bibkey: barlev2026blockretrieval authors: Daniella Bar-Lev year: 2026 title: “Coded Information Retrieval for Block-Structured DNA-Based Data Storage” doi: 10.48550/arXiv.2603.17154 url: https://arxiv.org/html/2603.17154v2 claim: “Conjecture 1 asserts the universal hyperbolic tradeoff s1/E1+s2/E2<=1; Conjecture 2 asserts weak Pareto improvement from length n to n+1. Both use actual minimum iid uniform whole-file retrieval and positive partitions with maximum size at least two.” strata_touched:

• D5/S3/Resource/MinimumRetrievalTime
• D5/S3/Resource/VandermondeHyperbolicRefutation license: citation-only triage: anchor

Block-structured coded retrieval

Verified locator

DOI: 10.48550/arXiv.2603.17154. URL: https://arxiv.org/html/2603.17154v2 Scope: arXiv v2, Section II-A, Section V-B Conjecture 1, and Section VII-A Conjecture 2.

Source contract

The source is arXiv:2603.17154v2, Section II-A, Definitions 1 and 2, equation (5), and Section VII-A, Conjecture 2. Its field is the finite field F_q, with q a prime power. A generator has rank k, and its physical column indices are sampled independently, uniformly, and with replacement. Recovering a file means that the sampled-column span contains every standard basis vector belonging to that file. The two files have positive dimensions s1+s2=k.

For n>=k and max(s1,s2)>=2, Conjecture 2 asks for a rank-k length-n+1 generator whose two expected minimum retrieval times are each at most those of any given length-n generator. The successor is arbitrary: it need not append the old generator, be systematic, or be file-dedicated. Repeated columns remain distinct sampling indices, and the stated rank condition does not exclude zero columns. Exact equality of expected pairs is not the target. Remark 10 excludes the partition (1,1); it does not exclude (1,2).

Formal interface

MinimumRetrievalTime.claim quantifies over finite field structures and all positive two-file sizes, legal lengths, and full-rank matrices, encoded by their indexed columns. Matrix.rank_eq_finrank_span_cols and Submodule.eq_top_of_finrank_eq connect matrix rank with full column span. The coordinate-file definitions use the standard coordinate basis and the exact index cuts <s1 and >=s1.

The expectation in the formal statement is the Bochner integral of the minimum stopping time under the infinite product of uniform finite-index measures. The probability bridge proves measurability, the exact tail identity, finite expectation, integrability, and equality with the nonnegative integral; no subset-count formula is taken as a definition or axiom.

VandermondeHyperbolicRefutation.claim uses the same finite-field model, coordinate files and actual stopping-time expectations for Conjecture 1. Its scope is every full-rank generator, not just systematic or file-dedicated generators. A source-faithful refutation therefore needs one admitted code and its actual file expectations, rather than a universal inequality for a restricted class.

Literature boundary

The source v2 retains Conjecture 2. The nearby Vlachos–Bar-Lev preprint arXiv:2609.36067v1 concerns hyperbolic bounds (Conjecture 1), not the length-monotonicity assertion. Exact-id and title/topic searches found no earlier Conjecture 2 resolution in the searched scope. This is bounded negative evidence, not an exhaustive priority claim.