Correct. Correct. Correct!
Purgatory only makes us completely pure and focus on God alone, any attachment to a sinful nature will be “purged”, i.e. made fit for heaven. The sins themselves are indeed forgiven.
The language has been a barrier. Is it called a “place”? Does it take “time”? It has never been defined. Throughout tradition these words have been used, no doubt, but in the afterlife, these things surely do not exist as we know them. Think of how Scripture refers to heaven “above”, but does it exist prepositionally above the surface of the earth, at some longitudinal coordinates? I hear many Catholics talk about more or less “time” in Purgatory, and I can’t be sure of how some of them mean it, but one must be careful to qualify it as follows:
As the concept has developed, we are at a point of more or less calling it a condition or encounter, or some other less temporal/spacial term. Prior to his Papacy, Cardinal Ratzinger said the following about Purgatory, which you may find interesting:"The transforming ‘moment’ of this encounter cannot be quantified by the measurements of earthly time. It is, indeed, not eternal but a transition, and yet trying to qualify it as of ‘short’ or ‘long’ duration on the basis of temporal measurements derived from physics would be naive and unproductive. The ‘temporal measure’ of this encounter lies in the unsoundable depths of existence, in a passing-over where we are burned ere we are transformed. To measure such Existenzzeit, such an ‘existential time,’ in terms of the time of this world would be to ignore the specificity of the human spirit in its simultaneous relationship with, and differentation from, the world.
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"[Purgatory] is the inwardly necessary process of transformation in which a person becomes capable of Christ, capable of God and thus capable of unity with the whole communion of saints.
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“Encounter with the Lord is this transformation.”
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Joseph Ratzinger, Eschatology: Death and Eternal Life, p. 230-231So, it therefore might be, as you say, “instantaneous” as far as we understand time. So a forgiven person dies, he is rendered no longer able to sin and his soul is made completely heavenly in nature, and then he can see God face to face. I would say that would be sound theology consistent with what Purgatory is.