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

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[Submitted on 14 Jul 2026]

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

View a PDF of the paper titled Capability from Access Structure, Not Scale: Lower Bounds and Pre-Registered Tests for Hybrid Sequence Models, by Wenhui Chen and 3 other authors

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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.

Comments: 41 pages, 16 figures. Pre-registered small-scale experiments (scorecard: 11 supported, 7 partial, 1 failed). Information-theoretic lower bounds plus controlled trained witnesses. Jianlin Chen and Ziyao Lin contributed equally. Corresponding author: Chi Man Vong ([email protected] http URL)

Subjects:

Artificial Intelligence (cs.AI); Information Theory (cs.IT)

ACM classes: I.2.6; I.2.7

Cite as: arXiv:2607.14144 [cs.AI]

(or arXiv:2607.14144v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2607.14144

arXiv-issued DOI via DataCite

Submission history

From: Wenhui Chen [view email] [v1] Tue, 14 Jul 2026 05:52:12 UTC (483 KB)

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