Is M^{8}H duality consistent with Fourier analysis at the level of M^{4}× CP_{2}?M^{8}H duality predicts that spacetime surfaces as algebraic surfaces in complexified M^{8} (complexified octonions) determined by polynomials can be mapped to H=M^{4}× CP_{2}. The proposal (see this) is that the strong form of M^{8}H duality in M^{4} degrees of freedom is realized by the inversion map p^{k}∈ M^{4}→ ℏ_{eff}×p^{k}/p^{2}. This conforms with the Uncertainty Principle. However, the polynomials do not involve periodic functions typically associated with the minimal spacetime surfaces in H. Since M^{8} is analogous to momentum space, the periodicity is not needed. In contrast to this, the representation of the spacetime surfaces in H obey dynamics and the Himages of X^{4}⊂ M^{8} should involve periodic functions and Fourier analysis for CP_{2} coordinates as functions of M^{4} coordinates. Neper number, and therefore trigonometric and exponential functions are padically very special. In particular, e^{p} is a padic number so that roots of e define finiteD extensions of padic 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 padic number fields. The map of the tangent space of the spacetime surface X^{4}⊂ M^{8} to CP_{2} involves the analog velocity missing at the level of M^{8} and brings in the dynamics of minimal surfaces. Therefore the expectation is that the expansion of CP_{2} coordinates as exponential and trigonometric functions of M^{4} 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 h_{eff} assignable to gauge interactions resp. to very large value of gravitational Planck constant ℏ_{gr} 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 M^{8}H duality consistent with Fourier analysis at the level of M^{4}× CP_{2}? or the chapter Breakthrough in understanding of M^{8}H duality.
