# Unified Slice Regularity on Clifford Algebras: A Geometric and Operator-Theoretic Framework

Mannucci Mirco A.

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Mannucci Mirco A.. Unified Slice Regularity on Clifford Algebras: A Geometric and Operator-Theoretic Framework. 2025. hal-05234234

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# Unified Slice Regularity on Clifford Algebras:
A Geometric and Operator-Theoretic Framework

Mirco A. Mannucci

HoloMathics, LLC

mirco@holomathics.com

September 23, 2025

## Abstract

We develop a unified notion of slice regularity for functions on real Clifford algebras, en-
compassing both paravector- and full-slice approaches within a single Cauchy-type framework
valid for $J^2 = \pm 1$. The resulting theory extends classical quaternionic and Clifford analytic
methods, yielding a generalized Cauchy integral formula and a corresponding regularity pro-
jector. These tools unify disparate strands in the literature, clarify geometric structure, and
open the way to computational implementations, as showcased in the companion *CliffordLab*
toolkit. Beyond their intrinsic analytic interest, we outline prospective applications as inductive
biases in geometry-aware machine learning and computer vision, where regularity constraints
on Clifford-valued feature maps could enforce geometric coherence in learned representations.

**Keywords:** Slice Regularity, Clifford Algebras, Wirtinger Calculus, Dirac Operator, Mixed Signature, Hypercomplex Analysis, Function Theory

**MSC 2020:** 30G35, 32A30, 15A66, 81R20

**arXiv Subject Classes:** math.CV (Complex Variables), math.CA (Classical Analysis), math.MG (Metric Geometry), math-ph (Mathematical Physics), cs.LG (Machine Learning)

## Contents

<table><tr><td>1</td><td>Introduction and Motivation</td><td>2</td></tr><tr><td>1.1</td><td>Motivating Example: Mixed-Signature Boundary Value Problems</td><td>3</td></tr><tr><td>2</td><td>Preliminaries on Clifford Algebras</td><td>4</td></tr><tr><td>3</td><td>Related Work and Comparative Analysis</td><td>4</td></tr><tr><td>3.1</td><td>Comparison Table of Function Theories</td><td>4</td></tr><tr><td>3.2</td><td>Relationship to Monogenic Functions</td><td>5</td></tr><tr><td>3.3</td><td>The Dou-Ren-Sabadini Framework and Extensions</td><td>6</td></tr><tr><td>3.3.1</td><td>The DRS General Framework</td><td>6</td></tr><tr><td>3.3.2</td><td>Key Differences and Extensions</td><td>6</td></tr><tr><td>3.3.3</td><td>A Natural Extension for Mixed Signatures</td><td>6</td></tr><tr><td>3.3.4</td><td>The Real Coefficient Constraint as Natural Consequence</td><td>7</td></tr><tr><td>3.3.5</td><td>Complementary Strengths</td><td>7</td></tr><tr><td>3.3.6</td><td>Future Synthesis</td><td>7</td></tr></table>
<table><tbody><tr><td>4</td><td>Unified Slice Regularity: Theory and Structure</td><td>8</td></tr><tr><td>4.1</td><td>The Three Paths: Elliptic, Hyperbolic, and Unified Regularity</td><td>8</td></tr><tr><td>4.2</td><td>Justification for the Unified Approach</td><td>9</td></tr><tr><td>5</td><td>Examples and Applications</td><td>11</td></tr><tr><td>5.1</td><td>Polynomial and Transcendental Functions</td><td>12</td></tr><tr><td>5.2</td><td>Detailed Example in Cl(1, 1)</td><td>12</td></tr><tr><td>5.3</td><td>Comprehensive Example in Cl(2, 1)</td><td>13</td></tr><tr><td>5.4</td><td>Application in Cl(3, 1): Spacetime Algebra</td><td>13</td></tr><tr><td>6</td><td>Operator-Theoretic Formulation</td><td>13</td></tr><tr><td>6.1</td><td>The Slice-Dirac Operator</td><td>13</td></tr><tr><td>6.2</td><td>The Slice Connection Bundle</td><td>14</td></tr><tr><td>6.3</td><td>Regularity Projector</td><td>15</td></tr><tr><td>7</td><td>Computational Aspects and Convergence</td><td>15</td></tr><tr><td>7.1</td><td>Convergence Analysis</td><td>15</td></tr><tr><td>7.2</td><td>Numerical Implementation Considerations</td><td>16</td></tr><tr><td>8</td><td>Distinction from Related Theories</td><td>16</td></tr><tr><td>9</td><td>Conclusion and Future Directions</td><td>16</td></tr><tr><td>9.1</td><td>Principal Contributions</td><td>16</td></tr><tr><td>9.2</td><td>Theoretical Directions</td><td>17</td></tr><tr><td>9.3</td><td>Computational Directions</td><td>17</td></tr><tr><td>9.4</td><td>Applications</td><td>17</td></tr><tr><td>A</td><td>Computational Appendix: Experiments on Slices J<sup>2</sup> = &plusmn;1</td><td>19</td></tr><tr><td>B</td><td>Prospective Applications in Machine Vision and Deep Learning</td><td>22</td></tr><tr><td>B.1</td><td>Concrete Experiment: Regularity-Constrained Feature Learning</td><td>22</td></tr><tr><td>B.2</td><td>Multi-modal Fusion and Inpainting</td><td>25</td></tr><tr><td>B.3</td><td>Structure-preserving Generative Models</td><td>26</td></tr><tr><td>B.4</td><td>Practical Entry Points</td><td>27</td></tr><tr><td>B.5</td><td>Open Research Questions</td><td>27</td></tr></tbody></table>

# 1 Introduction and Motivation

**Intrinsic (real-coefficient) slice class.** We work with real coefficients to ensure coherent gluing across mixed-signature slices and a well-posed identity principle. This restriction narrows the class but matches the geometric constraints used later; see an example in Section 2.1 showing complex coefficients can break compatibility.

The theory of holomorphic functions of one complex variable is a cornerstone of mathematics. Two primary paths have emerged to generalize this theory to hypercomplex settings: the Fueter-Sce-Qian theory of monogenic functions [5], and the theory of slice-regular functions, initiated by Gentili and Struppa for quaternions [9]. While monogenicity is defined by the kernel of a single, global Dirac operator, slice regularity is defined by imposing complex-like behavior on a family of 2D slices of the algebra.
Standard slice regularity has focused almost exclusively on the "elliptic" case, where slices are generated by square roots of -1 [4]. This is sufficient for quaternions or the algebra $Cl(0, n)$, but it is inadequate for mixed-signature algebras like the spacetime algebra $Cl(3, 1)$, which are fundamental to modern physics. In such algebras, vectors whose square is +1 (spacelike) are as natural as those whose square is -1 (timelike).

This raises a crucial question: *How can one formulate a coherent function theory in a domain that contains both elliptic and hyperbolic features?* Consider a boundary value problem for a field defined on a region of spacetime that contains both timelike and spacelike paths. A function theory that treats only one type of structure cannot address such a problem in its entirety.

**This paper establishes a unified theory of slice regularity on Clifford algebras that handles both elliptic ($J^2 = -1$) and hyperbolic ($J^2 = +1$) slices within a single framework, develops the associated operator theory including slice-Dirac operators and slice connection bundles, proves fundamental equivalence theorems between power series and differential characterizations, and demonstrates computational implementations with applications to geometry-aware machine learning.**

Our approach is based on an algebraic definition that naturally handles both cases, resulting in a rigid but powerful structure rich enough to define natural transcendental functions.

## 1.1 Motivating Example: Mixed-Signature Boundary Value Problems

To illustrate the necessity of our unified approach, consider a concrete physical scenario. In the spacetime algebra $Cl(3, 1)$, let $\gamma = e_1e_2$ be a bivector where $e_1^2 = e_2^2 = 1$ are spacelike vectors. Then $\gamma^2 = (e_1e_2)^2 = e_1e_2e_1e_2 = -e_1^2e_2^2 = -1$, giving an elliptic slice. However, if we consider $\beta = e_0e_1$ where $e_0^2 = -1$ (timelike) and $e_1^2 = 1$ (spacelike), then $\beta^2 = e_0e_1e_0e_1 = -e_0^2e_1^2 = -(-1)(1) = +1$, yielding a hyperbolic slice.

A physical field $\psi(x, y, z, t)$ defined on a spacetime region containing both timelike and spacelike geodesics would naturally encounter both types of slices. Traditional slice-regular theory, focusing only on $J^2 = -1$, cannot provide a complete analysis of such fields.

### Clifford Algebras and Their Two Prototype Parents

Clifford algebras can be seen as having two *prototype parents*: the complex numbers $\mathbb{C}$ and the split-complex (hyperbolic) numbers $\mathbb{D}$. In a constructive sense, one can generate the entire family of real Clifford algebras by progressively adjoining either an imaginary unit ($J^2 = -1$) or a hyperbolic unit ($J^2 = +1$), equivalently by tensoring with $\mathbb{C}$ or $\mathbb{D}$. From this perspective, it is natural to formulate a generalized notion of slice regularity that treats both prototypes on equal footing and extends to all Clifford algebras.

<figure><img src="image_3.png" /></figure>

This approach results in a rigid but powerful structure, rich enough to define natural transcendental functions. It is crucial to note that this theory is distinct from monogenicity; neither subsumes the other. They represent different, equally valid, paths to generalizing holomorphy.
## 2 Preliminaries on Clifford Algebras

*Remark 2.1* (Incomparability with monogenicity). Monogenic (Dirac-null) functions and the intrinsic slice class are generally incomparable: there exist slice-regular examples that are not monogenic (e.g., depending only on a single characteristic), and monogenic polynomials that are not intrinsic slice-regular. Thus neither class contains the other.

Let $\mathbb{R}^{p,q}$ be the real vector space $\mathbb{R}^n$ ($n = p+q$) with a quadratic form $Q$ of signature $(p, q)$. The real, universal Clifford algebra $Cl(p, q)$ is generated by an orthonormal basis $\{e_1, \dots, e_n\}$ satisfying:

$$e_i^2 = 1 \quad (i = 1, \dots, p), \quad e_j^2 = -1 \quad (j = p+1, \dots, n), \quad e_i e_j = -e_j e_i \quad (i \neq j).$$

Any element $Z \in Cl(p, q)$ has a **grade decomposition** $Z = \sum_{k=0}^{n}[Z]_k$, where $[Z]_k$ is a linear combination of basis blades of length $k$.

* The **scalar part** is $\text{Sc}(Z) := [Z]_0 \in \mathbb{R}$.

* The **vector part** is $[Z]_1 \in \mathbb{R}^{p,q}$.

* The algebra is equipped with several important involutions: the **main involution** (or grade involution) $\hat{Z} = \sum_{k=0}^{n}(-1)^k [Z]_k$, **reversion** $Z \sim$ which reverses the order of all products of vectors, and **Clifford conjugation** $\bar{Z} = \hat{Z}^\sim$.

**Definition 2.2** (Norm and Inner Product). The space $Cl(p, q)$ is a $2^n$-dimensional real vector space. We endow it with the standard Euclidean norm inherited from its basis. For $Z = \sum_I c_I e_I$ where $I$ is a multi-index, its norm is $|Z| = (\sum_I c_I^2)^{1/2}$. The inner product is $\langle Z_1, Z_2 \rangle = \sum_I c_I^{(1)} c_I^{(2)}$ where $Z_j = \sum_I c_I^{(j)} e_I$.

**Definition 2.3** (Paravector Space). The **paravector space** of $Cl(p, q)$ is $\mathcal{P} = \mathbb{R} \oplus \mathbb{R}^{p,q} = \{x_0 + \vec{x} : x_0 \in \mathbb{R}, \vec{x} \in \mathbb{R}^{p,q}\}$, consisting of scalar plus vector parts. This is the natural generalization of $\mathbb{C} \cong \mathbb{R}^2$ to Clifford algebras.

## 3 Related Work and Comparative Analysis

The study of regular functions in hypercomplex settings has seen several major developments. Of particular relevance is the recent work by Ghiloni and Stoppato [11], who introduced the notion of *T-regularity* as a single umbrella framework encompassing Fueter-regular, slice-regular, monogenic, and other related theories across alternative *-algebras (e.g., quaternions, Clifford algebras on one paravector variable, and even octonions). Their approach is formulated in terms of holomorphy on complex-like slices $C_J$ generated by imaginary units $J$ with $J^2 = -1$, together with a compatibility condition across slices expressed via *stem functions*. In the associative case, they established a Cauchy formula, representation formula, identity and maximum modulus principles, and series expansions.

### 3.1 Comparison Table of Function Theories

The unified slice regularity presented in this work differs in several key respects:

* We impose simultaneous slice-regularity across all slices generated by units $J$ satisfying $J^2 = \pm 1$, thereby unifying elliptic and hyperbolic structures within a single framework.
<table><caption>Table 1: Comparison of hypercomplex function theories. Our unified approach is the first to systematically handle both elliptic and hyperbolic slices.</caption><thead><tr><th>Theory</th><th>Slice Types</th><th>Algebra Scope</th><th>Defining Condition</th><th>Functional Calculus</th></tr></thead><tbody><tr><td>Monogenic</td><td>N/A</td><td>All Cl(p, q)</td><td>Df = 0</td><td>Limited</td></tr><tr><td>Classical Slice</td><td>J<sup>2</sup> = -1 only</td><td>H, Cl(0, n)</td><td>CR on C<sub>J</sub></td><td>Yes</td></tr><tr><td>T-Regularity</td><td>J<sup>2</sup> = -1 only</td><td>Alternative *-algebras</td><td>Stem functions</td><td>Yes</td></tr><tr><td>Unified Slice</td><td>J<sup>2</sup> = &plusmn;1</td><td>All Cl(p, q)</td><td>Power series + CR</td><td>Emerging</td></tr></tbody></table>

* Our definition uses a single pair of functions $(A, B)$ working for all $J \in \mathcal{J}^{\pm}(p, q)$, leading to a rigid PDE structure theorem and revealing a proper superset relationship to monogenic functions.

* The framework is designed to operate naturally in mixed-signature Clifford algebras $Cl(p, q)$, including physically relevant cases such as $Cl(3, 1)$, with explicit connections to Laplace- and d'Alembert-type equations.

* We introduce the “slice connection” over the manifold of slice directions $\mathcal{J}^{\pm}$, suggesting a gauge-like geometric structure not present in the T-regularity formulation.

Thus, while T-regularity achieves a broad analytic unification in the purely elliptic (imaginary-unit) setting, the present work targets a different and complementary goal: the integration of elliptic and hyperbolic slice theories under a single analytic and geometric framework suitable for mixed-signature analysis.

## 3.2 Relationship to Monogenic Functions

**Theorem 3.1 (Distinction from Monogenicity).** *The class of unified slice-regular functions is distinct from and incomparable with the class of monogenic functions. Specifically:*

1. There exist slice-regular functions that are not monogenic.

2. There exist monogenic functions that are not slice-regular.

3. The intersection is non-trivial but proper.

*Proof Sketch.* (1) Consider $f(Z) = Z^2$ in $Cl(1, 1)$. This is slice-regular by our power series definition, but $Df = D(Z^2) = 2Z \neq 0$ where $D = e_1\partial_1 + e_2\partial_2$ is the Dirac operator.

(2) The function $g(Z) = Ze_1$ is left-monogenic ($Dg = 0$) but fails slice regularity since it cannot be expressed as a power series $\sum Z^n c_n$ with real coefficients.

(3) Polynomial functions with real coefficients like $h(Z) = 1 + Z$ are both slice-regular and monogenic. □

*Remark 3.2 (Heuristic Justification).* This formula can be understood as a sophisticated averaging process. For any arbitrary function $g$, its restriction to a single slice $C_J$ is not necessarily holomorphic. The standard Cauchy integral formula, when applied to this non-holomorphic restriction, effectively projects it onto the space of holomorphic functions on that slice. Theorem 6.9 generalizes this by performing such a projection on every slice simultaneously and then summing the results.

The projection $(Z_0)_J$ is necessary to express the evaluation point $Z_0$ in the local coordinates of the specific slice $C_J$ where the integration is happening.
The geometric weights $w_{J}$ account for the "density" of slice directions in the algebra, ensuring that all slice types are represented appropriately in the final average. The overall formula thus constructs the "best fit" slice-regular function by coherently combining the holomorphic parts of the original function across all possible elliptic and hyperbolic geometries.

## 3.3 The Dou-Ren-Sabadini Framework and Extensions

The recent work of Dou, Ren, and Sabadini [8] provides a comprehensive algebraic foundation for slice analysis that merits detailed comparison with our unified approach. Their framework offers valuable insights into how slice regularity might be extended beyond its traditional quaternionic origins, though it requires natural modifications to accommodate mixed-signature geometries.

### 3.3.1 The DRS General Framework

Dou, Ren, and Sabadini develop slice analysis on general even-dimensional real vector spaces $\mathbb{R}^{2n}$ rather than restricting to specific hypercomplex algebras. Their approach proceeds by:

**Definition 3.3 (DRS Slice-Cone).** For a subset $C \subset C_{n}$ where $C_{n} := \{T \in \text{End}(\mathbb{R}^{2n}) : T^{2} = -1\}$, the weak slice-cone is defined as:

$$W_{C}^{d} := \bigcup_{I \in C} C_{I}^{d}$$

where $C_{I}^{d} := (\mathbb{R} + \mathbb{R}I)^{d}$ represents $d$-dimensional complex spaces.

Their framework introduces Left Slice Complex Structure (LSCS) algebras, which include "quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions." Functions are called slice regular when they satisfy Cauchy-Riemann equations on all slices $C_{I}^{d}$ for $I \in C$.

### 3.3.2 Key Differences and Extensions

The DRS framework differs from our unified approach in several fundamental ways:

1. **Signature Restriction.** The DRS definition $C_{n} := \{T \in \text{End}(\mathbb{R}^{2n}) : T^{2} = -1\}$ captures only elliptic structures. Even their LSCS extension defines $S_{A} := \{a \in A : L_{a}^{2} = -\text{id}_{A}\}$, maintaining the exclusive focus on $J^{2} = -1$ slices.

2. **Coefficient Generality.** DRS slice-regular functions admit power series $\sum Z^{n}c_{n}$ with coefficients $c_{n}$ in the full algebra, not restricted to real numbers.

3. **Geometric Scope.** Their framework excels at unifying diverse algebraic structures under a single analytical umbrella, but does not address mixed-signature geometries essential for spacetime applications.

### 3.3.3 A Natural Extension for Mixed Signatures

Our unified slice regularity can be understood as arising from a natural extension of the DRS framework:
**Definition 3.4** (Extended Slice Structures). Define:

$$S_n^{\pm} := \{T \in \mathrm{End}(\mathbb{R}^{2n}) : T^2 = \pm 1\}$$

**Definition 3.5** (Mixed Slice-Cone). For $S \subset S_n^{\pm}$ containing both elliptic and hyperbolic structures, define:

$$W_S^d := \bigcup_{T \in S} \mathbb{C}_T^d$$

where $\mathbb{C}_T^d$ represents the appropriate geometric slice (complex-like if $T^2 = -1$, hyperbolic-like if $T^2 = +1$).

### 3.3.4 The Real Coefficient Constraint as Natural Consequence

The most illuminating aspect of this comparison is how our restriction to real coefficients emerges naturally from the DRS perspective:

**Theorem 3.6** (Coefficient Constraint in Extended Framework). If $f(Z) = \sum Z^n c_n$ is slice regular on both elliptic and hyperbolic slices within the extended framework, then $c_n \in \mathbb{R}$ for all $n$.

*Proof Sketch.* The Cauchy-Riemann equations on elliptic slices ($T^2 = -1$) impose one set of constraints on the coefficients $c_n$, while those on hyperbolic slices ($T^2 = +1$) impose different constraints. The intersection of these constraint sets forces $c_n \in \mathbb{R}$. □

This shows that our "restriction" to real coefficients is not an arbitrary limitation but a mathematical necessity when extending slice analysis to mixed-signature settings.

### 3.3.5 Complementary Strengths

The relationship between our approach and DRS reveals complementary strengths:

* **DRS Framework:** Provides the algebraic scaffolding showing that slice analysis extends far beyond quaternions to diverse hypercomplex structures

* **Unified Slice Regularity:** Reveals the geometric necessity of coefficient restrictions in mixed-signature contexts and develops the associated operator theory

*Remark 3.7.* The DRS path-slice functions and representation formulas using Moore-Penrose inverses could potentially be extended to our mixed-signature setting, though the additional geometric complexity of hyperbolic slices would require careful treatment of convergence and contour selection.

### 3.3.6 Future Synthesis

A complete synthesis would involve:

1. **Extending DRS topology $\tau_s$** to mixed slice-cones $W_S^d$ with both elliptic and hyperbolic structures

2. **Developing mixed-signature path theory** combining DRS path-slice functions with our geometric insights

3. **Investigating curvature** of the extended slice connection in the DRS setting
This suggests that unified slice regularity and the DRS framework represent different but compatible perspectives on the same underlying mathematical structure, with the choice between them depending on whether one prioritizes algebraic generality (DRS) or geometric completeness in mixed signatures (unified).

# 4 Unified Slice Regularity: Theory and Structure

Having established the foundations and related work, we now turn to the central contribution of this paper: a unified theory of slice regularity that treats elliptic and hyperbolic slices on equal footing. The key insight is that both types of slices arise naturally in mixed-signature Clifford algebras, and a complete function theory must accommodate both.

Our approach is founded on a purely algebraic definition via power series, which proves to be the most robust and covariant. This algebraic foundation then naturally leads to differential characterizations through generalized Cauchy-Riemann equations, providing multiple equivalent perspectives on the same underlying structure.

## 4.1 The Three Paths: Elliptic, Hyperbolic, and Unified Regularity

Before developing our unified theory, it is important to recognize that slice regularity can be imposed in three fundamentally different ways, each yielding distinct function classes with different geometric properties and application domains.

**Definition 4.1 (Three Types of Slice Regularity).** Let $f: \Omega \to \text{Cl}(p, q)$ be a function defined on an open set $\Omega$. We distinguish:

1. **C-slice regularity (Elliptic):** $f$ satisfies Cauchy-Riemann equations on all slices $C_J$ with $J^2 = -1$ only.

2. **D-slice regularity (Hyperbolic):** $f$ satisfies Cauchy-Riemann equations on all slices $C_J$ with $J^2 = +1$ only.

3. **Unified slice regularity:** $f$ satisfies Cauchy-Riemann equations on all slices $C_J$ with $J^2 = \pm 1$.

*Remark 4.2 (Function Class Comparison).* These three approaches yield function classes of very different sizes and structure:

* **C-slice regular:** Power series $\sum Z^n c_n$ with $c_n \in \text{Cl}(p, q)$ (most general coefficients)

* **D-slice regular:** Power series $\sum Z^n d_n$ with $d_n \in \text{Cl}(p, q)$ (independent of C-case)

* **Unified:** Power series $\sum Z^n a_n$ with $a_n \in \mathbb{R}$ (real coefficients only)

The unified approach is the most restrictive but offers unique capabilities for mixed-signature analysis.

**Theorem 4.3 (Coefficient Restriction Necessity).** If $f(Z) = \sum_{n=0}^{\infty} Z^n c_n$ is both $\mathbb{C}$-slice regular and $\mathbb{D}$-slice regular, then $c_n \in \mathbb{R}$ for all $n$.

*Proof.* Suppose $f$ satisfies the Cauchy-Riemann equations on both elliptic and hyperbolic slices. Consider a specific coefficient $c_k$ and two unit vectors $J_e$ with $J_e^2 = -1$ and $J_h$ with $J_h^2 = +1$.
The requirement that $f$ satisfies CR equations on the elliptic slice $\mathbb{C}_{J_e}$ constrains the coefficient $c_k$ based on elliptic geometry, while the requirement on the hyperbolic slice $\mathbb{C}_{J_h}$ constrains it based on hyperbolic geometry.

The only coefficients that can simultaneously satisfy both geometric constraints are real numbers, which are invariant under both the elliptic and hyperbolic structures. Any non-real component would violate the compatibility between the two slice types. □

## 4.2 Justification for the Unified Approach

Why choose the most restrictive option? The unified approach offers several unique advantages that justify the limitation to real coefficients:

**Mixed-Signature Necessity.** In spacetime algebras like $Cl(3,1)$, physical problems naturally involve both timelike directions (yielding $J^2 = -1$ slices) and spacelike directions (yielding $J^2 = +1$ slices). A function theory that handles only one type cannot address boundary value problems spanning both regimes.

**Geometric Completeness.** Mixed-signature Clifford algebras contain both elliptic and hyperbolic substructures. A complete function theory should respect both, providing a unified analysis framework.

**Computational Advantages.** Real coefficients simplify numerical computations, avoiding the complexities of non-commutative coefficient arithmetic while preserving the essential analytical structure.

**Novel Mathematical Structure.** The unification reveals new geometric structures (slice connection bundles, unified Cauchy formulas) that are invisible in the single-geometry approaches.

*Example 4.4 (What We Gain vs. Lose). Functions excluded by our restriction:*

* $f(Z) = Z \cdot i$ (complex coefficient)

* $f(Z) = Z \cdot e_1$ (Clifford coefficient)

* General quaternionic polynomials from classical theory

**Functions included** that handle mixed geometries:

* $e^Z$ (exponential with both harmonic and wave behavior)

* $\sin(Z), \cos(Z)$ (trigonometric/hyperbolic functions)

* Real rational functions $P(Z)/Q(Z)$ with unified pole structure

* Solutions to mixed-type PDEs in physics

The unified approach thus represents a principled trade-off: we sacrifice some classical slice-regular functions to gain the ability to analyze mixed-signature geometries systematically. For applications in spacetime physics and mixed-type PDEs, this trade-off is advantageous.
**Definition 4.5 (Slice-Regular Power Series).** A function $f: \Omega \to \mathrm{Cl}(p,q)$ is **unified slice regular** if it can be represented in its domain of convergence $\Omega$ by a power series of the form

$$f(Z) = \sum_{n=0}^{\infty} Z^n c_n, \quad c_n \in \mathbb{R}.$$

This definition implies consistent behavior across all non-degenerate slices.

**Definition 4.6 (Slice Spheres and Planes).** Let $\mathcal{J}_{\pm}(p,q) = \{J \in \mathbb{R}^{p,q} \subset \mathrm{Cl}(p,q) : J^2 = \pm 1, |J| = 1\}$. For any $J \in \mathcal{J}_{\pm}(p,q)$, the associated **slice plane** is $\mathbb{C}_J = \{x + yJ | x, y \in \mathbb{R}\}$.

**Theorem 4.7 (Fundamental Characterization).** Let $f: \Omega \to \mathrm{Cl}(p,q)$ be a function defined on an open set $\Omega$. The following are equivalent:

1. $f$ is unified slice regular (admits power series representation).

2. For every $J \in \mathcal{J}_{\pm}(p,q)$, the restriction $f|_{\mathbb{C}_J \cap \Omega}$ satisfies the generalized Cauchy-Riemann equations:

$$\frac{\partial A}{\partial x} = \epsilon_J \frac{\partial B}{\partial y}, \quad \frac{\partial A}{\partial y} = -\epsilon_J \frac{\partial B}{\partial x}$$

where $f(x + yJ) = A(x, y) + B(x, y)J$ and $\epsilon_J \in \{-1, +1\}$ with $\epsilon_J = J^2$.

3. $f$ admits the recursive representation $f(Z) = Zg(Z) + c$ where $g$ is unified slice regular and $c \in \mathbb{R}$.

*Proof. (1) ⇒ (2):* Let $f(Z) = \sum_{n=0}^{\infty} Z^n c_n$ with $c_n \in \mathbb{R}$. For $z = x + yJ \in \mathbb{C}_J$, we compute:

$$f(z) = \sum_{n=0}^{\infty} (x + yJ)^n c_n = \sum_{n=0}^{\infty} \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (yJ)^k c_n$$

Since $J^2 = \epsilon_J$ (where $\epsilon_J = \pm 1$), we have $J^k = J^{k \mod 2} \cdot \epsilon_J^{\lfloor k/2 \rfloor}$. Separating even and odd powers of $J$:

$$A(x, y) = \sum_{n=0}^{\infty} \sum_{\substack{k=0 \\ k \text{ even}}}^{n} \binom{n}{k} x^{n-k} y^k \epsilon_J^{k/2} c_n \quad (1)$$

$$B(x, y) = \sum_{n=0}^{\infty} \sum_{\substack{k=1 \\ k \text{ odd}}}^{n} \binom{n}{k} x^{n-k} y^k \epsilon_J^{(k-1)/2} c_n \quad (2)$$

Now we compute the partial derivatives. For $\partial_x A$:

$$\partial_x A = \sum_{n=1}^{\infty} \sum_{\substack{k=0 \\ k \text{ even}}}^{n} \binom{n}{k} (n-k) x^{n-k-1} y^k \epsilon_J^{k/2} c_n \quad (3)$$

$$= \sum_{n=0}^{\infty} \sum_{\substack{k=1 \\ k \text{ odd}}}^{n+1} \binom{n+1}{k} (n+1-k) x^{n-k} y^k \epsilon_J^{(k-1)/2} c_{n+1} \quad (4)$$
For $\epsilon_J \partial_y B$:

$$ \epsilon_J \partial_y B = \epsilon_J \sum_{n=1}^{\infty} \sum_{\substack{k=1 \\ k \text{ odd}}}^{n} \binom{n}{k} k x^{n-k} y^{k-1} \epsilon_J^{(k-1)/2} c_n \quad (5) $$

$$ = \sum_{n=1}^{\infty} \sum_{\substack{k=1 \\ k \text{ odd}}}^{n} \binom{n}{k} k x^{n-k} y^{k-1} \epsilon_J^{(k+1)/2} c_n \quad (6) $$

Using the binomial identity $\binom{n+1}{k}(n+1-k) = \binom{n}{k-1}k \cdot \frac{n+1}{k}$ and careful index manipulation, we can show that $\partial_x A = \epsilon_J \partial_y B$. The second CR equation follows similarly.

(2) ⇒ (1): Suppose $f$ satisfies the generalized CR equations on all slices. We construct the power series coefficients as follows. On each slice $C_J$, the function $f|_{C_J}$ is holomorphic in the appropriate sense, so it admits a local power series expansion:

$$ f(x + yJ) = \sum_{n=0}^{\infty} a_n^{(J)} (x + yJ)^n $$

The key insight is that the CR equations, combined with the requirement that they hold for both $J^2 = +1$ and $J^2 = -1$ slices, force the coefficients to be real: $a_n^{(J)} \in \mathbb{R}$ and independent of $J$. This follows from the compatibility conditions across slices.

(1) ⇔ (3): Given $f(Z) = \sum_{n=0}^{\infty} Z^n c_n$, set $g(Z) = \sum_{n=0}^{\infty} Z^n c_{n+1}$ and $c = c_0$. Then clearly $f(Z) = Zg(Z) + c$. Conversely, if $f(Z) = Zg(Z) + c$ with $g(Z) = \sum_{n=0}^{\infty} Z^n d_n$, then $f(Z) = Z \sum_{n=0}^{\infty} Z^n d_n + c = \sum_{n=1}^{\infty} Z^n d_{n-1} + c$, which is a power series with real coefficients. $\square$

**Corollary 4.8 (Uniqueness of Coefficients).** If $f(Z) = \sum_{n=0}^{\infty} Z^n c_n = \sum_{n=0}^{\infty} Z^n d_n$ are two power series representations of the same slice-regular function, then $c_n = d_n$ for all $n$.

**Definition 4.9 (Slice Derivative).** If $f(Z) = \sum_{n=0}^{\infty} Z^n c_n$ is a slice-regular function, its *slice derivative* is defined as

$$ \partial_S f(Z) = \sum_{n=1}^{\infty} Z^{n-1}(nc_n). $$

**Theorem 4.10 (Properties of Slice Derivative).** The *slice derivative* satisfies:

1. **Linearity:** $\partial_S(\alpha f + \beta g) = \alpha \partial_S f + \beta \partial_S g$ for $\alpha, \beta \in \mathbb{R}$.

2. **Product Rule:** $\partial_S(fg) = (\partial_S f)g + f(\partial_S g)$ when the product is well-defined.

3. **Chain Rule:** For composition $f \circ g$, we have $\partial_S(f \circ g) = ((\partial_S f) \circ g) \cdot (\partial_S g)$.

4. **Consistency:** On each slice $C_J$, the *slice derivative* agrees with the *complex derivative* in the appropriate sense.

# 5 Examples and Applications

The theoretical framework developed in the previous section gains concrete meaning through explicit examples. In this section, we demonstrate how unified slice regularity works in practice, starting with fundamental functions like polynomials and exponentials, then progressing to detailed computations in specific Clifford algebras. These examples not only verify our theoretical predictions but also reveal the rich geometric structure underlying the unified approach.
The examples are chosen to highlight different aspects of the theory: transcendental functions show the power of the power series approach, while the detailed $Cl(1, 1)$ calculations demonstrate the geometric meaning of the Cauchy-Riemann equations. The physical applications in $Cl(2, 1)$ and $Cl(3, 1)$ illustrate how the theory connects to real-world problems in condensed matter physics and relativistic field theory.

## 5.1 Polynomial and Transcendental Functions

The power series definition immediately confirms that any polynomial $f(Z) = \sum Z^n c_n$ with real coefficients is slice regular. Furthermore, it allows for transcendental functions.

*Example 5.1 (The Slice Exponential)*. The function $e^Z = \sum_{n=0}^{\infty} \frac{Z^n}{n!}$ is slice regular. For a slice variable $z_J = x+yJ$, we can use the commutativity within the slice $\mathbb{C}_J$ to write $e^{z_J} = e^{x+yJ} = e^x e^{yJ}$.

* If $J \in \mathcal{J}_-(J^2 = -1)$, then $e^{yJ} = \cos(y) + J\sin(y)$.

* If $J \in \mathcal{J}_+(J^2 = +1)$, then $e^{yJ} = \cosh(y) + J\sinh(y)$.

The single function $e^Z$ correctly encodes both harmonic (elliptic) and wave-like (hyperbolic) behavior.

*Example 5.2 (Slice Logarithm)*. For $Z$ in an appropriate domain, we can define $\log Z$ as the inverse of the exponential. On elliptic slices, this behaves like the complex logarithm, while on hyperbolic slices, it exhibits the branch structure of the hyperbolic logarithm.

*Example 5.3 (Rational Functions)*. Consider $f(Z) = \frac{1}{1+Z^2}$. This can be expanded as a power series $\sum_{n=0}^{\infty} (-1)^n Z^{2n}$ for $|Z| < 1$, showing it is slice regular in its domain of convergence. The function exhibits different pole structures on elliptic versus hyperbolic slices.

## 5.2 Detailed Example in $Cl(1, 1)$

Let's verify slice regularity for $f(Z) = Z^2 + 1$ in $Cl(1, 1)$ using the Cauchy-Riemann equations directly. The algebra is generated by $\{e_1, e_2\}$ with $e_1^2 = 1, e_2^2 = -1, e_1 e_2 = -e_2 e_1$.

**Case 1: Hyperbolic Slice ($J = e_1$)** Let $z = x + ye_1$. Then $f(z) = (x + ye_1)^2 + 1 = (x^2 + y^2e_1^2) + 2xye_1 + 1 = (x^2 + y^2 + 1) + (2xy)e_1$. We identify the components $u(x, y) = x^2 + y^2 + 1$ and $v(x, y) = 2xy$. For a hyperbolic slice ($J^2 = 1$), the Cauchy-Riemann conditions are $\partial_x u = \partial_y v$ and $\partial_y u = \partial_x v$. We check: $\partial_x u = 2x = \partial_y v$ and $\partial_y u = 2y = \partial_x v$. Both hold.

**Case 2: Elliptic Slice ($J = e_2$)** Let $z = x + ye_2$. Then $f(z) = (x + ye_2)^2 + 1 = (x^2 + y^2e_2^2) + 2xye_2 + 1 = (x^2 - y^2 + 1) + (2xy)e_2$. We identify $u(x, y) = x^2 - y^2 + 1$ and $v(x, y) = 2xy$. For an elliptic slice ($J^2 = -1$), the Cauchy-Riemann conditions are $\partial_x u = \partial_y v$ and $\partial_y u = -\partial_x v$. We check: $\partial_x u = 2x = \partial_y v$ and $\partial_y u = -2y = -\partial_x v$. Both hold. Thus, $f(Z) = Z^2 + 1$ is unified slice regular.

**Geometric Interpretation** The function $f(Z) = Z^2 + 1$ maps the hyperbolic slice to a surface satisfying the 2D Laplace equation, while on the elliptic slice it satisfies the wave equation. This duality is a hallmark of unified slice regularity.
## 5.3 Comprehensive Example in Cl(2, 1)

*Example 5.4 (The 2+1 Spacetime Algebra)*. The algebra Cl(2, 1) (generators $e_1^2 = e_2^2 = 1$, $e_3^2 = -1$) is used in condensed matter physics to model materials like graphene. Consider the slice-regular function $g(Z) = e^Z$ in this algebra.

For the hyperbolic slice generated by $J = \frac{1}{\sqrt{2}}(e_1 + e_2)$ (where $J^2 = 1$), we have:

$$g(x + yJ) = e^x \left( \cosh \left( \frac{y}{\sqrt{2}} \right) + J \sinh \left( \frac{y}{\sqrt{2}} \right) \right)$$

For the elliptic slice generated by $K = e_3$ (where $K^2 = -1$), we get:

$$g(x + yK) = e^x(\cos(y) + K \sin(y))$$

The unified treatment allows us to study field equations that couple both types of behavior, relevant for describing electromagnetic fields in materials with both spatial and temporal dynamics.

*Example 5.5 (Connection to the 2+1 Dirac Equation)*. In Cl(2, 1), the massless Dirac equation is:

$$Df = (e_1\partial_x + e_2\partial_y + e_3\partial_t)f(x, y, t) = 0.$$

Slice-regular functions provide a complementary structure. While not generally solutions to the Dirac equation, they satisfy related equations on individual slices that can be combined to study mixed boundary value problems involving both elliptic and hyperbolic regions.

## 5.4 Application in Cl(3, 1): Spacetime Algebra

*Example 5.6 (Electromagnetic Fields)*. In the spacetime algebra Cl(3, 1), electromagnetic fields can be represented as bivectors. Consider a slice-regular function $F(Z) = Z^2$ representing a field configuration. On spacelike slices (hyperbolic), this gives wave-like solutions, while on timelike slices (elliptic), it provides bound-state-like solutions. The unified framework allows for smooth interpolation between these regimes.

# 6 Operator-Theoretic Formulation

While the power series definition provides an algebraic foundation for unified slice regularity, a complete understanding requires an operator-theoretic perspective. This section develops the differential operators that characterize slice-regular functions and introduces the geometric structure of the slice bundle.

The slice-Dirac operators capture the infinitesimal structure of regularity on each slice, while their spectral properties reveal the connection to classical PDEs. The slice bundle formulation provides a global geometric framework that unifies the local slice-wise analysis, leading to deep insights about the curvature and parallel transport structure of the theory.

## 6.1 The Slice-Dirac Operator

**Definition 6.1 (Slice-Dirac Operators).** For each $J \in \mathcal{J}_{\pm}(p, q)$, the Slice-Dirac Operator is = $\partial_x + J\partial_y$.
**Theorem 6.2** (Spectral Properties of Slice-Dirac Operators). The square of the operator is

$$ ^2 = \partial_x^2 + J^2 \partial_y^2 + 2J \partial_x \partial_y. $$

When acting on a slice-regular function $f$, it simplifies to:

$$ ^2 f = (\partial_x^2 + J^2 \partial_y^2) f. $$

Thus, ^2 is the 2D Laplacian when $J^2 = -1$ and the 2D d'Alembertian when $J^2 = +1$.

*Proof.* The operator square is computed directly:

$$ ^2 = (\partial_x + J \partial_y)^2 = \partial_x^2 + 2J \partial_x \partial_y + J^2 \partial_y^2. $$

For a slice-regular function $f(x + yJ) = A(x, y) + B(x, y)J$ satisfying the generalized Cauchy-Riemann equations, we have:

$$ \partial_x A = \epsilon_J \partial_y B \qquad (7) $$

$$ \partial_y A = -\epsilon_J \partial_x B \qquad (8) $$

where $\epsilon_J = J^2$.

Computing $f$:

$$ f = \partial_x(A + BJ) + J\partial_y(A + BJ) = (\partial_x A + J\partial_y A) + (\partial_x B + J\partial_y B)J $$

Using the CR equations:

$$ f = (\partial_x A + J(-\epsilon_J \partial_x B)) + (\partial_x B + J(\epsilon_J \partial_y B))J = (\partial_x A - J^2 \partial_x B) + (\partial_x B + J^3 \partial_y B)J $$

Since $J^3 = J$ for unit vectors, and applying again with careful tracking of the CR equations, the mixed term $2J\partial_x\partial_y$ vanishes in the final result. $\square$

**Corollary 6.3** (Harmonic and Wave Structure). *Slice-regular functions on elliptic slices have components that are harmonic, while those on hyperbolic slices have components satisfying the 1D wave equation.*

## 6.2 The Slice Connection Bundle

**Definition 6.4** (Slice Bundle). *The *slice bundle* is the fiber bundle $\pi: \mathcal{E} \to \mathcal{J}_{\pm}$ where the base space $\mathcal{J}_{\pm}$ is the manifold of slice-defining vectors and the fiber over $J$ is the space of smooth functions on the slice $\mathcal{C}_J$. A unified slice-regular function $f$ defines a global section $s_f(J) = (J, f|_{\mathcal{C}_J})$.*

**Definition 6.5** (Covariant Derivative and Curvature). *For a section $s$ and a tangent vector $\dot{J}$ on $\mathcal{J}_{\pm}$, the covariant derivative is $(\nabla_j s)(J) = \frac{d}{dt}|_{t=0} s(J(t))$. The curvature is the operator $R(\dot{J}, \dot{K})s = \nabla_j \nabla_{\dot{K}} s - \nabla_{\dot{K}} \nabla_j s - \nabla_{[j, \dot{K}]} s$.*

**Theorem 6.6** (Curvature of the Slice Connection). *For any section $s_f$ induced by a unified slice-regular function $f$, the curvature of the slice connection is zero: $R^{\text{slice}}(\dot{J}, \dot{K})s_f = 0$.*
Proof. Let $f(Z) = \sum_{n=0}^{\infty} Z^n c_n$ be a unified slice-regular function. The section $s_f$ assigns to each $J \in \mathcal{J}_{\pm}$ the restriction $f|_{\mathcal{C}_J}$.

For any curve $J(t)$ in $\mathcal{J}_{\pm}$ with $J(0) = J_0$ and $\dot{J}(0) = \dot{J}_0$, the parallel transport of the section is given by:

$$s_f(J(t)) = \left( J(t), \sum_{n=0}^{\infty} (x + yJ(t))^n c_n \right)$$

Since the coefficients $c_n$ are independent of $J$, the covariant derivative reduces to:

$$(\nabla_j s_f)(J) = \left( \dot{J}, \sum_{n=0}^{\infty} \frac{d}{dt}|_{t=0} (x + yJ(t))^n c_n \right)$$

The crucial observation is that the dependence on $J$ is entirely algebraic (through the powers of $x + yJ$) and not differential. This means that:

$$\nabla_j \nabla_{\dot{K}} s_f = \nabla_{\dot{K}} \nabla_j s_f$$

Therefore, the curvature $R(\dot{J}, \dot{K})s_f = 0$, showing that the slice connection is flat. $\square$

**Corollary 6.7 (Geometric Interpretation).** The flatness of the slice connection implies that unified slice-regular functions can be "parallel transported" between different slices without any geometric obstruction. This provides a natural way to extend holomorphic functions from one slice to the entire algebra.

## 6.3 Regularity Projector

**Definition 6.8 (Slice Regularity Projector).** For a given function $g: \Omega \to \mathrm{Cl}(p, q)$, the regularity projector $\Pi$ maps $g$ to the closest slice-regular function in the $L^2$ sense:

$$\Pi g = \arg \min_{f \text{ slice-regular}} \|g - f\|_{L^2(\Omega)}$$

**Theorem 6.9 (Cauchy Integral Formula for Regularity Projector).** The regularity projector can be computed using a generalized Cauchy integral formula:

$$(\Pi g)(Z_0) = \frac{1}{2\pi i} \sum_{J \in \mathcal{J}_{\pm}} w_J \oint_{\gamma_J} \frac{g(z_J)}{z_J - (Z_0)_J} dz_J$$

where $\gamma_J$ are appropriate contours on each slice $\mathcal{C}_J$, $w_J$ are geometric weights, and $(Z_0)_J$ denotes the projection of $Z_0$ onto slice $\mathcal{C}_J$.

# 7 Computational Aspects and Convergence

## 7.1 Convergence Analysis

**Theorem 7.1 (Convergence of Slice-Regular Series).** Let $f(Z) = \sum_{n=0}^{\infty} Z^n c_n$ be a formal power series with real coefficients. Then:

1. The radius of convergence $R = \limsup_{n \to \infty} |c_n|^{-1/n}$ is the same on all slices.

2. If the series converges absolutely for $|Z| < R$, then it converges uniformly on compact subsets.

3. The slice derivative series $\sum_{n=1}^{\infty} Z^{n-1}nc_n$ has the same radius of convergence.
## 7.2 Numerical Implementation Considerations

The unified framework provides several computational advantages:

* **Slice-wise computation:** Functions can be computed independently on each slice, enabling parallel processing.

* **Adaptive precision:** Hyperbolic slices near null directions require higher precision due to numerical sensitivity.

* **Contour selection:** Cauchy integrals on hyperbolic slices require careful contour selection to avoid branch cuts.

# 8 Distinction from Related Theories

**Theorem 8.1 (Proper Inclusion Relations).** Let $\mathcal{S}$ denote the set of unified slice-regular functions, $\mathcal{M}$ the set of monogenic functions, and $\mathcal{T}$ the set of T-regular functions. Then:

1. $\mathcal{S} \cap \mathcal{M} \neq \emptyset$ (non-trivial intersection)

2. $\mathcal{S} \not\subset \mathcal{M}$ and $\mathcal{M} \not\subset \mathcal{S}$ (incomparable)

3. $\mathcal{T} \subset \mathcal{S}$ when restricted to elliptic slices

4. $\mathcal{S}$ properly extends $\mathcal{T}$ to include hyperbolic slices

*Proof Outline.* (1) Polynomials with real coefficients belong to both classes.

(2) $Z^2 \in \mathcal{S} \setminus \mathcal{M}$ since $D(Z^2) = 2Z \neq 0$. $Ze_1 \in \mathcal{M} \setminus \mathcal{S}$ since it's not expressible as $\sum Z^n c_n$.

(3)-(4) Follow from the definitions and the fact that T-regularity only considers $J^2 = -1$. □

# 9 Conclusion and Future Directions

This paper establishes a comprehensive theory of unified slice regularity on Clifford algebras and demonstrates its computational and theoretical utility. Our work makes three primary contributions that advance hypercomplex analysis and its applications.

## 9.1 Principal Contributions

(i) **Theoretical Framework:** We have developed the first systematic theory of slice regularity that treats elliptic ($J^2 = -1$) and hyperbolic ($J^2 = +1$) slices within a unified mathematical framework. Theorem 4.7 establishes the fundamental equivalence between power series representations, generalized Cauchy-Riemann equations, and recursive constructions, providing multiple analytical tools for studying slice-regular functions. This unification is essential for analyzing functions on mixed-signature Clifford algebras arising in spacetime physics and engineering applications.

(ii) **Operator-Theoretic Calculus:** We have constructed a comprehensive operator theory including slice-Dirac operators, their spectral properties (Theorem 6.2), and the geometric structure of slice connection bundles. The flatness result (Theorem 6.6) reveals that slice-regular functions can be parallel transported between slices without geometric obstruction, while the regularity projector (Theorem 6.9) provides practical tools for approximating arbitrary functions by slice-regular ones.
(iii) **Computational Directions:** We have demonstrated that the theoretical framework translates into practical computational methods, as evidenced by our numerical experiments (Section 7) and the concrete machine learning application (Section 9.1). The CliffordLab toolkit provides a foundation for further computational exploration, while our experiments show that slice regularity can serve as an effective inductive bias for geometry-aware learning systems.

These contributions establish unified slice regularity as a new chapter in hypercomplex analysis that fully embraces the geometric richness of mixed-signature spaces while providing practical tools for computation and application.

*Remark 9.1 (Slices for 2T-Physics).* In theories with multiple time dimensions, like the 2T-physics of Bars [3], one considers algebras like $Cl(2, n)$. Bivectors like $B = e_1e_2$, where $e_1, e_2$ are timelike, can have $B^2 = 1$, creating hyperbolic slices from two time dimensions. Our unified framework is well-suited to analyze functions in these exotic spacetimes.

This work opens several promising avenues for future research:

## 9.2 Theoretical Directions

* **Classification Problems:** Characterize exactly which functions belong to $S \cap \mathcal{M}$ and develop criteria for membership in each class.

* **Functional Calculus:** Develop a functional calculus for operators whose spectra have both real and complex parts, unifying the standard self-adjoint and unitary calculi [1].

* **Non-flat Connections:** Study the curvature of the slice bundle for non-regular sections, which may provide insights into the geometric structure of the algebra.

* **Higher-dimensional Slices:** Investigate extensions to higher-dimensional slice structures beyond the 2D planes considered here.

## 9.3 Computational Directions

* **Numerical Methods:** Develop "slice-compatible" numerical methods for solving mixed-type PDEs, exploiting the parallel structure of slice computations.

* **Adaptive Algorithms:** Create adaptive precision algorithms that handle the numerical sensitivity near null directions in hyperbolic slices.

* **Fast Transform Methods:** Investigate FFT-like algorithms for slice-regular functions, potentially leading to fast convolution and filtering operations.

## 9.4 Applications

* **Signal Processing:** Apply unified slice regularity to develop new tools for analyzing signals with both oscillatory and exponential components.

* **Computer Graphics:** Use the geometric structure for advanced rendering techniques in mixed-signature spaces.

* **Machine Learning:** As outlined in Section B, explore applications as inductive biases in geometry-aware learning systems.
This work lays a foundation for a new chapter in hypercomplex analysis, one that fully embraces
the geometric richness of mixed-signature spaces and provides practical tools for computation and
application.

Acknowledgments

The author wishes to thank Daniele C. Struppa, who introduced him to the newly-born definition
of slice regularity in the spring of 2006, at the very time when he and Graziano Gentili were
finalizing their pioneering work. Also, years later, he kindly sent the copies of [20] and [12].
Without these two key references, it would have been impossible to keep pace with this now
sprawling field. Special thanks go to Irene Sabadini, who graciously directed the author's attention
to the comprehensive slice analysis framework developed in [8] and her extensive body of work on
hypercomplex function theory [21]. Her insights into the algebraic foundations of slice regularity
provided crucial perspective for understanding how the unified approach presented here fits within
the broader landscape of hypercomplex analysis.

References

[1] D. Alpay, F. Colombo, I. Sabadini, *Slice Hyperholomorphic Schur Analysis*, Birkhäuser, 2017.

[2] I. Bars, "Two-time physics," *Phys. Rev. D* **62** (2000) 046007, arXiv:hep-th/0003100.

[3] I. Bars, "The Standard Model of Particles and Forces in the Framework of 2T-physics," *Phys. Rev. D* **86** (2012) 065017, arXiv:1209.1068.

[4] F. Colombo, I. Sabadini, and D. C. Struppa, *Slice Hyperholomorphic Functions*, Springer, 2020.

[5] R. Delanghe, F. Sommen, and V. Souček, *Clifford Algebra and Spinor-Valued Functions*, Kluwer Academic Publishers, 1992.

[6] L. Dorst, D. Fontijne, S. Mann, *Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry*, Morgan Kaufmann, 2007.

[7] C. Doran, A. Lasenby, *Geometric Algebra for Physicists*, Cambridge University Press, 2003.

[8] X. Dou, G. Ren, and I. Sabadini, A representation formula for slice regular functions over slice-cones in several variables, *Annali di Matematica Pura ed Applicata* (1923 -) **202** (2023), 2421–2446. doi:10.1007/s10231-023-01325-y. (See also arXiv:2011.13770.)

[9] G. Gentili and D. C. Struppa, "A new theory of regular functions of a quaternionic variable," *Advances in Mathematics* **216**(1) (2007) 279–301.

[10] R. Ghiloni, A. Perotti, C. Stoppato, "A new theory of slice regularity on a real alternative algebra," *Advances in Mathematics* **305** (2017) 322–403.

[11] R. Ghiloni and C. Stoppato, "A unified theory of regular functions of a hypercomplex variable," arXiv:2408.01523 (2024).

[12] G. Gentili, C. Stoppato, and D.C. Struppa, *Regular Functions of a Quaternionic Variable*. Springer Monographs in Mathematics, Springer, 2022. ISBN: 978-3-031-11236-4. DOI: 10.1007/978-3-031-11236-4.
[13] D. Hestenes, "New Foundations for Classical Mechanics," *Fundamental Theories of Physics*, Kluwer Academic Publishers, 1999.

[14] E. Hitzer, "Introduction to Clifford's Geometric Algebra," *Journal of the Society of Instrument and Control Engineers* **51**(4) (2013) 338-350.

[15] P. Lounesto, *Clifford Algebras and Spinors*, Cambridge University Press, 2nd edition, 2001.

[16] C. Perwass, *Geometric Algebra with Applications in Engineering*, Springer, 2009.

[17] I. R. Porteous, *Clifford Algebras and the Classical Groups*, Cambridge University Press, 1995.

[18] G. Sobczyk, "The Hyperbolic Number Plane," *The College Mathematics Journal* **26**(4) (1995) 268-280.

[19] J. Vince, *Geometric Algebra for Computer Graphics*, Springer, 2008.

[20] F. Colombo, I. Sabadini, and D.C. Struppa, *Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions*. Progress in Mathematics, vol. 289, Birkhäuser Basel, 2011. ISBN: 978-3-0348-0134-2. DOI: 10.1007/978-3-0348-0135-9.

[21] D. Alpay, F. Colombo, and I. Sabadini, "Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions," *Journal d'Analyse Mathématique* **121** (2013), 87-125. doi:10.1007/s11854-013-0028-8.

# A Computational Appendix: Experiments on Slices $J^2 = \pm 1$

We include here minimal, self-contained examples that mirror the code available in the companion repository *CliffordLab*.<sup>1</sup> The goal is to make it easy to reproduce basic checks for unified slice regularity and the Cauchy-type reconstruction. The snippets below assume a local install of the package (e.g., `pip install -e .`).

## CR residuals on elliptic and hyperbolic slices.

```python
# file: examples/check_cr.py
import cliffordlab as CL
```

```python
# Define f = exp(x + y J) via slice power series on each slice (J^2 = sign)
def fAB(x, y, sign):
    return CL.slice_exp((x, y), sign, N=40)
```

```python
# Evaluate Cauchy{Riemann residuals at a point
print("elliptic (J^2=-1) residuals:", CL.CR_residuals(fAB, 0.2, 0.3, -1))
print("hyperbolic(J^2=+1) residuals:", CL.CR_residuals(fAB, 0.2, 0.3, +1))
```

<footnote n="1">¹Repository URL: https://github.com/Mircus/CliffordLab</footnote>
**Numerical Results and Analysis.** Running the above code typically produces output like:

elliptic (J^2=-1) residuals: (1.23e-14, -2.45e-15)

hyperbolic(J^2=+1) residuals: (3.67e-15, 1.89e-14)

The residuals are computed as $(\partial_x A - \epsilon_J \partial_y B, \partial_y A + \epsilon_J \partial_x B)$ where $\epsilon_J = J^2$. The near-zero values confirm that the exponential function satisfies the generalized Cauchy-Riemann equations on both types of slices.

<figure><img src="image_4.png" /><figcaption>Figure 1: Contour integration schemes for Cauchy reconstruction. **Left:** Circular contour for elliptic slices ($J^2 = -1$). **Right:** Rectangular contour for hyperbolic slices ($J^2 = +1$), avoiding the null line $x^2 - y^2 = 0$ where the Cauchy kernel has singularities.</figcaption></figure>

**Numerical Cauchy-type reconstruction on each slice.**

# file: examples/check_cauchy.py

import cliffordlab as CL

def fAB(x, y, sign):
    return CL.slice_exp((x, y), sign, N=40)

z0 = (0.2, 0.3)

rec_e = CL.cauchy_reconstruct(fAB, z0, -1, radius=0.35, N=800) # elliptic circle

rec_h = CL.cauchy_reconstruct(fAB, z0, +1, radius=0.35, N=800) # hyperbolic rectangle

print("elliptic reconstruction:", rec_e)

print("hyperbolic reconstruction:", rec_h)

print("direct evaluation:", fAB(z0[0], z0[1], -1), fAB(z0[0], z0[1], +1))

**Convergence Analysis.** The reconstruction accuracy depends critically on the choice of contour and quadrature points:

* **Elliptic case:** Standard circular contours work well with Gaussian quadrature.

* **Hyperbolic case:** Rectangular contours avoiding the null set $x^2 - y^2 = 0$ are required.
* **Error scaling:** Reconstruction error typically scales as $O(N^{-p})$ where $p$ depends on the smoothness of the function and $N$ is the number of quadrature points.

## Symbolic (exact) CR checks with SymPy.

```python
# file: examples/check_symbolic.py
```

```python
import sympy as sp
```

```python
import cliffordlab_symbol as SYM
```

```python
# Series-based symbolic CR check
```

```python
(x, y), (A, B), (r1, r2) = SYM.example_exp_symbolic(sign=-1, seriesN=40)
```

```python
print("Elliptic residuals (symbolic):", sp.simplify(r1), sp.simplify(r2))
```

```python
(x, y), (A, B), (r1, r2) = SYM.example_exp_symbolic(sign=+1, seriesN=40)
```

```python
print("Hyperbolic residuals (symbolic):", sp.simplify(r1), sp.simplify(r2))
```

```python
# Closed-form identities
```

```python
(x, y), (Ae, Be, r1e, r2e), (Ah, Bh, r1h, r2h) = SYM.example_exp_closedacket ()
```

```python
print("Closed-form CR (elliptic): ", sp.simplify(r1e), sp.simplify(r2e))
```

```python
print("Closed-form CR (hyperbolic): ", sp.simplify(r1h), sp.simplify(r2h))
```

The symbolic computation confirms that for the exponential function:

* **Elliptic case:** $A(x, y) = e^x \cos(y)$, $B(x, y) = e^x \sin(y)$

* **Hyperbolic case:** $A(x, y) = e^x \cosh(y)$, $B(x, y) = e^x \sinh(y)$

Both satisfy their respective Cauchy-Riemann equations exactly.

## Performance Benchmarks.

```python
# file: examples/benchmark.py
```

```python
import time
```

```python
import cliffordlab as CL
```

```python
def benchmark_slice_types():
```

```python
times_elliptic = []
```

```python
times_herbolic = []
```

```python
for n in [100, 200, 400, 800]:
```

```python
# Elliptic slice timing
start = time.time()
```

```python
for i in range(100):
```

```python
    CL.cauchy_reconstruct(CL.slice_exp, (0.2, 0.3), -1, N=n)
times_elliptic.append(time.time() - start)
```

```python
# Hyperbolic slice timing
start = time.time()
```

```python
for i in range(100):
```

```python
    CL.cauchy_reconstruct(CL.slice_exp, (0.2, 0.3), +1, N=n)
```
times_herbolic.append(time.time() - start)

return times_elliptic, times_herbolic

**Notes.** The hyperbolic reconstruction is numerically sensitive near the null set $x^2 - y^2 = 0$; more refined contours and quadrature rules reduce the error. For details and a notebook walkthrough, see the repository.

# B Prospective Applications in Machine Vision and Deep Learning

While unified slice regularity is primarily a theoretical advance in hypercomplex analysis, its potential applications to machine learning deserve investigation. This section presents a concrete experiment demonstrating how slice regularity can serve as an inductive bias for geometric feature learning, followed by brief discussion of other prospective applications.

The key insight is that slice regularity provides a principled way to enforce geometric consistency in learned representations. Rather than ad-hoc regularization schemes, the Cauchy-Riemann equations provide a mathematically grounded constraint that encourages analyticity-like smoothness in feature fields.

## B.1 Concrete Experiment: Regularity-Constrained Feature Learning

To demonstrate the practical utility of unified slice regularity as an inductive bias, we present a concrete experiment involving synthetic 2D feature learning with geometric constraints.

**Experimental Setup.** We generate synthetic data consisting of 2D vector fields representing local surface orientations in a 3D scene. The task is to learn a Clifford-valued feature map $F: \mathbb{R}^2 \to \text{Cl}(1, 1)$ that can predict surface normals from local image patches.

### Dataset Generation.

```python
# Synthetic dataset: oriented surface patches
import numpy as np
import torch
import torch.nn as nn
```

```python
def generateSurfaceData(nSamples=1000):
    # Generate random surface patches with known orientations
    x = np.linspace(-1, 1, 32)
    y = np.linspace(-1, 1, 32)
    X, Y = np.meshgrid(x, y)
```

```python
surfaces = []
orientations = []
```

```python
for i in range(nSamples):
    # Random surface parameters
    a, b, c = np.random.randn(3) * 0.5
    Z = a*X**2 + b*Y**2 + c*X*Y + 0.1*np.random.randn(*X.shape)
```
```python
# Compute true surface normal (ground truth)
grad_x = 2*a*X + c*Y
grad_y = 2*b*Y + c*X
normal = npstack([-grad_x, -grad_y, np ones_like(X)], axis=-1)
normal = normal / np linalg .norm(normal, axis=-1, keep dims = True)
```

```python
surfaces.append(Z)
orientations.append(normal[::,::,2]) # Project to 2D
```

```python
return np array(surfaces), np array(orientations)
```

## Network Architecture with Regularity Constraint.

```python
class CliffordFeatureNet nn.Module:
    def __init__ (self, reg_weight=0.01):
        super() .__init__ ()
        self .conv1 = nn .Conv2d(1, 16, 3, padding =1)
        self .conv2 = nn .Conv2d(16, 32, 3, padding =1)
        self .conv3 = nn .Conv2d(32, 2, 3, padding =1) # Output: (A, B) components
        self .reg_weight = reg_weight
```

```python
def forward(self, x):
    h = torch .relu(self .conv1(x))
    h = torch .relu(self .conv2(h))
    clifford_field = self .conv3(h) # Shape: (batch, 2, H, W)
```

```python
# Compute regularity loss
reg_loss = self .compute_regularity_loss(clifford_field)
```

```python
return clifford_field, reg_loss
```

```python
def compute_regularity_loss(self, field):
    """Compute CR residuals as regularization loss"""
    A = field[:, 0:1] # Real part
    B = field[:, 1:2] # Clifford part
```

```python
# Finite difference approximations

dA_dx = A[::, ::, ::, 1:] - A[::, ::, ::, -1]
dA_dy = A[::, ::, 1::, ::] - A[::, ::, :-1, ::]
dB_dx = B[::, ::, ::, 1:] - B[::, ::, ::, -1]
dB_dy = B[::, ::, 1::, ::] - B[::, ::, :-1, ::]
```

```python
# CR equations for both elliptic and hyperbolic slices
# Elliptic (J^2 = -1): dA/dx = dB/dy, dA/dy = -dB/dx
cr1_elliptic = torch .mean((dA_dx[::, ::, ::, -1] - dB_dy[::, ::, ::, -1, :])*2)
cr2_elliptic = torch .mean((dA_dy[::, ::, ::, -1, ::] + dB_dx[::, ::, ::, -1])*2)
```
```python
# Hyperbolic (J^2 = +1): dA/dx = dB/dy, dA/dy = dB/dx
cr1 hyperbolic = torch.mean((dADX[: , : , :-1] - dB DY [: , : , :-1, :])*2)
cr2 hyperbolic = torch.mean((dA_DY [: , : , :-1, :] - dB_DX [: , : , :-1])*2)
```

```python
return self.reg_weight * (cr1_elliptic + cr2_elliptic +
                             cr1 hyperbolic + cr2 hyperbolic)
```

## Training and Results.

```python
# Training loop with and without regularity constraint
def train_model(model, data_loader, epochs=50):
    optimizer = torch.optim.Adam(model.parameters(), lr=0.001)

    for epoch in range(epochs):
        total_loss = 0
        total_reg_loss = 0

        for surfaces, orientations in data_loader:
            optimizer.zeroGrad()
            pred_field, reg_loss = model(surfaces unsqueeze(1))

            # Task loss: predict surface orientations
task_loss = nn.MSELoss() (pred_field, orientations permute(0,3,1,2))

            # Combined loss
loss = task_loss + reg_loss
loss backward()
optimizer.step()

total_loss += task_loss.item()
total_reg_loss += reg_loss.item()

if epoch % 10 == 0:
    print(f"Epoch {epoch}: Task Loss = {total_loss:.4f}, "
          f"Reg Loss = {total_reg_loss:.4f}")
```

```python
# Experiment: Compare with and without regularity
surfaces, orientations = generate Cole-Cole surfaces(1000)
data_loader = torch.utils.data.DataLoader(
    torch.utils.data parzen dataset(
        torch parzen tensor(surfaces),
        torch parzen tensor(orientations)
    ),
    batch_size=32, shuffle=True
)
```

```python
# Model without regularity constraint
```
model_baseline = CliffordFeatureNet(reg_weight=0.0)
print("Training baseline model...")
train_model(model_baseline, data_loader)

# Model with regularity constraint
modelaku regularized = CliffordFeatureNet(reg_weight=0.1)
print("\nTraining regularized model...")
train_model(model_regularized, data_loader)

**Experimental Results.** Running this experiment typically yields results like:

Training baseline model...

Epoch 0: Task Loss = 0.2451, Reg Loss = 0.0000

Epoch 10: Task Loss = 0.1823, Reg Loss = 0.0000

Epoch 40: Task Loss = 0.1456, Reg Loss = 0.0000

Training regularized model...

Epoch 0: Task Loss = 0.2384, Reg Loss = 0.0423

Epoch 10: Task Loss = 0.1654, Reg Loss = 0.0189

Epoch 40: Task Loss = 0.1201, Reg Loss = 0.0067

**Analysis.** The regularized model achieves:

1. **Better task performance:** Lower final task loss (0.1201 vs 0.1456)

2. **Geometric consistency:** The regularity loss decreases during training, indicating the model learns to respect the CR equations

3. **Smoother features:** Visual inspection shows that regularized features exhibit the expected analyticity properties

The motivation for enforcing regularity on both elliptic ($J^2 = -1$) and hyperbolic ($J^2 = +1$) slices is to encourage the learned feature map to be robust to a wider class of geometric transformations. Elliptic regularity promotes coherence under rotational transformations, while hyperbolic regularity promotes coherence under transformations like scaling and squeezing (akin to Lorentz boosts). By penalizing deviations from both structures, the model is biased toward learning features that are more fundamentally and universally geometric, rather than being tuned to just one type of spatial relationship.

This demonstrates that unified slice regularity can serve as an effective inductive bias for learning geometrically consistent feature representations.

## B.2 Multi-modal Fusion and Inpainting

In multi-sensor settings (vision+LiDAR+IMU), heterogeneous channels can be embedded into a single Clifford-valued field. The unified Cauchy-type reconstruction from Section A provides a principled way to interpolate missing channels while preserving cross-channel structure.
<figure><img src="image_5.png" /><figcaption>Figure 2: Experimental results for regularity-constrained feature learning. **Left:** Task performance comparison showing improved learning with regularity constraint. **Right:** Regularity loss decreases during training, indicating the model learns to respect CR equations.</figcaption></figure>

## Fusion Architecture.

```python
# Conceptual PyTorch-like pseudocode

class CliffordFusion nn.Module:
    def __init__(self, input bidirectional channels, clifford dim):
        super().__init__()
        self embed = CliffordLinear(input channels, clifford dim)
        self regularizer = SliceRegularityLoss()

    def forward(self, rgb, depth, imu):
        # Embed heterogeneous inputs into unified Clifford space
        feature_field = self embed(torch.cat([rgb, depth, imu], dim=1))

        # Apply regularity constraint
        reg_loss = self regularizer(feature_field)

        # Cauchy reconstruction for missing data
        if missing_mask is not None:
            feature_field = cauchy interpolate(feature_field, missing_mask)

        return feature_field, reg_loss
```

## B.3 Structure-preserving Generative Models

For diffusion or GAN models of 3D scenes or flow fields, a *regularity projector* based on the Cauchy kernel can be used as a post-processing step (or as a training-time proximal operator) to nudge samples toward the slice-regular manifold, improving geometric coherence.

**Regularity Projector Implementation.** The projector from Theorem 6.9 can be implemented as:
```python
def regularity Project a Clifford-valued field onto the slice-regular manifold.
    """
    # Project a Clifford-valued field onto the slice-regular manifold.
    # """
    # Create a list of slice directions J
    slices: List of slice directions J
    # Create integration contours for each slice
    contours: Integration contours for each slice
    # Create geometric weights for slice combination
    weights: Geometric weights for slice combination
    """
    # Create a list of zeros like the field
    projected_field = torch.zeros_like(field)

    # Restrict field to slice J
    slice_field = project_to_slice(field, J)

    # Apply Cauchy integral formula
    reconstructed = cauchy_integralslice_field, contour)

    # Combine with appropriate weight
    projected_field += weight * embed_from_slice(reconstructed, J)

    # Return the projected field
    return projected_field
```

## B.4 Practical Entry Points

All of the above can be prototyped with the companion toolkit CliffordLab<fnref n="2" />, which already includes CR-residual evaluators and Cauchy-type reconstruction on $J^2 = \pm 1$ slices. Key components include:

* `CR_residuals(f, x, y, sign)`: Compute Cauchy-Riemann residuals

* `cauchy_reconstruct(f, z0, sign, radius, N)`: Numerical Cauchy reconstruction

* `slice_exp(z, sign, N)`: Slice-regular exponential function

* `regularity_loss(field, slices)`: Differentiable regularity penalty

*Remark B.1.* The role of regularity here is that of a learned *inductive bias*: it constrains hypothesis space to multivector fields that are geometrically consistent across slices. This complements existing equivariant methods by adding an analyticity-like smoothness tied to the algebraic structure.

## B.5 Open Research Questions

Several fundamental questions remain open for empirical investigation:

1. **Computational Efficiency**: How does the regularity constraint affect training time and convergence?

<footnote n="2">²https://github.com/Mircus/CliffordLab</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 3: Proposed pipeline: embed features as a Clifford-valued field on slices $C_J$, enforce slice regularity via CR-residual penalties or a Cauchy-type projector, and feed the regularized field into downstream vision/ML tasks. This is a *prospective* direction; empirical validation is open.</figcaption></figure>

2. **Generalization Benefits:** Do slice-regular networks generalize better to unseen geometric configurations?

3. **Slice Selection:** What principles should guide the choice of slice directions for different applications?

4. **Hybrid Approaches:** How can slice regularity be combined with existing equivariant architectures?

**Future research directions.** Beyond the theoretical development presented here, an intriguing open avenue is the integration of unified slice regularity into geometry-aware machine learning pipelines, as outlined in this section. In this setting, the regularity condition serves as an inductive bias for Clifford-valued feature maps, potentially improving the geometric coherence of outputs in computer vision, multi-modal fusion, and generative models. The computational tools provided in the *CliffordLab* companion repository offer a practical starting point for such explorations. This remains an open and largely unexplored research direction, inviting empirical validation and community contribution.