Why Catalan's constant G is irrational

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Why Catalan's constant G is irrational

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Take the determinant proof that Catalan's constant G is irrational (Theorem 1.1) and explain why it works. Assume G is rational, then show that two bounds on the same determinant cannot both hold.

Why the usual route failed

The criterion [0:15]: if G = a/b, every nonzero integer form AG − B is at least 1/b in size. So a proof needs nonzero integer forms that tend to zero.
Example and non-example: Apéry's forms for ζ(2) and ζ(3) keep their decay after the denominators are cleared. Zudilin's Catalan approximations converge fast, but their integer forms do not tend to zero.
Family results prove that some even beta value is irrational, but they never single out β(2) = G.

The determinant argument

Rationality [1:14]: Δ_N has size 48N, and its entries are moments that combine 1, G and ζ(2). Chebyshev rows with high Taylor contact cancel ζ(2). So if G ∈ ℚ, Δ_N ∈ ℚ.
Floor [1:50]: for a nonzero rational, log|Δ| = Σ v_p(Δ) log p, so bounding the denominators bounds |Δ| from below. Three things produce the bound:
- an estimate at the prime 2;
- two denominator layers (p², p) at the odd primes;
- the prime number theorem, which adds up the losses.
Nonvanishing comes from arithmetic, not positivity: for N = p prime, Frobenius reduces the matrix to three fixed matrices, and an exact certificate shows they are invertible. The result is liminf 𝓛_N > −2.29084.
Toy check [2:56]: suppose x is a nonzero rational whose denominator divides 72, so |x| ≥ 1/72. If an integral shows |x| < 1/100, there is no room left for x.
Ceiling [3:23]: with no hypothesis at all, Δ_N is an integral of Vandermonde products with logarithmic interactions. A Chebyshev expansion and certified rational trial coefficients give limsup 𝓛_N ≤ −2.290939875.
Collision [3:57]: along the same sequence of primes, the floor sits about 10⁻⁴ above the ceiling, so G is irrational. This gives no irrationality measure.

Reuse it [4:37]

When you read another determinant irrationality proof, look for four parts:

rationality under the hypothesis
a nonzero determinant
a denominator floor
an integral ceiling

All four live on the n² scale. Then check that the floor beats the ceiling.

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