2026-08-16
Multilevel electron splitting
Origin of spin 1/2 and transition to photon
The usual Wave Electricity operator separates a state into two complementary idempotent components. This suffices to describe the primary internal phase and external motion, but not to retain the electron’s entire multilevel structure. In particular, one revolution of the primary wave returns the finite operator to its initial value, whereas the full electron cycle must distinguish two successive traversals of the internal contour.
Deeper splitting is introduced for this purpose. The first binary decomposition allows a two-sheeted extension factor \(Q_\Gamma\) to be constructed, while recursively resolving the remainder produces a chain of algebraically orthogonal projectors \(P_n\). After a separate metric mapping and the assumption of self-similarity, this chain defines the state \(\boldsymbol\Gamma_\beta\), whose norm equals the Lorentz factor. The two-sheeted extension, spin representation, and motion metric share an algebraic foundation, but have different definitions and physical mappings.
The relative phase of two fixed channels can have an integer winding number on a chosen closed parameter cycle. This index is preserved under continuous deformation as long as the cycle remains closed, both coefficients are nonzero, and the channels retain their identification. Identifying it with physical spatial closure and assuming its conservation in interactions require additional conditions; normalization of the full state alone does not provide them.
\[\tag{1} \boxed{ J(a,b) \longrightarrow \{P_n\} \longrightarrow \begin{cases} Q_\Gamma(a),&\text{two-sheeted structure and half-angle phase},\\ \boldsymbol\Gamma_\beta,&\text{motion metric and }\gamma(\beta). \end{cases} } \] 1. The Finite State Operator
The first level of the algebra is defined by two complementary idempotents:
\[\tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Their difference defines the hyperbolic unit \(\j=\ep-\em\), for which \(\j^2=1\). The finite two-parameter operator has the form
\[\tag{3} \boxed{ J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a}. } \] The parameter \(a\) describes internal periodicity, while \(b\) describes the external motion of the center. In the example of a uniform internal cycle, \(t\) denotes time in the rest frame of the center, and the frequency is defined with respect to that time:
\[\tag{4} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac vc. \] The finite phase operator satisfies algebraic unitarity:
\[\tag{5} J(a,b)\overline{J(a,b)}=1. \] Equality (5) means that multiplication by \(J\) preserves a state’s norm in a separately specified compatible Hilbert-space metric. It does not identify the algebraic product \(J\overline J\) with the scalar norm of the vector \(J\), does not by itself constitute an energy conservation law, and does not determine the metric depth of the state.
2. Two Periodicities of the Electron State
In the rest frame, \(b=0\). The primary internal phase repeats under \(a\to a+2\):
\[\tag{6} J(a+2,0)=J(a,0). \] In the adopted two-sheeted model, the deeper-level factor still differs from its initial value after one primary revolution. For the full operator with a fixed external parameter and charge sector, the periodicity is specified as
\[\tag{7} \boxed{\mathcal J_{\sigma,\eta}(a+2,b)\ne\mathcal J_{\sigma,\eta}(a,b),\qquad\mathcal J_{\sigma,\eta}(a+4,b)=\mathcal J_{\sigma,\eta}(a,b).} \] Thus, the primary wave frequency does not decrease. Only the period required to restore the entire internal configuration doubles. A single variable \(a\) can describe both periodicities if a factor is added whose phase evolves at half the rate of the primary phase.
3. The First Step of Recursive Splitting
Let the first deeper level correspond to an independent commuting hyperbolic unit \(\j_1\). It defines a pair of projectors
\[\tag{8} p_1^+=\frac{1+\j_1}{2}, \qquad p_1^-=\frac{1-\j_1}{2}. \] Denote the first extracted channel and the first residual branch by
\[\tag{9} P_0=p_1^-, \qquad R_1=p_1^+, \qquad 1=P_0+R_1, \qquad P_0R_1=0. \] On the first binary decomposition, introduce the closure factor and its formal reverse orientation. The sign \(\nu=\pm1\) denotes the direction in which the relative phase accumulates for a fixed orientation of the parameter cycle. It is neither a spin realization nor a charge label:
\[\tag{10} \boxed{Q_\Gamma^{(\nu)}(a)=P_0+R_1e^{i\nu\pi a/2},\qquad\nu=\pm1.} \] To match the notation used in the other articles, set
\[\tag{11} P_0\equiv\pmp,\qquad R_1\equiv\pme,\qquad\jp=\pmp-\pme=-\j_1. \] The canonical closure factor is chosen with positive orientation:
\[\tag{12} Q_\Gamma(a):=Q_\Gamma^{(+1)}(a)=\jp^{a/2}=\pmp+\pme e^{i\pi a/2}, \] The inverse factor has the opposite direction of phase accumulation:
\[\tag{13} Q_\Gamma^{(-1)}(a)=Q_\Gamma^{-1}(a)=\jp^{-a/2}=\pmp+\pme e^{-i\pi a/2},\qquad Q_\Gamma Q_\Gamma^{-1}=1. \] The mathematics of this splitting is presented here.
4. The General Recursive Chain of Projectors
The first binary decomposition need not be the last. Let each subsequent level correspond to an independent commuting unit \(\j_n\) and the projectors
\[\tag{14} p_n^+=\frac{1+\j_n}{2}, \qquad p_n^-=\frac{1-\j_n}{2}. \] The channels extracted within the recursive residual branch are
\[\tag{15} \boxed{ P_n =\left(\prod_{k=1}^{n}p_k^+\right)p_{n+1}^-, \qquad P_nP_m=\delta_{nm}P_n. } \] After \(N+1\) steps, the identity decomposes as
\[\tag{16} 1=\sum_{n=0}^{N}P_n+R_{N+1}, \qquad R_{N+1}=\prod_{k=1}^{N+1}p_k^+. \] The recursive extension of the first remainder has the form
\[\tag{17} R_1=P_1+R_2, \qquad R_2=P_2+R_3, \qquad\ldots \] Thus, the pair \(\pmp,\pme\) is not a construction separate from the general concept. It represents the first physically selected decomposition \(P_0+R_1\). Further resolution of the residual branch yields the channels \(P_1,P_2,\ldots\).
The algebra specifies \(P_nP_m=0\) for distinct channels and preserves the identity in a finite decomposition with a remainder. Metric orthogonality requires a separate inner product. The physical meaning of each \(P_n\)—an internal sheet, a spatial projection, or another attribute—arises only after an additional mapping.
5. The Full Electron Operator
In the adopted electron realization, the common factor \(Q_\Gamma=Q_\Gamma^{(+1)}\) is used independently of the spin and charge attributes. Write the full particle operator in the consistent form:
\[\tag{18} \boxed{\mathcal J_{\sigma,\eta}(a,b)=J_\sigma(a,b)Q_\Gamma(a)J_{q,\eta}\cdots,\qquad\sigma=1,2,\quad\eta=\pm1.} \] The two realizations of the primary dynamical channels have the form
\[\tag{19} \begin{aligned}J_1(a,b)&=\ep e^{i\pi b}+\em e^{i\pi a},\\ J_2(a,b)&=\ep e^{i\pi a}+\em e^{i\pi b}.\end{aligned} \] In both cases, the parameter \(b\) specifies the same center velocity. The external mapping functional must read the channel containing \(b\), taking account of the realization \(\sigma\):
\[\tag{20} \mathcal P_{\mathrm{ext}}^{(1)}[J_1]=\mathcal P_{\mathrm{ext}}^{(2)}[J_2]=\beta=\sin(\pi b). \] The index \(\sigma=1,2\) denotes the two realizations used in the spin description, while \(\eta=\pm1\) specifies a separate charge sector: \(q_\eta=-e\eta\), with \(e>0\). The factor \(Q_\Gamma\) specifies two-sheeted periodicity and is the same for both realizations and both charges in the adopted canonical notation. The formal sign \(\nu\) should not automatically be identified with \(\sigma\) or \(\eta\). The independent factors act on separate spaces and are multiplied after extension by identity operators on the remaining factors; states of different particles are combined by a tensor product.
6. Geometric Mapping of the Two Sheets
The first deeper-level decomposition can be mapped onto two nearby branches of a single closed contour:
\[\tag{21} P_0\longleftrightarrow C_+, \qquad R_1\longleftrightarrow C_-, \qquad R_\pm=R\pm\frac{\Delta R}{2}. \] A single wave traverses both branches sequentially. To leading order, each branch has length \(2\pi R\), so the full two-sheeted path length is
\[\tag{22} L_1=L_2=2\pi R, \qquad \boxed{L_J=L_1+L_2=4\pi R.} \] For the fundamental mode of the full path, the phase \(2\pi\) is distributed over the length \(L_J\). Denote the internal geometric angle by \(\theta_{\mathrm{int}}=s/R\). Then
\[\tag{23} k_J=\frac{2\pi}{L_J}=\frac1{2R},\qquad\boxed{\chi=k_Js=\frac{\theta_{\mathrm{int}}}{2}.} \] The two-revolution cycle, rather than the radial separation of the branches, produces the factor \(1/2\). The quantity \(\Delta R\), the shape of the transitions between the branches, and their precise physical arrangement are separate geometric hypotheses.
7. Two Sheets and the Spin Mapping
The projectors \(P_0\) and \(R_1\) denote two internal channels, rather than spin-up and spin-down states. Mapping them onto successive sheets and selecting two spin realizations are different operations. The formal orientations of the relative phase have the form
\[\tag{24} Q_\Gamma^{(\nu)}(\theta_{\mathrm{int}})=P_0+R_1e^{i\nu\theta_{\mathrm{int}}/2},\qquad\theta_{\mathrm{int}}=\pi a,\quad\nu=\pm1. \] When the internal angle changes by \(2\pi\), only the second component of \(Q_\Gamma\) changes sign, rather than the entire operator. For the canonical orientation:
\[\tag{25} \boxed{\begin{aligned} Q_\Gamma(\theta_{\mathrm{int}}+2\pi)&=P_0-R_1e^{i\theta_{\mathrm{int}}/2}\ne-Q_\Gamma(\theta_{\mathrm{int}}),\\ Q_\Gamma(\theta_{\mathrm{int}}+4\pi)&=Q_\Gamma(\theta_{\mathrm{int}}). \end{aligned}} \] A physical rotation about a laboratory axis must be described by a separate angle \(\Theta\), without identifying it with the internal traversal unless that correspondence is proved. In the spinor mapping, adopt normalized basis states \(|1\rangle,|2\rangle\) and representatives \(\Psi_1(\Theta)=e^{-i\Theta/2}|1\rangle\) and \(\Psi_2(\Theta)=e^{i\Theta/2}|2\rangle\). The generator of rotation about the chosen axis is defined as
\[\tag{26} \widehat S_{\mathbf n}=i\hbar\frac{\partial}{\partial\Theta}, \] In this additionally specified representation, it has the eigenvalues
\[\tag{27} \boxed{\widehat S_{\mathbf n}\Psi_1=+\frac{\hbar}{2}\Psi_1,\qquad\widehat S_{\mathbf n}\Psi_2=-\frac{\hbar}{2}\Psi_2.} \] The representatives \(\Psi_\sigma\) change their overall sign after a rotation through \(2\pi\) and are restored after \(4\pi\). This is a property of the adopted spinor mapping, rather than a direct consequence of (25). A complete derivation requires a mapping of internal states into spinors and consistent rotations about different axes. Related constructions are presented in the articles on electron spin and the Dirac equation in splitting operators.
8. Metric Mapping of the Deeper-Level Chain
To pass from algebraic projectors to metric depth, associate an orthonormal vector with each channel:
\[\tag{28} P_n\longleftrightarrow\boldsymbol\xi_n, \qquad \boldsymbol\xi_n\boldsymbol\cdot\boldsymbol\xi_m =\delta_{nm}. \] Hilbert-space completion yields a space of square-summable coefficients of the internal channels:
\[\tag{29} \mathcal H_J=\overline{\operatorname{span}\{\boldsymbol\xi_0,\boldsymbol\xi_1,\boldsymbol\xi_2,\ldots\}}\cong\ell^2. \] The vectors \(\boldsymbol\xi_n\) are not additional coordinates of observable three-dimensional space. They serve as metric representatives of orthogonal levels of the internal state.
9. The Self-Similar State \(\boldsymbol\Gamma_\beta\)
Let the linear shift \(S\boldsymbol\xi_n=\boldsymbol\xi_{n+1}\) preserve the metric norm. It is not automatically assigned the properties of an automorphism of the projector algebra. With a separate assumption of self-similarity and a common transition coefficient \(\beta\), the state satisfies the equation
\[\tag{30} \boxed{ \boldsymbol\Gamma_\beta =\boldsymbol\xi_0 +\beta S\boldsymbol\Gamma_\beta. } \] For \(|\beta|<1\), it expands into the series
\[\tag{31} \boldsymbol\Gamma_\beta =\sum_{n=0}^{\infty}\beta^n\boldsymbol\xi_n. \] By orthogonality of the channels,
\[\tag{32} \|\boldsymbol\Gamma_\beta\|^2 =1+\beta^2\|\boldsymbol\Gamma_\beta\|^2, \] hence,
\[\tag{33} \boxed{ \|\boldsymbol\Gamma_\beta\| =\frac1{\sqrt{1-\beta^2}} =\gamma(\beta). } \] With the physical identification \(\beta=v/c\) and the principal-branch choice \(b=\arcsin\beta/\pi\), \(|\beta|<1\), the relation to the external parameter is as follows. Ordinary scalar sine and cosine functions are used here:
\[\tag{34} \beta=\sin(\pi b),\qquad\cos(\pi b)=\sqrt{1-\beta^2},\qquad\gamma=\frac1{\cos(\pi b)}. \] 10. Why \(Q_\Gamma\) and \(\boldsymbol\Gamma_\beta\) Are Different Objects
The two-sheeted factor and the self-similar state are constructed on the same architecture of deeper-level channels, but they must not be identified:
\[\tag{35} \boxed{ Q_\Gamma(a)\ne\boldsymbol\Gamma_\beta. } \] The factor \(Q_\Gamma\) uses the first decomposition \(P_0+R_1\) and contains the relative phase \(e^{i\pi a/2}\). The state \(\boldsymbol\Gamma_\beta\) uses the recursive chain with weights \(\beta^n\). A common algebraic foundation does not make them the same physical object.
Their norms also have different meanings:
\[\tag{36} Q_\Gamma\overline{Q_\Gamma}=1, \qquad J\overline J=1, \qquad \|\boldsymbol\Gamma_\beta\|=\gamma. \] The first two equalities are algebraic identities of phase operators. The third uses a separately specified Hilbert-space metric of the deeper-level channels. Algebraic unitarity of the finite factors therefore does not contradict the growth of \(\gamma\) and does not imply that the particle’s physical energy is constant.
11. Closure of a Localized Wave
The algebraic factor describes internal periodicity. Spatial closure requires separately selecting the internal channel and specifying a geometric mapping. Introduce the phase-geometric quantity
\[\tag{37} V_J(t)=cJ_\sigma(t), \] Let \(\Pi_{\mathrm{int}}\) select the internal component, and \(\mathcal P_{\mathrm{geom}}\) map it into the spatial kinematics of the internal contour. Define the closure functional through this projection:
\[\tag{38} \mathbf v_{\mathrm{int}}(t)=\mathcal P_{\mathrm{geom}}[\Pi_{\mathrm{int}}V_J(t)],\qquad \mathcal C_{\mathrm{int}}(T)=\int_0^T\mathbf v_{\mathrm{int}}(t)\,dt. \] A necessary condition for return of the internal spatial structure is
\[\tag{39} \boxed{\mathcal C_{\mathrm{int}}(T_e)=0\quad\text{(condition for internal spatial return)}.} \] The integral of the full \(cJ\) cannot automatically be used in place of (38): in the rest frame, \(J(a,0)=\ep+\em e^{i\omega t}\), and the constant component produces the nonzero contribution \(cT\ep\). It is the selected internal spatial projection that must close. Zero net integrated displacement does not, by itself, prove localization, stability, or absence of radiation; these properties require dynamics. The center’s worldline need not close.
12. The Topological Index of Internal Closure
In this section, the symbols \(q_0,q_1,q_{\mathrm{rel}}\) denote phase coefficients, rather than electric charge. The index concerns their loop on the parameter cycle \(\Gamma\), rather than directly describing the center’s trajectory or a spatial contour. Its sign changes when the cycle orientation is reversed; comparing indices requires a common convention and a fixed channel basis.
To investigate phase topology, consider the coefficients of the formally oriented factor. The channels are fixed, and \(\nu\) is independent of \(\sigma\) and \(\eta\):
\[\tag{40} Q_\Gamma^{(\nu)}=P_0q_0+R_1q_1,\qquad q_0=1,\qquad q_1=e^{i\nu\chi}. \] Their relative phase is
\[\tag{41} q_{\mathrm{rel}}=\frac{q_1}{q_0}=e^{i\nu\chi}. \] On the fixed oriented parameter cycle \(\Gamma\), the angle \(\chi\) increases from zero to \(2\pi\). If \(q_0,q_1\) are continuous and nonzero, and the relative phase returns uniquely to its initial value, define its integer winding number:
\[\tag{42} \boxed{N_\Gamma[Q_\Gamma^{(\nu)}]=\frac1{2\pi i}\oint_\Gamma q_{\mathrm{rel}}^{-1}\,dq_{\mathrm{rel}}\in\mathbb Z.} \] For the two directions of accumulation of the deeper-level phase, we obtain
\[\tag{43} N_\Gamma[Q_\Gamma^{(+1)}]=+1,\qquad N_\Gamma[Q_\Gamma^{(-1)}]=-1. \] For a constant relative coefficient on the same auxiliary cycle,
\[\tag{44} Q_{\mathrm{free}}=1=P_0+R_1, \qquad q_{\mathrm{rel}}=1, \qquad N_\Gamma[Q_{\mathrm{free}}]=0. \] Thus, the chosen phase loops have different winding numbers. Their correspondence to a physically closed particle and a free wave is an additional mapping, rather than a topological theorem:
\[\tag{45} \boxed{\text{phase loop with winding: }N_\Gamma=\pm1,\qquad\text{constant loop: }N_\Gamma=0.} \] 13. Conditions for Conservation of the Phase Index
The winding number is preserved under a homotopy \(q_{\mathrm{rel}}(u,\lambda)\) of closed loops in \(\mathbb C\setminus\{0\}\), where \(u\) is the parameter of the fixed cycle and \(\lambda\) is the deformation parameter. For the diagonal factor
\[\tag{46} Q_\Gamma^{(\nu)}\overline{Q_\Gamma^{(\nu)}}=1,\qquad|q_0|=|q_1|=1. \] both coefficients have unit modulus. This ensures that they are nonzero within this diagonal model, but index conservation additionally requires the cycle to remain closed and the channel identification to remain unchanged. Ordinary Hilbert-space normalization \(|q_0|^2+|q_1|^2=1\) allows one component to vanish and does not, by itself, guarantee these conditions.
\[\tag{47} \boxed{ \begin{gathered} q_{\mathrm{rel}}(\cdot,\lambda): S^1\to\mathbb{C}\setminus\{0\}, \\ \text{continuous homotopy in fixed channels} \\ \Longrightarrow\quad N_{\Gamma}=\operatorname{const}. \end{gathered} } \] Under the stated conditions, a continuous deformation cannot connect phase loops with different indices:
\[\tag{48} N_\Gamma=\pm1\, \not\longleftrightarrow\, N_\Gamma=0\quad \text{within the deformation class of (47)}. \] This is a conditional mathematical result concerning the stability of phase winding numbers. Connecting it to the physical stability of the electron requires proving that the allowed dynamics preserve the channels used, the nonzero coefficients, and the closed cycle. Transitions that change these conditions are not excluded by the theorem.
14. The Physical Meaning of the Inverse Operator
The algebraic equality
\[\tag{49} Q_\Gamma Q_\Gamma^{-1}=1, \] is not an equation for converting an electron into a photon. With fixed channels and a single common cycle, the inverse factor has the opposite phase index:
\[\tag{50} N_\Gamma[Q_\Gamma^{-1}] =-N_\Gamma[Q_\Gamma]. \] For two diagonal factors with nonzero coefficients, defined on the same oriented cycle and in the same channel basis, the relative-phase indices add:
\[\tag{51} N_\Gamma[Q_AQ_B] =N_\Gamma[Q_A]+N_\Gamma[Q_B]. \] Under these conditions, algebraic cancellation gives a zero index for the product:
\[\tag{52} \boxed{ N_\Gamma[Q_{\Gamma}Q_{\Gamma}^{-1}] =N_\Gamma[Q_{\Gamma}]+N_\Gamma[Q_{\Gamma}^{-1}] =0. } \] For the full operator in (18), \(\mathcal J_{\sigma,\eta}Q_\Gamma^{-1}=J_\sigma J_{q,\eta}\cdots\) holds. The charge sector is preserved. Such cancellation does not physically open a spatial contour or define a free-photon state. The source of the inverse factor and the allowed transition of the full state must follow from a separate interaction operator.
15. Phase Compensation and Physical Interaction
Additivity of the index under ordinary phase multiplication does not prove conservation of the sum of indices of different particles: their states are combined by a tensor product, and their internal cycles may differ. If a global index \(N_{\mathrm{tot}}\) can be defined for the full interacting state and its conservation by the dynamics can be proved, a conditional balance can be written:
\[\tag{53} \boxed{N_{\mathrm{tot}}^{\mathrm{in}}=N_{\mathrm{tot}}^{\mathrm{out}}\quad\text{(additional law, if justified)}.} \] For two phase factors on a single common cycle, an algebraic example of compensation is possible:
\[\tag{54} N_\Gamma[Q_A]=+1, \qquad N_\Gamma[Q_B]=-1, \qquad N_\Gamma[Q_AQ_B]=0. \] Equality (54) does not determine which particles can realize these factors. Electron–positron annihilation must be written as a transformation of a joint state:
\[\tag{55} \boxed{\mathcal S_{e^-e^+}=\mathcal J_{e^-}\otimes\mathcal J_{e^+} \xrightarrow{\;\mathcal U_{\mathrm{int}}\;} \mathcal S_{\gamma\gamma},\qquad Q_{\mathrm{tot}}^{\mathrm{in}}=Q_{\mathrm{tot}}^{\mathrm{out}}=0.} \] In the adopted canonical notation, the electron and positron use the same \(Q_\Gamma\), so each has \(N_\Gamma=+1\) on an individual cycle with the same orientation. Opposite indices cannot be assigned merely because their charges are opposite. A naive sum would give \(2\), rather than \(0\); this demonstrates the need for a separate definition of the full-state index, rather than prohibiting observed annihilation. Possible changes of cycles, channels, or their relation to a common state require a dynamical description.
The reverse process is likewise described by a full interaction operator and the usual physical balances:
\[\tag{56} \mathcal S_{\gamma\gamma}\xrightarrow{\;\mathcal U_{\mathrm{pair}}\;}\mathcal J_{e^-}\otimes\mathcal J_{e^+},\qquad Q_{\mathrm{tot}}^{\mathrm{in}}=Q_{\mathrm{tot}}^{\mathrm{out}}=0. \] Pair creation and annihilation are consistent with conservation of charge, energy, momentum, and angular momentum. They cannot be regarded as already derived from the compensation \((+1)+(-1)\): the correspondence between local phase indices and the full state has not yet been constructed.
16. Energy and the Closure Index
Topological stability of a phase loop and physical energy conservation concern different structures. For a coordinate-space amplitude with the convention \(e^{-i\omega t}\), the energy correspondence has the form
\[\tag{57} \widehat E=i\hbar\frac{\partial}{\partial t}, \qquad E=\hbar\omega. \] For a closed system with Hamiltonian dynamics, energy conservation is related to invariance under time translations. It does not follow from the identity \(Q_\Gamma\overline{Q_\Gamma}=1\). Winding-number conservation, in turn, requires the homotopy conditions and does not imply an energy balance.
\[\tag{58} \begin{aligned} \text{homotopy conditions of (47)}&\Longrightarrow N_\Gamma=\operatorname{const},\\ \text{time-translation invariance of the full Hamiltonian dynamics}&\Longrightarrow E_{\mathrm{total}}=\operatorname{const}. \end{aligned} \] Physical interactions require separate laws of conservation of energy, momentum, charge, and angular momentum. A possible additional topological law needs its own definition and proof. Only an invariant of a phase loop under a restricted class of deformations has been established here.
17. Why Substituting \(\beta=1\) into \(\boldsymbol\Gamma_\beta\) Does Not Produce a Photon
The self-similar state \(\boldsymbol\Gamma_\beta\) describes the deeper-level motion metric of an already formed massive particle. Its norm diverges as the speed approaches the speed of light:
\[\tag{59} \|\boldsymbol\Gamma_\beta\|^2 =1+\beta^2+\beta^4+\cdots \longrightarrow\infty, \qquad |\beta|\to1. \] A photon therefore cannot be obtained simply by substituting \(\beta=1\) into the massive-particle chain. This limit indicates the boundary of applicability of the state \(\boldsymbol\Gamma_\beta\), rather than a mechanism for photon creation.
For a free wave, a trivial internal closure factor can be chosen. Its index on the auxiliary cycle is zero. The photon velocity is specified separately by the massless propagation law; setting \(b=1/2\) merely encodes \(\beta=1\) formally, rather than extending the massive state:
\[\tag{60} Q_{\mathrm{free}}=1,\qquad N_\Gamma[Q_{\mathrm{free}}]=0,\qquad|v_\gamma|=c,\qquad b_\gamma=\frac1\ext{(formal external label)}. \] This regime arises after restructuring the full interacting state, rather than as an infinitely accelerated electron. The primary wave phase may be retained and continued in the free regime, but its deeper-level topology and metric organization are different.
18. Possible Physical Consequences of Deeper Splitting
Mapping the first level onto two nearby branches provides a basis for investigating small corrections to the internal path length, magnetic moment, and external energy gradient. These quantities, however, do not follow merely from the existence of the projectors \(P_0\) and \(R_1\).
For the anomalous magnetic moment, the lengths of the transitions between the branches and the relation between changes in the full path and the magnetic frequency require additional justification. This is discussed in “Geometric Origin of the Electron’s Anomalous Magnetic Moment.”
Coulomb’s law requires a separate mapping of the internal difference into the external potential energy \(U(R)\). The phase operator itself contains no external separation and cannot produce the dependence \(1/R\) without an additional rule. This problem is considered in “Geometric Origin of the Electric Force.”
19. What Follows from the Construction and What Remains Hypothetical
The algebra directly implies: algebraic orthogonality of \(P_n\), decomposition of the identity with a remainder, the form of \(Q_\Gamma\), its invertibility and algebraic unitarity, and different periods of the primary and deeper-level phases. Additivity of winding numbers additionally requires a fixed common cycle and nonzero relative coefficients.
The metric mapping and self-similarity imply: a Hilbert space of internal channels, the series \(\boldsymbol\Gamma_\beta\), and its norm \(\gamma(\beta)\) for \(|\beta|<1\). The metric, isometric shift, weights \(\beta^n\), and identification \(\beta=v/c\) are additional assumptions.
The topological result: the winding number of a continuous nonzero relative phase on a fixed closed cycle is preserved under homotopy in fixed channels. Without additional conditions, this implies neither physical particle stability nor conservation of the sum of individual indices during interactions.
The following remain physical hypotheses: the mapping of channels onto spatial branches, their separation \(\Delta R\), dynamical localization and stability, the connection between internal states and spinor rotations, the specific structure of the charge sector, a global topological index, and the interaction dynamics. Transforming the joint spin state into photon polarizations also requires a separate derivation.
Conclusion
Multilevel splitting defines an architecture of internal channels. The finite operator \(J_\sigma(a,b)\) encodes the internal phase and external motion; the common factor \(Q_\Gamma\) specifies the two-sheeted extension; and \(J_{q,\eta}\) specifies the independent charge sector. The formal orientations \(\nu=\pm1\) allow the winding number of the deeper-level phase to be investigated, but do not replace the spin or charge label. After metric mapping and the assumption of self-similarity, the recursive chain yields \(\boldsymbol\Gamma_\beta\).
\[\tag{61} \boxed{1=P_0+R_1\longrightarrow\begin{cases} Q_\Gamma^{(\nu)}=P_0+R_1e^{i\nu\theta_{\mathrm{int}}/2},\quad\nu=\pm1,\\ R_1=P_1+R_2\longrightarrow\{P_n\}\longrightarrow\boldsymbol\Gamma_\beta\quad\text{(metric and self-similarity)}. \end{cases}} \] The two-sheeted factor and \(\boldsymbol\Gamma_\beta\) share an algebraic foundation but serve different purposes. The former describes relative-phase periodicity; the latter describes the adopted metric structure of motion. Physical localization and the spin representation require independent mappings.
The exact topological result concerns the phase loop under the stated conditions:
\[\tag{62} \boxed{N_\Gamma=\frac1{2\pi i}\oint_\Gamma q_{\mathrm{rel}}^{-1}dq_{\mathrm{rel}}=\operatorname{const}\quad\text{under conditions (47)}.} \] The inverse factor carries the opposite winding number on a single common cycle, so the product \(Q_\Gamma Q_\Gamma^{-1}\) has zero index. This is algebraic compensation. A claim of conservation of a global index in physical interactions can be introduced only after \(N_{\mathrm{tot}}\) is defined and the allowed dynamics are justified:
\[\tag{63} \boxed{N_\Gamma[Q_\Gamma Q_\Gamma^{-1}]=(+1)+(-1)=0,\qquad \Delta N_{\mathrm{tot}}=0\quad\text{only with an additional global law}.} \] The current notation distinguishes phase topology, spatial closure, the spin realization, and the charge sector. A physical transformation between particles and radiation requires restructuring the full joint state. Its mechanism cannot be replaced by cancellation of invertible factors or by extending the theorem about phase loops to an arbitrary interaction.