<p>This paper studies the inverse problem in information geometry: given a Hessian manifold, is it derived from an open exponential family? As a motivating case, we prove that for the natural exponential family generated by the generalized hyperbolic secant distribution (NEF-GHS) with shape parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the dual Hessian manifold is not derived from any open exponential family, thereby extending a result of Letac for the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Our proof exploits a local obstruction coming from branch points in the Hessian potential. Inspired by this observation, we prove a broad result: when a Hessian metric admits a real-analytic global Hessian potential, “derivedness” is determined by its restrictions to open submanifolds.</p>