The proposal (see this) is that the strong form of M8-H duality in M4 degrees of freedom is realized by the inversion map pk∈ M4→ ℏeff×pk/p2. This conforms with the Uncertainty Principle. However, the polynomials do not involve periodic functions typically associated with the minimal space-time surfaces in H. Since M8 is analogous to momentum space, the periodicity is not needed.
In contrast to this, the representation of the space-time surfaces in H obey dynamics and the H-images of X4⊂ M8 should involve periodic functions and Fourier analysis for CP2 coordinates as functions of M4 coordinates.
Neper number, and therefore trigonometric and exponential functions are p-adically very special. In particular, ep is a p-adic number so that roots of e define finite-D extensions of p-adic numbers. As a consequence, Fourier analysis extended to allow exponential functions required in the case of Minkowskian signatures is a number theoretically universal concept making sense also for p-adic number fields.
The map of the tangent space of the space-time surface X4⊂ M8 to CP2 involves the analog velocity missing at the level of M8 and brings in the dynamics of minimal surfaces. Therefore the expectation is that the expansion of CP2 coordinates as exponential and trigonometric functions of M4 coordinates emerges naturally.
The possible physical interpretation of this picture is considered. The proposal is that the dimension of extension of rationals (EQ) resp. the dimension of the transcendental extension defined by roots of Neper number correspond to relatively small values of heff assignable to gauge interactions resp. to very large value of gravitational Planck constant ℏgr originally introduced by Nottale.
Also the connections with the quantum model for cognitive processes as cascades of cognitive measurements in the group algebra of Galois group (see this and this) and its counterpart for the transcendental extension defined by the root of e are considered. The geometrical picture suggests the interpretation of cognitive process as an analog of particle reaction emerges.
See the article Is M8-H duality consistent with Fourier analysis at the level of M4× CP2? or the chapter Breakthrough in understanding of M8-H duality.
For a summary of earlier postings see Latest progress in TGD.