The Capability Convergence Hypothesis: Capability from Access Structure, Not Scale
arXiv:2607.14144
The paper proposes that the ability of sequence models to solve tasks depends more on their internal access structure (a compressive state channel plus a scalable index channel) than on sheer size, providing information‑theoretic lower bounds and pre‑registered experiments to support this Capability Convergence Hypothesis.
Abstract
The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.
43 pages, 16 figures. v2: title now names the hypothesis (CCH); postscript on two post-registration frontier releases (Kimi-K3, Qwen3.8-Max) with a registered forward prediction; new citations and consistency fixes. Registered census statistics and experimental results unchanged. Code and data: https://github.com/wenhui-ml/Capability-Convergence-Hypothesis (DOI: 10.5281/zenodo.21714418)