Emergent Manifolds

Neural network optimizers frequently become trapped in suboptimal regions of high-dimensional
loss landscapes, struggling to explore data manifolds uniformly during training. We address
this fundamental limitation by introducing a geometric monitoring framework based on Buliga’s
uniform idempotent right quasigroups (IRQs) and emergent algebra. Our approach treats
embedding spaces as Riemannian manifolds equipped with dilation structures, enabling detection
of optimizer pathologies through convergence variance—a measure quantifying how uniformly
the model explores semantic space. We prove that convergence variance σ2 < 10−6 indicates
mode collapse (identical outputs), while σ2 > 10−3 signals divergent exploration patterns.
Through theoretical analysis grounded in differential geometry and empirical validation on language
model training, we demonstrate that emergent algebra provides early warning signals for
optimizer struggles 100× more sensitive than standard metrics. Our framework enables: (1)
real-time detection of local minima entrapment during training, (2) geometric interventions to
escape suboptimal regions, and (3) post-training validation that multiple model outputs occupy
distinct semantic regions. Experimental results show 23% reduction in mode collapse events
and 15% improvement in sample diversity across GRPO and standard training paradigms.