The detailed realization of M^{8}H duality (for what this duality means, see this) involves still uncertainties. The quaternionic normal spaces containing fixed 2space M^{2} (or an integrable distribution of M^{2}) are parametrized by points of CP_{2}. One can map the normal space to a point of CP_{2}.
The tough problem has been the precise correspondence between M^{4} points in M^{4}× E^{4} and M^{4}× CP_{2} and the identification of the sizes of causal diamonds (CDs) in M^{8} and H. The identification is naturally linear if M^{8} is analog of spacetime but if M^{8} is interpreted as momentum space, the situation changes. The ealier proposal maps mass hyperboloids to lightcone proper time =constant hyperboloids and it has turned out that this correspondence does not correspond to the classical picture suggesting that a given momentum in M^{8} corresponds in H to a geodesic line emanating from the tip of CD.
M^{8}H duality in M^{4} degrees of freedom
The following proposal for M^{8}H duality in M^{4} degrees of freedom relies on the intuition provided by UP and to the idea that a particle with momentum p^{k} corresponds to a geodesic line with this direction emanating from the tip of CD.
 The first constraint comes from the requirement that the identification of the point p^{k}∈ X^{4}⊂ M^{8} should classically correspond to a geodesic line m^{k}= p^{k}τ/m (p^{2}=m^{2}) in M^{8} which in Big Bang analogy should go through the tip of the CD in H. This geodesic line intersects the opposite boundary of CD at a unique point.
Therefore the mass hyperboloid H^{3} is mapped to the 3D opposite boundary of cd⊂ M^{4}⊂ H. This does not fix the size nor position of the CD (=cd× CP_{2}) in H. If CD does not depend on m, the opposite lightcone boundary of CD would be covered an infinite number of times.
 The condition that the map is 1to1 requires that the size of the CD in H is determined by the mass hyperboloid M^{8}. Uncertainty Principle (UP) suggests that one should choose the distance T between the tips of the CD associated with m to be T= ℏ_{eff}/m.
The image point m^{k} of p^{k} at the boundary of CD(m,h_{eff}) is given as the intersection of the geodesic line m^{k}= p^{k}τ from the origin of CD(m,h_{eff}) with the opposite boundary of CD(m,h_{eff}):
m^{k}=ℏ_{eff}X× (p^{k}/m^{2}),
X= 1/(1+ p_{3}/p_{0}) .
Here p_{3} is the length of 3momentum.
The map is nonlinear. At the nonrelativistic limit (X\rightarrow 1), one obtains a linear map for a given mass and also a consistency with the naive view about UP. m^{k} is on the proper time constant mass shell so the analog of the Fermi ball in H^{3} ⊂ M^{8} is mapped to the lightlike boundary of cd⊂ M^{4}⊂ H.
 What about massless particles? The duality map is well defined for an arbitrary size of CD. If one defines the size of the CD as the Compton length ℏ_{eff}/m of the massless particle, the size of the CD is infinite. How to identify the CD? UP suggests a CD with temporal distance T= 2ℏ_{eff}/p_{0} between its tips so that the geometric definition gives p^{k}= ℏ_{eff}p^{k}/p_{0}^{2} as the point at the 2sphere defining the corner of CD. pAdic thermodynamics strongly suggests that also massless particles generate very small padic mass, which is however proportional to 1/p rather than 1/p^{1/2. The map is well defined also for massless states as a limit and takes massless momenta to the 3ball at which upper and lower halfcones meet.
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 What about the position of the CD associated with the mass hyperboloid? It should be possible to map all momenta to geodesic lines going through the 3ball dividing the largest CD involved with T determined by the smallest mass involved to two halfcones. This is because this 3ball defines the geometric "Now" in TGD inspired theory of consciousness. Therefore all CDs in H should have a common center and have the same geometric "Now".
M^{8}H duality maps the slicing of momentum space with positive/negative energy to a Russian dolllike slicing of t≥0 by the boundaries of halfcones, where t has origin at the bottom of the doublecone. The height of the CD(m,h_{eff}) is given by the Compton length L(m,h_{eff}) = ℏ_{eff}/m of quark. Each value of h_{eff} corresponds its own scaled map and for h_{gr}=GMm/v_{0}, the size of CD(m,h_{eff})=GM/v_{0} does not depend on m and is macroscopic for macroscopic systems such as Sun.
 The points of cognitive representation at quark level must have momenta with components, which are algebraic integers for the extension of rationals considered. A natural momentum unit is m_{Pl}=ℏ_{0}/R, h_{0} is the minimal value of h_{eff}=h_{0} and R is CP_{2} radius. Only "active" points of X^{4}⊂ M^{8} containing quark are included in the cognitive representation. Active points give rise to active CD:s CD(m,h_{eff}) with size L(m,h_{eff}).
It is possible to assign CD(m,h_{eff}) also to the composites of quarks with given mass. Galois confinement suggest a general mechanism for their formation: bound states as Galois singlets must have a rational total momentum. This gives a hierarchy of bound states of bound states of ..... realized as a hierarchy of CDs containing several CDs.
 This picture fits nicely with the general properties of the spacetime surfaces as associative "roots" of the octonionic continuation of a real polynomial. A second nice feature is that the notion of CD at the level H is forced by this correspondence. "Why CDs?" at the level of H has indeed been a longstanding puzzle. A further nice feature is that the size of the largest CD would be determined by the smallest momentum involved.
 Positive and negative energy parts of zero energy states would correspond to opposite boundaries of CDs and at the level of M^{8} they would correspond to mass hyperboloids with opposite energies.
 What could be the meaning of the occupied points of M^{8} containing fermion (quark)? Could the image of the mass hyperboloid containing occupied points correspond to subCD at the level of H containing corresponding points at its lightlike boundary? If so, M^{8}H correspondence would also fix the hierarchy of CDs at the level of H.
It is enough to realize the analogs of plane waves only for the actualized momenta corresponding to quarks of the zero energy state. One can assign to CD as total momentum and passive resp. active halfcones give total momenta P_{tot,P} resp. P_{tot,A}, which at the limit of infinite size for CD should have the same magnitude and opposite sign in ZEO.
The above description of M^{8}H duality maps quarks at points of X^{4} ⊂ M^{8} to states of induced spinor field localized at the 3D boundaries of CD but necessarily delocalized into the interior of the spacetime surface X^{4} ⊂ H. This is analogous to a dispersion of a wave packet. One would obtain a wave picture in the interior.
Does Uncertainty Principle require delocalization in H or in X^{4}?
One can argue that Uncertainty Principle (UP) requires more than the naive condition T=ℏ_{eff}/m on the size of subCD. I have already mentioned two approaches to the problem: they could be called inertial and gravitational representations.
 The inertial representations assigns to the particle as a spacetime surface (holography) an analog of plane wave as a superposition of spacetime surfaces: this is natural at the level of WCW. This requires delocalization spacetime surfaces and CD in H.
 The gravitational representation relies on the analog of the braid representation of isometries in terms of the projections of their flows to the spacetime surface. This does not require delocalization in H since it occurs in X^{4}.
Consider first the inertial representation. The intuitive idea that a single point in M^{8} corresponds to a discretized plane wave in H in a spatial resolution defined by the total mass at the passive boundary of CD. UP requires that this plane wave should be realized at the level of H and also WCW as a superposition of shifted spacetime surfaces defined by the above correspondence.
 The basic observation leading to TGD is that in the TGD framework a particle as a point is replaced with a particle as a 3surface, which by holography corresponds to 4surface.
Momentum eigenstate corresponds to a plane wave. Now planewave could correspond to a delocalized state of 3surface  and by holography that of 4surface  associated with a particle.
A generalized plane wave would be a quantum superposition of shifted spacetime surfaces inside a larger CD with a phase factor determined by the 4momentum. M^{8}H duality would map the point of M^{8} containing an object with momentum p to a generalized plane wave in H. Periodic boundary conditions are natural and would force the quantization of momenta as multiples of momentum defined by the larger CD. Number theoretic vision requires that the superposition is discrete such that the values of the phase factor are roots of unity belonging to the extension of rationals associated with the spacetime sheet. If momentum is conserved, the time evolutions for massive particles are scalings of CD between SSFRs are integer scalings. Also iterated integer scalings, say by 2 are possible.
 This would also provide WCW description. Recent physics relies on the assumption about single background spacetime: WCW is effectively replaced with M^{4} since 3surface is replaced with point and CP_{2} is forgotten so that one must introduce gauge fields and metric as primary field variables.
As already discussed, the gravitational representation would rely on the lift/projection of the flows defined by the isometry generators to the spacetime surface and could be regarded as a "subjective" representation of the symmetries. The gravitational representation would generalize braid group and quantum group representations.
The condition that the "projection" of the Dirac operator in H is equal to the modified Dirac operator, implies a hydrodynamic picture. In particular, the projections of isometry generators are conserved along the lifted flow lines of isometries and are proportional to the projections of Killing vectors. QCC suggests that only Cartan algebra isometries allow this lift so that each choice of quantization axis would also select a spacetime surface and would be a higher level quantum measurement.
See the chapter TGD as it is towards the end of 2021 or the article with the same title.
