In chapter 4 of the Book of Amos, verses 1-3 are talking about people who pervert the truths of the church. They will fall into falsities in outermost things.

In the Bible, fish represent "lower" things than mammals, so we can interpret the fishhooks in verse 2 as meaning being caught and held fast in natural or lower things.

Verses 4-6 are about acts of worship such as tithes and sacrifices. These look similar to genuine worship, but are only external sorts of things. We can tell because ‘teeth’ (in verse 6) represent ultimates or outermost things (see Secrets of Heaven 6380). It follows that “cleanness of teeth” would mean outermost things that look good but only imitate genuine worship. The Lord exhorts, “Yet you have not returned to me.”

Verses 7-8. Some things true will remain, when where there are too many false ideas, the truths don't get through. This can be seen where the Lord says, “I made it rain on one city; I withheld rain from another city... where it did not rain the part withered.” Again the Lord exhorts, “Yet you have not returned to me.”

Verse 9. Afterward all things of the church are falsified, shown by blight attacking the gardens, vineyards, fig tree and olive trees. The last three represent spiritual, natural and celestial things, or all the things of spiritual life. “Yet you have not returned to me,” says the Lord.

Verses 10-11. The Lord explains the devastating things he allowed to happen: plague in Egypt, death of young men by swords, stench in the camps, Sodom and Gomorrah. This is because they are profaned by sensual knowledges. Profanation means the mixing of good and evil together. (See Secrets of Heaven 1001[2]).

This extends to all things of the church, with the church being the Lord’s kingdom on earth (Secrets of Heaven 768[3]).

With profanation “as soon as any idea of what is holy arises, the idea of what is profane joins immediately to it,” (Secrets of Heaven 301).

Now there is hardly anything left. “Yet you have not returned to Me,” says the Lord again.

Verses 12-13: Because people adamantly remain in their profane ways, they are warned, “Prepare to meet your God!”. This is the God powerful and mighty, “who forms mountains, and creates the wind,” and even more close to home, “Who declares to man what his thought is.” As intimately a knowing as that is, the Lord’s love for all humanity is contained in His exhortations for them to turn themselves to Him.

See, for example, Luke 6:44-45, and True Christian Religion 373.

395. 'Anyone killing Cain, vengeance will be taken on him sevenfold' means that to violate faith thus separated was utterly forbidden. This is clear from the meaning of 'Cain' as separated faith, and from the meaning of 'seven' as utterly inviolable. As is well known, the number seven was considered holy on account of the six days of creation and of the seventh, which is the celestial man in whom peace, rest, and the Sabbath exist. This is the reason why the number seven occurs so many times in the religious ceremonies of the Jewish Church, in every instance standing for that which is holy. It is also the reason why periods of time, long as well as short, were divided into seven, and were called weeks, as in the case of the long time intervals leading up to the time when the Messiah was to come, Daniel 9:24-25. A period of seven years is also called a week by Laban and Jacob, Genesis 29:27-28. Consequently wherever the number seven occurs it stands either for something holy or else for something utterly inviolable, as in David,

Seven times in the day I praise You. Psalms 119:164.

In Isaiah,

The light of the moon will be as the light of the sun, and the light of the sun will be sevenfold, as the light of seven days. Isaiah 30:26.

Here 'the sun' is love, and 'the moon' faith deriving from love, a faith which is to be as love.

[2] Just as the periods of man's regeneration divide into six before he reaches the seventh, which is the celestial man, so also do the periods of vastation even until nothing celestial is left. This was represented by the many captivities of the Jews, and by the last of them, the Babylonian, which was a captivity of seven decades, that is, seventy years. And it was several times stated that the land had to rest during its Sabbaths. It was also represented by Nebuchadnezzar, of whom it is said in Daniel,

His heart will be changed from man, and the heart of a beast given him until seven times will pass over him. Daniel 4:16, 25, 32.

In reference to the vastation of the last times it is recorded in John,

I saw another sign in heaven, great and wonderful - seven angels holding the seven Revelation 15:1, 6-7.

They will trample over the holy city for forty-two months (that is, six times seven). Revelation 11:2.

In the same book,

I saw a book [written] within and on the back, sealed with seven seals. Revelation 5:1.

Severe and increased penalties were for the same reason expressed by the number seven, as in Moses,

If you will not obey Me in this matter, I will chastise you seven times worse for your sins. Leviticus 26:18, 21, 24, 28.

In David,

Return to our neighbours sevenfold into their bosom. Psalms 79:12.

So then, it is because it was utterly forbidden to violate faith - since it had a use to serve, as has been stated - that the statement is made about 'vengeance being taken sevenfold on him who killed Cain'.