Determine whether the preset-history sandwich factor 2 is attainable

Determine whether the factor 2 in the sandwich inequality relating adaptive covering complexity with a single external reference to non-overlapping preset-history LZ77 phrase count is attained by any reference string and target string.

Background

Theorem 3 establishes that, for a single reference string R, the adaptive covering complexity with dictionary {R} lies between the corresponding preset-history non-overlapping LZ77 phrase count and twice that count. The paper does not determine whether the upper factor 2 is sharp. It reports only that exhaustive searches over binary references of length at most 5 and targets of length at most 8 found no ratio exceeding 5/3.

References

We do not know whether the factor~$2$ is attained; in an exhaustive search over binary $R$ and $s$ with $|R|\le5$ and $|s|\le8$ the ratio never exceeded $5/3$.

— On the Adaptive Covering Complexity of Strings: Unifying Relative and Self-Referential Compression  (2610.26021 - Marteau, 19 Oct 2026) in Remark following Theorem 3 ("Sandwich with preset-history LZ77"), Section 4.4

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