All we have to determine what 390 and 40 mean is that Judah was freed about 40 year later, therefore we can assume that Israel was freed after 390 years of captivity. However in regard to Israel we don't know who was their captors all those years, and how many were actually in captivity. We just have to guess.
I'm still not following your logic
The northern kingdom was taken captive in around 722 BC, 390 years later would be around 332 BC, so you believe the northern kingdom was freed in 332 BC?
The last vision (ezekiel 40-48) given to Ezekiel was given in the 25th year of exile, 14 years after Jerusalem was destroyed
Ezekiel 40:1
In the twenty-fifth year of our exile, at the beginning of the year, on the tenth day of the month—
That means there was 45 years of exile for Judah when Ezekiel received his final vision, not 40.
Additionally, Ezekiel does not say the punishment is to last 390 years. Ezekiel was to lay on his side for 390 days to represent the 390 years of Israel's INIQUITY.
Ezekiel 4:5
For I have assigned you the years of their iniquity according to the number of days you lie down, 390 days; so you will bear the iniquity of the house of Israel.
The point is that God kept Israel ethnically pure while scattered among the nations.
We have no historical record of this. so this would be under the assumption that your interpretation is correct.
That's what the function of the sieve was, the final cleaning and preparation of the grain for milling.
I agree, the function of the sieve was for separating the loose chaff, straw, and dirt from the grain. Larger "pebbles" and grain would stay in the sieve, while the loose chaff, straw, and dirt would fall to the ground.
I'm not sure what this has to do with "maintaining ethnic purity" as you claim. Regardless of "ethnic purity" or not, Ephraim became as gentiles through their divorce from God. Especially considering Paul doesn't make any distinction between Ephraim and gentiles in romans 9. Paul quotes hosea 1:10 and 2:23 as fulfilled with the inclusion of the gentiles with the jews in the vessels or mercy.
The term birthright only appears a few times, but can be discerned by the context and by description. The birthright can also be a 'blessing', though perhaps not in the same way it pertained to Jacob and Esau.
The birthright is the double portion inheritance. It is not the same as the prophetic blessings.
Read the accounts.
Matthew 15
22 And, behold, a woman of Canaan came out of the same coasts, and cried unto him, saying, Have mercy on me, O Lord, thou son of David; my daughter is grievously vexed with a devil.
23 But he answered her not a word. And his disciples came and besought him, saying, Send her away; for she crieth after us.
24 But he answered and said, I am not sent but unto the lost sheep of the house of Israel.
The interchange indicates that she knew who he was and he knew who she was...one of the lost sheep he was 'sent only' to.
The other woman (the woman at the well) also indicated that she was a descendant of Jacob.
The Samaritans are not necissarily the lost sheep of Israel, for Jesus tells the disciples not to go the town of the Samaritans but only the lost sheep of Israel.
Matthew 10:5-6 These twelve Jesus sent out, instructing them, “Go nowhere among the Gentiles and
enter no town of the Samaritans, but go rather to the lost sheep of the house of Israel.
Jesus is not admitting that the Canaanite is of the lost sheep of Israel. In fact, that is why he does not answer her in the first place. The woman begs jesus and He states "it's not right to take the children's bread and throw it to the dogs". Israel is the children, the Canaanite woman is the dog.
Matthew 15:22-28 And behold, a Canaanite woman from that region came out and was crying, “Have mercy on me, O Lord, Son of David; my daughter is severely oppressed by a demon.”
But he did not answer her a word. And his disciples came and begged him, saying, “Send her away, for she is crying out after us.” He answered, “I was sent only to the lost sheep of the house of Israel.” But she came and knelt before him, saying, “Lord, help me.” And he answered, “
It is not right to take the children’s bread and throw it to the dogs.” She said, “Yes, Lord, yet even the dogs eat the crumbs that fall from their masters’ table.” Then Jesus answered her, “O woman, great is your faith! Be it done for you as you desire.” And her daughter was healed instantly.
The other woman (the woman at the well) also indicated that she was a descendant of Jacob.
When Ephraim was divorced, it became no longer God's people and God was no longer their God (hosea 1:9). Thus, descendants of Jacob through the northern kingdom's side that remained "ethnically pure" as you put it or mixed with the gentiles, would still be not God's people, until their inclusion in new covenant. Therefore, even if the samaritan woman did descend from Jacob through Ephraim, under the old covenant, she was not recognized as Israel.
Hence when the disciples return, they are surprised Jesus was taking with her.
John 4:27
27Just then his disciples came back. They marveled that he was talking with a woman, but no one said, “What do you seek?” or, “Why are you talking with her?”
Notice what Josephus states of the Samaritans of that time: Josephus, Antiquities, 9.14,3:
" and are called in the Hebrew tongue
Cutheans, but in the Greek tongue
Samaritans. And when they see the Jews in prosperity, they pretend that they are changed, and allied to them; and call them kinsmen: as though they were derived from Joseph, and had by that means an original alliance with them. But when they see them falling into a low condition, they say they are no way related to them: and that the Jews have no right to expect any kindness or marks of kindred from them: but they declare that they are sojourners, that come from other countries."
The 'lost sheep' were the descendants of the northern kingdom that remained or returned to Samaria after the Assyrian conquest.
I agree. And scripture supports this. Scripture supports those from the northern kingdom living in Samarian and Jerusalem, after Israel had been removed and Judah was still a kingdom (2 chronicles 30). Scripture supports those from the northern kingdom living in the southern kingdom post Babylonian exile (1 chronicles 9).