Hi! My name is Grisha Pochuev. This is a beautiful and inspiring conjecture, but unfortunately I have found a counterexample to its dimension-free linear formulation.
The construction is quite simple: let (X_1=S) be a uniformly random (n)-bit string, and let (X_2) reveal the whole string except with probability (\varepsilon), when the entire string is erased. Every individual bit (S_j) is then an admissible (\varepsilon)-redund. Approximate maximality forces (\Omega) to retain information about all (n) bits, including on the erasure event, which requires an error of order (n\varepsilon), contradicting any universal bound of the form (C\varepsilon).
The proof rules out every pair of fixed finite constants for commonality and maximality. The approximate-determinism requirement is not needed, so the obstruction is somewhat stronger than required.
I would be very grateful if you could take a look and let me know whether you agree with the argument and how you assess it under the counterexample-bounty provision.
@johnswentworth @David Lorell
Hi! My name is Grisha Pochuev. This is a beautiful and inspiring conjecture, but unfortunately I have found a counterexample to its dimension-free linear formulation.
The construction is quite simple: let (X_1=S) be a uniformly random (n)-bit string, and let (X_2) reveal the whole string except with probability (\varepsilon), when the entire string is erased. Every individual bit (S_j) is then an admissible (\varepsilon)-redund. Approximate maximality forces (\Omega) to retain information about all (n) bits, including on the erasure event, which requires an error of order (n\varepsilon), contradicting any universal bound of the form (C\varepsilon).
The proof rules out every pair of fixed finite constants for commonality and maximality. The approximate-determinism requirement is not needed, so the obstruction is somewhat stronger than required.
Preprint: https://doi.org/10.5281/zenodo.21478853
Code and verification materials: https://github.com/Grisha-Pochuev/wentworth-lorell-common-information-counterexample
Email: n_854@mail.ru
Telegram: https://t.me/just_impossible
I would be very grateful if you could take a look and let me know whether you agree with the argument and how you assess it under the counterexample-bounty provision.