We present a unifying statistical formulation for many fundamental problems in genome science and develop a reference-free, highly efficient algorithm that solves it. Sequence diversification - nucleic acid mutation, rearrangement, and reassortment - is necessary for the differentiation and adaptation of all replicating organisms. Identifying sample-dependent sequence diversification, e.g. adaptation or regulated isoform expression, is fundamental to many biological studies, and is achieved today with next-generation sequencing. Paradoxically, current analyses begin with attempts to align to or assemble necessarily incomplete reference genomes, a step that is at odds with detecting the most important examples of sequence diversification. In addition to being computationally expensive, reference-first approaches suffer from diminished discovery power: they are blind to unaligned or mis-aligned sequences. We provide a unifying formulation for detecting sample-dependent sequence diversification that subsumes core problems faced in diverse biological fields. This formulation allows us to construct an algorithm that performs inference on raw reads, avoiding references completely. We illustrate the power of our approach for new data-driven biological discovery with examples of novel single-cell resolved, cell-type-specific isoform expression, including expression in the major histocompatibility complex, and de novo prediction of viral protein adaptation including in SARS-CoV-2.