This paper proves a finite-dimensional structural theorem for recursive holographic descent and source recognition. The theorem begins with a declared finite source carrier whose states are passed through a specified chain of completely positive trace-preserving channels and compared through the distinctions accessible at a terminal level. It determines which source distinctions survive composed descent, which become terminal-invisible, and when terminal data recognize source-level structure. The main theorem constructs composed accessibility quotients, survivor spaces, terminal-invisible source kernels, and a recursive compression invariant. Accessibility quotients form a quotient tower, survivor spaces decrease under further descent, terminal-invisible source kernels increase, and compression is monotone. Cumulative compression is not a sum of unrelated one-step losses; it is the codimension of the composed survivor range, with increments given by successive losses of already surviving source distinctions. The source-recognition theorem gives the factorization criterion. A source observable is recognizable at a terminal level exactly when it lies in the survivor space. A general source invariant on a state class is recognizable exactly when it factors through the composed accessibility quotient. A terminal value identifies a declared source identity class only when the recognized invariant is source-complete for that identity relation. Thus terminal data acquire source significance only when they survive the composed channel chain, factor through the composed accessibility quotient, and are complete for the declared source identity relation. The theorem belongs to a broader sequence of structural results: upstream carrier and admissibility results supply finite source carriers and lawful descendant structures from fixed carrier, completion, comparison, and realization data, while the companion one-step boundary-compression theorem identifies what survives a single boundary channel. Here the zero-compression case is full source separation, equivalently informational completeness of the composed terminal law; physical completely positive trace-preserving recovery requires additional recovery data. The theorem therefore provides a finite-dimensional criterion for recursive source recognition and a carrier-level audit condition for boundary, measurement, selection, holographic, and source-identification claims. License note: Distributed under CC BY-NC-ND 4.0.
Salimah Meghani (Sun,) studied this question.