What could the interpretation for the events in which the dimension of the extension of rationals increases? Galois extension is extensions of an extension with relative Galois group Gal(rel)= Gal(new)/Gal(old). Here Gal(old) is a normal subgroup of Gal(new). A highly attractive possibility is that evolutionary sequences quite generally (not only in biology) correspond to this kind of sequences of Galois extensions. The relative Galois groups in the sequence would be analogous to conserved genes, and genes could indeed correspond to Galois groups (see this). To my best understanding this corresponds to a situation in which the new polynomial Pm+n defining the new extension is a polynomial Pm having as argument the old polynomial Pn(x): Pm+n(x)=Pm(Pn(x)).
What about the interpretation at the level of conscious experience? A possible interpretation is that the quantum jump leading to an extension of an extension corresponds to an emergence of a reflective level of consciousness giving rise to a conscious experience about experience. The abstraction level of the system becomes higher as is natural since number theoretic evolution as an increase of algebraic complexity is in question.
This picture could have a counterpart also in terms of the hierarchy of inclusions of hyperfinite factors of type II1 (HFFs). The included factor M and including factor N would correspond to extensions of rationals labelled by Galois groups Gal(M) and Gal(N) having Gal(M)⊂ Gal(M) as normal subgroup so that the factor group Gal(N)/Gal(M) would be the relative Galois group for the larger extension as extension of the smaller extension. I have indeed proposed (see this) that the inclusions for which included and including factor consist of operators which are invariant under discrete subgroup of SU(2) generalizes so that all Galois groups are possible. One would have Galois confinement analogous to color confinement: the operators generating physical states could have Galois quantum numbers but the physical states would be Galois singlets.
For a summary of earlier postings see Latest progress in TGD.