Exponential memory lifetime for the alternative three-dimensional construction

Prove exponential memory lifetime for the alternative three-dimensional Euclidean passive-memory construction with blow-up factor R greater than or approximately equal to 8^4, thereby independently verifying its self-correcting properties and establishing the associated contraction, locality, and noise-stability guarantees.

Background

The paper discusses the three-dimensional passive-memory construction of Ref. \cite{balasubramanian2026passive}, whose probabilistic embedding argument requires an impractically large blow-up factor. That reference proposes an explicit alternative with a much smaller blow-up factor, R greater than or approximately equal to 8^4, corresponding to subdivision scale 8.

A complete proof that this alternative has exponential memory lifetime is not provided. The paper identifies independent verification of its self-correction, including transparent proofs of contraction, locality, and noise stability, as a concrete unresolved task.