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arXiv · 2608.24971

HCC+: Hyperbolic Guarding for Certified Attention Retrieval

Liangchen Ge
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Abstract

We study the Lipschitz stability of attention retrieval in hyperbolic spaces. Existing methods lack deterministic guarantees on attention-weight preservation under finite-precision representations. We introduce HCC+, a theoretical framework exploiting three properties of the Poincar\'e ball: exponential volume growth enabling query-independent boundary truncation; logarithmic covering radius of hyperbolic 1-centers enabling dimension-independent critical-key identification; and a packing bound with constants independent of the embedding dimension. We prove two deterministic guarantees: for exact retrieval, the per-layer attention deviation is bounded by 10\% of its ideal value; for soft attention, the total variation distance decays as $O(1/\sqrt{n})$, the rate of finite-sample variance. As a consequence of the guarding mechanism, the framework achieves a storage reduction factor of $6.1\times$ relative to FP16. We provide the first deterministic, query-independent retrieval certificate in non-Euclidean geometry.

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Liangchen Ge. 2026-08-25. HCC+: Hyperbolic Guarding for Certified Attention Retrieval. https://arxiv.org/abs/2608.24971

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