Nonreciprocal Spherical Attention and Retrieval: Sharp Coarsening Clocks and Certified Score Correction
Abstract
We develop nonreciprocal spherical attention as a mathematical framework for coarsening and compressed nearest-neighbor retrieval. For two shrinking acute blocks whose centers have zero skew pairing and whose total mass ratio is subexponential in inverse temperature, we prove limβ→∞β−1log Tβ=1−cosϕ, without restricting particle-count growth. An explicit three-body flow has two rigorously separated metastable exponents. For finite compression, mass centering yields a sharp O(r2) attention-output error bound at fixed temperature, with an explicit spherical-normalization correction. We formulate skew query-side score calibration as a strictly convex moment problem and derive a unique normal equation, residual-based optimization certificates, and bounded low-rank projections. A norm-aware margin objective calibrates native squared-distance rankings. Experiments use semantic retrieval data, one million visual descriptors with three codebook seeds, and a ten-million-vector FAISS IVF-PQ index with recomputed exact cosine ground truth. On SIFT1M, skew margin calibration raises native-L2 hit@10 from 0.8926 to 0.8948; symmetric calibration reaches 0.8996. The ten-million-vector index reaches 0.8827 hit@10 at 0.1794 ms per query in a four-thread batch benchmark, while validation selects no skew update. Exact streamed attention over ten million keys evaluates compressed memory on 1,024 held-out queries. Independent matrix solves, automatic differentiation, finite differences, interval arithmetic, full-flow tests, and nine Rocq-checked algebra lemmas support reproducibility. Directionality alone does not establish retrieval superiority.
Related
- Beyond Multiplicity Correction: Sharp Retrieval Thresholds for Attention with Correlated Distractors
- Efficient estimation and the cost of complete-case coarsening under monotone sequential MAR
- Formation of clusters and coarsening in weakly interacting diffusions
- An Energy-Stable Variable-Step SAV Finite Element Method for a Droplet Thin-Film Coarsening Model with a Lennard–Jones Singular Potential
- Graph Coarsening Enables Escaping Local Optima in Compute-Constrained Nonconvex MAX-CUT
BibTeX
@misc{alpay2026nonreciprocal,
title = {{Nonreciprocal Spherical Attention and Retrieval: Sharp Coarsening Clocks and Certified Score Correction}},
author = {Faruk Alpay and Taylan Alpay},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02237},
eprint = {2611.02237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02237},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonreciprocal Spherical Attention and Retrieval: Sharp Coarsening Clocks and Certified Score Correction
AU - Faruk Alpay
AU - Taylan Alpay
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02237
UR - https://arxiv.org/abs/2611.02237
AB - We develop nonreciprocal spherical attention as a mathematical framework for coarsening and compressed nearest-neighbor retrieval. For two shrinking acute blocks whose centers have zero skew pairing and whose total mass ratio is subexponential in inverse temperature, we prove lim_β→∞β^−1log T_β=1−cosϕ, without restricting particle-count growth. An explicit three-body flow has two rigorously separated metastable exponents. For finite compression, mass centering yields a sharp O(r^2) attention-output error bound at fixed temperature, with an explicit spherical-normalization correction. We formulate skew query-side score calibration as a strictly convex moment problem and derive a unique normal equation, residual-based optimization certificates, and bounded low-rank projections. A norm-aware margin objective calibrates native squared-distance rankings. Experiments use semantic retrieval data, one million visual descriptors with three codebook seeds, and a ten-million-vector FAISS IVF-PQ index with recomputed exact cosine ground truth. On SIFT1M, skew margin calibration raises native-L2 hit@10 from 0.8926 to 0.8948; symmetric calibration reaches 0.8996. The ten-million-vector index reaches 0.8827 hit@10 at 0.1794 ms per query in a four-thread batch benchmark, while validation selects no skew update. Exact streamed attention over ten million keys evaluates compressed memory on 1,024 held-out queries. Independent matrix solves, automatic differentiation, finite differences, interval arithmetic, full-flow tests, and nine Rocq-checked algebra lemmas support reproducibility. Directionality alone does not establish retrieval superiority.
ER -