Entries
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lax-100001Reconstructing Binary Words of Unknown Length with #-Subword Queries
A deterministic guesser reconstructs a hidden binary word of unknown length n from counts of scattered subword occurrences. Unrestricted queries suffice in O(sqrt(log n log log n)) queries. Queries restricted to Lyndon words for 0 < 1 suffice in O(log n) queries. Both bounds count the initial letter queries and allow unrestricted computation between queries. A generic reduction reconstructs a word over a finite alphabet by querying each letter and running the binary strategy on every pair projection. It preserves the binary query restrictions under relabelling and gives the same asymptotic bounds for each fixed ordered alphabet. The proofs are self-contained and formalized in Lean.
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lax-100002Lower Bounds for the Real Blocky Rank of Boolean Matrices
Boolean matrices can require quadratically many real blocky summands in the ratio of their logarithm to their iterated logarithm. For every integer N ≥ 2^512, we construct a Boolean N × N matrix whose real blocky rank exceeds (log₂ N / log₂ log₂ N)² / 65536. The proof combines a distance obstruction for short blocky decompositions, polynomial evaluation over a binary finite field, and a transfer of distance from binary to real inputs through integer minors. A deterministic field tower gives a computable entry algorithm. Lean checks the field representation, the algorithm's correctness, and its matrix rank bound. It also checks a polynomial bound for complete entry queries from binary inputs in a model that counts bit operations.