Toy Combinatorial Interpretability Models Reveal Lottery Tickets in Early Feature Space

Reports on how sparsification, compression, and lottery tickets change model behavior have been mixed in the prior literature, with beneficial effects observed in some studies and adverse effects in others. Moreover, prior work has not considered actual deployment conditions, where decision logic is already fixed for the incumbent. To assess these mixed findings from a practical standpoint, we study the production-replacement question at the deployment level, namely whether an accuracy-matched lottery ticket or another sparse challenger can replace an incumbent dense model without reconfiguring downstream decision logic. We therefore audit a broad, protocol-specific panel of deployment-relevant behaviors spanning calibration, OOD response, class-level reliability, representations, and downstream policy decisions, and summarize clean-accuracy-excluded deviations with a behavioral-compatibility distance. Across extensive experiments, sparse candidates repeatedly recover dense-reference accuracy yet remain behaviorally different; in several study-band-matched settings, LTs also show lower corruption accuracy. In small-gap settings with fixed-threshold policy diagnostics, lottery-ticket replacement changes 7% to 10% of accept--review decisions. This churn creates precisely the burden that drop-in replacement is meant to avoid: reconfiguring and revalidating downstream decision logic. These findings establish the limits of clean-accuracy certification: Establishing compatibility with a fixed incumbent is distinct from attributing churn uniquely to sparsity or treating every measured deviation as harmful. Our theory explains the routing result: Even exact pointwise top-1 agreement cannot bound fixed-threshold decision changes, and small confidence shifts near the operating boundary can generate first-order routing churn.

Neural networks acquire structured representations at specific moments during training, yet identifying these transitions typically relies on retrospective, label-dependent metrics. We introduce a bifurcation theory of representation dynamics to detect these moments in real time. Analyzing a passive GMM probe attached to the evolving encoder, we show the onset of structure corresponds to a supercritical pitchfork bifurcation driven by the loss Hessian. The system exhibits a theoretically predictable zero-crossing ($β_c$) that, compared to the network's current state ($β$), yields a dynamic ratio $β(t)/β_c(t)$: a universal, label-free phase coordinate for representation dynamics, computable entirely from hidden states. We empirically validate four distinct transition regimes predicted by this coordinate across diverse settings: SAEs on language models (Pythia), SSL (CIFAR), and grokking (modular arithmetic). Crucially, under finite dissipation, macroscopic symmetry-breaking can lag the initial zero-crossing by orders of magnitude, which providing a rigorous dynamical account of the delayed escape observed in grokking. Microscopically, the bifurcation creates a shared unstable subspace, forcing collective symmetry breaking. We term this the "feature lottery" in SAE training: a feature's terminal interpretability becomes predictable remarkably early. By only 5% of training, early atom purity robustly predicts final convergence purity, with top-decile early atoms achieving over 12x the baseline purity at convergence. Beyond explaining concept emergence, $β/β_c$ provides a practical early-warning indicator for training health, detecting the onset of usable structure, the crystallization of feature identity, and representational collapse epochs before downstream metrics react.

The Strong Lottery Ticket Hypothesis (SLTH) asserts that sufficiently overparameterized, randomly initialized neural networks contain sparse subnetworks that, even without any training, can match the performance of a small trained network on a given dataset. A key mathematical tool in the theoretical study of SLTH has been the Random Subset Sum Problem (RSSP). The SLTH has recently been extended to the quantized setting, where the network weights are sampled from a discrete set rather than from a continuous interval. These new results are however far from those in arbitrary-precision setting in several ways. In this work, we provide an analysis of the RSSP in the discrete setting, and use it to derive tight SLTH guarantees in the quantized case. Our analysis obtain tight bounds on the failure probability of finding a strong lottery ticket in the quantized regime, providing an exponential improvement over previous results. Most importantly, it unifies the literature by showing that both approximate representations in the continuous setting and exact representations in quantized settings naturally emerge as limiting cases of our results. This perspective not only sharpens existing bounds but also provides a cohesive framework that simultaneously handles approximation and rounding errors.