# Proof that the Blank Tape Halting Problem is undecideable [duplicate]

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:
1. Write $$w$$ on the input tape.
3. Transfer control to $$T$$.
It is not too hard to construct $$R$$ explicitly from $$T$$.