Admission or Denial of a Core You

Q: Do you [admit or] deny that there is a core 'you' [e.g., a soul] after you have renounced everything?

A: Practicing the Eight Renunciations, one works with renouncing behaviors that are not supportive or conducive to establishing a meditation practice. Renouncing these behaviors may also help one 'get along' in society so that they may continue their practice unimpeded. 

Having developed a meditation practice and having progressed along the path, the Asheyana practitioner applies personal experiential knowledge of the Insubstantial Basis of Being as a reference point to evaluate for themselves proposed Substantial Bases of Being and proposed Substantial Bases of Concept, or interpretations, beliefs, and worldviews on how to regard their own experience.

Having evaluated these for oneself, the Asheyana practitioner may find it useful to renounce Substantial Bases of Being or Substantial Bases of Concept due to conviction in these having a known outcome, namely Joyless Anxiety.

Looking then at what is meant by "soul" :

"In many religious and philosophical traditions, the soul is the spiritual essence of a person, which includes one's identity, personality, and memories, an immaterial aspect or essence of a living being that is believed to be able to survive physical death. The concept of the soul is generally applied to humans, though it can also be applied to other living or even non-living entities, as in animism." - Soul, Wikipedia

In the Asheyana view, the "soul," its assignment of one's "spiritual essence," and its attributes of "one's identity, personality, and memories" can all be experienced, yet these are regarded as transient and impermanent appearances--They appear and have meaning, yet when looking, one finds no solidity behind them.

In the Asheyana view, when regarded as a visceral experience, what is behind the appearance of a "soul", its assignments, and its attributes is the Insubstantial Basis of Being. When regarded as a belief, what is behind the appearance of "soul," its assignments, and its attributes is the Insubstantial Basis of Concept.

As for what happens before human birth and after human death, or what is carried from mortal lifetime to mortal lifetime..? 

Presented as a secular path, Asheyana view makes no assertion about what happened before an individual's human birth and after an individual's human death, or what, if anything, is carried from mortal lifetime to mortal lifetime. In the Asheyana view, behind speculation of such topics is simply the Insubstantial Basis of Concept.

Formalizing such metaphysical speculation could be useful for enforcing moral order in human society or for explaining human justice in a broad, archetypal way. However, given the reason above (arriving at Joyless Anxiety) that might lead one to renounce Substantial Bases of Concept, such an approach would appear to be not risk-free.

Is Insubstantial Basis itself some kind of Immortal or Eternal "soul"..?

In the Asheyana view, all possibilities of appearance arise from the Insubstantial Basis. A being's arising from the Insubstantial Basis in a particular place and time is regarded as unique and irreproducible. At the time of death, a being then dissolves back into the Insubstantial Basis. Retention of an individual's spiritual essence, of their identity, personality, or memories is not asserted.

In addition, although compassionate attempts have been made to present conceptual analogies for what the personal experience of dissolution back into Insubstantial Basis would be like, a reliable report of what one can personally expect to experience during this process of dissolution back into the Insubstantial Basis is necessarily unavailable, and thus, not forthcoming.

If one is to inevitably dissolve back into the Insubstantial Basis, instead of speculating upon ideas of what this would be like, it would seem worthwhile to familiarize oneself with the experience of the Insubstantial Basis during this very life.

In traveling the Asheyana path, one develops familiarity with the ever-present Insubstantial Basis gradually, over time. Finding home in the Insubstantial Basis is how one comes to rest at ease in one's own life.