By innermost kendra bindu, you mean the innermost core, right?
Yes. Kendra = center Bindu = Point
And does this Atman transmigrates as per advaita?
I think you answered this yourself but here is what I can add :
AtmA is One, the original One, in reality , especially when referred to as the innermost core, point, kendra-bindu
OR
The fundamental , pure , the foundation - on which the world appears stacked. (This is why Shri MahaVishNu is in the ksheersagar - He is the foundation.)
What transmigrates is the layered-one - that has been carrying the layers over years and lifetimes. We can say it is "
the sUkshma sharIra propelled and powered by AtmA".
The important point is that the sUkshma sharIra cannot do anything without the AtmA. AtmA is intrinsic to it. The sUkshma is the covering. Like saying the seed is in the tree,
Bhagavad Geeta chapter 2 has the word "
dehi" at one point - DEHI = the one to whom this deha (body) belongs / the one who is [trapped] in the deha.
Then in chapter 15 , a shloka says the One carries samskaar (imprints) , vikaar (flaws) , vAsanA (desires) just as the wind/air (vAyu) carries fragrance (gandha) from the flowers.
Now this air has a qualifier - the fragrance. It left the garden long ago, but is still carrying the fragrance.
This verse refers to the One who is carrying the samskaar (imprints) as
Ishwar!
BG 15.8 shariram yad avApnoti
yach cha-apy utkramatishvarah:
grhitvaitAni samyati
vaAur gandhanivaAshayAt
Just think - Ishwar? Isn't Ishwar omnipresent, omniscient, omnipotent? Why would He carry imprints from one body to another?
=>All beings belong to Him and each local-jiva needs their due time to iterate.
=>Ishwar means controller, and
in this context, the controller that made the body sentient is leaving the body now, and carrying the imprints (sUkshma sharIra) at death of body.
So is the case with AtmA.
Stepping aside in knowledge, the jnAni understands that I am not these layers.
I am the pure consciousness which is the most fundamental to all existence.
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The argument about Swamijis using the word Soul :
1. soul with a small s can be jivAtmA
2. We can say that this entire kaal chakra (wheel of time) is governed by the One Soul (Capital S).
In this context , Soul = AtmA = Brahman = Purushottam
3. Dada Bhagwan taught advaita, but his website uses the word 'Soul' with a capital S all the way to Siddha-Lok (dimension).
The same being is identifying as Me, what changes is the understanding and identification.
The pointer changes.
Another point: While you are understand it, the purpose of the Upanishads was to conduct the experiment and derive the conclusion, or arrive at it.