Space

Beeng 01:00

«…Everywhere am I, and within Me all is.
Great Vast is My Name.»

Space is primary and absolute.

Its primacy and absoluteness lies in the fact that it cannot be redefined in terms of any other concept. At most, what presently seems possible to say is that space is:
– that which is everywhere,
– that within which everything is located.

Space Quantum (SQ). Discreteness

Space (macrospace) consists of elementary quanta—fundamental Planck-scale hypercubes, which we call the Space Quantum (SQ) [1]. Each SQ has an internal microspace with its own dimension d. This dimension can be arbitrarily large; it is integer-valued and local.

For symbols and abbreviations used, see Appendix 7. Notation Conventions.

Note 1

Hypothesis: in the course of further research, other classes of topology for the “carrier cell” may be identified—with their own face combinatorics F(d) and, consequently, their own C(d) = 2·F(d) (simplicial (d-simplex), orthoplex (cross-polytope), etc.). This would yield different mathematics.

Changes in microdimension d. SER

The dimension of SQ cells can change between individual dimension states d.

Transitions follow the SER (Space Expansion Rule):

Degradation: SQ_d → 2×d×SQ_d−1
Reverse: 2×(d+1)×SQ_d → SQ_d+1.

Geometry and topology

Each SQ is a hypercube with edge length ℓₚ.

Its internal volume is V_d = ℓₚᵈ.

An SQ connects to its neighbors via its (d−1)-faces; one SQ has 2ᵈ such connections.

The faces of the hypercube label discrete orthogonal coordinate directions that define the local frame of space.

A network of such connected cells forms macrospace with mean dimension D.

Сеть объединяет SQ разной размерности d.

Space discreteness implies a requirement of “unboundedness”, which is possible only under a condition of “closedness”. The best implementation of these principles is a Tᴰ hypertorus with periodic boundaries [2]. The rules and specifics of the geometry of such a space are described in Appendix 1. Geometry of Discrete Multidimensional Space.

Due to variants of “connectivity”, SQ with higher local dimension d can form macrospaces with mean dimension D < d (by selecting sub-networks). The case D > d is impossible. This also sets an observational constraint on the current macro-dimension: it is certainly not less than 3.

Note 2

The family of such spaces is broader, but topologically the Tᴰ hypertorus is the best observational match.

SQ charge

Each SQ contains charge q.

SQ charge is equivalent to mass (hereafter measured in kg).

SQ charge is a discrete quantity.

SQ charge can:
- be emitted from SQ into space, forming substance (emission),
- be absorbed back into SQ when substance is destroyed (absorption);
- transitions occur when local configuration conditions of the environment (“windows”) coincide; integrality and balances are preserved (charge conservation).

SQ charge governs the expansion dynamics of the space of the observable Universe and the full gravitational “connectedness” of Space.

Further in the text, the following terms will/may be used:
- the cell “charge”, the “metric” (spatial) charge—the cell’s actual charge;
- for computational convenience, the elementary charge may be set equal to 1 (as may distance, time, volume, and mass).

The total charge of a cell of dimension d: N_d = 2 × d × d!—always an integer number of elementary charges—q_d = N_d × Q_abs. After substance emission, the SQ charge becomes non-full, but it still remains an integer number of Q_abs.

Charge quantum (ChQ)

SQ charge consists of elementary charges = the charge of a zero-dimensional cell Q_abs = 1.82865×10⁻¹³⁴ kg.

Charge Quantum (ChQ).

Charge interaction

SQ charges interact with each other. Charge interaction between SQ is fundamental and is realized in an arbitrary bubble of the Universe through the gravitational constant G—a meta-parameter of the Universe that reflects the coupling between charges (masses). The G constant is fixed and invariant.

Operationally, G is a single coefficient that maps the masses (charges) of the interacting objects and the distance between them into the observable attractive force:

F = G × m₁ × m₂ / r^(d-1)

d – space dimension

Interaction quantum (IQ)

Fundamental interaction (force quantum) is the attractive force between two mass quanta (elementary charges) at a distance of one space quantum.

Interaction Quantum (IQ).

Mass (charge) attraction is a more fundamental interaction of the Universe than the interactions of the Standard Model, and it does not require a carrier particle (graviton).

Interaction range

Each (space) cell of dimension d has a fixed charge Qd, consisting of 2^d elementary charges. This charge acts on the surrounding cells (in the broad sense—on the medium). Nominally, the force at a discrete distance r (the number of orthogonal steps) is described as

F_d(r)=Q_d / r^d−1

The force quantum is indivisible: at a node, the interaction either exists (one full quantum or more) or it does not. Therefore, there is a maximum interaction range R_d—the last step after which R_d(r) < 1 and the interaction becomes zero.

Hierarchy of radii (top to bottom):

d ≥ 3. The radius is stable for any d and equals 2.
d = 2. Phenomenon. Doubling of the radius: R₂ = 4. This creates a “support” regime and helps the 3D framework maintain connectedness (“props up” connectedness) under degradations. This radius staircase is one of the reasons why 3D becomes the first stable volumetric regime at minimal cost.
d = 1. The interaction range returns to R₁ = 2.
d = 0. Phenomenon. Formally, the force increases with distance (F₀ ∝ r). Physical interpretation: zero-dimensional states are unstable in isolation and spontaneously tend to coalesce into d = 1 (a seed of dimension growth).

Disclaimer. Everything above is a direct consequence of the model’s discreteness. Nevertheless, the radius staircase provides a natural mechanism that may be a significant factor when SER is switched into reverse mode.

Note on “4/3” (2D node, r = 3). The formal nominal value F₂(3) = 4/3 conflicts with quantum indivisibility. We record this unique case as an open technical node: an integer scheme or a physical / geometric interpretation is needed.

At present, the most acceptable explanation of the phenomenon appears to be that dimension 2 is an unstable state (simultaneously: (i) the interaction range jumps to 4 and (ii) quanta cannot be at a distance equal to 3), and that within it Space Quanta SQ undergo a phase transition from scattered elements to connected space (spontaneous unification) / from connected space to scattered elements (spontaneous localization).

Quantumness

Space is discrete; it consists of SQ.
SQ states are discrete: microdimension changes sequentially in integer steps between individual dimension states d.
The minimal time interval for a transition is the time quantum t_abs.
Transitions follow SER (Space Expansion Rule):
- Degradation: SQ_d → 2×d×SQ_d−1
- Reverse: 2×(d+1)×SQ_d → SQ_d+1.

This process leads to an increase in the number of cells and an increase in the radius of space – see Space Expansion Rule (SER).

The cell state is described by the “wave function” Ψ_d: a probability vector for which allowed event will occur at the next SER step (see Appendix 2. SQ wave function: Fate ):

Fate:

(1) decrease in dimension (Down: D→D−1),
(2) holding the current dimension (Hold: D→D),
(3) increase in dimension (Up: D→D+1),

Burden:

(4) emission of charge (substance) into macrospace,
(5) absorption of such charge (substance) from space.

Heredity

The base rule SQ_d → 2×d×SQ_d−1 is preserved at all steps.
The parent charge is split equally among the offspring, preserving the overall balance. The system has no arbitrary “smearing” of charge—it is discrete and quantized. Under SER reverse transition, the charges of all SQ_d sum into SQ_d+1.
The SQ edge always remains Planck-length ℓ_p; under any transition it does not stretch or shrink.
Face orientation is inherited: the offspring replicate the parent’s frame system, ensuring consistency of the local space lattice.
Connectedness is preserved: the children inherit the parent’s boundary connections through the corresponding faces.

Information. SQ SSD

Hypothetically, SQ faces can be treated as discrete state registers (state carriers) with information capacity proportional to the face area of the corresponding dimension (“pixels”, “memory slots”) [3]. For now, assume 1 “pixel” = 1 bit.

If SQ faces are treated as discrete state registers, it is hypothetically consistent that one could “write” into them the “algorithmics” of our Universe bubble (physics, interaction rules, local constants, rules of dynamic homeostasis, local coefficients, etc.).

For a cell of dimension d:

number of faces: 2 × d
“pixels” per face: 2^d−1
total slot grid: B_d= d × 2^d.

As dimension grows, “available memory” grows exponentially, while the requirement to “encode” charge grows logarithmically in content (polynomially in arguments). Therefore, higher dimensions create an increasing surplus of information carriers.

The critical point is d = 3: from this point downward, “memory” is strictly sufficient only for charge distribution, while the “algorithmics of the world” (complex interaction rules) no longer “fits”. This can serve as an additional explanation of the behavior of Space and Matter at low dimensions (see Curtain of the Curtain). The multidimensional world is rich and diverse, but entropy advances and wins.

Note 3

Within the present text, this assumption is not used in any calculations and yields no physical consequences (no anisotropy, chromaticity, birefringence, or scattering). It is introduced as an auxiliary abstraction for possible future sections (for example, on encoding neighbor links / the history of SER steps / entropy).

Properties of local dimensions

d	Faces 2×d	Pixels per face 2^d−1	Total pixels d×2^d, bits on faces	Charge Q=2^d×d!	Long-range interaction law	r_max* [4]	Fate (ii)
0	1	1	1	1	1/ r^-1	Force increases with distance (i)	Up
1	2	1	2	2	1/r⁰	2	Up, Down, Hold
2	4	2	8	8	1/r¹	4	Up, Down, Hold
3	6	4	24	48	1/r²	2	Up, Down, Hold
4	8	8	64	384	1/r³	2	Up, Down, Hold
5	10	16	160	3840	1/r⁴	2	Up, Down, Hold
n	2n	2ⁿ⁻¹	n×2ⁿ	2ⁿ×n!	1/r ⁿ⁻¹	2^n/(n−1)	Up, Down, Hold

(i) This is, obviously, a paradox. But zero-dimensionality is paradoxical in itself, so for now we accept the picture dictated by the mathematics.

(ii)
– Up – find another SQ_n → SQ_n+1
– Down – decay back into SQ_n → SQ_n-1
– Hold – remain unchanged

Note 4

Since this is a discrete interaction (elementary charge / elementary distance / elementary time), the interaction range is measured only in integer units. This means there can be no “fractional” forces or distances. Therefore, the interaction range always has a finite radius. Moreover, for d ≥ 3 this radius is always equal to 2.

d = 0. Faces: 1. Face area: 1. Charge: 1. Direction: degenerate—the only Fate option is to join with an identical object and form d = 1. “Wave function”: Ψ₀ is degenerate (one configuration, probability 1). Step outcome: only Up (0→1); Hold/Down are not defined. Emission is impossible.

d = 1. Event: two SQ₀ merged into SQ₁. Faces: 2. Total face area: 2. Charge: 2. Burden: from two SQ₀ is transferred to the new 1D cell. Charge conservation holds. Fate: the probability of three outcomes appears (up to SQ₂—Up, down to SQ₀—Down, unchanged—Hold). Ψ₁: a single uniform configuration (with point symmetry “left–right”); there is no migration of quanta—any permutation is identical to the original. Outcomes: Hold is unstable; Up is possible under external “pushing apart”, Down—under “pulling in” (but there is no wandering).

d = 2. Event: four SQ₁ merged into SQ₂. Faces: 4. Total face area: 8. Charge: 4. Burden: from four SQ₁ is transferred to the new 2D cell. Charge conservation holds. Fate: the probability of three outcomes (up to SQ₃—Up, down to SQ₁—Down, unchanged—Hold) is preserved.

d = 3. Event: six SQ₂ merged into SQ₃. Faces: 6. Total face area: 24. Charge: 8. Burden: from six SQ₂ is transferred to the new 3D cell. Fate: the probability of three outcomes (up to SQ₄—Up, down to SQ₂—Down, unchanged—Hold) becomes a genuinely probabilistic phenomenon.

From here on, everything follows the same typical scenario. However, dimensions 1 and 2 clearly stand out among the others. Starting from dimension 3 and above, the “interaction range” of charges stabilizes at 2. At level 2 it equals 4, and at level 1 it equals 2. I will return to this phenomenon once the Background Process gains a “support” of stability and, more broadly, to “why 3D is so special” …

d = 4. With all other processes remaining analogous, another phenomenon appears at this dimension level (and persists for all subsequent dimensions): starting from dimension 4, the interaction range becomes constantly 2.

Beeng 04:00

«…And Promised Hour shall come –
The scrip shall be rent;
And Vast shall let fall
A little of Her Burden…»

Space

Quantum of quanta

support me

matter