Context. One of the key questions of the star formation problem is whether brown dwarfs (BDs) form in the manner of stars directly from the gravitational collapse of a molecular cloud core (star-like) or whether BDs and some very low-mass stars (VLMSs) constitute a separate population that forms alongside stars comparable to the population of planets, for example through circumstellar disk (peripheral) fragmentation. Aims: For young stars in Taurus-Auriga the binary fraction has been shown to be large with little dependence on primary mass above ≈ 0.2 M⊙, while for BDs the binary fraction is < 10%. Here we investigate a case in which BDs in Taurus formed dominantly, but not exclusively, through peripheral fragmentation, which naturally results in small binary fractions. The decline of the binary frequency in the transition region between star-like formation and peripheral formation is modelled. Methods: We employed a dynamical population synthesis model in which stellar binary formation is universal with a large binary fraction close to unity. Peripheral objects form separately in circumstellar disks with a distinctive initial mass function (IMF), their own orbital parameter distributions for binaries, and small binary fractions, according to observations and expectations from smoothed particle hydrodynamics (SPH) and grid-based computations. A small amount of dynamical processing of the stellar component was accounted for as appropriate for the low-density Taurus-Auriga embedded clusters. Results: The binary fraction declines strongly in the transition region between star-like and peripheral formation, exhibiting characteristic features. The location of these features and the steepness of this trend depend on the mass limits for star-like and peripheral formation. Such a trend might be unique to low density regions, such as Taurus, which host binary populations that are largely unprocessed dynamically in which the binary fraction is large for stars down to M-dwarfs and small for BDs. Conclusions: The existence of a strong decline in the binary fraction - primary mass diagram will become verifiable in future surveys on BD and VLMS binarity in the Taurus-Auriga star-forming region. The binary fraction - primary mass diagram is a diagnostic of the (non-)continuity of star formation along the mass scale, the separateness of the stellar and BD populations, and the dominant formation channel for BDs and BD binaries in regions of low stellar density hosting dynamically unprocessed populations.