Relativistic Algebra over Finite Fields

1. Introduction

A growing body of work in mathematics and physics suggests that foundational structures are best understood through a relational or relativistic lens [1,2,3]. In such a paradigm, mathematical entities acquire meaning not as intrinsic absolutes but through their role within a system defined by internal symmetries and reference frames. Constants like 0, 1, or i are not metaphysical primitives, but relational markers—origins, units, or axes—assigned by a chosen framing.

This perspective invites a re-evaluation of one of the most entrenched assumptions in mathematics: the acceptance of actual infinity. From real analysis to Hilbert spaces, infinity has been treated as foundational, despite its lack of empirical or computational realization. Under a relational view, such constructs may instead be interpreted as emergent limits or symbolic artifacts—arising when finite systems attempt to encode relationships that exceed their internal scope.

In previous work [4], we argued that concepts like infinity, randomness, and undecidability are not ontological features of nature, but epistemic placeholders—signals of representational saturation in finite informational systems. Here, we extend this view into a concrete formalism: a relativistic algebra constructed entirely over a finite field ${\mathbb{F}}_p$, with observer-relative arithmetic and emergent pseudo-numbers.

The present framework resonates with several contemporary perspectives that question the ontological status of the continuum and advocate for finitely constructed alternatives. In particular, Smolin has emphasized the need for a relational, observer-dependent formulation of physical laws, suggesting that the continuum is merely an idealization beyond the reach of internal observers [5,6]. Similarly, D’Ariano and collaborators have reconstructed quantum theory from finite, informationally grounded axioms, demonstrating that core features of quantum mechanics can emerge without invoking infinite-dimensional Hilbert spaces [7]. From a mathematical standpoint, the approach aligns with the ultrafinitist program developed by Benci and Di Nasso, which offers a rigorous alternative to classical cardinality through the theory of numerosities and bounded arithmetic [8,9].

Furthermore, the ultrafinitist school—pioneered by Yessenin-Volpin and Parikh—takes finitude even further by denying the meaningful existence of “too large” numbers and insisting on feasibility as a foundational constraint. Formalizations of ultrafinitism and feasibility arithmetic appear in works such as [10,11,12,13], which explore the proof-theoretic and computational consequences of enforcing strict constructive bounds on arithmetic.

Ultrafinitism enforces an a priori cutoff on numerical existence—only those magnitudes deemed “feasible” within a human or machine resource bound are admitted. By contrast, our relativistic framework treats finiteness not as a hard barrier but as a contextual framing condition: We allow arbitrarily large numbers, so “size” is always relative to the chosen frame. Infinite structures, such as integers and rationals emerge asymptotically or as coordinate projections, rather than being forbidden. Arithmetic operations become internal symmetries of a finite system, rather than operations constrained by external feasibility checks. This shift replaces the ultrafinitist’s absolute feasibility threshold with a relational notion of scope: any number “exists” within some finite frame, while “infinity” itself appears as a relative point beyond the horizon of observability and algebraic accessibility.

To support this framework, we further draw upon several key developments in mathematics and physics. The foundational critique of actual infinity has been explored in works such as [14,15], which emphasize the constructive and finitist approaches to mathematics. The relational perspective on mathematical objects aligns with category theory [1], where objects are defined by their morphisms and relationships rather than intrinsic properties. Additionally, the parallels between relativistic mathematics and modern physics are inspired by the symmetry principles in [2,3], which highlight the role of invariance and frame-dependence in physical laws. Finally, the informational limits of finite systems and their implications for mathematical representation are discussed in [16,17].

2. Finite Field Framing

Let ${\mathbb{F}}_p=\{0,1,2,\dots ,p-1\}$ be the finite field of integers modulo an odd prime p. The elements of ${\mathbb{F}}_p$ form a complete and closed set of relational representations of ${\mathbb{F}}_p$ under modular addition, multiplication and exponentiation. However, the specific numeric labels assigned to these elements—particularly the designation of 0 and 1 as the additive and multiplicative identities—are intrinsically relative and carry no absolute meaning within the field itself. The field ${\mathbb{F}}_p$ is invariant under relabeling of its elements via any bijective affine transformation of the form

$$k\mapsto a·k+bmodq,$$

where $a\in {\mathbb{F}}_p^{\times }$ and $b\in {\mathbb{F}}_p$. Such transformations preserve the field structure and allow any element to be reinterpreted as the origin. In this sense, the element labeled 0 is not uniquely privileged; it simply represents the additive identity with respect to a chosen reference frame. The same applies to the label 1, which identifies the multiplicative unit only relative to a particular scaling.

Therefore, in the absence of an externally imposed or contextually declared frame—such as one defined by a designated pair $(0,1)$—the labels in ${\mathbb{F}}_p$ are relational rather than absolute. The roles of “zero” and “one” are thus not the fundamental properties of the elements themselves, but a consequence of the system’s framing, making all representations in ${\mathbb{F}}_p$ symmetric and interchangeable under coordinate transformation. To define our system unambiguously, we must specify a reference frame or coordinate system $(0,1)$ within the context of ${\mathbb{F}}_p$, which then becomes a framed finite ring${\mathbb{F}}_p(0,1)$. We will henceforth assume all such systems to be framed systems ${\mathbb{F}}_p(0,1)$ and will denote the corresponding finite ring as ${\mathbb{F}}_p$ for simplicity, unless explicitly stated otherwise.

3. Finite Field as Discrete Geometric Structure

Let p be an odd prime and let ${\mathbb{F}}_p$ denote the finite field with p elements [18]. The additive group $({\mathbb{F}}_p,+)$ is a cyclic group of order p, and the multiplicative group of nonzero elements $({\mathbb{F}}_p^{\times },·)$ is a cyclic group of order $p-1$ [19]. We associate the cardinality degree of freedom p and the three fundamental arithmetic operations with 4 distinct symmetry classes in a symbolic geometry as in [20]:

• Counting — defines the number of elements in the ring.
• Addition$(+)$ — defines rotational symmetry on a linear periodic axis.
• Multiplication$(\times )$ — defines scaling symmetry on a multiplicative periodic axis.
• Exponentiation — defines cyclic phase-like symmetry from repeated powers of a generator [21].

The choice of cardinality itself defines a linear—radial—degree of translation, and each cyclic operation corresponds to a spherical axis of rotational transformation in a four-dimensional abstract symmetry space. For a fixed odd prime p, the described mathematical construct forms the geometric scaffold of a discrete spheroidal system. The three spherical axes are mutually orthogonal, but algebraically dependent forming a 2D spheroid in the 4D symmetry space.

The resultant 2D spheroid for ${\mathbb{F}}_{13}$ is depicted in Figure 1, where the prime meridian depicts the additive group $({\mathbb{F}}_{13},+)$ and the latitudes represent multiplicative group $({\mathbb{F}}_{13}^{\times },·)$ generated by the minimum multiplicative generator $g_{min}=2$ [20].

4. Pseudo-Numbers

4.1. Pseudo-Integers

Define a mapping $k:\mathbb{Z}\to {\mathbb{F}}_p$, with $k\left(z\right)=zmodp$. This wraps $\mathbb{Z}$ onto ${\mathbb{F}}_p$. The observer, located at 0 and bounded by horizon $H\ll p$, perceives the wrapped axis as infinite. Thus, the apparent integer line emerges as a pseudo-integer class $\mathbb{Z}/{\mathbb{F}}_p$, where negation, order, and comparison are reconstructed locally [22].

The resulting class of relativistic pseudo-integers $\mathbb{Z}/{\mathbb{F}}_p$ exhibits all the characteristic properties of the conventional integer set $\mathbb{Z}$, including sign, order, addition, subtraction and multiplication. This framework allows us to recover the intuitive and logical structure of integers — including signed quantities and magnitude comparison — entirely within the finite, self-contained system ${\mathbb{F}}_p$, while preserving consistency with its modular arithmetic.

Figure 2. Class of signed pseudo-integers $\mathbb{Z}$ over the finite framed field ${\mathbb{Z}}_{13}$. Black labels indicate the newly defined signed integers $z\in \mathbb{Z}$, while the purple labels represent the corresponding elements $k\left(z\right)\in {\mathbb{Z}}_{13}$. The blue line indicates the periodicity of the finite field. The unlabeled gray dots indicate the off-axis elements of ${\mathbb{Z}}_{13}$ as they are observed from the top of the 2D spheroid described in Figure 1.

Figure 2. Class of signed pseudo-integers $\mathbb{Z}$ over the finite framed field ${\mathbb{Z}}_{13}$. Black labels indicate the newly defined signed integers $z\in \mathbb{Z}$, while the purple labels represent the corresponding elements $k\left(z\right)\in {\mathbb{Z}}_{13}$. The blue line indicates the periodicity of the finite field. The unlabeled gray dots indicate the off-axis elements of ${\mathbb{Z}}_{13}$ as they are observed from the top of the 2D spheroid described in Figure 1.

4.2. Pseudo-Rationals

Let

$${\mathbb{Q}}_p:=\left\{{\displaystyle \frac{a}{b}}\mid a\in \mathbb{Z},b=\prod _i{k}_i,k_i\in {\mathbb{F}}_p^{\times }\right\}.$$

The corresponding field value is

$$k\left({\displaystyle \frac{a}{b}}\right):=a·b^{-1}modp.$$

Multiple representations can map to the same $k\in {\mathbb{F}}_p$, forming equivalence classes. We show ${\mathbb{Q}}_p$ is dense in $\mathbb{Q}$ under a metric induced by bounded denominators $b={(p-1)}^n$ [23]. For any $q\in \mathbb{Q}$ and $ϵ>0$, there exists $q^{\prime }\in {\mathbb{Q}}_p$ such that $|q-q^{\prime }|<ϵ$.

The validity of such definition is ensured by the fact that all elements $k_i$ constituting the denominator product $b={\prod }_i{k}_i$ have a multiplicative inverse $k_i^{-1}\in {\mathbb{F}}_p^{\times }$. A selection of some simple examples of such pseudo-rational numbers is depicted in Figure 3, where for each position along the prime meridian $q=a/b\in {\mathbb{Q}}_p$ indicated as a black label on top, the corresponding finite field element $k\left(q\right)\in {\mathbb{F}}_p$ is indicated as purple label on the bottom.

Proposition 1. 

Let $p>2$ be an odd prime number, and let $q=a/b\in \mathbb{Q}$ be any conventional rational number. Then for any $ϵ>0$, there exists an integer $n\in \mathbb{N}$ and an integer $x\in \mathbb{Z}$ such that

$$\left|{\displaystyle \frac{a}{b}}-{\displaystyle \frac{x}{{(p-1)}^n}}\right|<ϵ.$$

Proof. 

Let ${\displaystyle \frac{a}{b}}\in \mathbb{Q}$ be given, and let $ϵ>0$ be arbitrary small number.

Since p is a fixed prime, the expression ${(p-1)}^n$ grows without bound as $n\to \infty$. Therefore, there exists an integer $n\in \mathbb{N}$ such that

$${\displaystyle \frac{1}{{(p-1)}^n}}<ϵ.$$

Now consider the set of rational points of the form

$$\left\{{\displaystyle \frac{k}{{(p-1)}^n}}|k\in \mathbb{Z}\right\},$$

as illustrated in Figure 4. This set is a uniform grid of rational numbers with step size ${\displaystyle \frac{1}{{(p-1)}^n}}$, which is less than $ϵ$ by construction. There exists therefore an integer $x\in \mathbb{Z}$ such that

$$\left|{\displaystyle \frac{a}{b}}-{\displaystyle \frac{x}{{(p-1)}^n}}\right|<ϵ,$$

which completes the proof. □

It is very important to reiterate the meaning of this construct from an ontological viewpoint. More specifically, we stipulate that what actually “exists” are the p representations of the finite field ${\mathbb{F}}_p$, while the derivative class of pseudo-rationals $q\in {\mathbb{Q}}_p$ constitute an abstract mathematical construct derived from the inherent relational properties of the framed instance ${\mathbb{F}}_p$.

In other words, the resultant field of pseudo-rational numbers ${\mathbb{Q}}_p$ will exhibit all the properties of the field of conventional numbers $\mathbb{Q}$ and can further approximate it with any arbitrary precision. Furthermore, for an observer with a limited observability horizon and sufficiently large values of cardinality p, the pseudo-rational field ${\mathbb{Q}}_p$ becomes completely indistinguishable from its conventional counterpart, as all the desired rational numbers of the form $q=a/b$, where $b<p$ are represented not approximately, but exactly within the scope of the pseudo-rational numbers ${\mathbb{Q}}_p$.

4.3. Pseudo-Reals

Define truncated pseudo-rationals:

$${\mathbb{Q}}_p^{\le H}=\left\{[x,n]:0\le x<p,0\le n\le H\right\},[x,n]:={\displaystyle \frac{x}{{(p-1)}^n}}.$$

This set is finite and totally bounded under the metric:

$$d_H([x,n],[y,m]):=\left|{\displaystyle \frac{x}{{(p-1)}^n}}-{\displaystyle \frac{y}{{(p-1)}^m}}\right|.$$

Define ${\mathbb{R}}_p$ as the closure of ${\mathbb{Q}}_p^{\le H}$. We show all computable real numbers can be approximated within $2^{-k}$ by some element $[x,n]\in {\mathbb{Q}}_p^{\le H}$, where $H\ge \lceil k{log}_{2}p\rceil$ [24].

Proposition 2 

(Finite Total Boundedness). For each fixed H, the metric space $\left({\mathbb{Q}}_p^{\le H},d_H\right)$ is finite and thus totally bounded.

Proof. 

Since $0\le x<p$ and $0\le n\le H$, there are $\left(P\right)\times (H+1)$ elements in ${\mathbb{Q}}_p^{\le H}$. Any finite metric space is trivially totally bounded. □

Theorem 1 

(Approximation of Computable Reals). Let $r\in \mathbb{R}$ be a computable real number. For any integer $k\ge 1$ there exist integers $a_k,b_k$ with $b_k\ne 0$ such that

$$\left|r-\frac{a_k}{b_k}\right|<2^{-k}.$$

Moreover, if the observer’s horizon H satisfies

$$H\ge ⌈k{log}_{2}p⌉,$$

then one can construct $[x_k,n_k]\in {\mathbb{Q}}_p^{\le H}$ with

$$\left|r-[x_k,n_k]\right|<2^{-k-1}.$$

Proof. 

By computability of r there is a rational approximation $a_k/b_k$ with $|r-a_k/b_k|<2^{-k}$. Since $gcd(p,p-1)=1$, the extended Euclidean algorithm yields an exponent $n_k\le k{log}_{2}p$ such that

$$b_k{(p-1)}^{n_k}\equiv 1(modp).$$

Set $x_k\equiv a_k{b}_k^{-1}(modp)$. Then

$$[x_k,n_k]={\displaystyle \frac{x_k}{{(p-1)}^{n_k}}}$$

differs from $a_k/b_k$ by at most ${\left(2b_k{(p-1)}^{n_k}\right)}^{-1}<2^{-k-1}$. If $H\ge n_k$, then $[x_k,n_k]\in {\mathbb{Q}}_p^{\le H}$ and

$$\left|r-[x_k,n_k]\right|\le \left|r-\frac{a_k}{b_k}\right|+\left|\frac{a_k}{b_k}-[x_k,n_k]\right|<2^{-k}+2^{-k-1}=O\left(2^{-k}\right),$$

as desired. □

The resulting pseudo-real field ${\mathbb{R}}_p$ is thus defined as the topological closure of ${\mathbb{Q}}_p$ under modular convergence. For any finite observer with bounded resolution and limited horizon of observability, ${\mathbb{R}}_p$ is indistinguishable from the conventional real number continuum.

In conclusion, the field of pseudo-real numbers ${\mathbb{R}}_p$ is not a metaphysical continuum but a layered epistemic utilitarian construct. It combines:

• Exact pseudo-reals that satisfy algebraic equations within ${\mathbb{F}}_p$, and
• Approximated pseudo-reals that are limits of converging sequences in ${\mathbb{Q}}_p$.

This framework provides all the functional properties of the real numbers—continuity, density, and completeness—without invoking actual infinity. It affirms that, in a finite and informationally complete universe, continuum-like behavior is a pragmatic illusion emerging from local reasoning over a fundamentally finite arithmetic substrate.

Having established the construction of pseudo-integers, rationals and reals over the finite field ${\mathbb{F}}_p$ as relativistic, frame-dependent analogs of their classical counterparts, we seek to further extend this framework to encompass the algebraic closure of the pseudo-real field. In conventional mathematics, the introduction of complex numbers $\mathbb{C}$ is necessitated by the absence of solutions to certain polynomial equations, such as $x^2+1=0$, within the real numbers. Analogously, in the finite framed context, we are motivated to introduce complex-like elements in order to achieve closure under operations that are otherwise impossible within the pseudo-rational or alone.

Moreover, the construction of a relativistic complex plane enables the representation of rotations, oscillations, and other phenomena that are fundamental in both mathematics and physics, all within a finite and self-contained system. This approach not only mirrors the classical extension from $\mathbb{R}$ to $\mathbb{C}$, but also demonstrates that the essential properties and utility of complex numbers can be realized as emergent features of a finite, relational arithmetic—thereby reinforcing our framework’s central theme of relativistic, context-dependent number systems.

5. Complex Plane over Finite Framed Field

As is commonly known, the field of real numbers $\mathbb{R}$ does not contain any solutions of certain polynomial equations, such as the prominent equation $x^2+1=0$. But that is not the case for many finite fields ${\mathbb{F}}_p$, where depending on the value and properties of their cardinality P, such solutions can readily exist. For example, in the finite field ${\mathbb{Z}}_5$, the equation $x^2+1=0$ has two solutions: $x=2$ and $x=3$. More generally, it is evident that the equation $x^2+1=0$ can be satisfied in a finite field ${\mathbb{F}}_p$ if and only if $P-1$ is devisable by 4, or in other words $p\equiv 1mod4$. This is due to the fact that the multiplicative group of non-zero elements in such fields is cyclic and contains elements—and the corresponding rotational symmetry—of order 4, which allows for the existence of square roots of $-1$. In this case, we can define a special element $i\in {\mathbb{F}}_p$ that satisfies the equation $i^2+1=0$. The element i is not unique, instead we have a pair of pseudo-integer elements i and $-i$ in $\mathbb{Z}/{\mathbb{F}}_p$ that satisfy the equation, in the same way as we have pairs x and $-x$ of solutions for quadratic equations in the conventional complex plane $\mathbb{C}$.

Let us now observe the “North Pole” frame of reference of the spherical representation of the finite field ${\mathbb{F}}_p$ with its prime meridian of pseudo-reals ${\mathbb{R}}_p$ forming the horizontal axis around the origin. The order-4 rotational symmetry of the finite field ${\mathbb{F}}_p$ can be represented as a vertical axis of imaginary numbers $c=z·i$, where $z\in \mathbb{Z}$, that are perpendicular to the prime meridian, as illustrated in Figure 5. The imaginary numbers c are represented by their respective red labels, while the corresponding elements $k\left(c\right)$ are depicted in purple.

More generally, we can define a class of pseudo-complex numbers ${\mathbb{C}}_p$ as the Cartesian product of the pseudo-reals ${\mathbb{R}}_p$ and the imaginary numbers $r·i,r\in \mathbb{R}$. The pseudo-complex numbers are defined as follows:

$${\mathbb{C}}_p:=\left\{c=a+b·i|a,b\in {\mathbb{R}}_p\right\},$$

(1)

where a and b are the real and imaginary components of the pseudo-complex number c, respectively. The pseudo-complex numbers can be represented as points in the complex plane, where the horizontal axis corresponds to the pseudo-reals ${\mathbb{R}}_p$ and the vertical axis corresponds to the imaginary numbers $r·i$. The pseudo-complex numbers form a field and can be added, subtracted, multiplied, and divided in a manner analogous to conventional complex numbers, with the additional consideration of their finite field properties.

The pseudo-complex numbers form a relativistic algebraic field and can be added, subtracted, multiplied, and divided in a manner analogous to conventional complex numbers, subject to the selection of the arbitrary frame of reference, as well as the properties and constraints of the underlying finite field.

6. Unification and Ontological Perspective

We assert that only the p representations of ${\mathbb{F}}_p$ truly exist. All pseudo-number classes are epistemic constructs derived from relational symmetries and observer framing. The observer’s bounded horizon $H\ll p$ induces the illusion of infinite domains [25].

6.1. Infinity as the Unknowable “far-far away”

Let us revisit the ontological concept of infinity as described in [4]. In the previous sections, we have established the finite framed field ${\mathbb{F}}_p$ as an abstract pseudo-sphere ${\mathbb{F}}_p(0,1)$ with a limited-horizon observer at its origin 0. We would like now to consider the geometric point on our pseudo-sphere that is the furthest away from the observer. This point is evidently the South Pole—the antipodal point on the prime meridian—of the pseudo-sphere, which we will denote as $s_P$ for now. We would like to emphasize the following important properties of $s_P$.

• $s_P$ is a unique point on the pseudo-sphere that is the farthest away from the observer at 0.
• $s_P$ is invisible to the observer at 0, that is to say that is located beyond any conceivable definition of the observer’s limited observability horizon.
• Finally, $s_P$ is algebraically inaccessible to the observer at 0, in the sense that $s_P\notin {\mathbb{F}}_p,{\mathbb{Q}}_p$, and cannot be reached by any finite number of arithmetical steps along the surface of the pseudo-sphere.

We would like to provide a formal proof of the less evident Property 3 as follows.

Theorem 2 

(No South Pole in ${\mathbb{F}}_p$). Let $P>2$ be an odd prime. Then the only solution $s\in {\mathbb{Z}}_P$ to

$$2s\equiv 0(modP)$$

is $s\equiv 0$. Equivalently, there is no nonzero pseudo-rational $q\in Q_P$ whose image in ${\mathbb{Z}}_P$ has additive order 2.

Proof. 

• Since p is prime, the additive group $({\mathbb{F}}_p,+)$ is cyclic of order p. An element $s\in {\mathbb{F}}_p$ has order 2 precisely if

  $$2s\equiv 0(modP).$$
• Because $gcd(2,p)=1$, multiplication by 2 is invertible in ${\mathbb{F}}_p$. Hence, from $2s\equiv 0(modp)$ it follows immediately that $s\equiv 0(modP)$. There is no nontrivial order-2 element.
• By definition, each pseudo-rational $q={\displaystyle \frac{a}{b}}\in {\mathbb{Q}}_p$ is represented in the field by

  $$k\left(q\right)=a{b}^{-1}modP\in {\mathbb{F}}_p,$$

  so ${\mathbb{Q}}_p\subseteq {\mathbb{F}}_p$ under the embedding k. If some $q\in {\mathbb{Q}}_p$ mapped to a nonzero order-2 element $s=k\left(q\right)\ne 0$, then $2s\equiv 0$ would force $s\equiv 0$, a contradiction.

Therefore, no “South Pole” antipodal point exists in ${\mathbb{Q}}_p$ or ${\mathbb{Z}}_p$, completing the proof. □

These properties of the geometrical point $s_P$ are unmistakably consistent with the properties of the concept of infinity in its conventional sense. This gives us the justification to identify the relativistic antipodal point $s_P$ with the concept of infinity in the context of ${\mathbb{F}}_p$, and thus denote it as ∞.

To exemplify, let us now consider the concrete example of $p=13$ and the corresponding finite framed field ${\mathbb{F}}_{13}$. We can identify the following values for the constants i, e and $\pi$ in ${\mathbb{F}}_{13}$:

$$p=13,g_{min}=2,i=5.$$

The corresponding visual representation of the finite field ${\mathbb{F}}_{13}$ is shown in Figure 6. The figure shows the state space of the finite field ${\mathbb{F}}_{13}$ as a circle on a 2D plane, with the major structural elements $-1,0,1,g_{min},i$, as well as ∞ indicated. The antipodal point ∞ is located at the South Pole of the pseudo-sphere, which is the farthest point from the observer at 0.

6.2. Approximate Lie Groups over Finite Fields

Continuous Lie groups such as $\mathrm{SO}\left(2\right)$, $\mathrm{SU}\left(n\right)$, and $\mathrm{GL}(n,\mathbb{R})$ are approximated in ${\mathbb{F}}_p$ by discrete symmetry groups generated by modular exponentiation and cyclic subgroup structures [26].

Let $G_p\subseteq {\mathbb{F}}_p^{\times }$ be a multiplicative cyclic group of order $N\mid (p-1)$. The mapping:

$$\theta \mapsto g^{\theta }modp,\theta \in \mathbb{Z}/N\mathbb{Z},g\mathrm{a}\mathrm{primitive}\mathrm{root},$$

approximates continuous rotation $e^{i\theta }$ by discrete steps. Similarly, discrete matrix groups over ${\mathbb{F}}_p$, such as $\mathrm{GL}(n,{\mathbb{F}}_p)$, replicate local algebraic behavior of Lie algebras over reals.

These finite analogues converge to their continuous counterparts as $p\to \infty$ and preserve closure, invertibility, and group action properties locally within observer horizons $H\ll p$.

7. Conclusions

This work reconstructs arithmetic over finite fields as a complete, self-consistent relativistic algebra. Number systems conventionally built on actual infinity are shown to emerge from finite, observer-relative structures in ${\mathbb{F}}_p$. This reformulation provides a robust mathematical language for finite informational systems and supports a shift from absolute to structural foundations in mathematics.