Function adjoint_output_gradient 

pub fn adjoint_output_gradient<R, O>(
    linearization: &AcceptedOutputLinearization<'_, R, O>,
    output_cotangent: &[f64],
    properties: LinearOperatorProperties,
    solver: LinearSolveRequest<'_>,
) -> Result<AdjointGradient, Diagnostic>
where
    R: LinearizedRelation<f64> + ?Sized,
    O: LinearizedOutput<f64> + ?Sized,

Pull one complete output cotangent back through an accepted implicit relation.

The output projection first forms paired O_w^T c and O_p^T c; the ordinary adjoint path then computes the total design cotangent. This keeps eliminated-boundary and other direct output terms paired with the same projection used by forward mode.

Errors

Returns EQ0704 if relation and output layouts differ, and preserves output, relation, and transposed-solver diagnostics.