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

Theorem: Regulator-stable failure of descent

The unchanged free-product obstruction › Spatial Fourier regularization · lines 835–856 · proof included or linked

III@2#thm:regulated-gap

For every family $G_{ρ} : P_{0} \to C$ ,

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

For fixed real $κ \neq = 0$ , the bound tends to $∣ κ ∣ J /2 > 0$ .

Proof. The symbol difference is $- κ J_{ρ}$ , while the quotient labels remain identical. Apply Lemma lem:two-point and the convergence lemma.

Displayed formulas in this passage

unlabeled · line 838
$max {∣ G_{ρ} ([0], ϕ) - F^{P}_{κ, ρ} (f; 0, ϕ) ∣, ∣ G_{ρ} ([0], ϕ) - F^{P}_{κ, ρ} (f; K ξ, ϕ) ∣} \geq \frac{∣ κ ∣∣ J _{ρ} ∣}{2} .$