Finite-Window Tests of Isolated-Dipole Representations for Fallback-Fed Magnetar Power Histories

Magnetars are neutron stars with the highest known magnetic fields. This characteristic makes them singular astrophysical objects from which unique information can be gathered. Magnetic fields of common neutron stars are usually described using isolated dipoles. On the other hand, an isolated-dipole interpretation of magnetar power requires distinguishing curve shape, parameter-domain admission, and recovery of birth properties. A fallbackfed magnetar power history can resemble an isolated dipole over a finite interval, but curve match does not establish that the fitted parameters lie within a chosen domain or recover the star’s birth state. We formulate an accuracy-dependent test that separates these questions. With a common time origin, every finite positive isolated-dipole solution obeys L0 = C/(t + τ)2, where C, τ > 0, and declines by less than 2 dex over a decade in time. Faster target decline gives a one-sided, optimizer-independent lower bound on the maximum logarithmic error. For the adopted inner-supply torque model, an exact decomposition separates equilibrium-envelope evolution from relative spin relaxation. We apply the test to six histories. Over 104–105 s and at a maximum-error tolerance of 0.05 dex, two histories with n = 4/3 give opposite outcomes: one is excluded by an endpoint floor of 0.0914 dex, whereas the other is matched on the grid by a finite dipole outside the primary period–field domain, with a sampled maximum residual of 0.0134 dex. A post-transition control admits an isolated continuation whose extrapolated period differs from the input birth period. Thus, finite-window representability, parameter-domain admission, and recovery of birth properties are distinct. These examples are conditional on the engine prescription and do not model disk transport or radiation.