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Two Squares Canonical Factorization

We present a new combinatorial structure in a string: a canonical factor- ization for any two squares that occur at the same position and satisfy some size restrictions. We believe that this canonical factorization will have application to re- lated problems such as the New Periodicity Lemma, Crochemore-Rytter Three Squares Lemma, and ultimately the maximum-number-of-runs conjecture.