This work presents applied case-study demonstrations of Constrained Recursive Hypothesis Inference (CRHI), a framework for structured exploratory reasoning under uncertainty. Building upon the foundational epistemology and operational architecture established in earlier CRHI papers, the present work examines how recursive exploratory reasoning behaves across historical anomalies, unresolved scientific systems, and artifact-analysis cases. The paper demonstrates how competing explanatory branches may be recursively organized, constrained, filtered, ranked, pruned, and compared through convergence analysis without assuming anomalous claims are necessarily valid. Historical examples illustrate delayed convergence and mechanism stabilization, modern unresolved systems demonstrate active uncertainty landscapes, and artifact-analysis cases demonstrate false convergence collapse under stronger constraint filtering and instrumentation analysis. Representative case studies include meteorites, continental drift, ball lightning, high-temperature superconductivity, turbulence, AI interpretability, the OPERA faster-than-light neutrino anomaly, and N-rays. These cases are used not as claims of equivalence, but as demonstrations of recursive exploratory reasoning under incomplete observability and evolving evidential structures. CRHI is proposed as a domain-agnostic methodology for navigating uncertainty in complex scientific environments while preserving falsifiability, evidential discipline, recursive constraint filtering, convergence analysis, and computational restraint. Taken together, the CRHI trilogy attempts to establish a unified framework for exploratory scientific reasoning spanning epistemological foundations, operational architecture, and applied recursive inference analysis.