A single event can have several possible outcomes, although only one of them is recorded in a specific experiment. For example, in an interferometer, a particle may arrive at one of two outputs, and the frequencies of these outcomes depend on the phase shift between the paths. Can such a situation be described as the splitting of a single state? A hypothesis is proposed below: the wave function represents the amplitudes of the channels of a split event, and the operator J determines their phase dynamics.

Let event A have probability p. Its splitting is understood as the identification of mutually exclusive realization variants A₊ and A₋. A possibility channel is a state component corresponding to a specific realization variant. For such a partition, the following holds:

In the Wave Electricity concept, splitting isolates the orthogonal components of a single whole. For an amplitude-based description, we additionally introduce orthonormal channel states:

We associate amplitudes with the probability weights. Let us write the initial state of branch A as

Then its squared norm equals the probability of the initial event:

When p=1, a complete normalized state is described. When p<1, this is merely a branch of event A; the complete state must also include the possibility of its absence. The probabilities themselves do not determine the relative phases of the amplitudes.

We use the operator adopted in the concept:

In the state space, the idempotents act as projectors: each preserves its own channel and nullifies the other. The parameter a retains the meaning of an internal state, while b represents external motion; their physical dynamics are determined by the model. Then

The operator alters the phases while preserving the norm in the introduced channel metric:

Thus, the splitting defines the structure of possibilities, the amplitudes determine their weights, and J describes the phase state. If the weights change, an additional law for amplitude redistribution is required: the diagonal phase operator J does not accomplish this on its own.

The internal state is not yet a scalar position wave function. To obtain this, a registration mapping must be specified. Let u₊(x) and u₋(x) denote the transition amplitudes from the internal channels to the result x. Then

If channels transition coherently into a single indistinguishable result, their amplitudes add up. According to the Born rule, the probability density of this result is given by \[\tag{9} \rho_A(x)=|\psi_A(x)|^2=p_+|u_+(x)|^2+p_-|u_-(x)|^2+2\sqrt{p_+p_-}\operatorname{Re}\!\left[e^{i\pi(a-b)}u_+^*(x)u_-(x)\right]. \] The last term describes interference. The detection mapping must preserve normalization across all results: for a complete continuous representation, the integral of the density is required to equal \(p\). If the paths remain distinguishable due to orthogonal tags, the cross term vanishes when these tags are ignored. Therefore, partitioning an event into mutually exclusive outcomes versus coherently combining paths into a common outcome requires different rules. A simple example involves equal weights \(p_+=p_-=1/2\) and detection in the two orthonormal output states \((\xi_++\xi_-)/\sqrt{2}\) and \((\xi_+-\xi_-)/\sqrt{2}\). The output probabilities then take the form \[\tag{10} P_1=\frac14\left|e^{i\pi b}+e^{i\pi a}\right|^2,\qquad P_2=\frac14\left|e^{i\pi b}-e^{i\pi a}\right|^2,\qquad P_1+P_2=1. \] When the phases coincide, the entire weight is directed to the first output; with a relative phase of \(pi\), it goes to the second. This is the mathematical scheme for phase control in an interferometer. Its physical application requires a specific correspondence between the parameters \(J\) and the phases of the experimental paths.

Each channel may possess its own subsidiary possibilities. If q(ℓ|k) is the conditional probability of a subsidiary outcome, then the amplitudes at the next level can be defined as follows:

The probability weight of the parent branch is conserved. Each level allows for its own phase dynamics, including the operator Jₖ. An individual branch becomes a wave function once its variables, evolution law, and registration mapping are specified.

One may propose an additional hypothesis: the number of state-splitting levels has no predetermined limit. Each successive level is capable of revealing new channels of possibility while preserving the probability weight of the parent branch. If such channels possess their own dynamics and admit a physical mapping, they may manifest as hitherto unstudied properties of particles or interactions. However, infinite splitting depth does not in itself prove the existence of an infinite number of physical properties; for each hypothesized property, it is necessary to define an observable manifestation and a method for experimental verification.

The resulting construction links the probabilistic description of events with the idempotent structure J. To arrive at a full-fledged physical theory, it remains to define the mapping of internal channels to observable outcomes, the dynamics of amplitudes, and the relationship between phases and experimental quantities. The Born rule is adopted here in the construction of amplitudes; however, the Schrödinger equation and the mechanism of individual realization do not yet follow from this construction.

The hypothesis of repeated splitting can be extended to observable reality itself. If a state allows for the unfolding of internal channels, one can also postulate the existence of a more general state of which the world accessible to us is but one component. Here, a "higher level" refers to the parent level of the structure, not an additional spatial dimension.

Let us denote the general state by \(\Psi_{\mathrm{gen}}\) and the projector for the channel of our reality by \(P_R\). Its splitting can then be represented schematically as

If these components are orthogonal, the squared norm of the general state equals the sum of their norms. However, a probabilistic interpretation of the weight of the entire reality requires a separate definition of the events or registration procedures to which such a weight corresponds. It cannot be automatically interpreted as the probability of our world's existence.

In this hypothesis, the channel of our reality encompasses not only individual particles and events but also observers, their instruments, and records of results. Space and accessible physical properties emerge as stable metric and dynamic manifestations of the state within the channel. Consequently, the observer perceives it as a complete world: all comparisons and measurements available to them are carried out via the structures of that same channel.

For the phase dynamics of the parent splitting, one might propose an operator \(J_R(a_R,b_R)\), constructed by analogy with J. However, the physical significance of its parameters must be defined separately; one cannot simply transfer the values ​​of a particle's internal state and velocity to them without further justification. Furthermore, the phase operator does not, in itself, facilitate exchange between channels; such exchange would require a distinct interaction.

For this hypothesis to acquire testable physical content, it is necessary to determine the law governing the formation of the metric within the channel, the conditions for its stability, and the potential observable consequences of its connection to the parent state. If the other channels are completely isolated and have no influence on measurement results, this construct remains an interpretation that is experimentally indistinguishable from a description of our reality alone.