Extend trace coupling to adaptive candidate sets

Establish whether adaptive candidate sets, including beam-search procedures, admit a martingale extension of the frozen-trace coupling argument.

Background

The trace-coupling theorem applies when the approximate replay uses the same candidate permutation as the exact execution, with the execution trace frozen. Adaptive candidate generation, such as beam search, changes the candidate set in response to earlier approximate decisions, so the paper leaves unresolved whether a martingale-based argument can extend the coupling framework to this setting.

References

Whether the trace-length saturation boundary can be characterized from representation geometry alone; whether adaptive candidate sets (beam search) admit a martingale extension of the coupling argument; and whether a pilot selector choosing rotation, bit width, and verification cutoff from covariance statistics alone recovers exact-score topology.

— When Does Low-Bit Quantization Preserve the Decisions of Vector Search?  (2609.09854 - Xiao et al., 9 Sep 2026) in Section 8, Discussion: Practical Guidance, paragraph “Open directions”