RBPD Theory): A Finite Infinite Dual-Branch, Outcome-Differentiated, Recursive Dynamic Framework

This paper presents the Reference-Boundary Projection-Divergence Theory (RBPDTheory) as a proposed general framework for analyzing how a system uses a currentlyavailable reference as a representation of a wider target domain, applies that representationbeyond its demonstrated support, encounters outcome-level discrepancies, and may revise itsreference, model, or criteria through repeated feedback.The primary case is the Temporal Epistemic Boundary-Projection and Recursive Recog-nition Case (TEBPRC), illustrated by a finite A-S reference projected onto an A-Z targetset. The framework has exactly two principal mathematical branches: a finite branch andan infinite branch. The infinite branch distinguishes set cardinality from measurable ortask-weighted coverage and does not treat an infinite target domain as a finite set with alarger label.A key addition is an explicit Boundary-Blending Operator, separated from the ProjectionOperator. Outcomes are differentiated into mismatch, quantitative divergence, logical con-tradiction, paradox, representational distortion, and explanatory insufficiency. A recursivestate-transition model represents diagnosis, boundary recognition, revision, retesting, localclosure, and reopening under new evidence.The manuscript supplies formal notation, conditional propositions, a proved residual-errorlemma, operational definitions, falsifiability conditions, a domain-mapping protocol, and aresearch roadmap. RBPD is proposed as a research framework, not asserted to be a verifieduniversal law. Its generality and originality remain hypotheses requiring formal comparison,domain-specific operationalization, empirical testing, and independent replication.