Super Higher-Teichm\"uller Geometry and Loop Amplitudes

We construct a supersymmetric extension of the Fock-Goncharov cluster ensemble associated with a split basic classical Lie supergroup $G$ and a marked bordered surface $S$. The resulting structure defines a super higher-Teichm\"uller geometry: a split super--thickening of $(\mathscr A_{G,S}, \mathscr X_{G,S})$ equipped with a mutation atlas preserving a canonical super log-symplectic form. Each super seed carries an integer weight matrix $W$ encoding Cartan weights of an abelian odd slice, transforming by the column $g$--vector rule and giving rise to a flat logarithmic superconnection and a canonical super volume form. On this geometric foundation we define a canonical logarithmic superform $\Omega_{\mathrm{super}}^{(L)}$ on a loop fibration $\pi_L : \mathscr X^{(L)}_{G,S} \!\to\! \mathscr X_{G,S}$ as the relative lift of the base super volume. For $G = PGL(4|4)$, the corresponding super period $P_{\mathrm{super}} = \int_{C} \Omega_{\mathrm{super}}^{(L)}$ encodes the loop amplitude data of planar $N = 4$ super Yang--Mills, expressed through a unified and triangulation-independent formula that satisfies Steinmann and cluster adjacency, with the even sector given by Chen iterated integrals and the odd sector captured by an invariant BCFW delta.