I’m sharing SÆLQR II, an information-theory preprint about recovering information from unordered, noisy fragments.

The model starts with a stored string. It is fragmented, some fragments disappear, and the surviving fragments are corrupted. The receiver gets their contents and lengths, but their original positions are hidden.

The central problem is accounting for the information cost of losing order, alongside the information lost through noise.

The manuscript reports three main developments:

• Capacity with memory inside fragments. For the stated weakly symmetric word channels and message-independent fragmentation laws, capacity depends on each fragment’s realized information after a location penalty. Noise may have arbitrary dependence inside a fragment, while noise between fragments remains independent. Hidden-quality examples show why replacing the information distribution with an average capacity can give an incorrect prediction.
• Exact behavior near the critical threshold. For shuffled binary symmetric channels, a uniform soft-threshold approximation yields a Gaussian crossover and an explicit operational leading coefficient. At the central critical point, the number of communicable bits scales as N/√log₂N, even though the normalized first-order rate vanishes. The manuscript also determines the optimal leading correction in a uniform entropy-cancellation bound.
• Constructive coding. For geometric fragmentation, independent whole-fragment loss, and binary substitutions, the release supplies a randomized polynomial-time capacity-approaching construction, finite reliability conditions, and reference encoding/decoding software.

The computational qualification matters: the algorithms are polynomial in blocklength and inverse capacity gap, but the proved sufficient blocklength is exponential in the inverse gap squared. Efficient codes attaining the exact critical coefficient, deterministic good-design certification, and practical finite-length optimization remain open.

The package includes a 69-page manuscript, editable source, reference software, twelve finite reproduction programs, and proof-dependency and novelty audits.

I’d particularly welcome technical scrutiny of the reverse-divergence covering argument, the internal-memory converse, and the critical-window derivation, along with pointers to closely related noisy-shuffling or DNA-storage literature.

Manuscript, code, and audits:
https://huggingface.co/datasets/PureOne/SAELQR-II

SÆLQR II: General Capacity Theorems and Exact Critical Laws for Unordered Noisy Fragments - Internal Memory, Sharp Repeated-Read Thresholds, and Constructive Coding | Zenodo