2026-09-14
Splitting of possibilities: channels and spatial ports
A New Language of Mathematics and Physics: How Spaces, Events, and States Emerge from a Unified Whole
Is it possible to provide a mathematical description not only of existing objects but also of the possible ways they might manifest? Typically, space, an event, or a state is treated as a pre-defined element of a model. This article examines a different construction sequence: first, a unified whole exists; then, it splits into independent channels, with each channel terminating in a "port" a distinct possibility for further manifestation.
In the most general terms, a port can be a space, a spacetime, an event, or a state. The specific meaning of the port is determined by the mathematical or physical model within which the splitting occurs.
A port does not create anything automatically; it merely designates a distinct possibility. For that possibility to acquire concrete content, an additional rule is required such as a mapping, a dynamic process, or a physical law. Thus, the structure of possibilities and its realization constitute two distinct levels of description.
The proposed structure is more than just an intuitive concept; it finds precise mathematical expression in idempotent algebra. Idempotents allow a unified whole to be represented as mutually separated channels that do not mix with one another yet collectively preserve the original unity. We will subsequently demonstrate how channels, ports, and a multi-level system of possibilities emerge from simple algebraic relations.
We begin by examining the principle itself, independent of physical space; we then introduce its idempotent foundation and analyze several examples: two system states, repeated splitting, a geometric mapping, fractal construction, and the representation of possible spacetime events.
1. What exactly is being split?
The word "splitting" typically evokes the image of an object divided into parts. If a material body is divided into two parts, each part receives a certain share of the original substance. Such a division can be formally written as
\[\tag{1} X=X_1+X_2. \] The splitting of possibilities carries a different meaning. The initial object is not necessarily physically fragmented. Instead, various ways in which it might continue or manifest are revealed:
\[\tag{2} X\longrightarrow\{X_{+},X_{-}\}. \] Here, \(X_{+}\) and \(X_{-}\) need not yet be two distinct, existing objects. They represent two distinguishable possibilities belonging to a single initial whole, \(X\). Depending on the model, these could be two states, two events, two regions of space, or two different modes of further construction.
Splitting of possibilities is an operation in which a single initial state unfolds into several independent channels of potential manifestation.
The existence of multiple possibilities does not necessitate an immediate choice of one over the others. Possibilities may remain available, coexist as components of a unified structure, or participate in transition dynamics. The question of which possibility is realized and under what conditions pertains not to the splitting itself, but to the subsequent laws of the model.
2. Channels and ports
To distinguish the structure of the splitting from its specific content, we introduce two terms: channel and port.
A splitting channel is an independent direction in which the initial state may potentially continue. Port the endpoint of a channel at the level under consideration, representing a distinct possibility of manifestation.
\[\tag{3} X\longrightarrow \begin{cases} C_{+}\longrightarrow P_{+},\\ C_{-}\longrightarrow P_{-}, \end{cases} \] In formula (3), the quantities \(C_{+}\) and \(C_{-}\) denote two channels, while \(P_{+}\) and \(P_{-}\) denote their corresponding ports. If this structure is depicted as a tree, the channels are represented by branches. Thus, a "branch" is a visual representation of a channel, not a separate mathematical entity.
| Term | Meaning |
|---|---|
| Initial state | A unified whole prior to splitting |
| Channel | An independent direction for the potential continuation of the initial state |
| Branch | A geometric or graphicchannel representation |
| Port | The endpoint of a channel at a given level, representing a distinct possibility for manifestation |
| Mapping | A rule defining the specific content of a port |
| Realization | The transformation of a port's potential into space, space-time, an event, or a state |
| Transition | The transfer of a state from one port or channel to another |
Channels describe how possibilities are differentiated, while ports indicate where they can find specific manifestation.
A channel and a port should not be completely equated. A channel defines the path or affiliation of a possibility, whereas a port designates the accessible outcome of that path. Upon further splitting, a single port can become the starting point for several new channels.
3. What a port can represent
A port is introduced as a maximally general concept. It need not be a point in ordinary geometric space from the outset. Its content is determined by the mapping \(\mathcal M\) selected for a specific task:
\[\tag{4} P\xrightarrow{\;\mathcal M\;} \begin{cases} \text{space},\\ \text{spacetime},\\ \text{event},\\ \text{state}. \end{cases} \] A space port can denote a region where an intrinsic geometry unfolds. A spacetime port can encompass not only spatial coordinates but also temporal ordering. An event port represents a potential change or interaction. A state port specifies one of the available modes of an object or system.
The same formal port can be interpreted in different ways across different theories. Therefore, its physical meaning is not automatically contained within the symbol \(P\); it emerges only after the mapping, postulates, and laws of realization are defined.
4. Splitting, mapping, and realization
To construct a model consistently, it is necessary to distinguish between three operations.
Splitting creates distinguishable channels and ports. Mapping defines the meaning of each port. Realization transforms the corresponding possibility into a specific space, event, or state.
\[\tag{5} X\xrightarrow{\;\mathcal R\;}P_{\sigma} \xrightarrow{\;\mathcal M\;}Y_{\sigma}, \qquad \sigma\in\{+,-\}, \] where \(\mathcal R\) is the splitting operation, \(P_{\sigma}\) is one of the ports, \(\mathcal M\) is a given mapping, and \(Y_{\sigma}\) is its specific manifestation.
The splitting itself does not select a single port, assign a probability, trigger a transition between ports, or create physical space. It merely forms an organized system of possibilities. These constraints are important: they prevent the conflation of the pure algebraic structure with its physical interpretation.
5. A simple example: two states
Let system \(X\) admit two states \(A\) and \(B\). First, two ports are created, and then a specific state is associated with each of them:
\[\tag{6} X\longrightarrow \begin{cases} P_A\xrightarrow{\;\mathcal M\;}A,\\ P_B\xrightarrow{\;\mathcal M\;}B. \end{cases} \] In one model, \(A\) and \(B\) might signify two operating modes of a device; in another, two directions; in a third, two polarizations or two spin orientations. The splitting structure remains the same, while the meaning of the ports changes.
Formula (6) does not yet assert that the system randomly selects only one state. Several modes are possible: one port is active; both ports form a single composite state; transitions occur between ports; different ports are observed in different projections. The choice of specific dynamics must be specified separately.
6. Idempotent basis of channels
We now show that the channel structure has a precise algebraic representation. Let us introduce a hyperbolic unit \(\j\) such that
\[\tag{7} \j^2=1, \quad \j \ne \pm 1. \] Two idempotents are defined using this unit:
\[\tag{8} \ep=\frac{1+\j}{2}, \qquad \em=\frac{1-\j}{2}. \] They satisfy the relations
\[\tag{9} \boxed{ \ep^2=\ep, \qquad \em^{\,2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1. } \] Each equality acquires a natural meaning within the structure of possibilities. The conditions \(\ep^2=\ep\) and \(\em^2=\em\) demonstrate the stability of each channel: reapplying the same projector leaves the channel unchanged. The condition \(\ep\em=0\) expresses the separation of the channels. Finally, the equality \(\ep+\em=1\) shows that the two channels together preserve the original whole.
The general state can be written as
\[\tag{10} \boxed{ X=\ep X_{+}+\em X_{-}. } \] The idempotents \(\ep\) and \(\em\) isolate two independent channels, while the quantities \(X_{+}\) and \(X_{-}\) define the content of the respective ports. Mixed products vanish:
\[\tag{11} (\ep X_{+})(\em X_{-})=0. \] This does not imply the destruction of the original object; both components remain parts of a single unit. It is precisely the combination of channel separation and the preservation of the whole that allows idempotent algebra to serve as the mathematical foundation for the splitting of possibilities.
7. The port as an output and a new input
A port is an endpoint only relative to the chosen depth of analysis. If a possibility allows for further refinement, the port becomes the starting point for a new system of channels:
\[\tag{12} P_{+}\longrightarrow \begin{cases} P_{++},\\ P_{+-}. \end{cases} \] If both ports of the first level are split, four ports emerge:
\[\tag{13} \{P_{++},P_{+-},P_{-+},P_{--}\}. \] A port serves as the output of the preceding splitting level and a potential input for the next level.
Subsequent splitting does not negate the previous structure but rather refines the possibilities contained within its ports. A detailed algebraic construction of multiple levels is presented in the article “Multilevel Idempotent Splitting of Phase Planes.”
8. Port Addressing
After several levels, each port is assigned an address consisting of a sequence of selected channels:
\[\tag{14} P_{\boldsymbol{\sigma}} =P_{\sigma_1\sigma_2\ldots\sigma_N}, \qquad \sigma_k\in\{+,-\}. \] For example, the address \(P_{+-+}\) indicates that the \(+\) channel is used at the first level, the \(-\) channel at the second, and the \(+\) channel again at the third. With \(N\) independent binary levels, the number of ports is
\[\tag{15} N\, \text{levels} \quad\Longrightarrow\quad 2^N\, \text{ports}. \] A port address is not a geometric coordinate. A coordinate answers the question of where a point is located within a pre-defined space. A port address answers a different question: through what sequence of splittings did this specific possibility arise?
9. How a port becomes a space
Splitting channels need not correspond to directions in ordinary space. Therefore, the transition from an algebraic tree of ports to geometry requires a specific mapping. Let a radius vector be assigned to each port address:
\[\tag{16} \mathcal M\!\left(P_{\boldsymbol{\sigma}}\right) =\mathbf r_{\boldsymbol{\sigma}}. \] The mapping \(\mathcal M\) can define coordinates, direction, step length, rotation angle, scale, or a local coordinate system. Depending on the chosen rule, the same system of ports can be mapped into a one-dimensional, two-dimensional, or three-dimensional space, or into a more general geometry.
Consequently, the dimensionality of the space cannot be determined simply by counting the channels. Two channels might be represented by two rays on a plane, two regions in three-dimensional space, or two independent internal states. It is not the number of ports per se that defines the geometry, but the rule governing their mapping.
10. A fractal as a mapping of repeated splitting
A fractal serves as a clear example of how an abstract system of ports can acquire a geometric expression. Each port from the previous level becomes the starting point for the next splitting:
\[\tag{17} P_{\boldsymbol{\sigma}}\longrightarrow \left\{ P_{\boldsymbol{\sigma}+}, P_{\boldsymbol{\sigma}-} \right\}. \] After this, each child port is assigned its own geometric transformation:
\[\tag{18} \mathbf r_{\boldsymbol{\sigma}+} =F_{+}\!\left(\mathbf r_{\boldsymbol{\sigma}}\right), \qquad \mathbf r_{\boldsymbol{\sigma}-} =F_{-}\!\left(\mathbf r_{\boldsymbol{\sigma}}\right). \] The transformations \(F_{+}\) and \(F_{-}\) can alter position, scale, and angle. By repeating them at each level, one can obtain a self-similar geometric structure.
A fractal isnot by the splitting itself, but by one of the geometric mappings of the repeatedly splitting system of possibilities.
This distinction is fundamental. The same algebraic system of addresses can generate different fractals if the functions \(F_{+}\) and \(F_{-}\) are altered. Consequently, the splitting defines the structure of possibilities, while the transformation algorithm determines its observable form.
11. Spacetime and event ports
If a port is associated not only with a position but also with a moment in time, it can represent a possible spacetime event:
\[\tag{19} \mathcal M\!\left(P_{\boldsymbol{\sigma}}\right) =\left(\mathbf r_{\boldsymbol{\sigma}}, t_{\boldsymbol{\sigma}}\right). \] In such a mapping, a channel connects possible events, and a sequence of channels forms a possible history of the system. However, the mathematical existence of a channel does not imply that the corresponding event is physically realizable. Permissible connections must satisfy causality, conservation laws, speed limits, and other laws of the chosen physical model.
Therefore, the complete structure of ports may be broader than the set of realizable events. Algebra reveals possible independent directions, while physics selects the permissible processes from among them.
12. Physical state ports
Ports can also represent distinguishable states of a particle or field:
\[\tag{20} P_{+}\mapsto\text{state }+, \qquad P_{-}\mapsto\text{state }-. \] For instance, in a specific model, these ports could be associated with two polarizations, two spin directions, or two internal modes. Idempotent splitting ensures the mathematical distinguishability of channels but does not assign them a physical meaning. Such an assignment constitutes a separate postulate and must be consistent with observations.
In the Wave Electricity model, multilevel channels can be used to describe the internal components of a unified wave object. However, the proposed concept of splitting possibilities extends beyond a single physical model: it can be applied wherever it is necessary to separate the structure of possible manifestations from the rules governing their realization.
13. Transition between ports
Splitting creates ports but does not, in itself, transfer the state from one port to another. A separate operator (S) is required for the transition:
\[\tag{21} P_{+}\xrightarrow{\;S\;}P_{-}. \] This yields an important distinction:
\[\tag{22} \boxed{ \text{existence of ports} \neq \text{transition between ports}. } \] Depending on the physical problem, transitions may be forbidden, singular, periodic, or dependent on external influence. All such properties pertain to dynamics. Idempotent algebra primarily addresses the question of which independent channels exist, but it does not define the law of motion between them.
14. What the new approach offers
In mathematics, the splitting of possibilities offers a common language for describing branching and multilevel structures. Ports are assigned unique addresses, computations within idempotent channels can proceed independently, and the geometric mapping is decoupled from the underlying algebra.
In physics, this approach allows for the separate consideration of three questions: what possibilities exist, what they can be mapped to, and the laws governing their realization. In this framework, space, spacetime, events, and states do not serve as mandatory foundational entities but rather as possible contents of the ports.
Preserving the original unity is particularly important. Split channels need not describe distinct objects; they can remain mutually distinguishable parts of a single, unified system:
\[\tag{23} \ep+\em=1, \qquad \ep\em=0. \] The first equality preserves the whole, while the second separates the channels. It is the combined action of these two conditions that encapsulates the fundamental logic of the approach.
15. Scope of Application
Idempotent algebra provides a rigorous mathematical foundation for independent channels; however, it does not, in itself, establish a specific physical interpretation for the ports. To designate a port as space, an event, or a particle state, one must define the corresponding mapping and demonstrate that the resulting model is consistent with experimental data.
Furthermore, splitting does not determine the probability of realization, replace equations of motion, or automatically explain transitions between ports. These elements must be introduced via additional laws. Such a distinction does not weaken the approach but rather clarifies its scope of applicability.
Idempotent algebra substantiates the channel structure. The mapping defines the port's content.Dynamics determines transitions and the conditions for realization. The experiment tests the physical interpretation.
Conclusion
The proposed approach begins not with a ready-made space or a pre-defined set of states, but with a unified whole and the structure of its possibilities:
\[\tag{24} \boxed{ \text{whole} \longrightarrow \text{splitting} \longrightarrow \text{channels} \longrightarrow \text{ports} \longrightarrow \text{mapping} \longrightarrow \text{realization}. } \] The splitting of possibilities is the representation of a single initial state as a system of independent channels, each terminating in a port. In the most general sense, a port can be a space, spacetime, an event, or a state. Its specific content is determined by the mapping, while the conditions for realization are determined by dynamics or a physical law.
If a port splits further, it becomes the starting point for a new system of channels. This gives rise to a multilevel structure of addressable possibilities. Idempotent algebra gives this structure a precise form: the channels remain distinct, yet together they preserve the original whole.
Splitting creates channels. Channels terminate in ports. Ports open up possibilities for the emergence of space, spacetime, events, and states.