Proposition: Single entry, with its sign

No other multiple of ${\mathcal{C}}_{3,\mathrm{D}}$ added to ${{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,0;\varphi )$ is of the form ${{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,k[\gamma ];\varphi )$ with $k[\gamma ]$ the gauge-fixed gravitational coefficient of Proposition prop:gfcoeff.

Proof. Lemma lem:affine with $k=0$, $r=r_{\mathrm{D}}[\gamma ]=-u_{\mathrm{D}}[\gamma ]$ and linearity of ${\mathcal{A}}_1$ give ${{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,r_{\mathrm{D}};\varphi )={{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,0;\varphi )+{\mathcal{A}}_1(-u_{\mathrm{D}};\varphi )={{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,0;\varphi )-{\mathcal{C}}_{3,\mathrm{D}}$. If ${{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,0)+c{\mathcal{C}}_{3,\mathrm{D}}={{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,k)$ then ${\mathcal{A}}_1(k+c{u}_{\mathrm{D}};\varphi )=0$ for all $f,g,\varphi$; with $k=-u_{\mathrm{D}}$ this forces $(c-1){\mathcal{A}}_1(u_{\mathrm{D}};\varphi )=0$, and ${\mathcal{C}}_{3,\mathrm{D}}$ is not identically zero (XII-D's admission theorem gives a nonzero kernel), so $c=1$ in ${\mathcal{A}}_1(-u_{\mathrm{D}})$, i.e.\ the coefficient of ${\mathcal{C}}_{3,\mathrm{D}}$ is $-1$.