Cannot use ref or out parameter in lambda expressions

You can but you must explicitly define all the types so

(a, b, c, ref d) => {...}

Is invalid, however

(int a, int b, int c, ref int d) => {...}

Is valid

Under the hood, the anonymous method is implemented by hoisting captured variables (which is what your question body is all about) and storing them as fields of a compiler generated class. There is no way to store a ref or out parameter as a field. Eric Lippert discussed it in a blog entry. Note that there is a difference between captured variables and lambda parameters. You can have "formal parameters" like the following as they are not captured variables:

delegate void TestDelegate (out int x);
static void Main(string[] args)
{
    TestDelegate testDel = (out int x) => { x = 10; };
    int p;
    testDel(out p);
    Console.WriteLine(p);
}

Lambdas have the appearance of changing the lifetime of variables that they capture. For instance, the following lambda expression causes the parameter p1 to live longer than the current method frame as its value can be accessed after the method frame is no longer on the stack

Func<int> Example(int p1) {
  return () => p1;
}

Another property of captured variables is that changes to the variables are also visible outside the lambda expression. For example, the following code prints out 42

void Example2(int p1) {
  Action del = () => { p1 = 42; };
  del();
  Console.WriteLine(p1);
}

These two properties produce a certain set of effects which fly in the face of a ref parameter in the following ways:

• ref parameters may have a fixed lifetime. Consider passing a local variable as a ref parameter to a function.
• Side effects in the lambda would need to be visible on the ref parameter itself. Both within the method and in the caller.

These are somewhat incompatible properties and are one of the reasons they are disallowed in lambda expressions.