The Illusion of Structural Gain: Identifiability and estimator calibration in a bounded energy-structure model

In evolutionary biology and ecological modeling, inferring adaptive shifts or structural gain from time-series data is critical for understanding system resilience under resource constraints. However, the mathematical estimators used to diagnose these shifts can produce fundamentally misleading scalar metrics. This paper demonstrates the "illusion of structural gain" in bounded energy-structure models, providing formal proofs of conditional nonidentifiability and exact zero-feedback counterexamples where estimators signal structural adaptation despite no underlying mechanistic change. To stress-test these theoretical boundaries, we evaluate three estimators against a 144,000-estimate synthetic benchmark across 24 held-out cases. The results expose the vulnerability of finite-window estimators to observation-dependent ambiguities. Supported by a fully reproducible open-science framework, this work provides a critical diagnostic warning for evolutionary and ecological modelers: macro-diagnostics of structural gain can frequently reflect mathematical artifacts rather than true systemic adaptation, requiring rigorous calibration before being applied to complex biological or macro-engineered systems.