@Hat
Carter beyond the Gate=/=Carter who went through the Gate. This is the realization he comes to. There is no single entity referred to as Randolph Carter, there are infinite stages of the personality of Randolph Carter which believe they experience things in a certain order, but in reality, perspective and change are illusions.
"He was in many places at the same time. On earth, on October 7, 1883, a little boy named Randolph Carter was leaving the Snake-Den in the hushed evening light and running down the rocky slope and through the twisted-boughed orchard toward his Uncle Christopher's house in the hills beyond Arkham—yet at that same moment, which was also somehow in the earthly year of 1928, a vague shadow not less Randolph Carter was sitting on a pedestal among the Ancient Ones in earth's trans-dimensional extension. Here, too, was a third Randolph Carter in the unknown and formless cosmic abyss beyond the Ultimate Gate. And elsewhere, in a chaos of scenes whose infinite multiplicity and monstrous diversity brought him close to the brink of madness, were a limitless confusion of beings which he knew were as much himself as the local manifestation now beyond the Ultimate Gate."
Carter realizes that all conceptions of these Carters are cut from one eternal, dimensionless "Carter".
"All descended lines of beings of the finite dimensions, continued the waves, and all stages of growth in each one of these beings, are merely manifestations of one archetypal and eternal being in the space outside dimensions. Each local being—son, father, grandfather, and so on—and each stage of individual being—infant, child, boy, young man, old man—is merely one of the infinite phases of that same archetypal and eternal being, caused by a variation in the angle of the consciousness-plane which cuts it. Randolph Carter at all ages; Randolph Carter and all his ancestors both human and pre-human, terrestrial and pre-terrestrial; all these were only phases of one ultimate, eternal "Carter" outside space and time—phantom projections differentiated only by the angle at which the plane of consciousness happened to cut the eternal archetype in each case."
The penchant for knowledge and discovery that comes from Carter's archetype is due to how it is cut from the Supreme Archetype, which is Yog-Sothoth.
"The archetypes, throbbed the waves, are the people of the ultimate abyss—formless, ineffable, and guessed at only by rare dreamers on the low-dimensioned worlds. Chief among such was this informing BEING itself . . . which indeed was Carter's own archetype. The glutless zeal of Carter and all his forbears for forbidden cosmic secrets was a natural result of derivation from the SUPREME ARCHETYPE. On every world all great wizards, all great thinkers, all great artists, are facets of IT."