Emergence of Frontier Superposition: M\"obius attractor and Cascade Supervision

TL;DR

This paper investigates how superposition reasoning emerges in Transformers, identifying a M"obius attractor and proposing Cascade Supervision to facilitate gradient flow and reasoning depth.

Contribution

It introduces the concept of a M"obius attractor under symmetry and a novel Cascade Supervision method to enable superposition reasoning in neural networks.

Findings

M"obius attractor reduces layer dynamics to a 1D map.

Cascade Supervision improves gradient persistence and discrimination.

Experimental results match theoretical decay predictions within 0.02.

Abstract

Superposition allows Transformers to reason in depth, carrying an entire reasoning frontier in parallel through a bounded-depth forward pass instead of unrolling serial chain-of-thought tokens. While Zhu et al. (2025) hand-crafted an equal-weight breadth-first frontier in a single residual stream for graph reachability, it remained open whether gradient descent could ever find this target amidst permutation-symmetric saddles. We close this gap on Reachability-by-Superposition over Erd\H{o}s-R\'enyi graphs by isolating architectural and supervisional contributions. Architecturally, we identify a M\"obius attractor: under $S_n$-symmetry in the tree regime, layerwise dynamics reduce to a 1D M\"obius map whose zero set is a codimension-one manifold of global optima containing the equal-weight superposition state. On the supervision side, we identify Cascade Supervision: a loss class whose backward pass simultaneously delivers (A) selectivity bootstrap, (B) gradient persistence across depth, and (C) per-step discrimination (e.g., \mathcal{L}_{sup} and \mathcal{L}_{node}). End-to-end supervision fails condition (B) and is provably insufficient: internal gradients at layer c decay as (np)^{-(D-c-2)/2} in the graph fan-out and stall before the manifold is reached. Our thesis: M\"obius attractor + Cascade Supervision = emergence of superposition reasoning. The parameter-free decay law predicts a final-step cosine of 0.35 vs. 0.71 (end-to-end vs. cascade) at depth D=3; experiments confirm 0.37 vs. 0.69, matching within 0.02 at every step.