Capability from Access Structure, Not Scale: Lower Bounds and Pre-Registered Tests for Hybrid Sequence Models

Researchers propose the Capability Convergence Hypothesis (CCH), arguing that under a fixed per-token inference budget, representational convergence from scaling does not guarantee capability convergence. Instead, capability depends on access structure, specifically hybrid architectures combining compressive and verbatim-index channels, as demonstrated through pre-registered tests on the Newton's-apple problem. The study provides lower bounds and experimental evidence showing a predicted "scissors gap" in exact-retrieval error, supporting the claim that capability must be purchased by access structure rather than scale alone.

arXiv:2607.14144v1 Announce Type: new 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.