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We have James son of Alphaeus, whose brother is Levi (also called Matthew).

No verse in Scripture calls James, son of Alphaeus, the brother of Matthew/Levi. In fact, only Mark 2 calls Levi/Matthew the "son of Alphaeus." (Alphaeus, incidentally, was a common name; Halpai in Aramaic. It's entirely possible that James and Matthew were the sons of two different men who happened to have the same name.) Both Matthew and Mark list Matthew close to James in their lists of the Twelve; don't you think they'd mention if they were brothers?

There were two James's. Only two. James the Greater, brother of John, and James the Lesser, also known as James the Just. That's why they have those names, "Greater and Lesser". You would not go around talking about "berned the greater" and "berned the lesser" and "berned the other berned" and "berned some fourth berned we haven't told you about yet".

James the Greater was "put to the sword" by Herod around AD 44, as noted in Acts. James the Less/James the Just, bishop of Jerusalem & "brother of the Lord," also author of the Epistle, and "brother" of Joses, Simon the Zealot, and Jude Thaddeus ... he was stoned to death by the Pharisees (one source says he was thrown from the parapet of the temple first). Josephus dates this as happening in AD 62. (That's not what he calls the year, but we can figure it out from the information he gives.)

None of the ancient sources, including the Bible, specifies a third James, or a fourth, or a fifth. Their relics are unknown. Their manner, dates, and places of death are unknown. Nothing is known about them, because they never existed.

There were two James's. Only two. James the Greater, brother of John, and James the Lesser, also known as James the Just. That's why they have those names, "Greater and Lesser".

I forgot about James the Greater! You may be correct about James the less being the same as James the Just. Hard to say though since James the less is used only once in the NT:

Mark 15:40
There were also women looking on afar off: among whom was Mary Magdalene, and Mary the mother of James the less and of Joses, and Salome;

and at this time, Judas and Simon are not mentioned, plus it would seem that it would be mentioned that Mary was the Lord's mother.

Do you know if there are any early church writings that would confirm that James the less is James the Just?