A: Great question. First of all let me start out by saying that I can't prove that statement to you. While I am willing to share with you why I believe what I believe there is no way I can prove those beliefs. So I hope that even if we disagree we can at least appreciate one another's faith.
Now I also want to tackle another statement you made.
"Please don't refer to any other writings besides the Bible to explain this. I believe that it is cults that use other writings besides the Bible to prove their doctrines (heresies)."
If this is true then all churches are cults that teach heresy. The reason why is because when we are dealing with apocalyptic literature such as Daniel and Revelation, there is just no way of fully understanding it without using extra-biblical historical resources. Adventists are Historicists in their interpretation of scripture and as such we use extra-biblical historical resources to help us understand the events that the prophecies are delineating. If this approach makes the SDA church a heretical cult then Victorinus, Arnuf of Orleans, Eberhard II (all Catholic) and Martin Luther, John Calvin, Isaac Newton (all protestant) are also heretics for they all followed the Historicists method of interpreting prophecy. In addition, even if you are not a Historicist but are instead a Futurist or a Prederist (the majority of Christendom) you still have to use extra-biblical historical sources to explain the prophecies. So I conclude that when the SDA church uses historical writings and calendars that are extra-biblical in order to arrive at their understanding of 1844 they are simply doing what every body else does and are thus not a cult and not heretical.
What would make us a cult is if, like the Mormons, we relied only on Ellen White or other pioneers such as Uriah Smith in order to teach the 1844 doctrine. But this is not the case. 1844 was understood by William Miller and many others without the help of Ellen White (she hadn't even begun her ministry yet) and the consequent revelations of the sanctuary being in heaven do not necessitate EGW to be understood. In short, SDA's don't need EGW for any of our doctrines. They stand on the Bible alone. In addition, the SDA church does not put EGW next to James, Paul, Peter or any other Biblical writer. We place her beneath them. She did the same when she was alive. While there are some SDA's who treat her as though she was scripture this goes against the church's stance and even against EGW's own position. We consider her writings to be authoritative and inspired but they do not interpret scripture for us, they do not replace scripture for us, and they sure don't have an equal standing with scripture.
Now onto your question. Like I said, I cant prove it but I will simply explain to you why I believe it is true.
Most Christians teach that the judgment took place on the cross. Others teach that it will take place at the second coming. There are numerous NT verses that show the apostles were looking forward to a future judgment. This was after the cross, therefore, biblicaly speaking the judgment did not take place at the cross. Then there are the passages that say that when Jesus comes he comes to award both the righteous and the wicked including those who have not seen death (others teach the judgment happens at death). This shows us that a decision was reached before the second coming. Then there is Revelation 14:6, a message that is pronounced before the second coming and says that the judgment has already begun. Thus, the judgment did not happen at the cross and it will not happen at the second coming but since Jesus comes with rewards it cant happen after the second coming either. So the message is clear, the judgment happens sometime between the cross and the second coming. 1844 happens to between those two events. (see below)
Now that doesnt prove the date per se, in order to come to the actual date October 22, 1844 we would have to interpret Daniels prophecies using the Historicist method of interpretation. To do that would take so long that Im not going to do it here. But at this point Adventists establish the start time of the prophecy which Gabriel gives to Daniel and we count the 2300 years and we arrive at 1844. Using more complex historical sources such as calendars etc. we arrive at the October 22 date. None of this necessitates EGW or any other Adventist pioneer. If you are really interested in reading about the evidence for this then I recommend Clifford Goldsteins "1844 Made Simple" and Marvin Moores "The Case for the Investigative Judgment." Not that they can prove it either, but at least you will be able to see two very well written books on the topic that do not use EGW at all.
Anyways, understanding that the sanctuary of Daniel is the sanctuary in heaven (of course we are not suggesting that there is some building in heaven that looks just like the OT sanctuary)we then conclude through a systematic study of the sanctuary, its type and anti-type etc. that Jesus began his final work of cleansing the sanctuary in 1844. If you want to read my views on the theology of the Investigative Judgment then I recommend my paper (The Investigative Judgment)...
Honestly K, I am not a stickler on October 22, 1844. While I agree its the best interpretation of the time line I don't get hung up on it. The judgment could have begun in 1922 for all I care. The date doesn't change the theology of the judgment at all. The point is that God is now doing his final work on behalf of man, he is trying to get as many people into heaven before time runs out for us, and Jesus is coming soon.