2026-09-03
Geometric formation of an electron pair and the Pauli exclusion principle
Why one atomic orbital does not allow three electrons
Why does one atomic orbital admit two electrons but not a third? In quantum mechanics, this fact is expressed by the Pauli exclusion principle and the antisymmetry of the many-electron state. In the geometric model of wave electricity, it becomes possible to divide this problem into two parts: first, obtain two internal states of a single orbital and then establish a geometric rule for their joint occupation.
In this paper, the following mechanism is proposed. The orthogonal states of the operator \(J_n(a,0)\) and \(J_n(0,a)\) define two opposite spin projections. Their ordinary product identifies a frequency-closed pair: identical orientations create a relative harmonic \(2\omega_n\), while the product of opposite orientations projectively reduces to a scalar. After this, the geometric gradient of the energy projections unfolds the spatial planes of the two allowed charge waves into a single common plane and establishes a phase of \(\pi\) between them. The strict prohibition of the third electron is no longer energetic, but algebraic in nature: the outer product of the three states in a two-dimensional internal space is identically zero.
The usual product verifies the frequency closure of the pair, the geometric gradient forms its spatial configuration, and the outer product prohibits the third state.
\[ \tag{1} \boxed{ \begin{gathered} \{J_{n,+},J_{n,-}\} \;\Longrightarrow\; J_{n,+}J_{n,-}\sim1 \;\Longrightarrow\; \chi=0,\quad\Delta\varphi=\pi, \\[2mm] \dim\mathcal S_{nlm}=2 \;\Longrightarrow\; \Lambda^3\mathcal S_{nlm}=0 \;\Longrightarrow\; N_{nlm}\leq2. \end{gathered} } \] 1. Spatial Orbital and Internal State
In the work on geometric splitting of atomic orbitals, the spatial structure of the state is associated with the sequential splitting of the operator. We denote the resulting orbital by \(\Phi_{nlm}\), where \(n,l,m\) play the role of spatial quantum numbers.
It is necessary to separate the spatial shape of the orbital from the internal orientation of the electron. For a fixed orbital, two states of the ground operator are possible:
\[ \tag{2} J_{n,+}=J_n(a,0), \qquad J_{n,-}=J_n(0,a). \] We write the complete one-electron states as
\[ \tag{3} \Psi_{nlm}^{(+)} = \Phi_{nlm}J_{n,+}, \qquad \Psi_{nlm}^{(-)} = \Phi_{nlm}J_{n,-}. \] Thus, deep splitting determines the spatial parameters \(n,l,m\), and the choice of \(J_{n,+}\) or \(J_{n,-}\) is preserved as an independent binary property in all resulting orbitals.
2. Two spin states of one orbital
The internal states belong to two orthogonal planes:
\[ \tag{4} \boxed{ J_{n,+}\perp J_{n,-} }. \] However, this orthogonality does not mean that the observed three-dimensional spin vectors are directed at right angles. It refers to the intrinsic geometry of \(J\), whereas the physical measurement projects two states onto one chosen axis \(\widehat{\mathbf n}\):
\[ \tag{5} \widehat S_{\mathbf n}J_{n,+} = +\frac{\hbar}{2}J_{n,+}, \qquad \widehat S_{\mathbf n}J_{n,-} = -\frac{\hbar}{2}J_{n,-}. \] Therefore, \(J_{n,+}\) and \(J_{n,-}\) are geometric candidates for two opposite spin states of the same spatial orbital.
3. Intrinsic Relative Frequency
To examine the composition of two orientations, we expand them in an idempotent basis. We denote the phase factor by
\[ \tag{6} z_n(t)=e^{i\pi a}=e^{i\omega_n t}. \] Then the two operators take the form
\[ \tag{7} J_{n,+} = \ep+\em z_n, \qquad J_{n,-} = \ep z_n+\em. \] For the general state \(J=A(t)\ep+B(t)\em\), we introduce the relative phase and its frequency:
\[ \tag{8} \vartheta_J = \arg\frac{B}{A}, \qquad \boxed{ \Omega[J] = \frac{d\vartheta_J}{dt} }. \] The general phase of both components cancels out in the ratio \(B/A\), so \(\Omega[J]\) characterizes not the absolute phase factor, but the observed relative rotation of two idempotent sectors. For states (7), we obtain:
\[ \tag{9} \Omega[J_{n,+}]=+\omega_n, \qquad \Omega[J_{n,-}]=-\omega_n. \] 4. Product of Identical Orientations
Let's consider an attempt to combine two identically oriented internal states in a single orbital. Thanks to the properties \(\ep^2=\ep\), \(\em^2=\em\) and \(\ep\em=0\) we have
\[ \tag{10} \boxed{ J_{n,+}^2 = (\ep+\em z_n)^2 = \ep+\em z_n^2 = J_n(2a,0) }. \] For the second orientation, we similarly obtain
\[ \tag{11} \boxed{ J_{n,-}^2 = (\ep z_n+\em)^2 = \ep z_n^2+\em = J_n(0,2a) }. \] Since \(z_n^2=e^{i2\omega_n t}\), identical orientations create an uncompensated second harmonic:
\[ \tag{12} \boxed{ \Omega[J_{n,+}^2]=+2\omega_n, \qquad \Omega[J_{n,-}^2]=-2\omega_n }. \] Such a product is not locked into the original relative phase regime of the orbital. It creates a new dynamic state with double the relative frequency.
5. Product of Opposite Orientations
For different orientations, the result is fundamentally different:
\[ \tag{13} \begin{aligned} J_{n,+}J_{n,-} &= (\ep+\em z_n)(\ep z_n+\em) \\ &= \ep z_n+\em z_n \\ &= z_n(\ep+\em) = e^{i\omega_n t}. \end{aligned} \] Both idempotent components have the same phase. In the design representation, the common phase factor does not change state, so
\[ \tag{14} \boxed{ J_{n,+}J_{n,-} = e^{i\omega_n t} \sim1, \qquad \Omega[J_{n,+}J_{n,-}]=0 }. \] Thus, the opposite orientations cancel out the relative internal rotation. The product contains no residual frequency between the \(\ep\) and \(\em\) planes, although the common phase factor is preserved.
6. Frequency Locking Criterion
The obtained result allows us to formulate an additional criterion for an electron pair: two electrons can form a consistent state in the same spatial orbital if the product of their internal operators is a projected scalar and does not create a new relative frequency.
\[ \tag{15} \boxed{ J_1J_2\sim1 \quad\Longleftrightarrow\quad \Omega[J_1J_2]=0 }. \] Among the basis orientations, only the mixed products \(J_{n,+}J_{n,-}\) and \(J_{n,-}J_{n,+}\) satisfy this condition. Identical orientations create a harmonic of \(2\omega_n\) and do not meet the frequency locking criterion.
7. Total Energy and Relative Mode Energy
Doubling the relative frequency cannot be equated with an unacceptable doubling of the total energy. For two electrons, the total energy naturally adds up: \(E_{\mathrm{tot}}=E_1+E_2=2E_n\). This is also true for the allowed pair.
The energy scale of the uncompensated relative mode can be determined separately:
\[ \tag{16} E_{\mathrm{rel}} = \hbar|\Omega[J_1J_2]|. \] Then opposite orientations yield \(E_{\mathrm{rel}}=0\), while identical orientations yield \(E_{\mathrm{rel}}=2\hbar\omega_n\). This is not the total energy of the two particles, but an additional internal scale indicating the presence or absence of uncompensated relative rotation.
Frequency closure selects the permissible internal orientations of the pair, but does not yet determine the relative positions of its spatial waves. This problem is solved by the geometric gradient.
8. Internal and Spatial Planes
For further construction, it is necessary to distinguish between the internal planes of the operator and the spatial planes of motion of the charge waves. The internal states remain orthogonal, but their observed orbital projections may coincide:
\[ \tag{17} \boxed{ \Pi_{\mathrm{orb}}[J_{n,+}] = \Pi_{\mathrm{orb}}[J_{n,-}] = \Phi_{nlm} }. \] In other words, when a pair is formed, the states \(J_n(a,0)\) and \(J_n(0,a)\) do not merge. Only the spatial planes in which their charge waves propagate are aligned. As a result, the two electrons share a common orbital but retain distinct internal states and opposite spins.
9. Geometry of Two Rotating Waves
Let \(\chi\) be the angle between the spatial planes of the two waves, and \(\delta=\Delta\varphi\) be their phase difference. We choose the first plane in coordinates \((x,y)\):
\[ \tag{18} \widehat{\mathbf u}_1(\theta) = \begin{pmatrix} \cos\theta\\ \sin\theta\\ 0 \end{pmatrix}. \] The second plane is rotated relative to the first by an angle \(\chi\), and the wave has an additional phase \(\delta\):
\[ \tag{19} \widehat{\mathbf u}_2(\theta) = \begin{pmatrix} \cos(\theta+\delta)\\ \cos\chi\,\sin(\theta+\delta)\\ \sin\chi\,\sin(\theta+\delta) \end{pmatrix}. \] We introduce an instantaneous non-negative measure of misalignment:
\[ \tag{20} m(\theta,\chi,\delta) = \frac12 \left| \widehat{\mathbf u}_1(\theta) + \widehat{\mathbf u}_2(\theta) \right|^2. \] It vanishes if and only if the two waves occupy opposite positions at any instant:
\[ \tag{21} m=0 \quad\Longleftrightarrow\quad \widehat{\mathbf u}_2=-\widehat{\mathbf u}_1. \] 10. Average Deviation Measure
Let's average expression (20) over a full revolution. The scalar product of two directions has the average value.
\[ \tag{22} \left\langle \widehat{\mathbf u}_1\cdot \widehat{\mathbf u}_2 \right\rangle = \frac{1+\cos\chi}{2}\cos\delta. \] Therefore, the average measure of the mismatch is
\[ \tag{23} \boxed{ \mathcal M(\chi,\delta) = 1+ \frac{1+\cos\chi}{2}\cos\delta }. \] If the spatial planes already coincide, \(\chi=0\), then
\[ \tag{24} \mathcal M(0,\delta) = 1+\cos\delta = 2\cos^2\frac{\delta}{2}. \] The minimum of this expression is achieved at \(\delta=\pi\). If we first fix the antiphase, we obtain
\[ \tag{25} \mathcal M(\chi,\pi) = \frac{1-\cos\chi}{2} = \sin^2\frac{\chi}{2}. \] Now the minimum is achieved at \(\chi=0\). Therefore, the joint condition for complete matching is
\[ \tag{26} \boxed{ \chi=0, \qquad \Delta\varphi=\pi, \qquad \mathcal M(0,\pi)=0 }. \] 11. Connection with the Geometric Coulomb Gradient
In the paper "Geometric Origin of Electric Force," the interaction was attributed to incomplete compensation of the projections of two close internal branches. For an external object, the difference in projections is equal to
\[ \tag{27} \Delta P(\ell) = \frac{r_+}{\sqrt{\ell^2+r_+^2}} - \frac{r_-}{\sqrt{\ell^2+r_-^2}}. \] In the far region, where \(\ell\gg r_+,r_-\), this formula becomes
\[ \tag{28} \Delta P(\ell) \approx \frac{\Delta r}{\ell}. \] For a pair of rotating waves, it is necessary to consider not only the radial distance, but also the relative orientation of the planes and phases. As the simplest angular extension of formula (27), we introduce
\[ \tag{29} \boxed{ \Delta P_{\mathrm{pair}} (\ell,\chi,\delta) = \Delta P(\ell) \mathcal M(\chi,\delta) }. \] Formula (29) is a new hypothesis of this work: the radial projection of the previous model is supplemented by a positive measure of angular and phase mismatch.
In the far field, the expression takes the form
\[ \tag{30} \Delta P_{\mathrm{pair}} \approx \frac{\Delta r}{\ell} \mathcal M(\chi,\delta). \] 12. Mismatch Energy
To ensure that the consistent configuration corresponds to a minimum, we introduce a positive mismatch energy:
\[ \tag{31} U_{\mathrm{mis}} (\ell,\chi,\delta) = E_*\Delta P(\ell) \mathcal M(\chi,\delta), \qquad E_*>0. \] In the article on the origin of the electric force, the quantity \(E_*\) is the energy scale accessible to the external projection; for an electron, it is associated with the rest energy. Since all factors in (31) are non-negative, we obtain
\[ \tag{32} \boxed{ U_{\mathrm{mis}}(\ell,0,\pi)=0 }. \] Thus, geometry does not select an arbitrary wave arrangement, but a state with coinciding spatial planes and opposite phases.
13. Rotation of Spatial Planes
The dependence of energy on the angle \(\chi\) creates a geometric torque:
\[ \tag{33} \tau_\chi = -\frac{\partial U_{\mathrm{mis}}}{\partial\chi}. \] Denoting \(C(\ell)=E_*\Delta P(\ell)>0\), we obtain
\[ \tag{34} \tau_\chi = \frac{C(\ell)}{2} \sin\chi\cos\delta. \] Near antiphase \(\delta=\pi\):
\[ \tag{35} \tau_\chi = -\frac{C(\ell)}{2}\sin\chi. \] When \(\chi\) is positive, the moment is directed toward decreasing the angle. Therefore,
\[ \tag{36} \boxed{ \chi\longrightarrow0 }. \] It is the geometric gradient of the energy projections that unfolds the two spatial planes into a single common orbital plane. The orthogonality of the internal states \(J_{n,+}\) and \(J_{n,-}\) does not disappear.
14. Establishing Antiphase
The dependence of energy on \(\delta\) creates a phase gradient:
\[ \tag{37} Q_\delta = -\frac{\partial U_{\mathrm{mis}}}{\partial\delta} = C(\ell) \frac{1+\cos\chi}{2} \sin\delta. \] Let the phase be slightly deviated from antiphase:
\[ \tag{38} \delta=\pi+\eta, \qquad |\eta|\ll1. \] Then
\[ \tag{39} Q_\delta \approx -C(\ell) \frac{1+\cos\chi}{2}\eta. \] The sign of the forcing is opposite to the deviation, so the phase gradient returns the system to the state
\[ \tag{40} \boxed{ \Delta\varphi\longrightarrow\pi }. \] As a result, the same pThe projection mechanism performs two functions:
\[ \tag{41} \boxed{ \begin{aligned} -\frac{\partial U_{\mathrm{mis}}}{\partial\chi} &\Longrightarrow \text{alignment of spatial planes}, \\ -\frac{\partial U_{\mathrm{mis}}}{\partial\Delta\varphi} &\Longrightarrow \text{establishing antiphase}. \end{aligned} } \] 15. Final Geometry of the Electron Pair
In a consistent configuration, we have
\[ \tag{42} \widehat{\mathbf u}_2(\theta) = \widehat{\mathbf u}_1(\theta+\pi) = -\widehat{\mathbf u}_1(\theta). \] For a circular orbit of radius \(R_n\):
\[ \tag{43} \mathbf r_2(t)=-\mathbf r_1(t), \qquad |\mathbf r_1-\mathbf r_2|=2R_n. \] After half a period, the waves spatially exchange places:
\[ \tag{44} \mathbf r_1\left(t+\frac{T}{2}\right) = \mathbf r_2(t), \qquad \mathbf r_2\left(t+\frac{T}{2}\right) = \mathbf r_1(t). \] Since the electrons are indistinguishable, the overall configuration of the pair after this exchange remains physically the same.
16. Two Gradient Compensation Options
Ideally, the geometric projections cancel each other at every instant:
\[ \tag{45} \boxed{ \Delta P_{12}(t)=0, \qquad F_{12}(t)=0 }. \] This is an instantaneously consistent state. It occurs if the effective separation of the branches in the direction of mutual projection is zero throughout the entire cycle.
In a more general dynamic version, the projection changes sign after half a period:
\[ \tag{46} \Delta P_{12}\left(t+\frac{T}{2}\right) = -\Delta P_{12}(t). \] Then, for a full cycle, we obtain
\[ \tag{47} \boxed{ \left\langle\Delta P_{12}\right\rangle_T=0, \qquad \left\langle F_{12}\right\rangle_T=0 }. \] The first option describes a rigidly matched pair, the second a dynamic pair with a periodic internal exchange of energy projections. The choice between them should follow from a more detailed geometry of the split branches.
17. Singlet State
Two electrons have the same spatial orbital, but opposite internal states. The antisymmetric part of the pair can be written as
\[ \tag{48} \Psi_{\mathrm{pair}} = \frac{\Phi_{nlm}(1)\Phi_{nlm}(2)}{\sqrt2} \left[ J_{n,+}(1)J_{n,-}(2) - J_{n,-}(1)J_{n,+}(2) \right]. \] For this state, the total spin of the electron pair is zero:
\[ \tag{49} \boxed{ \widehat{\mathbf S}_{\mathrm{tot}} \Psi_{\mathrm{pair}}=0, \qquad S_{\mathrm{pair}}=0 }. \] Thus, the geometric orthogonality of the internal states, together with the common spatial projection, leads to the singlet pair structure.
18. Exterior Product and Exchange Geometry
To strictly prohibit reoccupancy, we introduce the exterior product of the internal states. The two-electron configuration is determined by the oriented area.
\[ \tag{50} \Omega_{nlm}^{(2)} = J_{n,+}\wedge J_{n,-}. \] When electrons are exchanged, the orientation of the area changes:
\[ \tag{51} J_{n,+}\wedge J_{n,-} = -J_{n,-}\wedge J_{n,+}. \] This rule is consistent with the spatial exchange of waves through half a period. If both electrons attempt to occupy the same state, the oriented area degenerates:
\[ \tag{52} J_{n,+}\wedge J_{n,+}=0, \qquad J_{n,-}\wedge J_{n,-}=0. \] The exterior product is a rule for the composition of identical electrons. Its geometric motivation is the change in orientation upon exchange, but a complete derivation of this rule directly from the J algebra remains a separate problem.
19. Why a third electron is impossible
For a fixed orbital, the internal space is two-dimensional:
\[ \tag{53} \mathcal S_{nlm} = \operatorname{span} \{J_{n,+},J_{n,-}\}, \qquad \dim\mathcal S_{nlm}=2. \] Therefore, any putative state of the third electron in the same orbital is a linear combination of the existing directions:
\[ \tag{54} J_3 = \alpha J_{n,+} + \beta J_{n,-}. \] The three-electron state would look like this:
\[ \tag{55} \Omega_{nlm}^{(3)} = J_{n,+}\wedge J_{n,-}\wedge J_3. \] Substituting (54), we obtain two terms, each containing a repeating direction:
\[ \tag{56} \begin{aligned} \Omega_{nlm}^{(3)} ={}& \alpha J_{n,+}\wedge J_{n,-}\wedge J_{n,+} \\ &+ \beta J_{n,+}\wedge J_{n,-}\wedge J_{n,-} =0. \end{aligned} \] Hence,
\[ \tag{57} \boxed{ \dim\mathcal S_{nlm}=2 \quad\Longrightarrow\quad \Lambda^3\mathcal S_{nlm}=0 \quad\Longrightarrow\quad N_{nlm}\leq2 }. \] The third electron lacks a new, independent internal direction. Therefore, it must move to a different spatial orbital, where a new one-electron state emerges.
20. Why the Coulomb Gradient Alone Is Not Enough
The Coulomb gradient cannot be considered a strict reason for the prohibition of a third electron. Three waves on a circle can be formally arranged with phases \(0\), \(2\pi/3\), and \(4\pi/3\). Such a configuration may be energetically unfavorable, but its existence is not nullified by the geometry of the distances alone.
The energy gradient selects the optimal configuration of the allowed pair, but strict prohibition only occurs when the three-electron state identically vanishes.
\[ \tag{58} \boxed{ \begin{aligned} \text{ordinary product} &\Longrightarrow J_{n,+}J_{n,-}\sim1, \\ \text{geometric gradient} &\Longrightarrow \chi=0, \quad \Delta\varphi=\pi, \\ \text{outer product} &\Longrightarrow N_{nlm}\leq2. \end{aligned} } \] 21. Why the frequency criterion is insufficient to prohibit the third electron
The ordinary product distinguishes between identical and opposite orientations and thus selects a frequency-closed pair. However, this alone does not invalidate the attempt to add a third electron. After the dot product of the pair is formed, formal multiplication by the third state yields
\[ \tag{59} J_{n,+}J_{n,-}J_3 \sim J_3. \] Therefore, the frequency criterion explains why opposite orientations form a consistent two-electron combination, but it does not yet imply a limit on the filling. A strict limitation arises only for the antisymmetric composition:
\[ \tag{60} \boxed{ J_{n,+}\wedge J_{n,-}\wedge J_3=0 }. \] Thus, the ordinary and external products serve different purposes and should not be interchanged.
As a result, the allowed electron pair simultaneously satisfies the frequency, spatial-geometric, and antisymmetric criteria.
| Criterion | Mathematical Notation | Physical Meaning |
|---|---|---|
| Frequency | \(J_{n,+}J_{n,-}sim1\) | No residual relative frequency |
| Geometric | \(\mathcal M(0,\pi)=0\) | Common plane and antiphase |
| Antisymmetric | \(J_{n,+}\wedge J_{n,-}\neq0\) | Allowed two-electron pair |
| Dimensional | \(\Lambda^3\mathcal S_{nlm}=0\) | A third electron is impossible |
22. Atomic Shell Capacity
Each spatial orbital corresponds to two internal states:
\[ \tag{61} (n,l,m,+), \qquad (n,l,m,-). \] If geometric splitting creates \(n\) spatial orbitals of magnitude \(n^2\) at the \(n\) level, then their total capacity is equal to
\[ \tag{62} \boxed{ N_n=2n^2 }. \] For example, the first orbital allows for the \(1s^2\) configuration. The third electron can no longer form a nonzero outer product in \(\mathcal S_{1s}\) space and must move to the next orbital. The capacities of \(2s^2\) and \(2p^6\) are obtained similarly.
23. Transition to Bound States
An antiphase configuration allows us to introduce a phase matching parameter:
\[ \tag{63} C_{12} = \left| \left\langle e^{i(\varphi_1-\varphi_2)} \right\rangle \right|. \] For independent waves with a random phase difference, we have \(C_{12}=0\). For a pair with a fixed phase \(\Delta\varphi=\pi\):
\[ \tag{64} \left\langle e^{i(\varphi_1-\varphi_2)} \right\rangle =-1, \qquad C_{12}=1. \] This state is phase-coupled. To prove the energetic coupling, it is necessary to further demonstrate that the total energy of the pair is lower than the energy of the separated states and that small deviations of \(\chi\) and \(\Delta\varphi-\pi\) create recurrent gradients. The expressions obtained above for \(\tau_\chi\) and \(Q_\delta\) already define a possible mechanism for such stability.
24. Obtained results and physical hypotheses
For rigor, we separate the algebraic results and the accepted physical rules. The equalities \(J_{n,+}^2=J_n(2a,0)\), \(J_{n,-}^2=J_n(0,2a)\), and \(J_{n,+}J_{n,-}=e^{i\omega_nt}\sim1\) follow directly from idempotent algebra. The two-dimensionality of the space \(\mathcal S_{nlm}\) implies the absence of a third independent internal direction. After adopting the external product as the composition rule for identical electrons, the third state is strictly zeroed out.
The function \(\mathcal M(\chi,\delta)\) accurately describes the chosen measure of misalignment of two circular waves and has a single minimum at \(\chi=0\), \(\delta=\pi\). However, its factorization with radial projection,
\[ \tag{65} \Delta P_{\mathrm{pair}} = \Delta P(\ell)\mathcal M(\chi,\delta), \] as well as the interpretation of \(E_*\Delta P\mathcal M\) as a positive misalignment energy, are new physical hypotheses. An additional physical principle is the criterion that the absence of a relative frequency determines the permissibility of pairing in a single orbital, while the harmonic \(2\omega_n\) denotes departure from the initial orbital regime. In the future, it is desirable to derive these rules directly from the full operator \(J(a,b)\), the geometry of its split branches, and the law of conservation of total energy.
Conclusion
One spatial orbital \(\Phi_{nlm}\) allows two orthogonal internal states \(J_n(a,0)\) and \(J_n(0,a)\). In observable space, they manifest as opposite spin projections. The usual product of these states is projected to a scalar and leaves no relative internal frequency. After such frequency locking, the geometric gradient of the energy projections rotates the spatial planes of the charge waves until they coincide and establishes a phase of \(\pi\) between them. This results in the formation of a matched singlet pair with zero total spin.
The strict prohibition of a third electron has a different nature. The interior of one orbital is two-dimensional, and the exterior product of its three elements is always zero. Therefore, the ordinary product selects frequency-compatible orientations, the geometric gradient explains the spatial dynamics of allowed pair formation, and the antisymmetric geometry of the exterior product limits the orbital's capacity to two electrons.
\[ \tag{66} \boxed{ \begin{gathered} \Phi_{nlm} \;\longrightarrow\; \{J_n(a,0),J_n(0,a)\} \;\longrightarrow\; J_{n,+}J_{n,-}\sim1, \quad \Omega_{\mathrm{rel}}=0, \\[1mm] \chi=0, \quad \Delta\varphi=\pi, \\[1mm] S_{\mathrm{pair}}=0, \qquad J_{n,+}\wedge J_{n,-}\neq0, \qquad \Lambda^3\mathcal S_{nlm}=0, \qquad N_{nlm}\leq2. \end{gathered} } \] Thus, geometric splitting, phase matching, spin compensation, and the Pauli exclusion principle become consistent parts of a single construction. The next step should be to derive the angular mismatch energy and the antisymmetric composition rule directly from the algebra of the global operator \(J\).