Unless u are aware that u are locked into and experiencing ur own interpretations of what is around u, u are caught in ur residual self. This is, the self caught in a digitised environment.
What we call real here is not real but electrical signals interpreted by the mind-brain. Or thus a neural interactive “simulation” (hologram), which is what the Matrix is.
So translated to more immediate psychological terms: at the level of projection u are caught in ur experience in the very images (u) project on the world. In this sense we are indeed caught in neurally constructed experience (hence construct) and thus not in a world as such independent of us out there. Hence construct thus in the sense that u are not in real immediate reality but in a constructed experience of reality. So we are also caught in a subjective self and not the real Self, which is the one in the real world.
If this hologram is not used to reflect but we are caught in reflex or first person, it is here that one can be caught in the hologram or matrix, and shut out of real self.
In this sense the construct or white empty room Neo is walking around in, in the video, is the world without the projections or thus without interpretations or colouring in of the world. A lucid dream state of sorts.
So the Neo walking around in the construct is the same Neo from the Matrix but now not in the matrix any more which is the equivalent of walking around in a state of suspended interpretations or projections. This would be the intermediate stage between the Matrix and being fully unplugged from the Matrix.
“”Your appearance now is what we call residual self image. It is the mental projection of your digital self.”―Morpheus to Neo [src]. Residual self image, or RSI for short is the subjective appearance of a human while connected to the Matrix.” (http://matrix.wikia.com/wiki/Residual_self_image).
So as long as we remain digitised or thus caught in either or state of the checkered board we remain caught in residual self.
In this sense Neo in the construct (and which is the same as the Neo self in the matrix) is the residual self or the mental projection of the digital self. A fake result or projection of being in a fake digitalised or discontinuous world/experience.
This is not the true core Self yet because that is the unplugged one. But the Neo which returns unplugged has upgraded from the residual self to the essential true Self which sees through the digital hologram.
This residual self in the Matrix is thus the self in resistance of its shadow (which is what sustained the matrix), and thus which is what keeps us in this fake digital self.