Circular and Hyperbolic Symmetry Unified in Hyper-Spacetime

1. Introduction

The full Lorentz group $O\left(1, 3\right)$ [1,2,3,4,5,6] encodes the fundamental symmetries of spacetime. It includes both continuous transformations - boosts and spatial rotations - as well as discrete reflections. Conventionally, Lorentz boosts are parameterized by the unbounded angle $\phi \in \left(-\infty ,+\infty \right)$ (rapidity). While effective, this parametrization leads to discontinuities at the lightlike boundary, and leaves the discrete symmetries disconnected from identity.

In this paper, we introduce a bounded representation of the full Lorentz group by embedding its transformations into a closed hyper-spherical geometry, which we term hyper-spacetime. The main step is to map the unbounded angle $\phi$ to a bounded angle $\beta$ using the Gudermannian function [7,8]. This mapping enables smooth passage across infinity, incorporates discrete transformations in a continuous manner, and eliminates algebraic singularities at the lightlike boundary.

The core of the paper (Section 2, Section 3, Section 4, Section 5 and Section 6) develops the mathematical framework. We first reformulate Lorentz boosts using a bounded angle parametrization that unifies circular and hyperbolic symmetry and applies naturally to special relativity, yielding harmonic relativistic proportionality factors [9]. Extending to four dimensions, we introduce a triplet of three-spheres that define hyper-spacetime and show how the full Lorentz group, with both continuous and discrete symmetries, emerges within its hyper-spherical coordinates. Spinor sets are then constructed and analyzed as eigen-spinors of geometric observables, reproducing the Dirac spectrum while confining singularities to scalar factors. The final part of the paper (Section 7 and Section 8) turns to interpretation and outlook, discussing the physical implications of hyper-spacetime, and concludes with a summary of its unifying features.

The mathematical formalism throughout is based on the geometric algebra (GA) of spacetime (STA) as developed by David Hestenes [10,11] (Appendix A). Foundations of geometric algebra where laid by Grassmann [12] and Clifford [13] in the 19th century, and the decisive motivation for its use lies in the generalization of rotations via rotors [14,15,16,17,18]. In STA [10], a spacetime inertial frame $\left\{t,\mathcal{x},\mathcal{y},\mathcal{z}\right\}$ is represented by four orthogonal basis vectors ${\gamma }_{\mu }=\left\{{\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3\right\}$, which satisfy the algebra of the Dirac gamma matrices [4].

2. Hyperbolic Rotation

Lorentz boosts are conventionally expressed as hyperbolic rotations, parameterized by the unbounded angle $\phi \in \left(-\mathrm{\infty },+\mathrm{\infty }\right)$ (rapidity) (Figure 1a).

A boost along the ${\gamma }_3$​-axis ($\mathcal{z}$-axis) is generated by the hyperbolic rotor $R_{\mathcal{z}}\left(\phi \right)$:

$$R_{\mathcal{z}}\left(\phi \right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\phi {\mathrm{ }\sigma }_3/2\right)=\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\phi /2\right)+\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\phi /2\right){\sigma }_3,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\phi \in \left(-\mathrm{\infty },+\mathrm{\infty }\right), R_{\mathcal{z}}{\tilde{R}}_{\mathcal{z}}=1$$

(1)

Where bivector ${\sigma }_3={\gamma }_3{\gamma }_0$ represents the temporal plane of rotation. Acting on the temporal basis vector ${\gamma }_0$​, the rotor generates the boosted spacetime vector $w\left(\phi \right)$:

$$w\left(\phi \right)=R_{\mathcal{z}}{\gamma }_0{\tilde{R}}_{\mathcal{z}}=\cosh \left(\phi \right){\gamma }_0+\sinh \left(\phi \right){\gamma }_3,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{‖w\left(\phi \right)‖}^2=1$$

(2)

However, rotor $R_{\mathcal{z}}\left(\phi \right)$ only covers the positive branch ${\mathbb{H}}_{+}^1$ of the hyperbola. The negative branch of the hyperbola ${\mathbb{H}}_{-}^1$ is not present (Figure 1a).

To cover the full hyperbolic symmetry in a single parameter domain, we map the unbounded angle $\phi \in \left(-\mathrm{\infty },+\mathrm{\infty }\right)$ onto the bounded angle $\beta \in \left[0, 2\pi \right]$ using the Gudermannian function $\phi ={\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}^{-1}\left(\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta \right)\right)$ (Appendix B) [7,8]. Substituting of this function into $R_{\mathcal{z}}\left(\phi \right)$ yields the rotor $L_z\left(\beta \right)$:

$$L_{\mathcal{z}}\left(\beta \right)=\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}^{-1}\left(\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta \right)\right){\sigma }_3/2\right)=\sqrt{\sec \left(\beta \right)}{L}_{u1}\left(\beta \right), L_{\mathcal{z}}{\tilde{L}}_{\mathcal{z}}=1\mathrm{br}-\mathrm{to}-\mathrm{break} L_{u1}\left(\beta \right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\sigma }_3, L_{u1}{\tilde{L}}_{u1}=\cos \left(\beta \right)$$

(3)

The scalar factor $\sqrt{\sec \left(\beta \right)}$​ ensures normalization, while $L_{u1}\left(\beta \right)$ acts as a temporal spinor (Appendix C). Applying rotor $L_{\mathcal{z}}\left(\beta \right)$ to ${\gamma }_0$ generates the boosted spacetime vector $w\left(\beta \right)$:

$$w\left(\beta \right)=L_{\mathcal{z}}{\gamma }_0{\tilde{L}}_{\mathcal{z}}=\sec \left(\beta \right){\gamma }_0+\tan \left(\beta \right){\gamma }_3,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\beta \in \left[0,\mathrm{ }2\pi \right],\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{‖w\left(\beta \right)‖}^2=1$$

(4)

Where the full hyperbolic symmetry is covered continuously (Figure 1b).

In summary, the bounded angle parametrization provides several advantages. It removes singularities by isolating them in the scalar density factor $\sqrt{\sec \left(\beta \right)}$​, while the associated spinor remains regular. At $\beta =\pm \pi /2$, the mapping passes smoothly through infinity, ensuring continuity across the lightlike boundary where crossing corresponds to a time reversal. The two disconnected hyperbolic branches ${\mathbb{H}}_{+}^1$ and ${\mathbb{H}}_{-}^1$​ are thereby closed into a single hyperbolic one-sphere ${\mathbb{S}}_H^1$​ (Figure 1b). In this way, hyperbolic symmetry is unified within a single angular cycle.

3. Harmonic Relativistic Proportionality Factors

In special relativity, the relative speed $v/c$ is conventionally mapped to the unbounded angle $\phi \in \left(-\mathrm{\infty },+\mathrm{\infty }\right)$ (rapidity). Using instead the bounded angle $\beta \in \left[0, 2\pi \right]$ introduced in Section 2, this mapping provides a harmonic set of relativistic proportionality factors:

$$sin\left(\beta \right)=\pm v/c,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }cos\left(\beta \right)=\pm \sqrt{1-{\left(v/c\right)}^2}\mathrm{ }\mathrm{br}-\mathrm{to}-\mathrm{break} \sec \left(\beta \right)={\displaystyle \frac{1}{\pm \sqrt{1-{\left(v/c\right)}^2}}},\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\tan \left(\beta \right)={\displaystyle \frac{\pm v/c}{\pm \sqrt{1-{\left(v/c\right)}^2}}}$$

(5)

These factors form a bridge between circular and hyperbolic symmetry:

$${sin}^2\left(\beta \right)+{cos}^2\left(\beta \right)=1,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{sec}^2\left(\beta \right)-{tan}^2\left(\beta \right)=1\mathrm{ }\mathrm{ }\mathrm{br}-\mathrm{to}-\mathrm{break} {{\mathbb{S}}^1\subset \mathbb{R}}^2\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }⟷\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{{\mathbb{S}}_H^1\subset \mathbb{R}}^{\mathrm{1,1}}$$

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Consequently, the bounded angle mapping unifies circular ${{\mathbb{S}}^1\subset \mathbb{R}}^2$ with hyperbolic ${{\mathbb{S}}_H^1\subset \mathbb{R}}^{\mathrm{1,1}}$ symmetry (Figure 1b). A complete cycle on one-sphere ${\mathbb{S}}^1{\subset \mathbb{R}}^2$ corresponds one-to-one with a complete cycle on hyperbolic one-sphere ${\mathbb{S}}_H^1{\subset \mathbb{R}}^{\mathrm{1,1}}$​. This includes a smooth passage through infinity at the lightlike boundaries $\beta =\pm \pi /2$.

To incorporate all the symmetries of the full Lorentz group $O\left(1, 3\right)$, the framework must extend from the two-dimensional (2D) relation ${\mathbb{R}}^2{⟷\mathbb{R}}^{\mathrm{1,1}}$ to a four-dimensional (4D) relation that unifies the Euclidean domain ${\mathbb{R}}^4$ with the Minkowski domain ${\mathbb{R}}^{\mathrm{1,3}}$.

4. One Unified Geometry Bridging ${\mathbb{R}}^4$

4.1. Definition of the Spinors $\left\{U_1,V_1\right\}$

, Rotor ${\Lambda }_1=\rho U_1$, and Vectors $\left\{p,q,u_r\right\}$$\left\{U_1,V_1\right\}$, and associated rotor ${\Lambda }_1=\rho U_1$, form a spinor structure that preserves the triplet of three-spheres $\left\{{\mathbb{S}}_H^3,{\mathbb{S}}_C^3,{\mathbb{S}}^3\right\}$ (Figure 2 and Figure 3). Thus, every three-sphere carries a natural spinor representation that encodes its symmetry. The Euclidean spinor $V_1$ (preserving ${\mathbb{S}}^3$) and spacetime spinor $U_1$ (preserving ${\mathbb{S}}_C^3$ and ${\mathbb{S}}_H^3$) are defined as:

$$V_1\left(\beta ,\theta ,\varphi \right)=S_1\left(\theta ,\varphi \right)L_{\mathrm{v}1}\left(\beta \right), L_{\mathrm{v}1}\left(\beta \right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\gamma }_3, V_1{V}_1^{-1}=1\mathrm{br}-\mathrm{to}-\mathrm{break} U_1\left(\beta ,\theta ,\varphi \right)=S_1\left(\theta ,\varphi \right)L_{u1}\left(\beta \right), L_{u1}\left(\beta \right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\sigma }_3, U_1{\tilde{U}}_1=\cos \left(\beta \right)\mathrm{br}-\mathrm{to}-\mathrm{break} {\mathsf{\Lambda }}_1\left(\beta ,\theta ,\varphi \right)=\sqrt{\sec \left(\beta \right)}{U}_1\left(\beta ,\theta ,\varphi \right), {\mathsf{\Lambda }}_1{\tilde{\mathsf{\Lambda }}}_1=1$$

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Here $S_1$ acts as a spatial spinor (Eq. 9), $L_{\mathrm{v}1}$ and $L_{u1}$​ act as temporal spinors (Appendix C), and the scalar factor $\sqrt{\sec \left(\beta \right)}$​ ensures the unitarity of rotor ${\mathsf{\Lambda }}_1$, the normalized form of $U_1$. Applying the spinors $U_1$ and $V_1$ yields the vectors spanning the three-spheres:

$$p\in {\mathbb{S}}_H^3 : p\left(\beta ,\theta ,\varphi \right)\mathrm{ }\mathrm{ }={\mathsf{\Lambda }}_1{\gamma }_0{\tilde{\mathsf{\Lambda }}}_1=\sec \left(\beta \right){\gamma }_0+\tan \left(\beta \right)e_3\left(\theta ,\varphi \right), {‖p‖}^2=1\mathrm{br}-\mathrm{to}-\mathrm{break} q\in {\mathbb{S}}_C^3\mathrm{ }:\mathrm{ }\mathrm{ }\mathrm{ }q\left(\beta ,\theta ,\varphi \right)\mathrm{ }\mathrm{ }=U_1{\gamma }_0{\tilde{U}}_1\mathrm{ }=\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\gamma }_0+\sin \left(\beta \right)e_3\left(\theta ,\varphi \right), {‖q‖}^2={\cos }^2\left(\beta \right)\mathrm{br}-\mathrm{to}-\mathrm{break} u_r\in {\mathbb{S}}^3:\mathrm{ }\mathrm{ }\mathrm{ }{u}_r\left(\beta ,\theta ,\varphi \right)=V_1{\gamma }_0{V}_1^{-1}=\cos \left(\beta \right){\gamma }_0+\sin \left(\beta \right)u_3\left(\theta ,\varphi \right), {‖u_r‖}^2=1$$

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Here, $p\in {\mathbb{S}}_H^3$ is a hyperbolic unit vector, $q\in {\mathbb{S}}_C^3$ is a causal vector with variable norm, and $u_r\in {\mathbb{S}}^3$ is a Euclidean unit vector. The causal three-sphere ${\mathbb{S}}_C^3$ covers the spacetime region bounded by the light cone of a past event (Figure 3a). Each vector $q\in {\mathbb{S}}_C^3$ is future pointing and its norm represents proper length, vanishing at the lightlike boundaries. In this way, ${\mathbb{S}}_C^3$ integrates a natural form of causality in the triplet of three-spheres.

In summary, the three-spheres $\left\{{\mathbb{S}}_H^3,{\mathbb{S}}_C^3,{\mathbb{S}}^3\right\}$ (Eq. 7) exhibit the following features:

• A common temporal axis ${\gamma }_0$ with a shared origin (spacetime event).

• Unit vectors $e_3\in {\mathbb{R}}^{\mathrm{1,3}}$ and $u_3{\in \mathbb{R}}^4$ that encode spatial direction and temporal orientation through the dual relation $\left\{e_3=u_3{\gamma }_0\leftrightarrow u_3=e_3{\gamma }_0\right\}$, expressing a geometric duality between domains.

• Domain-independent spatial bivector planes $\mathbb{i}{u}_3\subset {\mathbb{R}}^3$, common to both domains.

A complete cycle on the Euclidean three-sphere ${\mathbb{S}}^3$ corresponds one-to-one with a complete cycle on hyperbolic three-sphere ${\mathbb{S}}_H^3$, thereby unifying ${\mathbb{R}}^4$ and ${\mathbb{R}}^{1,3}$. The apparent boundary at infinity ($\beta =\pm \pi /2$) is absorbed smoothly, making the entire structure closed and traversable. We therefore define the unified geometry constructed from the triplet of three-spheres $\left\{{\mathbb{S}}_H^3,{\mathbb{S}}_C^3,{\mathbb{S}}^3\right\}$ as hyper-spacetime $\mathcal{B}$:

$$\mathcal{B}=\left\{{\mathbb{S}}_H^3,{\mathbb{S}}_c^3,{\mathbb{S}}^3\right\}$$

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Its coordinates are given by the hyper-spherical set $\left(\beta ,\theta ,\varphi \right)$, where the spatial angles $\left(\theta ,\varphi \right)$ encodes direction and the temporal angle $\beta$ encodes relative speed (Eq. 5). Hence, the familiar relativistic parameters of direction and speed are reformulated as angular variables in a closed domain, and continuity at the lightlike boundaries.

5. Full Lorentz Group

The full Lorentz group $O\left(\mathrm{1,3}\right)$ [19,20,21] contains the symmetries of spacetime that preserve the Minkowski metric. It includes: 1) continuous transformations as boosts and spatial rotations, and 2) spacetime event reflections $\left\{I,P,T,PT\right\}$: identity (I), parity (P), time-reversal (T), and spacetime-reversal (PT). Thus, the full Lorentz group is defined as:

$$\mathrm{O}\left(\mathrm{1,3}\right)=\left\{I,P,T,PT\right\}.S{O}^{+}\left(\mathrm{1,3}\right)$$

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where $S{O}^{+}\left(\mathrm{1,3}\right)$ is the proper orthochronous Lorentz subgroup connected to identity.

In the conventional formalism: boosts in $S{O}^{+}\left(\mathrm{1,3}\right)$ are parameterized by the unbounded angle $\phi \in \left(-\infty ,+\infty \right)$, leading to singularities at the lightlike boundaries. Moreover, the discrete reflections $\left\{P,T,PT\right\}$ remain disconnected from identity. The standard way to extend the full Lorentz group $O\left(\mathrm{1,3}\right)$ is through its double cover $\mathrm{P}\mathrm{i}\mathrm{n}\left(\mathrm{1,3}\right)$, as it includes both proper and improper transformations [21]. However, even within $\mathrm{P}\mathrm{i}\mathrm{n}\left(\mathrm{1,3}\right)$, the discrete transformations remain disconnected from identity, leaving a gap between reflections and continuous transformations. This gap is closed by hyper-spacetime, in which continuous and discrete symmetries arise naturally within a unified geometry.

5.1. A Unified Geometry

In the hyper-spherical coordinates $\left(\beta ,\theta ,\varphi \right)$ of hyper-spacetime (defined in Section 4), the spacetime event reflections $\left\{I,P,T,PT\right\}$ manifest naturally as smooth angular transformations:

$$\mathrm{Identity} I\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }:\left(\pm \beta ,\theta , \varphi \right)\mathrm{Parity} P\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }:\left(\mp \beta ,\theta \pm \pi , \varphi \right)\mathrm{Time}-\mathrm{reversal} T\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }:\left(\mp \beta \pm \pi ,\theta ,\varphi \right)\mathrm{Spacetime}-\mathrm{reversal} PT\mathrm{ }:\left(\pm \beta \pm \pi ,\theta \pm \pi ,\varphi \right)$$

(15)

These transformations correspond to changes in temporal orientation (sign changes or $\pm \pi$ shifts in $\beta$) and spatial orientation ($\pm \pi$ shifts in $\theta$). Crucially, all four reflections are smoothly connected to identity, eliminating discontinuities.

5.2. Spinor Representations

The spinor structure of hyper-spacetime provides explicit representations of the $\left\{I,P,T,PT\right\}$ transformations. Starting from the spacetime spinor $U_1\left(\beta ,\theta ,\varphi \right)$, variations in the hyper-spherical coordinates $\left(\beta ,\theta ,\varphi \right)$ generate the full set of spacetime spinors $\left\{U_{\mathrm{j}}\right\}$, where each of the spinors represents one of the $\left\{I,P,T,PT\right\}$ transformations:

$$I: U_1\left(\beta ,\theta ,\varphi \right)=U_1\left(+\beta ,\theta ,\varphi \right)=S_1\left(\theta ,\varphi \right)\left(+\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\sigma }_3\right)\mathrm{br}-\mathrm{to}-\mathrm{break} P: U_2\left(\beta ,\theta ,\varphi \right)=U_1\left(-\beta , \theta +\pi , \varphi \right)=S_2\left(\theta ,\varphi \right)\left(+\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\sigma }_3\right)\mathrm{br}-\mathrm{to}-\mathrm{break} T: U_3\left(\beta ,\theta ,\varphi \right)=U_1\left(-\beta +\pi ,\theta ,\varphi \right)=S_1\left(\theta ,\varphi \right)\left(-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right)-\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right){\sigma }_3\right)\mathrm{br}-\mathrm{to}-\mathrm{break} PT: U_4\left(\beta ,\theta ,\varphi \right)=U_1\left(+\beta +\pi ,\theta +\pi ,\varphi \right)=S_2\left(\theta ,\varphi \right)\left(-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right)+\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right){\sigma }_3\right)$$

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With $U_{\mathrm{j}}{\tilde{U}}_{\mathrm{j}}=\cos \left(\beta \right)\left\{+1,+1,-1,-1\right\}$. Similarly, changes in $V_1\left(\beta ,\theta ,\varphi \right)$ generate the full set of Euclidean spinors $\left\{V_{\mathrm{j}}\right\}$:

$$I: V_1\left(\beta ,\theta ,\varphi \right)=V_1\left(+\beta ,\theta ,\varphi \right)=S_1\left(\theta ,\varphi \right)\left(+\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\gamma }_3\right)\mathrm{br}-\mathrm{to}-\mathrm{break} P: V_2\left(\beta ,\theta ,\varphi \right)=V_1\left(-\beta , \theta +\pi , \varphi \right)=S_2\left(\theta ,\varphi \right)\left(+\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right)-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right){\gamma }_3\right)\mathrm{br}-\mathrm{to}-\mathrm{break} T: V_3\left(\beta ,\theta ,\varphi \right)=V_1\left(-\beta +\pi ,\theta ,\varphi \right)=S_1\left(\theta ,\varphi \right)\left(-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right)-\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right){\gamma }_3\right)\mathrm{br}-\mathrm{to}-\mathrm{break} PT: V_4\left(\beta ,\theta ,\varphi \right)=V_1\left(+\beta +\pi ,\theta +\pi ,\varphi \right)=S_2\left(\theta ,\varphi \right)\left(-\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta /2\right)+\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta /2\right){\gamma }_3\right)$$

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With $V_{\mathrm{j}}{V}_j^{-1}=\left\{+1,+1,+1,+1\right\}$. Note the algebraic closure: each of the spinors $U_{\mathrm{j}}$ and $V_{\mathrm{j}}$ can be a basis to generate the full set of transformations. The spatial spinors $\left\{S_1,S_2\right\}$ in $U_{\mathrm{j}}$ and $V_{\mathrm{j}}$ are generated from spatial spinor $S_1$ (Figure 2a) via a parity transformation:

$$P: {S_1\left(\theta +\pi ,\varphi \right)=S}_2\left(\theta ,\varphi \right), S_k{\tilde{S}}_i={\delta }_{ki}$$

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This orthonormal set $\left\{S_k\right\}$ corresponds to the spatial Pauli spinors [22].

Each set $\left\{U_{\mathrm{j}}\right\}$ and $\left\{V_{\mathrm{j}}\right\}$ forms an orthonormal basis within its respective metric, and the discrete transformations remain continuously connected to identity. The components of both sets are identical, differing only in their temporal bivector plane: $\left({\sigma }_3={\gamma }_3{\gamma }_0\right)\in {\mathbb{R}}^{\mathrm{1,3}}$​ in the Minkowski domain versus $\left({\gamma }_3={\sigma }_3{\gamma }_0\right)\in {\mathbb{R}}^4$​ ​in the Euclidean domain.

In summary, hyper-spacetime provides:

• A smooth, closed, continuous angular domain for the full Lorentz group $O\left(\mathrm{1,3}\right)$.

• Discrete reflections $\left\{P,T,PT\right\}$, continuously connected to identity.

• Spinor sets $\left\{U_{\mathrm{j}}, V_j\right\}$ forming orthonormal bases in their respective metrics, unifying the Euclidean domain ${\mathbb{R}}^4$ with the Minkowski domain ${\mathbb{R}}^{\mathrm{1,3}}$.

• A closed realization of the double cover $\mathrm{P}\mathrm{i}\mathrm{n}\left(\mathrm{1,3}\right)$ through the spinor sets $\left\{U_{\mathrm{j}}, V_j\right\}$.

• Isolation of infinities at the lightlike boundaries $\beta =\pm \pi /2$ into the scalar density factor $\sqrt{\sec \left(\beta \right)}$​ (Eq. 11), while the spinors themselves remain regular.

This prepares the ground for Section 6, where the eigen-spinor and eigenvalue structure of physical observables in hyper-spacetime is developed.

6. Eigen-Spinors and Eigenvalues in the Geometry of Hyper-Spacetime

Analogous to how the observables of $\mathrm{O}\left(2\right)$ lie on the surface of the one-sphere ${\mathbb{S}}^1$, the observables of hyper-spacetime reside on the surfaces of the three-spheres $\left\{{\mathbb{S}}_H^3,{\mathbb{S}}_C^3,{\mathbb{S}}^3\right\}$, each preserved by the spinor sets $\left\{U_{\mathrm{j}},V_{\mathrm{j}}\right\}$. These spinor sets act not only as symmetry generators, but also as eigen-spinors of the corresponding observables $\left\{p,q,u_r\right\}$ (Eq. 12). This follows directly from the geometric algebra eigen-spinor eigenvalue relation (Appendix D):

$$\mathcal{O}{\mathsf{\Omega }}_k={\lambda }_k{\mathsf{\Omega }}_k{e}_1 \mapsto \mathcal{O}={\displaystyle \frac{{\lambda }_k}{{\mathsf{\Omega }}_k{\tilde{\mathsf{\Omega }}}_k}}{\mathsf{\Omega }}_k{e}_1{\tilde{\mathsf{\Omega }}}_k$$

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Here, $\mathcal{O}$ is an observable, $e_1$ is a basis vector (reference observable, e.g., ${\gamma }_0,{\sigma }_3, \mathrm{o}\mathrm{r} {\gamma }_3$), ${\mathsf{\Omega }}_k$ are the eigen-spinors of $\mathcal{O}$, and ${\lambda }_k$ the corresponding eigenvalues. In hyper-spacetime, however, the spinor sets $\left\{U_{\mathrm{j}},V_{\mathrm{j}}\right\}$ are generated geometrically by variations in the hyper-spherical coordinates $\left(\beta ,\theta ,\varphi \right)$ (Eq. 15), rather than being obtained by solving an eigen-spinor eigenvalue equation explicitly (Appendix D).

6.1. Spatial Eigen-Spinors

The spatial spinor set $S_k$ (Eq. 18), generated from a parity transformation, act as eigen-spinors of the spatial unit vectors $e_3\in {\mathbb{S}}_{\mathrm{0,2}}^2\subset {\mathbb{R}}^{\mathrm{1,3}}$ (Figure 2a) and $u_3\in {\mathbb{S}}^2\subset {\mathbb{R}}^4$ (Eq. 10). They satisfy the GA eigen-spinor eigenvalue relations (Appendix D):

$$e_3\in {\mathbb{S}}_{\mathrm{0,2}}^2{ \mapsto e}_3{S}_k={\lambda }_k{S}_k{\gamma }_3, {\lambda }_k\in \left\{\pm 1\right\}\mathrm{br}-\mathrm{to}-\mathrm{break} {u_3\in {\mathbb{S}}^2 \mapsto u}_3{S}_k={\lambda }_k{S}_k{\sigma }_3, {\lambda }_k\in \left\{\pm 1\right\}$$

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So, $e_3$ and $u_3$ are observables with eigen-spinors $S_k$ and eigenvalues ${\lambda }_k=\pm 1$. Physically, these correspond to spin-up or spin-down measurements along the spatial directions of $u_3\left(\theta ,\varphi \right)$ and $e_3\left(\theta ,\varphi \right)$.

6.2. Spacetime Eigen-Spinors

The spacetime spinor set $\left\{U_{\mathrm{j}}\right\}$ (Eq. 16) and Euclidean spinor set $\left\{V_j\right\}$ (Eq. 17) act as eigen-spinors of the hyper-spacetime observables $\left\{p,q,u_r\right\}$ (Eq. 12):

$$p\in {\mathbb{S}}_H^3 \mapsto p{U}_j={\eta }_j{U}_j{\gamma }_0, {\eta }_j\in \left\{+1,+1,-1,-1\right\}\mathrm{br}-\mathrm{to}-\mathrm{break} q\in {\mathbb{S}}_C^3 \mapsto q{U}_j={\kappa }_j{U}_j{\gamma }_0, {\kappa }_j\in \cos \left(\beta \right)\left\{+1,+1,-1,-1\right\}\mathrm{br}-\mathrm{to}-\mathrm{break} u_r\in {\mathbb{S}}^3 \mapsto u_r{V}_j={\eta }_j{V}_j{\gamma }_0, {\eta }_j\in \left\{+1,+1,-1,-1\right\}$$

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Here, the eigenvalues ${\eta }_j$ encodes two constant spin states combined with positive/negative temporal orientation in ${\mathbb{S}}_H^3$ and ${\mathbb{S}}^3$, while the eigenvalues ${\kappa }_j$ encodes two variable $\left(\pm \cos \left(\beta \right)\right)$ spin states combined with positive/negative temporal orientation in ${\mathbb{S}}_c^3$. Unlike ${\eta }_j$​, the values of ${\kappa }_j$​ depend on relative speed via angle $\beta$ (Eq. 5). Hence, a spin state in ${\mathbb{S}}_C^3$​ vanishes at the lightlike boundaries $\beta =\pm \pi /2$.

6.3. Null Spinors

At the lightlike boundaries $\beta =\pm \pi /2$ ($v=\pm c$), the spacetime spinors $U_{\mathrm{j}}$​ reduce to null spinors:

$$U_{\mathrm{j}}\left(\beta ,\theta ,\varphi \right)=S_{\mathrm{k}}\left(\theta ,\varphi \right)L_{u\mathrm{j}}\left(\beta \right) \mapsto U_{\mathrm{j}}\left(\pm {\displaystyle \frac{\pi }{2}},\theta ,\varphi \right)={\displaystyle \frac{1}{\sqrt{2}}}{S}_{\mathrm{k}}\left(\theta ,\varphi \right)\left(\pm 1\pm {\sigma }_3\right), U_{\mathrm{j}}{\tilde{U}}_{\mathrm{j}}=0$$

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These spinors have zero intensity but preserve direction, lying on the light cone of ${\mathbb{S}}_C^3$​ and ${\mathbb{S}}_H^3$​. For the Euclidean spinors $V_j$​, the corresponding boundary form is:

$$V_{\mathrm{j}}\left(\beta ,\theta ,\varphi \right)=S_{\mathrm{k}}\left(\theta ,\varphi \right)L_{\mathrm{v}\mathrm{j}}\left(\beta \right) \mapsto V_{\mathrm{j}}\left(\pm {\displaystyle \frac{\pi }{2}},\theta ,\varphi \right)={\displaystyle \frac{1}{\sqrt{2}}}{S}_{\mathrm{k}}\left(\theta ,\varphi \right)\left(\pm 1\pm {\gamma }_3\right), V_{\mathrm{j}}{V}_1^{-1}=1$$

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Thus, the Euclidean spinors $V_{\mathrm{j}}$ keep their unit intensity, while the intensities of spacetime spinors $U_{\mathrm{j}}$ become zero. Each spinor set pass smoothly through the lightlike boundary.

6.4. Correspondence with the Dirac Equation

The eigen-spinors $U_{\mathrm{j}}$ (Eq. 21) and eigenvalues ${\eta }_j$ of the observable $p\in {\mathbb{S}}_H^3$ reproduce those of the Dirac equation [14] (two spin states combined with positive/negative temporal orientation, i.e., matter/antimatter) (Appendix E). This agreement is remarkable, because in hyper-spacetime these eigen-spinors are generated geometrically (Eq. 16), rather than being obtained by solving an eigen-spinor eigenvalue equation (Appendix D). This establishes a direct geometric origin for the Dirac spectrum, demonstrating that the fundamental structure of spacetime eigen-spinors emerges naturally within the hyper-spacetime framework.

In summary, the eigen-spinor structure of hyper-spacetime reproduces the Dirac spectrum, isolates singularities into a scalar density, and reveals null spinors as natural lightlike states.

7. Discussion

The unified hyper-spacetime framework $\mathcal{B}=\left\{{\mathbb{S}}_H^3,{\mathbb{S}}_C^3,{\mathbb{S}}^3\right\}$ extends the full Lorentz group $O\left(\mathrm{1,3}\right)$ into a continuous, bounded, and closed representation that reproduces the Dirac eigen-spinor spectrum while exposing structural features obscured in conventional formulations.

The agreement between Dirac eigen-spinors and the spinors of hyper-spacetime is striking: whereas Dirac spinors encode intrinsic particle properties, in hyper-spacetime they emerge directly from geometry through variations in the hyper-spherical coordinates $\left(\beta ,\theta ,\varphi \right)$ (Eq. 15). This correspondence shows that the algebraic structure of the Dirac equation can be understood as a geometric property.

In hyper-spacetime, boosts and spatial rotations combine seamlessly with the discrete symmetries $\left\{I,P,T,PT\right\}$, which appear as smooth angular transformations in the closed domain of $\left(\beta ,\theta ,\varphi \right)$. The replacement of the unbounded rapidity $\phi$ by the bounded angle $\beta$ closes the hyperbola ${\mathbb{S}}_H^1$​, incorporating the lightlike boundaries as regular points. This generalizes naturally to the four-dimensional (4D) hyperbolic three-sphere ${\mathbb{S}}_H^3$​, where infinities are absorbed into a smooth boundary.

The causal three-sphere ${\mathbb{S}}_C^3$ covers the spacetime region bounded by the light cone of a past event, a four-dimensional causality volume containing all information accessible to observation (Figure 3a). It is spanned by future-pointing vectors $q$ of variable norm ${\cos }^2\left(\beta \right)$. The Euclidean three-sphere ${\mathbb{S}}^3$, with its positive-definite metric, ensures completeness. Together, the three-spheres share a common temporal axis ${\gamma }_0$​, origin (spacetime event), and temporal bivector plane $\left(e_3=u_3{\gamma }_0\leftrightarrow u_3=e_3{\gamma }_0\right)$ (Eq. 10). This bivector plane expresses the geometric duality of the aligned spatial vectors, as $e_3$ and $u_3$ act as vector $\leftrightarrow$ bivector counterparts in opposite domains. They are linking spatial direction with temporal orientation and thereby unifying the Euclidean $\left\{{\gamma }_0,{\sigma }_k\right\}{\subset \mathbb{R}}^4$ and Minkowski $\left\{{\gamma }_0,{\gamma }_k\right\}\subset {\mathbb{R}}^{\mathrm{1,3}}$ domains. Hyper-spacetime thus provides a natural setting for Wick rotations and periodic time in path integrals [23,24], without the need to introduce imaginary time.

A direct manifestation of this duality is the emergence of temporal spinors (Appendix C). Both spacetime spinors $U_{\mathrm{j}}$​ and the Euclidean spinors $V_{\mathrm{j}}$​ decompose into spatial and temporal spinors (Eq. 16, 17). The spatial spinors encode orientation in the spatial bivector plane $\mathbb{i}{u}_3\left(\theta ,\varphi \right)\subset {\mathbb{R}}^3$, which lies in the common three-dimensional even subalgebra and is domain-independent, while the temporal spinors encode orientation in the shared temporal bivector plane. In this way, the spinor decomposition mirrors the dual geometric structure of hyper-spacetime.

The physical status of hyper-spacetime must ultimately be tested experimentally. We therefore challenge experimentalists [25,26] to search for phenomena that could reveal the predicted link between particle properties and spacetime geometry. Such confirmation would show that hyper-spacetime is not only mathematically consistent but also offers a new perspective on nature. This article has focused on establishing hyper-spacetime as a consistent framework: a closed representation of the full Lorentz group enriched by new geometric elements - temporal spinors, null spinors, and a shared bivector plane - that provide a foundation for future work.

8. Conclusion

The hyper-spacetime framework provides a bounded, continuous, and closed representation of the full Lorentz group $O\left(\mathrm{1,3}\right)$. Through a unified three-sphere geometry and hyper-spherical coordinates $\left(\beta ,\theta ,\varphi \right)$, it removes discontinuities, incorporates both continuous and discrete transformations in a single smooth domain, and reproduces the Dirac eigen-spinor spectrum with singularities isolated in scalar factors. Beyond these established results, hyper-spacetime introduces new elements - temporal spinors, null spinors, and a shared bivector plane - that offer a fresh perspective on spacetime symmetry. Together, these features define a consistent framework and a foundation for future exploration of geometric structure in nature.