Close the constant-factor gap in preset-history bounds

Establish whether the constant-factor gap in the preset-history LZ77 sandwich bounds can be reduced or closed for adaptive covering complexity with a single external reference.

Background

The paper proves a factor-2 upper bound between adaptive covering complexity with a single reference and preset-history non-overlapping LZ77 phrase count, but does not establish that this factor is optimal. The conclusion explicitly identifies closing this constant-factor gap as an open direction.

References

Several avenues remain open for future work: closing the gap on the constant factor in preset-history bounds, extending the sandwich theorems to multi-word dictionaries, and investigating the precise bit-cost analogues of the logarithmic law under optimized pointer representations.

— On the Adaptive Covering Complexity of Strings: Unifying Relative and Self-Referential Compression  (2610.26021 - Marteau, 19 Oct 2026) in Conclusion, Section 6