A No-Go Theorem for Introspective Prediction in Computational Machines

This paper presents two distinct but related contributions. First, we introduce a formal model, the Introspective Predictive Machine (IPM), designed to study the limits of systems that perform online prediction while attempting to assess their own predictive performance via internal subroutines. For this model, we prove a novel limit theorem, Theorem 1 (The No-Go Theorem for Perfect Introspection). Using Kleene’s Recursion Theorem, we demonstrate rigorously that any IPM’s introspective subroutine must fail in one of three specific ways: it may be incorrect, non-halting, or its execution must perturb the very state it aims to measure. This theorem constitutes a self-contained result in the theory of self-referential computation. Second, building upon this foundation, we articulate a separate and speculative research program. The program’s core challenge is to investigate whether the structure of the limit formalized in Theorem 1 can yield a novel perspective on, or reformulation of, Gödel’s Second Incompleteness Theorem. We outline a formalization roadmap that makes the significant technical hurdles—particularly the challenge of mapping the introspective task onto a consistency proof—explicit. This work provides a precise computational model, a proven theorem, and a clear, falsifiable framework for future research into the architecture of self-referential limits.