Appendix 1
Geometry of Discrete Multidimensional Space
Observed Space:
- Continuous and unbroken at accessible scales; without discontinuities or overlaps.
- Connected: from any region one can reach any other along a finite trajectory.
- Locally Euclidean: inertial “straight” exists; small regions are described by lines and planes.
- Orientable: “left/right” do not mix under loops.
- Homogeneous and isotropic on average on large scales; local granularity is averaged out.
- Causal: there is a finite signal propagation speed; “influence” propagates stepwise and consistently.
- Boundaries between media (dense/rarefied regions) do not break integrity: fields and trajectories continue across interfaces.
- Observations do not imply the existence of “dimension jumps” in a single step along a path; transitions appear smooth.
For such a macro-reality to be possible on a discrete substrate, SQ must tile without gaps or kinks, preserve local straightness, and allow smooth (yet discrete) transitions between different “media”. This corresponds to tiling by equal hypercubes with a Planck-scale edge, where differences between regions are set by the internal state of the cells (dimension and metric charge), and connections occur only via fully matching faces. Then macro-level continuity is a consequence of correct micro-level tiling.
Rules for constructing space from hypercubes
- Full face matching. Joining is possible only with full matching of faces of the same dimension. Partial, diagonal, and other “insets” are geometrically forbidden.
- Dimensional adjacency. Neighbors can only be cells whose dimensions differ by one (d with d±1). Any larger jump is implemented by a finite “staircase” of adjacent levels.
- Two-sidedness of internal faces. Any internal shared face separates exactly two volume cells—one on each side. One neighbor yields a gap, three or more yield overlap; both cases are excluded.
- Absence of kinks. Edges are always of Planck length; connections do not change the edge length and do not introduce “skew” joints.
- Staircase gradient. A transition from a region of dimension d to a region of dimension d+Δ is implemented as a finite sequence of layers …→(d)→(d+1)→…→(d+Δ). This is how smoothness at the macro-level is achieved under discreteness at the micro-level.
- Orthogonality of directions. Within each cell, axes are pairwise orthogonal; this guarantees an unambiguous “straight” (entry through a face—exit through the opposite face).
Where is the “edge”?
Birth from a single SQ cell naturally suggests the intuition that the Universe’s “bubble” should be a literal bubble—some region with uniform “rules of the game”, but:
- interactions are discrete yet continuous; mass motion does not imply “hitting” something or “flying out” somewhere—meaning the “rules of the game” cannot have a “side out-of-bounds line” and must be causally continuous;
- neither visually nor analytically are any preferred directions or regions identified—on the contrary, homogeneity and isotropy are observed;
- locally we observe Euclidean behavior (lines are straight, planes are flat) and inertial “straight”, and this holds throughout R*;
- moreover, SQ join only by full face matching; a “cut” in the lattice produces an unmatched face and breaks connectedness. An “edge” in the literal sense is incompatible with the rule of joining without gaps or kinks.
Conclusion: to reconcile a discrete cubic lattice with the absence of an “edge”, opposite faces of some finite existing SQ block must be identified pairwise (periodically) via:
- Full face matching. Gluing only “face-to-face” via Planck-scale faces of the same dimension; diagonal or curvilinear joints are excluded.
- Dimensional adjacency. Transitions between regions of different dimension are implemented as a “staircase” (…→d→d+1→…), without jumps or breaks.
- Two-sidedness of internal faces. Each internal face separates exactly two cells—no “hanging” and no “multiple” joints.
- Absence of kinks. Edges are strictly of Planck length; no “bending” of faces to close the shape.
- Orthogonality of directions. The local frame preserves mutual axis orthogonality in each cell; inertial “straight” is “entered through a face → exited through the opposite face”.
- Periodicity in integer steps. Any “loop” along a closed direction is an integer number of SQ along each axis; this naturally sets the alignment scales R*.
These conditions are best satisfied by a D-dimensional hypertorus (with periodic boundaries):
- it does not require bending faces or changing edge length (it preserves discrete geometry);
- it is compatible with the “dimension staircase” and with SER (no breaks under degradation/reverse);
- it preserves locally Euclidean kinematics (inertial “straight”);
- it allows arbitrary integer periods along the axes (alignment scales R*);
- homogeneity / isotropy on average is maintained, and local Euclidean behavior is not violated. There are no special directions or edges;
- topology is observationally silent. If the periodicities are larger than the causal diameter, repeated objects (“multi-images”) do not have time to appear—we see an ordinary “boundless” cosmos without “mirrors”; [18]
- it is compatible with lower-dimension sub-networks. Stable 3D sub-networks (and others) can exist within it without breaking overall connectedness or violating the joining rules. This will be useful slightly below (see Stable lower-dimension sub-networks);
- stability under SER. Periodic boundaries do not hinder dimension degradation and the “growth” of the number of SQ, nor the SER reverse; closedness is preserved automatically.
Note 18
And what if they are “equal”? We still would not see “the right as the left”. And this does not exclude that the causally connected region is the entire Universe we have.
Motion “straight” and “through” under alternating dimensions
- Straight-trajectory rule: within a cell, entry through a face continues as exit through the opposite face; at a junction of adjacent dimensions, the geometric continuation is chosen.
- Here “through” is a combinatorial abstraction (a transition through an opposite face under continuous cell gluing), not a physical passage like a tunnel/film. In our world there are no objects on the scale of a fundamental cell; even a photon spans on the order of 10⁴⁸ cells. Therefore, “through” describes lattice connectivity, not motion through a “hole” in matter or space.
Stable lower-dimension sub-networks
- In a multidimensional network, stable lower-dimension sub-networks can form consistently by simply selecting the required number of mutually orthogonal directions and fixing them across the entire domain.
- Their boundaries obey the same two requirements: full face matching and adjacency only of neighboring dimensions.
- Their boundaries obey the same two requirements: full face matching and adjacency only of neighboring dimensions.
Significance of the impact
The geometry of joining is strict but neutral: it ensures continuity, unbrokenness, and correct transitions without gaps or overlaps. Planck-scale discreteness does not generate its own trajectory fluctuations and does not require additional prohibitions on speeds, directions, or “topography”.
This is important for several reasons:
- it allows relying on the familiar macro-picture (inertial “straight”, Euclidean behavior of small regions, isotropy on average);
- it keeps the distribution of cell metric charge as the key physical factor (rather than joining specifics);
- it does not forbid the formation of stable lower-dimension sub-networks without breaking space integrity;
- the discrete substrate reproduces the real continuous picture.
Geometry of Discrete Multidimensional Space
Appendix 1