Capability from Access Structure, Not Scale: Lower Bounds and Pre-Registered Tests for Hybrid Sequence Models
Abstract
The Capability Convergence Hypothesis argues that representational convergence does not guarantee capability convergence, which instead requires hybrid architectures combining compressed state and verbatim indexing to overcome information-theoretic and architectural limits.
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.
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Capability Convergence Hypothesis (CCH): a sequel and boundary to the Platonic Representation Hypothesis.
PRH says that with scale, heterogeneous networks converge in representation. We ask where capability goes under a fixed per-token inference budget, and argue the two come apart: 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 this on a witness task (Newton's apple in an infinite stream) and name three resource walls:
a Shannon wall barring any o(Nb)-state architecture (information-theoretic, unconditional),
a horizon wall barring any fixed window,
a circuit wall barring fixed-depth attention-only composition (conditional on TC⁰ ≠ NC¹).
We separate what we prove (lower bounds + pre-registered experiments) from what we conjecture (the field-level convergence trend). We report the first pre-registered small-scale tests under criteria frozen before the data existed:
the predicted scissors gap is measured — exact-retrieval error 0.994 → 0.000 once the same 64-scalar state gains one global-attention layer;
the S5 state-tracking bifurcation lands at the registered boundary;
alignment rises with scale while capability stratifies by access structure;
a conjunction witness shows an irreducibly two-channel solution.
One prediction (channel commensurability) failed, with its direction reversed — reported as such. Full scorecard: 11 supported / 7 partial / 1 failed of 19.
Code, pre-registration protocol (frozen 2026-07-10), and result summaries:
https://github.com/wenhui-ml/Capability-Convergence-Hypothesis
Feedback and adversarial replication of the frozen falsification clauses are very welcome.
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