In this work, we evaluate the Shannon-like entropic measure of spatially-localized functions for a five-dimensional braneworld generated by a double sine-Gordon (DSG) potential. The differential configurational entropy (DCE) has been shown in several recent works to be a configurational informational measure (CIM) that selects critical points and brings out phase transitions in confined energy models with arbitrary parameters. We select the DSG scenario because it presents an energy-degenerate spatially localized profile where the solutions to the scalar field demonstrate critical behavior that is only a result of geometrical effects. As we will show, the DCE evaluation provides a method for predicting the existence of a transition between the phases of the domain wall solutions. Moreover, the entropic measure reveals information about the model that is capable of describing the phase sector where we obtain resonance modes on the massive spectra of the graviton. The graviton resonance lifetimes are related to the existence of scales on which 4D gravity is recovered. Thus, we correlate the critical points defined by the {CIMs} with the existence of resonances and their lifetimes. To extend our research regarding this system, we calculate the corrections to Newtons Law coming from the graviton modes.
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