Intrinsic Tangential Hadamard Differentiability of Rough-BSDE Solution Maps

We study first-order sensitivity of a scalar backward stochastic differential equation with a deterministic rough driver. The driver belongs to the nonlinear space of step-two weakly geometric p-rough paths, 2 < p < 3, so an ordinary Banach-space difference quotient is not available. At a fixed rough path x, we use a weakly geometric, finite-(p,p/2)-variation tensor realization of the Qian-Tudor tangent structure, represented intrinsically by a first-level direction h and a compatible second-level direction kappa. Admissible rough-path secants are required to converge in a strong levelwise variation topology. Under bounded smooth rough vector fields, a bounded terminal condition, and a globally Lipschitz generator, we construct a continuous linear map from the tangent fibre into the bounded-solution and BMO-integrand spaces. The proof first establishes a uniform four-jet expansion for reset rough flows along bounded realizations of full rough tangents. A Doss-Sussmann transformation transfers this expansion to quadratic generators. Uniform BMO and reverse-Holder estimates then yield a local difference-quotient theorem, which is propagated over a fixed deterministic partition and reconstructed in the original coordinates. The resulting derivative is independent of the joint lift, central decomposition, and radial realization. Consequently, solution difference quotients converge for varying directions and arbitrary admissible secants with strong levelwise variation contact. This is intrinsic tangential Hadamard differentiability on that tensor-coordinate tangent class. Locally bounded ray-homogeneous selections give Frechet-differentiable chart pullbacks at the parameter origin. The latter statement is chartwise; it is not Frechet differentiability of the solution map in the homogeneous rough-path metric.