Discretization in dimension D in terms of pinary cutoff means division of the manifold to cubelike objects. What suggests itself is homology theory defined by the measurement resolution and by the fluxes assigned to the induced Kähler form.
 One can introduce the decomposition of nD submanifold of the imbedding space to ncubes by n1planes for which one of the coordinates equals to its pinary cutoff. The construction works in both real and padic context. The hyperplanes in turn can be decomposed to n1cubes by n2planes assuming that an additional coordinate equals to its pinary cutoff. One can continue this decomposition until one obtains only points as those points for which all coordinates are their own pinary cutoffs. In the case of partonic 2surfaces these points define in a natural manner the ends of braid strands. Braid strands themselves could correspond to the curves for which two coordinates of a lightlike 3surface are their own pinary cutoffs.
 The analogy of homology theory defined by the decomposition of the spacetime surface to cells of various dimensions is suggestive. In the padic context the identification of the boundaries of the regions corresponding to given pinary digits is not possible in purely topological sense since padic numbers do not allow wellordering. One could however identify the boundaries submanifolds for which some number of coordinates are equal to their pinary cutoffs or as inverse images of real boundaries. This might allow to formulate homology theory to the padic context.
 The construction is especially interesting for the partonic 2surfaces. There is hierarchy in the sense that a square like region with given first values of pinary digits decompose to p square like regions labelled by the value 0,...,p1 of the next pinary digit. The lines defining the boundaries of the 2D square like regions with fixed pinary digits in a given resolution correspond to the situation in which either coordinate equals to its pinary cutoff. These lines define naturally edges of a graph having as its nodes the points for which pinary cutoff for both coordinates equals to the actual point.
 I have proposed earlier kenociteallb/categorynew what I have called symplectic QFT involving a triangulation of the partonic 2surface. The fluxes of the induced Kähler form over the triangles of the triangulation and the areas of these triangles define symplectic invariants, which are zero modes in the sense that they do not contribute to the line element of WCW although the WCW metric depends on these zero modes as parameters. The physical interpretation is as nonquantum fluctuating classical variables. The triangulation generalizes in an obvious manner to quadrangulation defined by the pinary digits. This quadrangulation is fixed once internal coordinates and measurement accuracy are fixed. If one can identify physically preferred coordinates  say by requiring that coordinates transform in simple manner under isometries  the quadrangulation is highly unique.
 For 3surfaces one obtains a decomposition to cube like regions bounded by regions consisting of square like regions and Kähler magnetic fluxes over the squares define symplectic invariants. Also Kähler ChernSimons invariant for the 3cube defines an interesting almost symplectic invariant. 4surface decomposes in a similar manner to 4cube like regions and now instanton density for the 4cube reducing to ChernSimons term at the boundaries of the 4cube defines symplectic invariant. For 4surfaces symplectic invariants reduce to ChernSimons terms over 3cubes so that in this sense one would have holography. The resulting structure brings in mind lattice gauge theory and effective 2dimensionality suggests that partonic 2surfaces are enough.
The simplest realization of this homology theory in padic context could be induced by canonical identification from real homology. The homology of padic object would the homology of its canonical image.
 Ordering of the points is essential in homology theory. In padic context canonical identification x=∑ x_{n}p^{n}→ ∑ x_{n}p^{n} map to reals induces this ordering and also boundary operation for padic homology can be induced. The points of padic space would be represented by ntuples of sequences of pinary digits for n coordinates. pAdic numbers decompose to disconnected sets characterized by the norm p^{n} of points in given set. Canonical identification allows to glue these sets together by inducing real topology. The points p^{n} and (p1)(1+p+p^{2}+...)p^{n+1} having padic norms p^{n} and p^{n1} are mapped to the same real point p^{n} under canonical identification and therefore the points p^{n} and (p1)(1+p+p^{2}+...)p^{n+1} can be said to define the endpoints of a continuous interval in the induced topology although they have different padic norms. Canonical identification induces real homology to the padic realm. This suggests that one should include canonical identification to the boundary operation so that boundary operation would be map from padicity to reality.
 Interior points of padic simplices would be padic points not equal to their pinary cutoffs defined by the dropping of the pinary digits corresponding p^{n}, n>N. At the boundaries of simplices at least one coordinate would have vanishing pinary digits for p^{n}, n>N. The analogs of n1 simplices would be the padic points sets for which one of the coordinates would have vanishing pinary digits for p^{n}, n>N. nksimplices would correspond to points sets for which k coordinates satisfy this condition. The formal sums and differences of these sets are assumed to make sense and there is natural grading.
 Could one identify the end points of braid strands in some natural manner in this cohomology? Points with n≤ N pinary digits are closed elements of the cohomology and homologically equivalent with each other if the canonical image of the padic geometric object is connected so that there is no manner to identify the ends of braid strands as some special points unless the zeroth homology is nontrivial. In kenociteallb/agg it was proposed that strand ends correspond to singular points for a covering of sphere or more general Riemann surface. At the singular point the branches of the covering would coincide.
The obvious guess is that the singular points are associated with the covering characterized by the value of Planck constant. As a matter fact, the original assumption was that all points of the partonic 2surface are singular in this sense. It would be however enough to make this assumption for the ends of braid strands only. The orbits of braid strands and string world sheet having braid strands as its boundaries would be the singular loci of the covering.
For background see the chapter Quantum Adeles.
