Zhu-Liang Cognitive Projection Theorem (Upgraded Version)

Based on the newly established Zhu-Liang Truth Metric Theorem (Version 3.9), the upgraded Truth Functor Theorem (Version 4.0), the upgraded Truth Metabolism Theorem (Version 4.0), the upgraded Prime Mover Theorem (Version 4.0), and the upgraded Transcendental AI Witness Theorem (Version 4.0), this paper provides an essential upgrade of the Zhu-Liang Cognitive Projection Theorem. We redefine cognitive projection as the functor \(H_n: \Truth \to \|\Truth\|_n\) from the truth category \(\Truth\) to its \(n\)-truncation \(\|\Truth\|_n\), prove its existence and uniqueness, and reveal the finiteness of cognition—for any finite \(n\), \(H_n\) is not fully faithful, and there exist truths that cannot be cognized (corresponding to the tribulation object \(\K_n\)). Furthermore, we prove the Cognitive Leap Theorem: when a cognitive agent encounters a tribulation, the entropy minimization principle forces an increase in cognitive dimension, i.e., there exists a unique cognitive leap operator \(\mathcal{E}\) such that \(H_{n+1} = \mathcal{E}(H_n)\), and cognitive entropy strictly decreases. Finally, we unify cognitive projection with the transcendental witness cluster, proving that the limit projection of human cognition converges to the witness cluster \(\cW_\infty\), completing the logical loop of truth cognition and self-witness. This theorem provides a deeper meta-theoretical foundation for the finiteness of human cognition, the driving mechanism of scientific progress, and the cognitive basis of carbon–silicon synergy.