"From the very beginning, and before time began, the eternal Father chose and prepared for his only-begotten Son a Mother in whom the Son of God would become incarnate and from whom, in the blessed fullness of time, he would be born into this world. Above all creatures did God so love her that truly in her was the Father well pleased with singular delight." - Ineffabilis Deus.Pope Pius IX, Ineffabilis Deus.
ALSO
"We declare, pronounce, and define that the doctrine which holds that the most Blessed Virgin Mary, in the first instant of her conception, by a singular grace and priviledge granted by Almighty God, in view of the merits of Jesus Christ, the Savior of the human race, was preserved free from all stain of original sin, is a doctrine revealed by God and therefore to be believed firmly and constantly by all the faithful." - Ineffabilis Deus. Pope Pius IX, Ineffabilis Deus.
The Church teaches that she was created without a sin nature and was "ever absolutely free of all stain of sin, all fair and perfect". - Pope Pius IX, Ineffabilis Deus.
And she remained "free of every personal sin her whole life long." The German Bishop's Conference, The Church's Confession of Faith (San Francisco: Ignatius Press, 1987), p. 46m quoting J.A. Mohler.
But of course you already know this. Her "Immaculate Conception" was formed "before time began". So it is no longer a "lineage" thing to consider. Only Mary and Jesus share the same DNA, according to her 'preparation before time began'.
But this is only layer two of the peeling back. The next layer is her "participation with Christ in redemption."
You misunderstand the Immaculate Conception. Mary was free of original sin, yes, but she was conceived and born *in the natural way* to her parents Anna and Joachim. She was conceived as a result of sexual intercourse.
She is a creature, not a divine being.