Theorem: Exact scaled-ball identification

For data in Specification spec:rich, put

The rule

is exactly ${B_{g_{s(D)},0}}^{\Gamma }$. Different $s$'s have different same-patch forms and cannot be identified by a patch-fixing isometry.

For the witness family of Proposition prop:nonempty, this rule recovers its parameter $s$.

Proof. For a constant metric scaling in dimension $n$,

The minimizing harmonic extension is unchanged because the energy has been multiplied by a positive constant. Thus in dimension three the DN energy is multiplied by $s$.

Choose a nonconstant smooth patch-supported boundary section. Its Euclidean energy is positive: a zero-energy harmonic extension would be constant. Therefore equality of the forms forces equality of $s$.

A diffeomorphism fixing the open patch has identity derivative on its tangent spaces. A patch-fixing isometry would therefore equate ${s_1}^2\delta$ and ${s_2}^2\delta$ on a nonzero tangential vector, forcing $s_1=s_2$. The formula for the witness family's maximum tag gives its claimed recovery.