The self-assembly of DNA-based monomers into higher-order structures has significant potential for realizing various biomimetic behaviors including algorithmic assembly, ultrasensitive detection, and self-replication. For these behaviors, it is desirable to implement high energetic barriers to undesired spurious nucleation, where such barriers can be bypassed via seed-initiated assembly. Joint-neighbor capture is a mechanism enabling the construction of such barriers while allowing for algorithmic behaviors, such as bit-copying. Cycles of polymerization with division could accordingly be used for implementing exponential growth in self-replicating materials. Previously, we demonstrated crisscross polymerization, a strategy that attains robust seed-dependent self-assembly of single-stranded DNA and DNA-origami monomers via joint-neighbor capture. Here, we expand the crisscross assembly to achieve autonomous, isothermal exponential amplification of ribbons through their concurrent growth and scission via toehold-mediated strand displacement. We demonstrate how this crisscross chain reaction, or 3CR, can be used as a detection strategy through coupling to single- and double-stranded nucleic acid targets and introduce a rule-based stochastic modeling approach for simulating molecular self-assembly behaviors such as crisscross-ribbon scission.