Actually, Collatz DOES require working under these constraints if staying in arithmetic. The conjecture itself needs universal quantification ("for ALL numbers...") to even state it. In pure arithmetic we can only verify specific cases: "4 goes to 2", "5 goes to 16 to 8 to 4 to 2", etc.

The proof requires:

1. The concept of rational numbers (not just natural numbers)
2. Proof by contradiction (logical structure above arithmetic)
3. Divisibility properties beyond basic operations

We can only use counting and basic operations (+,-,×,÷) in pure arithmetic. Any examples that stay within those bounds?

I am looking for a counter example - one that doesn't go above the arithmetic level for both system and level of proof - can you name any?

While you're correct that arithmetic operations can be derived recursively from succession, the paper's core insight isn't about formal derivability. Rather, it suggests some mathematical behaviors are "irreducible" - they arise directly from how properties interact rather than from deeper patterns waiting to be discovered. This may explain why certain simple-looking conjectures resist proof: we're seeking deeper explanations when the behaviour itself IS the fundamental interaction.