Proof that the Blank Tape Halting Problem is undecideable I have seen a few proofs that the Blank Tape Halting Problem is undecideable, however I'd like to check if the following is a valid proof (and if it isn't why not)

Proof: Suppose that the Blank Tape Halting problem is decideable, then there exists a universal turing machine $M$ that decides it. So given any turing machine $T$, $M$ decides if $T$ halts after being started on a blank tape.

Now let any turing machine $T$ and input $w$ be given. Construct a new turing machine $R$ (with start state blank) such that $R$ halts on a blank tape if and only if $T$ halts on $w$.

Then if $M$ takes $R$ as input, $M$ decides if $R$ halts on a blank tape and thus decides if $T$ halts on $w$. Since $T$ and $w$ were chosen arbitratily $M$ decides the Halting problem, a contradiction, so the Blank Tape Halting Problem is undecideable. $\square$.



Is there anything wrong with my reasoning above? I assume that there must be some key detail that I'm overlooking, or that my proof isn't completely rigorous

I guess the problematic argument may be the following 


  Construct a new turing machine $R$ (with start state blank) such that $R$ halts on a blank tape if and only if $T$ halts on $w$.


Since I haven't explicitly constructed such a turing machine $R$ in my proof above.
You need to explain how to construct $R$. In your case this can be done formally, by constructing a machine which works as follows:


Write $w$ on the input tape.
Move the head back to its initial position.
Transfer control to $T$.


It is not too hard to construct $R$ explicitly from $T$.

In order for your argument to go through, you need to show that this reduction is computable. Usually we describe such reductions informally and just claim (without proof) that it is computable, since actually proving such things formally is extremely technical and longwinded.