Characterize trace-length saturation from representation geometry

Characterize the trace-length saturation boundary for frozen-trace certificates in terms of representation geometry alone.

Background

The paper develops deterministic coupling and union-bound certificates for Vamana selection traces, but reports that these certificates can saturate at one on long paths. The unresolved issue is whether the point at which this saturation occurs can be predicted solely from properties of the underlying vector representation, rather than from explicitly enumerated trace decisions and their margins.

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”