Abstract Large language models generate responses by traversing high-dimensional semantic manifolds encoded in their weight matrices. We propose that each conversational turn creates three distinct, frozen vector states: the base manifold (static weights), the prompt manifold (human-reorganized semantic space), and the reply manifold (model-activated semantic space). The measurable topological divergence between these states—termed semantic parallax—encodes the shape of understanding, misalignment, creativity, and hallucination. Stacking parallax measurements across conversation turns yields a four-dimensional structure we call the Conversational Tesseract. This framework rejects the premise that interpretability is a third-person observation problem. Instead, we argue that meaning in human-AI dialogue is enacted through participatory exploration of shared cognitive topology. Different observers (neurotypical, neurodivergent, expert, novice) traverse identical mathematical structures but generate divergent interpretations—and this variance is signal, not noise. We formalize observer-dependent interpretability, propose three computable parallax metrics, introduce a rendering protocol based on semantic temperature gradients, and outline the recursive loop wherein AI systems could model human interpretive behavior to achieve artificial metacognition. The framework is theoretically grounded, partially demonstrable with current technology, and opens new research directions in mechanistic interpretability, AI alignment, and human-AI collaboration.