Catalan’s constant G is the alternating sum 1/1^2 − 1/3^2 + 1/5^2 − 1/7^2 + ···. Whether this number is irrational has been an open question for more than a century. In the preprint summarized here, Zhi‑Wei Sun (School of Mathematics, Nanjing University) presents a proof that G is irrational. The paper says the proof uses specially chosen ‘‘weights’’ built from the tails of the defining series.

At a high level the author builds finitely many linear combinations of the tail sums of the Catalan series. For an index m the tail T_m is the remainder of the alternating sum starting at term m, and the weighted tail u_m is T_m divided by 2m+1. Those weighted tails are arranged into a finite matrix of ‘‘residual’’ sums that encodes the basic recurrence enjoyed by the tails. A key step is to show that a suitably chosen matrix minor has full column rank. From that nonvanishing minor the paper constructs a rational scalar and an associated integer ‘‘integerizer’’ that tie the matrix data to a number built from G.

The proof then combines exact determinant identities with arithmetic estimates. Using a Cauchy–Binet expansion, each determinant summand factors into familiar objects: Pascal-type factors, Vandermonde factors, and Cauchy determinants. This algebraic factorization isolates the arithmetic content. The paper next studies prime power contributions directly. A ‘‘local saturation’’ type estimate compares the denominator coming from those determinant factors with a test layer coming from the residual matrix. Putting together local p‑adic lower bounds for every odd prime power and global asymptotic estimates, the author derives a contradiction under the assumption that G is rational. That contradiction is the heart of the claimed irrationality proof.

If correct, the result settles a long-standing classical question about the number G. The approach is notable because it mixes linear algebra on explicit finite matrices, exact determinant identities, and fine prime‑by‑prime arithmetic (p‑adic) estimates. These are concrete algebraic and arithmetic ingredients rather than probabilistic or transcendence tools commonly seen in related problems.