Fragility Spectrum: Measuring Resilience in Model-Theoretic Properties under Language Expansions

Abstract

We introduce the fragility spectrum, a quantitative framework to measure the resilience of model-theoretic properties (e.g., stability, NIP, NTP$_2$, decidability) under language expansions. The core is the fragility index $\operatorname{frag}(T, P \to Q)$, quantifying the minimal expansion needed to degrade from property $P$ to $Q$. We axiomatize fragility operators, prove stratification theorems, identify computational, geometric, and combinatorial collapse modes, and position it within Shelah's hierarchy. Examples include ACF$_0$ (infinite fragility for stability) and $\operatorname{Th}(\mathbb{Q}, +)$ (fragility 1 for $\omega$-stability). Connections to DOP, ranks, and external definability refine classifications. Extended proofs, applications to other logics, and open problems enhance the discourse.