Abstract
The Platonic Representation Hypothesis (PRH) posits 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:
- Shannon wall barring any o(Nb)-state architecture;
- Horizon wall barring any fixed window;
- 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.
Blogger's Review: The Capability Convergence Hypothesis (CCH) presents a fresh perspective on the relationship between capability and structure in hybrid sequence models, clarifying how to achieve capability enhancement under resource constraints. The validation through pre-registered experiments enhances the credibility of the theory and merits further attention regarding its potential in practical applications.