Monico (2024) · Theorem 3.4

# UAT Contradiction Machine

Step through the proof by contradiction: assume a gap \(\alpha > 0\) exists, then watch it get defeated.

Theorem 3.4: \(\mathcal{N}_4\) is dense in \(C(K)\). If \(\inf_{f \in \mathcal{N}_4} \|f - T\|_u = \alpha > 0\), we build \(f = \hat{f} - \alpha H + \alpha/2\) with \(\|f - T\|_u < \alpha\) — contradiction.
Target T(x) and approximation \(\hat{f}(x)\)
Separator H(x) and corrected f(x)
Target function T
Approximation quality
Neurons = 4
\(\alpha = \inf \|f - T\|_u\) (assumed gap)
—
Proof steps
Phase 1: Pick \(\hat{f} \in \mathcal{N}_4\) with \(\alpha \le \|\hat{f} - T\|_u < \frac{4\alpha}{3}\).
\(\|f - T\|_u < \alpha\) but \(f \in \mathcal{N}_4\) — CONTRADICTION!