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Catalan Conjecture 7 printed-clause refutation

Abstract

Refutes only the printed last clause of Conjecture 7 in arXiv:1811.00248v2 at t = 2, n = 1. The values are literature-attested; new_counterexample_claimed = false. No claim is made against the intended proposition, its proof, or Cigler’s original arXiv:1801.05608 statement, whose two terms both use the fifth power here. No priority is claimed for the values or identification of the printing error.

Wang and Xin, Hankel determinants for convolution powers of Catalan numbers, arXiv:1811.00248v2, printed page 3, Conjecture 7, has second argument 2t in the second Hankel determinant of its last line. The other power in that line is 2t + 1. The original PDF was visually checked on 2026-09-10. The scope sentence gives odd positive r = 2t + 1 and no lower bound on t or n. The formal claim universally quantifies over positive t and all natural n, and retains the printed second argument 2t.

Theorem 1.1 (The printed last clause is false).

Lean statement: D5/S0/Certificates/CatalanConjectureSevenRefutation.result

Proof. Machine-checked in Lean as D5/S0/Certificates/CatalanConjectureSevenRefutation.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

At t = 2, n = 1 the left side is H4 of the fifth Catalan power plus H7 of the fourth Catalan power. Their exact values are 5 and minus 4, so the left side is 1 whereas the printed right side is minus 5. The proof reuses the frozen unshifted Hankel definition and defines Catalan powers by Cauchy multiplication. All finite tables are local to the proof. A kernel-checked integer identity L times A equals U, with triangular determinants minus 6 and 24, certifies H7 equals minus 4. Independent integer computations by convolution and Bareiss, and by recurrence and Leibniz expansion, agree; all 14 positive controls from the same paper’s Theorems 2 and 3 agree as well. These numerical values are literature-attested: Theorems 2 and 3 on printed page 2 already imply them. Reference 9 is Cigler, Catalan numbers, Hankel determinants and Fibonacci polynomials, arXiv:1801.05608. In its v3, printed page 26, Conjecture 7.2, equation 7.6, both powers are 2k + 1. At k = 2, n = 1 both terms therefore use the fifth power, and 5 plus minus 10 equals minus 5, consistently. The refutation concerns only the 1811.00248v2 printed last clause at t = 2, n = 1. It does not assert that the intended proposition is false or that its proof has a gap, nor that Cigler’s corresponding original statement is false. No first discovery of these determinant values or first identification of the printing error is claimed; new_counterexample_claimed = false.

References