τ-Structural Incompleteness: Reflexive Compression, Structural Lag, and Gödel/Rosser as the Proof-Theoretic Subclass

This paper develops τ-Structural Incompleteness as a general class of structural non-closure. A regime becomes structurally incomplete when its generative/licensing architecture can produce admissible reflexive content about its own compression or completed-readout mechanism, while that same mechanism cannot globally absorb the self-generated content into completed readout. The governing mechanism is that generative complexity makes content available, anti-complexity attempts compression, diagonalization can make the compression mechanism itself generated content, and incompleteness appears when same-regime compression cannot globally complete that self-generated compression-content. The central formal item is a theorem-schema under declared reflexive compression-expression conditions. A τ-regime is represented as R=(X_R,C_R,A_R,E_R,⌜·⌝,B_R), where X_R is the generated/licensed domain, C_R is generative/licensing architecture, A_R is the licensed anti-complexity compression mechanism, E_R is completed readout when compression succeeds, ⌜·⌝ is a name/code/internal handle when available, and B_R is a boundary predicate concerning A_R or E_R. If generation/licensing, local compression/readout, structural downstreamness, reflexive compression-expression, readout-fidelity, and non-collapse hold, then E_R is not globally positively exhaustive over the reflexive domain generated by C_R. This is positive completed-readout non-exhaustion, not automatically settlement non-exhaustion. Gödel/Rosser incompleteness is recovered as the proof-compression subclass. In that instance, X_R=Sent_T; C_R is syntax, formation, coding, proof-predicate representation, substitution, and diagonalization; A_R is finite proof-compression, represented by a proof relation; and E_R is theoremhood-readout. Diagonalization makes proof-compression itself into generated sentence-content. Rosser's theorem supplies the consistency-only result that proof-zero is not structural-zero: for the Rosser sentence R_T, T⊬R_T and T⊬¬R_T, while the licensing structure of R_T remains active. The Gödel/soundness route supplies the truth-positive result only under the extra semantic assumption of soundness in the intended model ℕ. The philosophical posture is positive but conservative. Incompleteness is not treated merely as a failure or defect of formalization. Under declared reflexive τ conditions, it is interpreted as a structural feature of regimes that can generate content about their own compression/readout mechanisms, within a τ-framework that treats identity-through-return as structurally primal. Anti-complexity is structurally paired with complexity in the declared τ-regime: it is the compression/return counterpart of generative complexity. Structural lag is dependency-asymmetry, not temporal delay: compression presupposes generated/licensed content. The classical Gödel/Rosser/Tarski machinery remains standard metamathematics. The τ contribution is a structural reconstruction and general-class theorem-schema under declared assumptions, with Gödel/Rosser preserved rather than displaced.