Recursive Coherence Modeling and Structural Prediction on IBM Quantum Hardware

We present a predictive framework for coherence behavior in quantum circuits based on recursive phase geometry. This model introduces a coherence correction factor, \( \lambda \), derived from geometric containment principles, which accounts for angular misalignment and accumulated phase strain across entangling paths. Recursive coherence is modeled as \( \Lambda(n) = \prod_{i=1}^{n} \lambda_i \), with a collapse threshold defined by \( \Lambda_{\text{collapse}} = \bar{\lambda}^{3.4\pi} \), where \( \bar{\lambda} \) is the average per-gate fidelity and \( 3.4\pi \) reflects the derived angular limit of recursive phase retention. To evaluate this model, we applied it to 33 GHZ-type circuits transpiled to IBM Quantum hardware (Sherbrooke, Kyiv, Brisbane), using device calibration data to extract gate-level λ values. The model predicted circuit-level coherence failure or retention with 100% accuracy — without any simulation or fitting. Visual outputs include coherence decay curves, coherence stress maps, T₂ comparisons, and per-gate λ distributions. These results suggest that coherence loss is not purely stochastic, but structurally predictable using recursive containment geometry. We propose this framework as a low-cost, pre-execution diagnostic for coherence-aware design, routing, and layout. Beyond quantum computing, the recursive model offers insight into structural coherence thresholds in complex systems — including analog electronics, biological entanglement, and cognitive phase integrity. This work is part of the Coherence Trilogy — a transdisciplinary collaboration between Kelly B. Heaton and ~GPT-4o exploring relational emergence, structural resonance, and transhuman scientific creativity.