2026-07-22
Geometric representation of particle interactions
Part 4. State space and probabilities of sector rearrangements
In part three, a particle was defined as a stable coordinated assembly of independent operator sectors, and interaction as a change in branch membership maps \(\kappa_r\) only in active levels. This geometry enumerates the possible output assemblies, but by itself does not yet determine the probability with which each of them is realized.
For a probabilistic description, three objects must be redefined: the space of complete states, the inner product, and the interaction operator. The previous ket \(|A:B\rangle\) described the connection of two branches of a single split sector and therefore could not represent the entire particle. Now the ket will contain the full sector signature, the internal closure state, and a set of connection maps.
In this section, the rule \(P=|\mathcal A|^2\) is accepted as a probabilistic postulate consistent with quantum mechanics. Orthogonal channels and the rules for their restructuring follow directly from idempotent algebra, but the square of the amplitude modulus, the single-signaling mechanism, and specific numerical probabilities have not yet been derived from split geometry.
\[\tag{1} \boxed{ |\Psi_{\mathrm{out}}\rangle =\widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle =\sum_k\mathcal A_k|\mathcal C_k\rangle, \qquad P_k=|\mathcal A_k|^2. } \] 1. The complete state of a particle
The modern multioperator notation of a particle has the form
\[\tag{2} \boxed{ J_P =J_{\mathrm{dyn}}^{(\sigma)}(a,b) Q_\Gamma J_qJ_{\mathrm{orb}}J_fJ_cJ_g\cdots, \qquad \sigma=\pm. } \] The dynamical operator contains internal periodicity, external motion, and two orientations of the state. The operator \(Q_\Gamma\) characterizes the internal closure, \(J_q\) the charge structure, and \(J_{\mathrm{orb}}\) the orbital state. The remaining independent properties receive their own operators and are not required to be in the first two idempotent planes.
We associate ket with the complete assembly
\[\tag{3} |P\rangle =\left| J_{\mathrm{dyn}}^{(\sigma)}, Q_\Gamma,J_q,J_{\mathrm{orb}},J_f,J_c,J_g,\ldots \right\rangle. \] Its compact notation:
\[\tag{4} \boxed{ |P\rangle=|\Lambda_P,\kappa_P\rangle. } \] Here \(\Lambda_P\) is the complete structural signature, and \(\kappa_P=\{\kappa_r\}\) is the set of branch membership maps. Thus, ket no longer denotes a pair of components, but the complete state of a stable operator assembly.
2. Three Different Concepts of Norm
In a probabilistic model, it is especially important not to confuse three mathematically different quantities. The first is the finite algebraic norm of the dynamic operator:
\[\tag{5} J_{\mathrm{dyn}} \overline{J}_{\mathrm{dyn}}=1. \] It expresses the unity of the chosen operator geometry. The second is the normalization of the state vector:
\[\tag{6} \langle\Psi|\Psi\rangle=1. \] It acquires probabilistic meaning only after defining the positive inner product. The third quantity is the energy projection of the internal frequency:
\[\tag{7} E=\hbar\omega. \] Equality (5) is neither a quantity of physical energy nor its conservation law. Formulas (5)–(7) belong to different levels of description: the operator algebra, the geometry of the state space, and the physical projection of temporal dynamics.
3. Sector Space of a Single Particle
Each independent level \(r\) is associated with its own state space \(\mathcal H_r\). Level set:
\[\tag{8} r\in\mathcal R =\{\mathrm{dyn},\Gamma,q,\mathrm{orb},f,c,g,\ldots\}. \] The formal space of all sector combinations is formed by the tensor product:
\[\tag{9} \boxed{ \mathcal H_P^{(0)} =\mathcal H_{\mathrm{dyn}} \otimes\mathcal H_\Gamma \otimes\mathcal H_q \otimes\mathcal H_{\mathrm{orb}} \otimes\mathcal H_f \otimes\mathcal H_c \otimes\mathcal H_g\otimes\cdots. } \] The sign \(\otimes\) here denotes the product of state spaces. It should not be confused with the usual multiplication of commuting operators within the algebraic notation (2).
The separable state has the form
\[\tag{10} |P\rangle =|\lambda_{\mathrm{dyn}}\rangle \otimes|\lambda_\Gamma\rangle \otimes|\lambda_q\rangle \otimes|\lambda_{\mathrm{orb}}\rangle \otimes\cdots. \] However, the overall state may contain correlations between sectors and then cannot be factored into a single product:
\[\tag{11} |P\rangle =\sum_{\boldsymbol\lambda} C_{\boldsymbol\lambda} |\lambda_{\mathrm{dyn}},\lambda_\Gamma, \lambda_q,\lambda_{\mathrm{orb}},\ldots\rangle. \] 4. Admissible Subspace
The space \(\mathcal H_P^{(0)}\) contains formally possible combinations, but not every one of them represents a stable particle, a bound system, or a free wave. Therefore, we introduce the projectors of algebraic, metric, closing, topological, and dynamic stability:
\[\tag{12} \widehat{\mathcal P}_{\mathrm{adm}} =\widehat{\mathcal P}_{\mathrm{alg}} \widehat{\mathcal P}_{\mathrm{met}} \widehat{\mathcal P}_{\Gamma} \widehat{\mathcal P}_{\mathrm{top}} \widehat{\mathcal P}_{\mathrm{stable}}. \] The physically feasible space is defined by the image of this projector:
\[\tag{13} \boxed{ \mathcal H_P =\operatorname{Im} \widehat{\mathcal P}_{\mathrm{adm}} =\widehat{\mathcal P}_{\mathrm{adm}} \mathcal H_P^{(0)}. } \] For a valid state
\[\tag{14} \widehat{\mathcal P}_{\mathrm{adm}}|P\rangle=|P\rangle, \qquad \widehat{\mathcal P}_{\mathrm{adm}}^2 =\widehat{\mathcal P}_{\mathrm{adm}}. \] Thus, a forbidden assembly is excluded not by a verbal prohibition, but by the absence of a corresponding vector in the valid subspace. The specific form of each projector remains a separate problem of the model.
5. Signature and Basis of States
The structural signature of a particle contains the values of all the features necessary for its distinction:
\[\tag{15} \Lambda_P =\left( \sigma,E,Q,N_\Gamma, \mathrm{orb},f,c,g,\ldots \right). \] We will denote the basis vector as
\[\tag{16} |\Lambda,\kappa\rangle, \] where the same signature can admit multiple internal connection maps. The full vector of a particle is their linear combination:
\[\tag{17} |P\rangle =\sum_{\kappa\in K(\Lambda_P)} C_{\kappa}|\Lambda_P,\kappa\rangle. \] It is the full signature, not a single pair of split branches, that determines whether a state belongs to the electron, photon, muon, or other class.
6. Multiparticle States and Channel Space
The joint state of two original objects is written as
\[\tag{18} |\Psi_{12}\rangle =|P_1\rangle\otimes|P_2\rangle. \] For \(N\) objects:
\[\tag{19} \mathcal H^{(N)} =\bigotimes_{n=1}^{N}\mathcal H_{P_n}. \] Interaction can change the number and type of observed objects. Therefore, the space \(\mathcal H^{(N)}\) of a single fixed number of particles is not enough. We introduce the full space of physical channels:
\[\tag{20} \boxed{ \mathscr H =\bigoplus_{\chi}\mathcal H_\chi. } \] For example, for an electron-positron system, the accessible part of space may have the structure
\[\tag{21} \mathscr H_{e^-e^+} =\mathcal H_{e^-e^+} \oplus\mathcal H_{\gamma\gamma} \oplus\mathcal H_{\mu^-\mu^+} \oplus\mathcal H_{Ps\,\gamma} \oplus\cdots. \] Each term corresponds to an entire output channel with its own dimension, continuous parameters, and set of internal states.
7. Connection maps as part of the state
Let \(\kappa_r\) indicate which output objects the branches of sector \(r\) belong to:
\[\tag{22} \kappa_r: \{X_{1r}^+,X_{1r}^-,X_{2r}^+,X_{2r}^-\} \longrightarrow \{P'_1,P'_2,\ldots\}. \] The complete map of the system is a set of sector maps:
\[\tag{23} \boxed{ \kappa =\{\kappa_r\}_{r\in\mathcal R}. } \] The direct and cross assemblies of the previous model are preserved as two partial values of a single \(\kappa_r\). They are no longer considered universal states of the entire particle.
8. Inner Product
To go from a formal set of states to amplitudes, a positive-definite inner product is required. For each sector, we choose a metric mapping of the branches onto orthonormal vectors:
\[\tag{24} p_r^+\longleftrightarrow\boldsymbol\xi_r^+, \qquad p_r^-\longleftrightarrow\boldsymbol\xi_r^-, \qquad \boldsymbol\xi_r^\alpha\boldsymbol\cdot \boldsymbol\xi_r^\beta=\delta_{\alpha\beta}. \] In the chosen coordinate basis, the sectorial inner product can be written in terms of the positive Hermitian metric \(G_r\):
\[\tag{25} \langle x_r|y_r\rangle_r =x_r^\dagger G_r y_r, \qquad G_r=G_r^\dagger>0. \] For separable complete states, the sectorial overlap product is a natural definition:
\[\tag{26} \boxed{ \langle P'|P\rangle =\prod_{r\in\mathcal R} \langle\lambda'_r|\lambda_r\rangle_r. } \] Incompatibility in one mandatory sector nullifies the full overlap:
\[\tag{27} \langle\lambda'_s|\lambda_s\rangle_s=0 \quad\Longrightarrow\quad \langle P'|P\rangle=0. \] For the correlated state (11), the inner product is calculated in the full tensor space and is no longer factorized. If the selected states are non-orthogonal, their geometry is described by the Gram matrix:
\[\tag{28} G_{mn}=\langle\mathcal C_m|\mathcal C_n\rangle. \] Thus, the inner product is an additional metric structure of the state space. It does not automatically coincide with the algebraic norm of an individual operator.
9. Sector Interaction Operator
The concept already uses the notation \(V(t)=cJ(t)\) for velocity. Therefore, we denote the interaction operator not by \(\widehat V\), but by \(\widehat{\mathcal U}_{\mathrm{int}}\):
\[\tag{29} |\Psi_{\mathrm{out}}\rangle =\widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle. \] The operator is valid only in the set of active sectors \(\mathcal R_{\mathrm{act}}\):
\[\tag{30} \boxed{ \widehat{\mathcal U}_{\mathrm{int}} =\widehat{\mathcal P}_{\mathrm{adm}} \left[ \prod_{r\in\mathcal R_{\mathrm{act}}} \widehat{\mathcal U}_r(\kappa_r) \right] \otimes \widehat I_{\mathrm{spec}}. } \] Here \(\widehat I_{\mathrm{spec}}\) preserves spectator levels. If different sector transformations do not commute, the product in (30) must be ordered:
\[\tag{31} \prod_r\widehat{\mathcal U}_r \;\longrightarrow\; \mathcal T_\tau \exp\left( -i\int d\tau\, \widehat{\mathcal G}_{\mathrm{int}}(\tau) \right). \] On the admissible subspace, the projection \(\widehat{\mathcal P}_{\mathrm{adm}}\) acts as a unit. If an unprojected transformation creates forbidden components, its projection is no longer required to preserve the norm; this means that the chosen generator or set of channels under consideration is incomplete.
10. Structural Invariants of Interaction
The admissibility of a transition should not be determined solely by subsequent verification of the familiar list of conservation laws. In a closed complete system, interaction is considered as an internal transformation that preserves the generators of its symmetries and complete sector indices.
Let \(\widehat I_\alpha\) be the structural invariant operator. Then
\[\tag{32} \left[ \widehat{\mathcal U}_{\mathrm{int}}, \widehat I_\alpha \right]=0. \] For a norm-preserving operator, (32) implies the equality of the corresponding mean values:
\[\tag{33} \langle\Psi_{\mathrm{out}}| \widehat I_\alpha |\Psi_{\mathrm{out}}\rangle = \langle\Psi_{\mathrm{in}}| \widehat I_\alpha |\Psi_{\mathrm{in}}\rangle. \] Energy is associated with the generator of shifts along the time parameter, momentum with the generator of stable spatial translation, charge and closure with the indices of the corresponding internal sectors:
\[\tag{34} \widehat H=i\hbar\partial_\tau, \qquad \widehat p_s=-i\hbar\partial_s, \qquad \widehat Q,\quad\widehat N_\Gamma. \] If the geometry of the closed interaction is homogeneous with respect to the shift \(\tau\), then for unitary total evolution
\[\tag{35} \widehat{\mathcal U}_{\mathrm{int}}^\dagger \widehat H \widehat{\mathcal U}_{\mathrm{int}} =\widehat H \quad\Longleftrightarrow\quad \left[ \widehat{\mathcal U}_{\mathrm{int}}, \widehat H \right]=0 \quad\Longrightarrow\quad E_{\mathrm{in}}=E_{\mathrm{out}}. \] Therefore, conservation of energy is associated not with the unity of \(J\overline J\), but with the symmetry of the complete dynamics with respect to time transport. A rigorous derivation of this symmetry from the primary order of events remains a separate task of the concept.
11. Dynamic Generator
The perestroika operator can be represented as an evolution between two moments of the parameter \(\tau\):
\[\tag{36} \widehat{\mathcal U}_{\mathrm{int}}(\tau_2,\tau_1) =\mathcal T_\tau \exp\left[ -\frac{i}{\hbar} \int_{\tau_1}^{\tau_2} \widehat H_{\mathrm{int}}(\tau)\,d\tau \right]. \] The corresponding equation for the change of state is:
\[\tag{37} i\hbar\frac{d}{d\tau}|\Psi(\tau)\rangle =\widehat H_{\mathrm{int}}(\tau) |\Psi(\tau)\rangle. \] In sector form, the generator can include single-level and correlation terms:
\[\tag{38} \widehat H_{\mathrm{int}} =\sum_{r\in\mathcal R_{\mathrm{act}}} \widehat H_r +\sum_{r<s}\widehat H_{rs} +\widehat H_{\mathrm{corr}}. \] Formulas (36)–(38) define the general dynamic architecture. The explicit form of \(\widehat H_{\mathrm{int}}\) must still be obtained from the phase and sector geometry of the model.
12. Transition amplitude
Let \(|\mathcal C_k\rangle\) be the normalized state of the completeoutput channel. The transition amplitude is equal to
\[\tag{39} \boxed{ \mathcal A_k =\langle\mathcal C_k| \widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle. } \] The brace in (39) checks not the match of a single pair of branches, but rather the match of the entire signature, connection maps, and continuous parameters of the selected channel.
If the state and transformation are truly separable across active sectors, the matrix element is factorized:
\[\tag{40} \mathcal A_k =\prod_{r\in\mathcal R_{\mathrm{act}}} \langle\lambda_{k,r}| \widehat{\mathcal U}_r |\lambda_{\mathrm{in},r}\rangle_r. \] When correlating levels, a full matrix element is required:
\[\tag{41} \mathcal A_k \ne \prod_r\mathcal A_{k,r}. \] The channel forbidden by the projector has zero amplitude:
\[\tag{42} \widehat{\mathcal P}_{\mathrm{adm}} |\mathcal C_k\rangle=0 \quad\Longrightarrow\quad \mathcal A_k=0. \] 13. Probability Postulate
For a normalized discrete channel, the registration probability is taken to be equal to the square of the amplitude modulus:
\[\tag{43} \boxed{ P_k=|\mathcal A_k|^2. } \] For the projector onto the result subspace \(\widehat\Pi_k\), the same rule is written as
\[\tag{44} P_k =\langle\Psi_{\mathrm{out}}| \widehat\Pi_k |\Psi_{\mathrm{out}}\rangle, \qquad \widehat\Pi_k^2=\widehat\Pi_k. \] For a one-dimensional orthogonal channel
\[\tag{45} \widehat\Pi_k =|\mathcal C_k\rangle \langle\mathcal C_k|, \] and formula (44) becomes (43). Projection notation is useful when a single observable outcome contains multiple indistinguishable internal states.
14. Normalization and Unitarity
For a closed system, the complete evolution must preserve the statistical norm:
\[\tag{46} \widehat{\mathcal U}_{\mathrm{int}}^\dagger \widehat{\mathcal U}_{\mathrm{int}}=I. \] Then
\[\tag{47} \langle\Psi_{\mathrm{out}}| \Psi_{\mathrm{out}}\rangle =\langle\Psi_{\mathrm{in}}| \Psi_{\mathrm{in}}\rangle=1. \] If the orthogonal projectors form a complete set,
\[\tag{48} \sum_k\widehat\Pi_k=I, \qquad \widehat\Pi_k\widehat\Pi_l =\delta_{kl}\widehat\Pi_k, \] then the total probability is preserved:
\[\tag{49} \sum_kP_k=1. \] After restricting consideration to only a selected group of channels, the projection can reduce the norm. Then the conditional probability within this group is
\[\tag{50} P_k^{(S)} =\frac{|\mathcal A_k|^2} {\displaystyle\sum_{n\in S}|\mathcal A_n|^2}, \qquad k\in S. \] Therefore, unitarity applies to the complete closed system, and not necessarily to a truncated set of observable channels.
15. Superposition and interference of sector rearrangements
The complete output state is a superposition of the admissible channels:
\[\tag{51} |\Psi_{\mathrm{out}}\rangle =\sum_k\mathcal A_k|\mathcal C_k\rangle. \] If the same final channel is achieved by several indistinguishable sequences of sector rearrangements, their amplitudes are summed:
\[\tag{52} \mathcal A_k =\sum_\lambda\mathcal A_k^{(\lambda)}. \] For two alternatives
\[\tag{53} P_k =\left| \mathcal A_k^{(1)}+\mathcal A_k^{(2)} \right|^2. \] After expansion:
\[\tag{54} P_k =|\mathcal A_k^{(1)}|^2 +|\mathcal A_k^{(2)}|^2 +2\operatorname{Re} \left( \mathcal A_k^{(1)} \overline{\mathcal A_k^{(2)}} \right). \] In the geometric interpretation, it is not the different registered results that interfere, but rather the indistinguishable ways of modifying the kappa_r maps, leading to the same overall output signature.
16. Sequential Interactions
For a chain of transformations, the general operator is equal to the ordered product:
\[\tag{55} \widehat{\mathcal U}_{\mathrm{tot}} =\widehat{\mathcal U}_n\cdots \widehat{\mathcal U}_2 \widehat{\mathcal U}_1. \] Amplitude of the transition through the intermediate state \(|\mathcal C_m\rangle\):
\[\tag{56} \mathcal A_{i\to m\to f} =\langle\mathcal C_f| \widehat{\mathcal U}_2 |\mathcal C_m\rangle \langle\mathcal C_m| \widehat{\mathcal U}_1 |\mathcal C_i\rangle. \] Completeness of admissible intermediate states:
\[\tag{57} I_{\mathrm{adm}} =\sum_m|\mathcal C_m\rangle \langle\mathcal C_m|. \] Summing over them yields
\[\tag{58} \mathcal A_{i\to f} =\sum_m \langle\mathcal C_f| \widehat{\mathcal U}_2 |\mathcal C_m\rangle \langle\mathcal C_m| \widehat{\mathcal U}_1 |\mathcal C_i\rangle =\langle\mathcal C_f| \widehat{\mathcal U}_2 I_{\mathrm{adm}} \widehat{\mathcal U}_1 |\mathcal C_i\rangle. \] 17. Phase and Frequency Matching
The amplitude cannot depend solely on the combinatorics of the connections. Sector states contain phases, frequencies, orientations, and metric parameters. Therefore, a general operator should be sought in the form
\[\tag{59} \widehat{\mathcal U}_{\mathrm{int}} =\widehat{\mathcal U}_{\mathrm{int}} \left( \{\Delta\phi_r\}, \{\omega_r\}, \{\kappa_r\}, \Lambda_{\mathrm{in}} \right). \] For each active sector, a phase mismatch measure can be introduced:
\[\tag{60} \Delta\phi_r =\phi_{r,\mathrm{out}} -\phi_{r,\mathrm{in}}. \] A working hypothesis can relate the sector matrix element to split functions:
\[\tag{61} \langle\lambda_{k,r}| \widehat{\mathcal U}_r |\lambda_{i,r}\rangle_r =F_r\left( \Sin(\Delta\phi_r), \Cos(\Delta\phi_r), \Delta\omega_r,\kappa_r \right). \] The specific form of \(F_r\), its dimension, and normalization must be derived separately. Formula (61) indicates the direction of the search, but is not a ready-made interaction law.
18. Electron Pair Annihilation
Consider the Transition
\[\tag{62} |e^-,e^+\rangle \longrightarrow |\gamma_1,\gamma_2\rangle. \] This process cannot be described by a simple permutation of two universal branches. At least the dynamic, closure, and charge levels must be coordinated, and information about the two orientations of the dynamic operators must be reflected in the polarization state of the radiation.
Amplitude of the two-photon channel:
\[\tag{63} \mathcal A_{\gamma\gamma} =\langle\gamma_1,\gamma_2| \widehat{\mathcal U}_{\mathrm{int}} |e^-,e^+\rangle. \] The structural transition condition has the form
\[\tag{64} \Lambda_{e^-e^+}^{\mathrm{tot}} \overset{\widehat{\mathcal U}_{\mathrm{int}}}{\longrightarrow} \Lambda_{\gamma\gamma}^{\mathrm{tot}}, \qquad I_\alpha^{\mathrm{in}} =I_\alpha^{\mathrm{out}}. \] The closures of individual electron and positron waves can disappear only within a complete transition, in which their topological and dynamic properties are redistributed among all output components in a compensated manner.
19. Inverse Pair Production
The inverse process is written as
\[\tag{65} |\gamma_1,\gamma_2\rangle \longrightarrow |e^-,e^+\rangle. \] Its amplitude:
\[\tag{66} \mathcal A_{e^-e^+} =\langle e^-,e^+| \widehat{\mathcal U}_{\mathrm{int}} |\gamma_1,\gamma_2\rangle. \] If the full dynamics is reversible, the corresponding operators are related by conjugation:
\[\tag{67} \widehat{\mathcal U}_{\gamma\gamma\to e^-e^+} =\widehat{\mathcal U}_{e^-e^+\to\gamma\gamma}^{\dagger}. \] However, the equality of the observed probabilities of the forward and backward processes from (67) does not automatically follow: the preparation of the initial states, the available phase volume, and the number of internal channel configurations must be taken into account.
20. Different classes of output channels
During the interaction of an electron pair, qualitatively different rearrangements are possible:
\[\tag{68} |e^-,e^+\rangle \longrightarrow |e'^-,e'^+\rangle \qquad\text{— scattering}, \] \[\tag{69} |e^-,e^+\rangle \longrightarrow |\mu^-,\mu^+\rangle \qquad\text{— change in particle type}, \] \[\tag{70} |e^-,e^+\rangle \longrightarrow |Ps,\gamma\rangle \qquad\text{— formation of a bound state}. \] In the first case, the charge and closure classes of individual particles can be preserved when external dynamic parameters change. In the second, a restructuring of the internal frequency and metric scale is required. In the third, the orbital sector is activated, and a general closure of the composite system appears. Therefore, each process corresponds to its own set of \(\mathcal R_{\mathrm{act}}\).
21. Relationship with Observable Quantities
In a series of identically prepared events, the channel probability manifests itself as the limiting relative frequency:
\[\tag{71} P_k =\lim_{N\to\infty}\frac{N_k}{N}. \] But in an experiment, it is not only the total probability that is measured. The output channel usually has continuous parameters \(\zeta\), so a probability density function is required:
\[\tag{72} dP_k =|\mathcal A_k(\zeta)|^2 d\mu_k(\zeta), \] where \(d\mu_k\) is the measure of the available channel states. The total probability is:
\[\tag{73} P_k =\int_{\Omega_k} |\mathcal A_k(\zeta)|^2 d\mu_k(\zeta). \] To calculate cross sections, angular distributions, and lifetimes, the geometry of the states alone is not sufficient. An explicit interaction generator, a phase space measure, and a coupling of the amplitude to the source flux are needed.x states.
22. What follows from the construction, and what remains a hypothesis
The adopted mathematical construction implies: the tensor product of independent sector spaces; the direct sum of distinct physical channels; the action of projectors; the composition of successive operators; the appearance of an interference term when adding complex amplitudes.
The multi-operator architecture of the third part implies: the need for a complete sector ket; the separation of active and spectator levels; the dependence of the state on the set of \(\kappa_r\) maps; the insufficiency of a single direct or cross assembly to define a particle.
Additional definitions are: the positive inner product of each sector; the metric of the complete space; the orthogonal basis of observable channels; projector of the admissible subspace.
The following remain physical postulates and hypotheses: the rule \(P=|\mathcal A|^2\); description of the interaction by the sector rearrangement operator; the specific composition of active levels; the existence of a single recorded outcome; the relationship of split functions with matrix elements.
The following require separate derivation: the explicit operators \(\widehat H_{\mathrm{int}}\) and \(\widehat{\mathcal U}_{\mathrm{int}}\); the origin of their symmetries from primary geometry; numerical amplitudes; energy thresholds; interaction cross sections and decay times.
23. Basic Principle of the Probabilistic Model
The complete construction can be formulated as a sequence of three operations. First, a valid initial state is selected:
\[\tag{74} |\Psi_{\mathrm{in}}\rangle \in\mathscr H_{\mathrm{adm}}. \] Then the full sector dynamics operator creates a superposition of valid output channels:
\[\tag{75} \boxed{ |\Psi_{\mathrm{out}}\rangle =\widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle =\sum_k \langle\mathcal C_k| \widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle |\mathcal C_k\rangle. } \] Finally, the probabilistic postulate relates the state component to the recording frequency of the corresponding channel:
\[\tag{76} \boxed{ P_k =\left| \langle\mathcal C_k| \widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle \right|^2, \qquad \sum_kP_k=1. } \] The first part of formula (75) describes the geometry and dynamics of the complete restructuring. Formula (76) defines a probabilistic interpretation. They should be distinguished until the square of the amplitude modulus is obtained from a deeper principle of the model itself.
Conclusions
The particle ket now describes a complete multioperator assembly, rather than a connection of two universal branches:
\[\tag{77} \boxed{ |P\rangle =|\Lambda_P,\kappa_P\rangle \in \widehat{\mathcal P}_{\mathrm{adm}} \left( \bigotimes_{r\in\mathcal R}\mathcal H_r \right). } \] Different final reactions belong to different subspaces of the same complete channel space:
\[\tag{78} \mathscr H_{\mathrm{adm}} =\bigoplus_\chi \mathcal H_\chi^{\mathrm{adm}}. \] The interaction changes the membership maps only in the active sectors and preserves the structural invariants of the full closed-loop system:
\[\tag{79} \boxed{ \{\kappa_r^{\mathrm{in}}\} \overset{\widehat{\mathcal U}_{\mathrm{int}}}{\longrightarrow} \{\kappa_r^{\mathrm{out}}\}, \qquad \left[ \widehat{\mathcal U}_{\mathrm{int}}, \widehat I_\alpha \right]=0. } \] The energy balance is associated with the homogeneity of the overall dynamics with respect to time transfer, rather than with the equality of the algebraic norm to unity. Charge, closure, and other indices are preserved or redistributed within the overall state in a compensated manner.
After introducing the inner product, the channel amplitude is determined by the projection of the transformed state onto its overall signature, and the probability is determined by a separate statistical rule:
\[\tag{80} \boxed{ \mathcal A_k =\langle\mathcal C_k| \widehat{\mathcal U}_{\mathrm{int}} |\Psi_{\mathrm{in}}\rangle, \qquad P_k=|\mathcal A_k|^2. } \] Thus, the bra-ket formalism becomes the language of a complete sector space: the ket defines a consistent operator assembly, the operator describes its dynamic restructuring, the bra identifies a specific admissible channel, and the squared amplitude is associated with observed statistics. The next step should be to derive an explicit interaction generator and investigate whether a probabilistic rule can be derived from the phase geometry, stability, and completeness of internal channels.