If we do not add to or take away from the words given then yes "physical death" entered into the world because of Adam's sin.
Scripture says there were two trees highlighted in the garden: the tree of life and the tree of the knowledge of good and evil. God said they could freely eat of *ANY* tree in the garden except the forbidden tree. Again, ANY tree was freely available to eat except for the forbidden tree. So the tree of life was freely available to be eaten as many times as Adam and Even chose.
If the tree of life provided "life" as the fruit, and they ate of the tree, they would in fact *physically* live forever. God even says this is the reason why he barred Adam and Eve from the garden after they sinned.
Genesis 3:22 [brackets are mine for implication. I'm not adding to scripture.]
"And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his [physical] hand, and take also of the [physical] tree of life, and [physically] eat, and [physically] live for ever..."
The passage doesn't say "lest he take also from the tree of life and forever be allowed in heaven when he inevitably dies".
No God was preventing Adam from physical immortality because he was now spiritually corrupt. So if this is the case that means the opposite must be true: In other words, that if Adam did not spiritually corrupt himself (i.e. sin) he would've been able to freely eat of the tree of life and physically live forever.
So the death that entered into the world because of Adam was physical death.
But what about the fact that Adam didn't immediately die? What about the fact that he lived to be 930 years old? So wasn't the death God was speaking about a spiritual death instead?
Remember that God measures time from his perspective not ours. Scripture says, "a day to the lord is as a thousand years and a thousand years is as a day", testified twice in Psalm 90:4 and 2 Peter 3:8 ("a testimony of two or more establishes a truth" - Deut 19:5, Matthew 18:16 and 2 Corinthians 13:1).
God told Adam "in the day" he eats of the forbidden try he would surely die. If a thousand years is as a day to God, then God was true to his word. "In the day" (i.e. within the 1000 years) that Adam ate of the forbidden tree he surely died. Truly when one is separated from God's presence it's spiritual death, but God's presence never left man after his first sin. He just made sure they wouldn't live forever until mankind was restored.
- God clothes sinful Adam and Eve.
- Next we read that his sons Cain and Abel present offerings to God and when Cain's offering wasn't accepted God talks with him encouraging him and also warning him of satan's temptation.
- Then when Cain kills Abel God is there speaking to him, punishing him but also having mercy on him in punishment by marking him.
- Then we follow the genealogy of Adam where we read that after Enosh was born men began to call upon God's name.
- Next Enoch was taken up to God, never seeing death, because he pleased Him.
It wasn't until the events of Genesis 6 when God said that his spirit wouldn't always continue with man (because all flesh had corrupted itself), eventually wiping out all mankind with a *physical* flood.
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So no all the evidence suggests that physical death would have never been a reality if Adam never sinned.