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-100002Superlogarithmic Real Blocky Rank
Boolean matrices can require more than logarithmically many blocky summands even when arbitrary real coefficients are allowed. We give a self-contained proof for a family obtained by polynomial evaluation over a binary finite field followed by binary Hadamard encoding. A distance argument, a count across disjoint degree intervals, and an integer-minor argument give real blocky rank of order at least log N times log log N along the constructed sequence of square dimensions N. The Lean formalization proves the finite bound and its superlogarithmic consequence without an external graph theorem. The field basis is chosen noncomputably, so the result makes no claim about efficient entry computation.