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One arithmetic aside: Apéry’s recurrence contains a nonclassical arithmetic universe over ℤ[i].
Start with Apéry’s elementary sequence from the proof that ζ(2) is irrational:
aₙ = ∑ₖ₌₀ⁿ C(n,k)² C(n+k,k)
1, 3, 19, 147, 1251, …
Its Cauchy square is another integer sequence:
bₙ = ∑ₖ₌₀ⁿ aₖaₙ₋ₖ = 1, 6, 47, 408, 3745, …
The recurrence has a canonical continuation between integers. At two half-integers:
b(−½) = 0.508836067461604848…
b(−3⁄2) = 1.005777774355768242…
These values are periods of a weight-4 Bianchi modular object over the Gaussian integers ℤ[i] = {a + bi : a, b ∈ ℤ}. Its L-function satisfies:
L(2) = π²⁄20 · L(1)
Every tested prime p ≡ 3 (mod 4) gives zero; split primes give two distinct integer fingerprints. The pattern survives hundreds of primes.
The identification is proved using exact recurrences and geometric cycle calculations; the reconstructed functional equation and prime-by-prime matches independently check it.