Manuscript III@2 · III.tex · corollary · 82dad3e5

Corollary: Nonvanishing approximation error

The unchanged free-product obstruction › A fixed nonzero gap · lines 752–770 · proof included or linked

III@2#cor:product-error

For the fixed data above and any $G : P_{0} \to C$ ,

max {∣ G ([0], ϕ) - F^{P}_{κ} (f; 0, ϕ) ∣, ∣ G ([0], ϕ) - F^{P}_{κ} (f; K ξ, ϕ) ∣} \geq \frac{∣ κ ∣ J}{2} .

Agreement to $o (∣ κ ∣)$ on both representatives is impossible.

Proof. Apply Lemma lem:two-point.

Displayed formulas in this passage

unlabeled · line 755
$max {∣ G ([0], ϕ) - F^{P}_{κ} (f; 0, ϕ) ∣, ∣ G ([0], ϕ) - F^{P}_{κ} (f; K ξ, ϕ) ∣} \geq \frac{∣ κ ∣ J}{2} .$