Emergence of Frontier Superposition: M\"obius attractor and Cascade Supervision
Mirrored from arXiv — Machine Learning for archival readability. Support the source by reading on the original site.
Computer Science > Machine Learning
arXiv:2605.18820 (cs)
[Submitted on 12 May 2026]
Title:Emergence of Frontier Superposition: Möbius attractor and Cascade Supervision
View a PDF of the paper titled Emergence of Frontier Superposition: M\"obius attractor and Cascade Supervision, by Hongyu Gu and 1 other authors
View PDF
HTML (experimental)
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 Erdős-Rényi graphs by isolating architectural and supervisional contributions. Architecturally, we identify a Möbius attractor: under $S_n$-symmetry in the tree regime, layerwise dynamics reduce to a 1D Möbius 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: Möbius 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.
| Comments: | 40 pages, 3 figures |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI) |
| Cite as: | arXiv:2605.18820 [cs.LG] |
| (or arXiv:2605.18820v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.18820
arXiv-issued DOI via DataCite (pending registration)
|
Full-text links:
Access Paper:
- View PDF
- HTML (experimental)
- TeX Source
View a PDF of the paper titled Emergence of Frontier Superposition: M\"obius attractor and Cascade Supervision, by Hongyu Gu and 1 other authors
References & Citations
Loading...
Bookmark
Bibliographic Tools
Code, Data, Media
Demos
Related Papers
About arXivLabs
Bibliographic and Citation Tools
Bibliographic Explorer Toggle
Bibliographic Explorer (What is the Explorer?)
Connected Papers Toggle
Connected Papers (What is Connected Papers?)
Litmaps Toggle
Litmaps (What is Litmaps?)
scite.ai Toggle
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv Toggle
alphaXiv (What is alphaXiv?)
Links to Code Toggle
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub Toggle
DagsHub (What is DagsHub?)
GotitPub Toggle
Gotit.pub (What is GotitPub?)
Huggingface Toggle
Hugging Face (What is Huggingface?)
ScienceCast Toggle
ScienceCast (What is ScienceCast?)
Demos
Replicate Toggle
Replicate (What is Replicate?)
Spaces Toggle
Hugging Face Spaces (What is Spaces?)
Spaces Toggle
TXYZ.AI (What is TXYZ.AI?)
Recommenders and Search Tools
Link to Influence Flower
Influence Flower (What are Influence Flowers?)
Core recommender toggle
CORE Recommender (What is CORE?)
IArxiv recommender toggle
IArxiv Recommender
(What is IArxiv?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
More from arXiv — Machine Learning
-
Dimensional Balance Improves Large Scale Spatiotemporal Prediction Performance
May 20
-
Robust Basis Spline Decoupling for the Compression of Transformer Models
May 20
-
HELLoRA: Hot Experts Layer-Level Low-Rank Adaptation for Mixture-of-Experts Models
May 20
-
UCCI: Calibrated Uncertainty for Cost-Optimal LLM Cascade Routing
May 20
Discussion (0)
Sign in to join the discussion. Free account, 30 seconds — email code or GitHub.
Sign in →No comments yet. Sign in and be the first to say something.