Finite Relativistic Algebra at Composite Cardinalities

1. Introduction

Our finite-relativistic programme has so-far advanced through a trilogy of companion manuscripts whose aims were, respectively, (i) establish an ontological motivation for the move from the conventional infinitude conjecture toward a framework designed on the principle of relational finitude [1], (ii) to rebuild classical algebra inside a single framed prime field [3], and (iii) to uncover a smooth geometric layer, and the fundamental constants already latent in that finite algebra [2]. The present article constitutes a natural continuation of this effort: it extends the framework from prime moduli to arbitrary composite cardinalities q and shows that the resulting arithmetic still supports the ensemble of canonical constants, harmonic structures and low-dimensional topology.

More specifically, in [3] we fixed a prime $p\equiv 1(mod4)$ and a framing $(0,1)$ inside the field ${\mathbb{F}}_p$. The paper thus reconstructed signed integers, rationals, reals and complexes as pseudo-numbers: relational shadows generated by the field’s internal symmetries of translation, scaling and exponentiation. Crucially, this finitary setting preserved all familiar algebraic laws and analytic operations while dispensing with actual infinity. Subsequently, in [2], we demonstrated that a single finite field already contains the structure of smooth geometry. By arranging the three symmetry operators orthogonally to a cardinality axis, one obtains a combinatorial 2-spheroid $N_p$ whose ultrapower lift $S_p\subset R_p^4$ is an internal $C^{\infty }$ surface of constant curvature. The cyclic order of the field singles out three frame-invariant elements $i_p,{\pi }_p,e_p$, finite-field analog of the classical $i,\pi ,e$, which underpin a unified Fourier kernel valid in both the finite and continuous regimes.

Building on those foundations, the current article tackles moduli

$$q=\prod _{k=1}^K{p}_k,p_k\equiv 1(mod4),$$

and establishes four main results:

• Canonical constants at non-prime q. The quarter-turn $i_q$, half-period ${\pi }_q$ and minimum-action base $e_q$ are defined by lifting their prime counterparts through Hensel’s lemma and gluing them via the Chinese Remainder Theorem. Each satisfies a projective stability property, and together they generate the full symmetry triple on ${\mathbb{Z}}_q$.
• Bouquet of prime spheroids. The Cartesian splitting ${\mathbb{Z}}_q={\mathbb{F}}_{p_1}\times \dots \times {\mathbb{F}}_{p_K}$ realizes the composite 4-dimensional spheroid as a bouquet of its prime-field pieces glued along the zero-divisor locus, yielding explicit cardinality formulas and nested visualizations.
• Seifert-fibred 3-orbifold ${\mathcal{S}}_q$. Interpreting translation, multiplication and exponentiation as commuting circle actions upgrades the bouquet into a Seifert manifold whose exceptional fibres encode the prime-power decomposition of q. The resulting “DNA-like” geometry provides a topological dictionary between arithmetic and 3-manifold invariants.
• Mixed-radix and fiber decompositions. A triplet of expansion modes for ${\mathbb{Z}}_q$—derived from the operations of addition, multiplication and exponentiation—deliver unique digit vectors that serve as coordinates for harmonic analysis and modal splitting on ${\mathcal{S}}_q$. The sets of units of the resultant vector decompositions contain abundant classes of elements that become stable for large values of cardinality q.

Together these results show that the proposed finite relativistic algebra retains its algebraic coherence, analytic versatility and geometric depth when the ambient cardinality is no longer prime. In doing so, the paper completes the transition from two-dimensional to three-dimensional finite geometry, setting the stage for applications of physical dynamics.

2. Composite-Modulus Finite Ring

2.1. Bi-Prime Composition

We commence by considering a simple example of the composite modulus $q=p·p^{\prime }=1(mod4)$ composed of some pair of prime factors $p,p^{\prime }$. The ring ${\mathbb{Z}}_q$ is the quotient of the integers by the ideal generated by q.

Definition 1

(CRT coordinates). Via the Chinese Remainder Theorem we fix an isomorphism

$${\mathbb{Z}}_q\cong {\mathbb{F}}_p\times {\mathbb{F}}_{p^{\prime }},x⟼\left({\overline{x}}_p,{\overline{x}}_{p^{\prime }}\right),$$

and denote the projections by ${\mathcal{P}}_p,{\mathcal{P}}_{p^{\prime }}:{\mathbb{Z}}_q\to {\mathbb{F}}_p,{\mathbb{F}}_{p^{\prime }}$.

Definition 2

(Prime-field 2-spheroids [2]). For a prime p set

$${\mathcal{S}}_p=\left\{\mathbf{v}=(v_1,\dots ,v_4)\in {\mathbb{F}}_p^4|Q\left(\mathbf{v}\right)=1\right\},Q\left(\mathbf{v}\right)=v_1^2+v_2^2+v_3^2+v_4^2.$$

Definition 3

(Composite spheroid).

$${\mathcal{S}}_q=\left\{\mathbf{x}\in {\mathbb{Z}}_q^4|Q\left(\mathbf{x}\right)=1\right\}.$$

Proposition 1

(Cartesian splitting). Under the CRT identification ${\mathbb{Z}}_q^4\cong {({\mathbb{F}}_p\times {\mathbb{F}}_{p^{\prime }})}^4$ we have

$${\mathcal{S}}_q={\mathcal{S}}_p\times {\mathcal{S}}_{p^{\prime }}.$$

Proof. 

Write $\mathbf{x}=(x_1,\dots ,x_4)$. Then

$$Q\left(\mathbf{x}\right)modq=\left(Q\left({\mathcal{P}}_p\left(\mathbf{x}\right)\right)modp,Q\left({\mathcal{P}}_{p^{\prime }}\left(\mathbf{x}\right)\right)mod{p}^{\prime }\right),$$

so $Q\left(\mathbf{x}\right)\equiv 1(modq)$ iff $Q\left({\mathcal{P}}_p\left(\mathbf{x}\right)\right)=1$ in ${\mathbb{F}}_p$ and $Q\left({\mathcal{P}}_{p^{\prime }}\left(\mathbf{x}\right)\right)=1$ in ${\mathbb{F}}_{p^{\prime }}$. □

Definition 4

(Prime lifts in ${\mathbb{Z}}_q^4$).

$${\mathcal{S}}_p^{\uparrow }:=\left\{\mathbf{x}\in {\mathcal{S}}_q:{\mathcal{P}}_{p^{\prime }}\left(\mathbf{x}\right)=\mathbf{0}\right\},{\mathcal{S}}_{p^{\prime }}^{\uparrow }:=\left\{\mathbf{x}\in {\mathcal{S}}_q:{\mathcal{P}}_p\left(\mathbf{x}\right)=\mathbf{0}\right\}.$$

Lemma 2.1.

${\mathcal{S}}_q={\mathcal{S}}_p^{\uparrow }\dot{\cup }{\mathcal{S}}_{p^{\prime }}^{\uparrow }$ and ${\mathcal{S}}_p^{\uparrow }\cap {\mathcal{S}}_{p^{\prime }}^{\uparrow }=\left\{\mathbf{0}\right\}$.

Proof. 

Immediate from 1: if both projections of $\mathbf{x}$ were non-zero, each would have to satisfy $Q=1$ in its own field, contradicting the definitions above. Hence, the only common point is the origin. □

Remark 2.2

(Nested visualisation). Realise ${\mathbb{Z}}_q$ as the interval $[-32,32]\subset \mathbb{R}$. Every non-zero coordinate of $N_5^{\uparrow }$ is a multiple of 13 (length $\le 26$), whereas coordinates in $N_{13}^{\uparrow }$ may reach $\pm 30$. Thus in the Euclidean lift the ${\mathbb{F}}_p\left[5\right]$-spheroid sits inside the ${\mathbb{F}}_p\left[13\right]$-spheroid, touching only at $\mathbf{0}$. For any square-free composite $q={\prod }_i{p}_i$ the global spheroid is likewise a bouquet of prime-field spheroids glued along the zero-divisor locus.

Corollary 1

(Cardinalities). Using $|N_p|=p^2-p+1$ (see [2]) we get

$$|N_q|=|N_5|+|N_{13}|-1=120+2184-1=2303.$$

3. Composite-Modulus Construction of the Canonical Constants

Building on the intuition obtained from the bi-prime case, we now extend the definitions of the three canonical constants $i_q$, ${\pi }_q$, and $e_q$ to a general composite modulus q that is a product of K distinct odd primes, each congruent to $1(mod4)$. Let

$$q=\prod _{k=1}^K{p}_k,$$

where each $p_k$ is an odd prime satisfying $p_k\equiv 1(mod4)$. Importantly, there is no assumption here that $\left\{p_k\right\}$ are distinct, i.e. there can be an arbitrary number of repetitions of such primes in any order. We note the Chinese Remainder decomposition

$${\mathbb{Z}}_q\cong \prod _{k=1}^K{\mathbb{F}}_{p_k}.$$

We will now combine the prime-modulus constants $i_{p_k}$, ${\pi }_{p_k}$, and $e_{p_k}\in {\mathbb{F}}_{p_k}$ for each $k=1,\dots ,K$ via the CRT to obtain the three constants $i_q,{\pi }_q,e_q\in {\mathbb{Z}}_q$.

3.1. Quarter-turn $i_q\in {\mathbb{Z}}_q$

(a) Existence of $i_{p_k}$. Since $p_k\equiv 1(mod4)$, $-1$ is a quadratic residue modulo $p_k$. Let

$$i_{p_k}=min\left\{x\in {\mathbb{F}}_{p_k}:x^2\equiv -1(mod{p}_k),1\le x<\frac{p_k-1}{2}\right\}.$$

Then $i_{p_k}^2\equiv -1(mod{p}_k)$, and by Hensel’s lemma [5] there is a unique lift

$$i_{p_k}\in {\mathbb{F}}_{p_k}\mathrm{such}\mathrm{that}{i}_{p_k}^2\equiv -1(mod{p}_k).$$

Denote its inverse in ${\mathbb{F}}_{p_k}$ by $i_{p_k}^{-1}$, so $i_{p_k}·i_{p_k}^{-1}\equiv 1(mod{p}_k)$.

(b) Lifting i to ${\mathbb{Z}}_q$. By CRT,

$${\mathbb{Z}}_q\cong \prod _{k=1}^K{\mathbb{F}}_{p_k}.$$

Hence, there is a unique element $i_q\in {\mathbb{Z}}_q$ whose projection to each factor ${\mathbb{F}}_{p_k}$ is $i_{p_k}$. Equivalently, in CRT notation,

$$i_q=\left(i_{p_1}mod{p}_1,i_{p_2}mod{p}_2,\dots ,i_{p_K}mod{p}_K\right)\in \prod _{k=1}^K{\mathbb{F}}_{p_k}\cong {\mathbb{Z}}_q.$$

By construction, each $i_{p_k}^2\equiv -1(mod{p}_k)$, so

$$i_q^2\equiv -1\mathrm{in}{\mathbb{Z}}_q.$$

Equivalently, one may express $i_q$ via the inverses in each prime component:

$$i_q={\left(i_{p_1}^{-1}\times i_{p_2}^{-1}\times \dots \times i_{p_K}^{-1}\right)}^{-1}.$$

(3.1)

Thus $i_q$ is the “quarter-turn” in ${\mathbb{Z}}_q$ whose reduction modulo $p_k$ is $i_{p_k}$ for each k.

3.2. Minimum Action $e_q\in {\mathbb{Z}}_q$

(a) Existence of $e_{p_k}$. For each prime $p_k$, let

$$e_{p_k}=argmin\left\{min\left(x,p_k-x\right):x\in {\left({\mathbb{Z}}_{p_k}\right)}^{\times },x\mathrm{a}\mathrm{primitive}\mathrm{root}\mathrm{modulo}{p}_k\right\},$$

i.e. the unique generator of ${\mathbb{F}}_{p_k}^{\times }$ closest to 1. Since primitive roots lift to primitive roots modulo $p_k$, there is (for each choice of residue class of $e_{p_k}$) a unique

$$e_{p_k}\in {\mathbb{F}}_{p_k}\mathrm{satisfying}{e}_{p_k}\mathrm{is}\mathrm{a}\mathrm{primitive}\mathrm{root}\mathrm{of}{\mathbb{F}}_{p_k}^{\times }.$$

(b) Lifting e to ${\mathbb{Z}}_q$. By CRT,

$${\mathbb{Z}}_q^{\times }\cong \prod _{k=1}^K{\mathbb{F}}_{p_k}^{\times }.$$

Each $e_{p_k}$ is the distinguished “closest-to-one” primitive root in ${\mathbb{F}}_{p_k}^{\times }$. Therefore, there is a unique

$$e_q\in {\mathbb{Z}}_q^{\times }$$

whose projection to each factor ${\mathbb{F}}_{p_k}^{\times }$ is $e_{p_k}$. Equivalently, in CRT form,

$$e_q=\left(e_{p_1}mod{p}_1,e_{p_2}mod{p}_2,\dots ,e_{p_K}mod{p}_K\right).$$

(3.2)

By construction, $e_q$ is a primitive root of ${\mathbb{Z}}_q^{\times }$, and its “distance from 1”,

$$\Delta {exp}_q\left(0\right)=e_q-1,$$

is minimal among all generators of ${\mathbb{Z}}_q^{\times }$.

3.3. Half Period ${\pi }_q\in {\mathbb{Z}}_q$

(a) Existence of ${\pi }_{p_k}$. For any prime$p_k$, one has $\phi \left(p_k\right)=p_k(p_k-1)$. In complete analogy with the prime case, we set

$${\pi }_{p_k}={\displaystyle \frac{\phi \left(p_k\right)}{2}}={\displaystyle \frac{p_k-1}{2}}.$$

Since $p_k\equiv 1(mod4)$, the integer ${\pi }_{p_k}$ is well-defined. If $e_{p_k}$ is the minimum action generator of ${\mathbb{F}}_{p_k}^{\times }$, then

$$e_{p_k}^{{\pi }_{p_k}}\equiv -1(mod{p}_k).$$

(b) Lifting $\pi$ to ${\mathbb{Z}}_q$. Since

$$\phi \left(q\right)=\prod _{k=1}^K\phi \left(p_k^{n_k}\right)=\prod _{k=1}^K\left[p_k^{n_k-1}(p_k-1)\right],$$

we define

$${\pi }_q={\displaystyle \frac{\phi \left(q\right)}{2}}={\displaystyle \frac{1}{2}}\prod _{k=1}^K\left[p_k^{n_k-1}(p_k-1)\right].$$

(3)

Choose the minimum action generator $e_q\in {\mathbb{Z}}_q^{\times }$. Under CRT, $e_q$ projects to primitive roots ${\tilde{g}}_{p_k}\in {\mathbb{F}}_{p_k}^{\times }$ for each k. Then

$${\tilde{g}}^{{\pi }_q}\equiv -1\mathrm{in}\mathrm{each}{\mathbb{F}}_{p_k},\mathrm{hence}{\tilde{g}}^{{\pi }_q}\equiv -1(modq).$$

Therefore ${\pi }_q$ is the unique “half-period” exponent sending a CRT-lifted generator $e_q$ to $-1$ in ${\mathbb{Z}}_q$.

3.4. Summary of the Three Composite Constants for $q={\prod }_k{p}_k$

Putting the above constructions together, for $q={\prod }_{k=1}^K{p}_k$ with each $p_k\equiv 1(mod4)$, the three canonical constants in ${\mathbb{Z}}_q$ are given precisely by the Chinese-remainder lifts of their prime-modulus counterparts:

(i)

Quarter-turn $i_q\in {\mathbb{Z}}_q$: For each $k,i_{p_k}\in {\mathbb{F}}_{p_k}$ is the unique Hensel-lift of $i_{p_k}$,

$$i_q=\left(i_{p_1}mod{p}_1,\dots ,i_{p_K}mod{p}_K\right)\in {\mathbb{Z}}_q,$$

equivalently, $i_q$ is the unique element with $i_q^2\equiv -1(modq)$.

(ii)

Exponential base $e_q\in {\mathbb{Z}}_q$: For each $k,e_{p_k}\in {\mathbb{F}}_{p_k}$ is the unique Hensel-lift of $e_{p_k}$,

$$i_q=\left(e_{p_1}mod{p}_1,\dots ,e_{p_K}mod{p}_K\right)\in {\mathbb{Z}}_q,$$

so that $e_q$ is a primitive root of ${\mathbb{Z}}_q^{\times }$ minimizing $e_q-1$.

(iii)

Half-period ${\pi }_q\in {\mathbb{Z}}_q$: For each $k,{\pi }_{p_k}={\displaystyle \frac{p_k-1}{2}}$, and

$${\pi }_q={\displaystyle \frac{1}{2}}\prod _{k=1}^K{p}_k(p_k-1)={\displaystyle \frac{\phi \left(q\right)}{2}}\in {\mathbb{Z}}_q,$$

and for any CRT-lifted generator $\tilde{g}$, ${\tilde{g}}^{{\pi }_q}\equiv -1(modq)$.

In each case, the element in ${\mathbb{Z}}_q$ is uniquely determined by its reductions modulo each prime factor $p_k$ and is obtained via the Chinese Remainder Theorem. This completes the generalization of the three canonical constants $i,\pi ,e$ from prime moduli to an arbitrary composite modulus $q={\prod }_k{p}_k$.

3.5. Profinite Stability of the Canonical Constants

The construction of canonical constants $i_q,{\pi }_q,e_q$ offers a finite, relational foundation for the analogs of $i,\pi ,e$ in ${\mathbb{Z}}_q$. However, for this framework to represent a truly coherent and scalable system, these constants must exhibit a form of consistency as q evolves. Proving their profinite stability demonstrates precisely this: it shows that these definitions are not ad-hoc, but rather are specific realizations of underlying, universal p-adic entities that project consistently across the entire family of admissible moduli. This stability ensures that as we consider increasingly complex (larger q) arithmetic systems, the “finite versions” of $i,\pi$ and e behave in a compatible and nested way, reflecting a deep structural integrity and pointing towards a unified description that transcends any single choice of modulus [5].

Write every modulus as an unordered product of primes

$$q=\prod _{k=1}^K{p}_k,p_k\equiv 1(mod4).$$

The multiset $\left\{p_k\right\}$ allows repetitions, so if the same prime appears r times the factor $p^r$ is present in q; we simply suppress the exponent in the notation.

For each admissible q the finite-relativistic framework assigns three canonical constants

$$i_q,{\pi }_q,e_q\in {\mathbb{Z}}_q^{\times },$$

obtained by Chinese remaindering the prime-power values $i_{p^r},{\pi }_{p^r},e_{p^r}$, where r is the multiplicity of p in q.

Lemma 3.1

(Hensel-compatible lifts). Fix a prime $p\equiv 1(mod4)$ and choose $c\in \{i,\pi ,e\}$. There exists a unique sequence ${\left(c_{p^r}\right)}_{r\ge 1}$ satisfying

(1)

$c_{p^1}=c_p$;

(2)

$c_{p^{r+1}}\equiv c_{p^r}(mod{p}^r)$ for all $r\ge 1$;

(3)

$c_{p^r}$ realises the same “nearest” minimization modulo $p^r$ that defines $c_p$ modulo p [5].

Hence $\left(c_{p^r}\right)$ is Cauchy in ${\mathbb{F}}_p$ and converges to a unit $C_p\in {\mathbb{F}}_p^{\times }$ [5].

(Sketch) .

The defining polynomials $x^2+1$, $x^{p-1}+1$ and the cyclotomic polynomial for primitive roots have derivatives coprime to p at $c_p$; Hensel’s lemma therefore yields a unique lift to $c_{p^{r+1}}$ that agrees with $c_{p^r}$ modulo $p^r$ and stays nearest to its reference point [5]. The congruence makes the sequence Cauchy. □

Theorem 3.2

(Projective stability of $i_q,{\pi }_q,e_q$). For each $c\in \{i,\pi ,e\}$ there exists a unit

$$C=\prod _{p\equiv 1\left(4\right)}{C}_p\in {\widehat{\mathbb{Z}}}^{\times },C_p=\underset{r\to \infty }{lim}{c}_{p^r},$$

such that for every modulus $q={\prod }_k{p}_k$ (with multiplicities understood) one has

$${\pi }_q\left(C\right)=c_q\left(\mathrm{equality}\mathrm{in}{\mathbb{Z}}_{\mathbf{q}}\right).$$

Consequently, whenever $m\mid q$ the congruence $c_q\equiv c_m(modm)$ holds. Thus the families ${\left(i_q\right)}_q$, ${\left({\pi }_q\right)}_q$, ${\left(e_q\right)}_q$ stabilise projectively across the entire directed system of moduli written in prime-product form.

Proof. 

Fix c. Lemma 3.1 provides a compatible tower ${\left(c_{p^r}\right)}_{r\ge 1}$ for each prime $p\equiv 1(mod4)$. Define $C:={\prod }_{p\equiv 1\left(4\right)}{C}_p\in {\widehat{\mathbb{Z}}}^{\times }$. For $q={\prod }_k{p}_k$ let $r_p$ denote the multiplicity of p in $\left\{p_k\right\}$. The projection ${\mathcal{P}}_q:\widehat{\mathbb{Z}}\twoheadrightarrow {\mathbb{Z}}_q$ acts coordinate-wise, sending the p-adic component $C_p$ to $c_{p^{r_p}}$ and ignoring all other primes. The Chinese Remainder Theorem re-assembles these images into $c_q$. If $m\mid q$ then $r_p\left(m\right)\le r_p\left(q\right)$ for every prime, so the same argument gives ${\pi }_m\left(C\right)=c_m$ and hence $c_q\equiv c_m(modm)$. □

4. Detailed Description of the Established Morphology

This section presents a comprehensive account of the geometric and topological structure arising from the three independent arithmetic motions on ${\mathbb{Z}}_q$ when

$$q=\prod _{k=1}^K{p}_k$$

is composite. We describe the total space as a Seifert-fibred 3-orbifold [8], explain how its 2-dimensional quotient recovers a pinched torus [9], and analyze the local fiber behavior along each zero-divisor seam.

4.1. Three Independent Circle Actions on ${\mathbb{Z}}_q$

On the set of residues ${\mathbb{Z}}_q$, one identifies three natural $S^1$-type symmetries:

• Translation (add 1):

  $$x⟼x+1(modq).$$
  Denote this action by $T:x\mapsto x+1$. It provides one global “longitude” circle.
• Multiplication by a unit (stretch/shrink):

  $$x⟼u·x,u\in {\mathbb{Z}}_q^{\times }.$$
  Because ${\mathbb{Z}}_q^{\times }$ is generally not cyclic when q is composite [6], varying u sweeps out a second circle.
• Exponentiation by a generator (twist):

  $$x⟼x^k,k\in \langle g\rangle \subset {\mathbb{Z}}_q^{\times }$$

  for a chosen “phase generator” g. When ${\mathbb{Z}}_q^{\times }$ is not cyclic, “multiply by u” and “raise to a power” become two distinct circle directions [7].

Because all three actions commute, their combined orbits form a 3-dimensional CW-complex. Topologically, the full space of orbits is a Seifert-fibred 3-orbifold

$$S_q=\left({\mathbb{Z}}_q\right){\times }_{(T,\times ,exp)}{S}^1,$$

endowed with a circle fiber over each base point in a 2-dimensional quotient.

4.2. The Seifert-Fibred 3-Orbifold Structure

Base 2-Orbifold.

When we quotient out the “twist” (exponentiation) circle, the remaining orbits of translation and multiplication form a 2-dimensional orbifold:

$$B_q=S_q/\langle \mathrm{exponentiation}\rangle ,$$

whose underlying topological surface is a genus-1 torus [4]. However, along each set of zero–divisor residues (those divisible by some prime $p_i\mid q$), the “multiply” and “twist” fibers collapse to a point, creating cone points (pinch loci) on the torus.

• Generic points: For any invertible residue $x\in {\left({\mathbb{Z}}_q\right)}^{\times }$, none of the three actions collapses. The projection of its orbit (under translation × multiplication) down to $B_q$ is a smooth point on the torus, carrying a full circle fiber upstairs.
• Zero-divisor loops: Fix a prime divisor $p_i\mid q$. The set

  $${\mathbb{Z}}_{p_k}=\{x\in {\mathbb{Z}}_q:p_k\mid x\}$$

  forms a closed loop (an embedded $S^1$) in the translation-multiplication quotient. Along $Z_{p_k}$, multiplication by $p_k$ (hence by any power of $p_k$) is no longer invertible, so one of the two “latitude” circles collapses. Consequently, each point of $Z_{p_k}$ becomes a cone point of order $m_k$ in the base orbifold $B_q$ [8].

Hence $B_q$ is precisely a torus with r cone points, one of order $m_k$ for each prime $p_i\mid q$. Equivalently,

$$B_q\cong T^2\left(p_1,p_2,\dots ,p_r\right)$$

as a 2-orbifold with cone points of multiplicities

$$m_k={\displaystyle \frac{q}{p_k}},1\le k\le r.$$

Seifert Fibers and Exceptional Fibers.

Over each smooth point of $B_q$, the “twist” circle remains intact, forming a regular Seifert fiber in $S_q$. Over each cone point (zero–divisor loop) $Z_{p_i}\subset B_q$, the circle action collapses so that:

$$\mathrm{fiber}\mathrm{over}{\mathbb{Z}}_{p_i}\cong S^1/\left(\mathrm{rotation}\mathrm{by}\mathrm{angle}2\pi /m_i\right),$$

which is a singular (exceptional) fiber of type $\left(m_i\right)$. Concretely, a small neighborhood of a cone point in $B_q$ is modeled on ${\displaystyle D^2/\left(e^{2\pi i/m_i}\right)}$, and the corresponding upstairs neighborhood is

$$\left(D^2/\left(e^{2\pi i/m_i}\right)\right)\times \left(S^1/\mathrm{Id}\right)\cong {\displaystyle \frac{D^2\times S^1}{(e^{2\pi i/m_i},\mathrm{Id})}},$$

which is the standard local model of a Seifert fibered neighborhood with exceptional fiber of order $m_i$ [8].

5. Modal Decomposition

Lemma 5.1.

For any natural number q and the corresponding set of prime divisors of ${\prod }_n{p}_n=q-1$, $p_n$ have multiplicative inverses in ${\mathbb{Z}}_q$ for all n.

Proof. 

It is sufficient to show that $\gcd (q,p_n)=1$ for all n, where $\gcd \left(\right)$ stands for Greatest Common Divisor. $p_n$ are primes so they don’t have any divisors greater than 1. Suppose that $p_n$ divides q, so that $q=m·p_n$ for some integer m. Since $p_n$ divides q and also divides $Q-1$ it has to divide the difference

$$(m·p_n)-(m·p_n-1)=1$$

(4)

But a prime number $p_n$ cannot divide 1 since primes are greater than 1. □

5.1. Mixed Radix Decomposition of ${\mathbb{Z}}_q$

Theorem 5.2

(Mixed-radix decomposition in ${\mathbb{Z}}_q$). Let an integer $q>2$ be written as an ordered product of (not necessarily distinct) primes

$$q=p_1{p}_2\dots p_r,p_i\ge 2.$$

Define mixed-radix weights

$$u_1:=1,u_i:=p_1{p}_2\dots p_{i-1}(2\le i\le r),⟹u_{i+1}=p_i{u}_i.$$

(a)

Set $\mathcal{K}:=[0,p_1-1]\times \dots \times [0,p_r-1].$ The map

$$\Phi :\mathcal{K}⟶{\mathbb{Z}}_q,\Phi (k_1,\dots ,k_r)=\sum _{i=1}^r{k}_i{u}_i(modq)$$

is a bijection. Consequently, every residue $k\in {\mathbb{Z}}_q$ has a unique ordered digit vector$(k_1,\dots ,k_r)={\Phi }^{-1}\left(k\right)$ with $0\le k_i<p_i$.

(b)

Let $\mathcal{P}$ be any permutation of $\{1,\dots ,r\}$. If one simultaneously permutes the primes and the weights,

$$p_i^{\left(\mathcal{P}\right)}:=p_{\mathcal{P}\left(i\right)},u_i^{\left(\mathcal{P}\right)}:=\prod _{\mathcal{P}\left(j\right)<\mathcal{P}\left(i\right)}{p}_{\mathcal{P}\left(j\right)},$$

then

$$k=\sum _{i=1}^r{k}_i{u}_i(modq)⟺k=\sum _{i=1}^r{k}_{\mathcal{P}\left(i\right)}{u}_i^{\left(\mathcal{P}\right)}(modq).$$

Thus, re-labelling the independent subsystems merely permutes the ordered list $(k_1,\dots ,k_r)$; the unordered multiset$\{k_1,\dots ,k_r\}$ is invariant.

Proof. 

Injectivity. Assume ${\sum }_i{k}_i{u}_i\equiv {\sum }_i{k}_i^{\prime }{u}_i(modq).$ Reducing modulo $p_1$ annihilates every term except $k_1-k_1^{\prime }$, hence $k_1=k_1^{\prime }$. Dividing the congruence by $p_1$ inside ${\mathbb{Z}}_q$ and repeating the argument inductively yields $k_i=k_i^{\prime }$ for all i.

Surjectivity. Given $k\in [0,q-1]$, perform successive divisions:

$$k_1:=kmod{p}_1,k^{\left(1\right)}:=⌊k/p_1⌋;k_2:=k^{\left(1\right)}mod{p}_2,k^{\left(2\right)}:=⌊k^{\left(1\right)}/p_2⌋;\dots$$

After r steps $k^{\left(r\right)}=0$ and $k={\sum }_{i=1}^r{k}_i{u}_i$, so k lies in the image of $\Phi$. Parts (a) and (b) follow immediately, completing the proof. □

The resultant ordered digit vector $(k_1,\dots ,k_r)$ provides a mixed radix decomposition of elements $k\in {\mathbb{Z}}_q$ as follows:

$$k=\sum _{i=1}^r{k}_i{u}_i(modq),$$

5.2. Logarithmic Mixed Radix Decomposition

We already established that for any non-zero element $k\in {\mathbb{Z}}_q$, it can be represented as power of a ring generator $k=e^m$, where $m\in \{0,\dots ,q-2\}$, and we denote the ring generator as e for brevity. Applying the Theorem 5.2 to the element m of the reduced ring ${\mathbb{Z}}_q^{\times }$ we obtain a unique ordered digit vector representation of m in terms of the prime factors of $q-1$

$$m=\sum _{n=1}^s{m}_n{v}_n(modq-1)$$

and can now decompose k into a so-called logarithmic mixed-radix decomposition as follows:

$$k=e^m=exp\left(\sum _{n=1}^s{m}_n{v}_n(modq-1)\right)=\prod _{n=1}^s{e}^{m_n{v}_n}.$$

5.3. Primary Component Decomposition of ${\mathbb{Z}}_q^{\times }$

Theorem 5.3

(Fibre decomposition of the unit group ${\mathbb{Z}}_q^{\times }$). Let $q>1$ have prime-power factorization

$$q=\prod _{i=1}^s{p}_i^{e_i},p_i\mathrm{prime},e_i\ge 1.$$

For each i put

$$M_i:={\displaystyle \frac{q}{p_i^{e_i}}},c_i\equiv M_i^{-1}(mod{p}_i^{e_i}),e_i^{*}:=c_i{M}_i(modq),$$

so that $e_i^{*}\equiv 1(mod{p}_i^{e_i}),e_i^{*}\equiv 0(mod{p}_j^{e_j})(j\ne i).$ Choose a generator $g_i$ of the cyclic unit group ${\left({\mathbf{Z}}_{p_i^{e_i}}\right)}^{\times }\left(:=\langle g_i\rangle \right)$; for $p_i=2$ and $e_i\ge 3$ one may take $g_i=5$. Define the fibre generators

$$G_i:=1-e_i^{*}+g_i{e}_i^{*}\in {\mathbb{Z}}_q,i=1,\dots ,s.$$

(i)

$G_i\equiv g_i(mod{p}_i^{e_i})$ and $G_i\equiv 1(mod{p}_j^{e_j})$ for $j\ne i$; hence the $G_i$ commute and each has multiplicative order $\phi \left(p_i^{e_i}\right)=p_i^{e_i-1}(p_i-1)$.

(ii)

Every unit $u\in {\left({\mathbb{Z}}_q\right)}^{\times }$ can be written

$$u=\prod _{i=1}^s{G}_i^{{\ell }_i},0\le {\ell }_i<\phi \left(p_i^{e_i}\right),$$

and the exponent tuple $({\ell }_1,\dots ,{\ell }_s)$ is unique.

(iii)

The map

$$({\ell }_1,\dots ,{\ell }_s)⟼\prod _{i=1}^s{G}_i^{{\ell }_i}(modq)$$

is a group isomorphism

$$\prod _{i=1}^s{\mathbb{Z}}_{\phi \left(p_i^{e_i}\right)}\cong {\mathbb{Z}}_q^{\times }.$$

(Sketch).  (i) By construction $e_i^{*}$ are orthogonal idempotents ($e_i^{*}{e}_j^{*}=0$ for $i\ne j$ and ${\sum }_i{e}_i^{*}=1$), so $G_i$ reduces to $g_i$ modulo its own prime power and to 1 modulo all others; hence $G_i$ has the claimed orders and commutes with $G_j$ ($i\ne j$).

(ii)

Given $u\in {\mathbb{Z}}_q^{\times }$, let ${\ell }_i:={log}_{g_i}\left(umod{p}_i^{e_i}\right)$ in ${\mathbf{Z}}_{\phi \left(p_i^{e_i}\right)}$. Then $u\equiv G_i^{{\ell }_i}(mod{p}_i^{e_i})$ and $u\equiv 1(mod{p}_j^{e_j})$ for $j\ne i$. Multiplying the $G_i^{{\ell }_i}$ reproduces u modulo every prime power, hence modulo q. Uniqueness of the tuple follows by comparing residues mod $p_i$.

(iii)

Parts (i)-(ii) show the map is bijective and respects multiplication.

  □

Consequently every non-zero-divisor of ${\mathbb{Z}}_q$ (unit) factors uniquely as a product of the fibre generators $\left\{G_i\right\}$; the exponents ${\ell }_i$ serve as “multiplicative digits” in the fibre decomposition of ${\mathbb{Z}}_q^{\times }$. Zero-divisors cannot be so decomposed, as at least one of their coordinates modulo $p_i$ vanishes.

6. Conclusions

The present manuscript further develops our finite-relativistic programme by extending every construction hitherto restricted to a single prime modulus to all composite cardinalities

$$q=\prod _{k=1}^K{p}_k,p_k\equiv 1(mod4).$$

Our results demonstrate that the algebraic, analytic and geometric machinery developed in the prime setting survives—and in fact acquires new richness—once Chinese–remainder structure and mixed radices are admitted.

• Universality of the canonical constants. The quarter-turn $i_q$, half-period ${\pi }_q$ and minimum-action base $e_q$ are obtained by a prime-wise Hensel lift followed by a CRT amalgamation. They form a projectively compatible family, so each is the reduction of a single profinite unit $i,\pi ,e\in {\mathbb{Z}}_q$, ensuring consistency across all moduli. This settles the logical question of whether “finite versions” of $i,\pi ,e$ can coexist in a single framework—they can and do.
• From bouquets to Seifert 3-orbifolds. The discrete 2-spheroid $N_q$ factors as a bouquet of its prime-field companions; equipping it with the commuting circle actions of translation, multiplication and exponentiation lifts the bouquet to a Seifert-fibred 3-orbifold $S_q$. Each prime divisor of q produces an exceptional fibre whose multiplicity records its power in q, furnishing a direct correspondence between arithmetic data and low-dimensional topology.
• Digitised arithmetic, harmonic and modal analysis. A mixed-radix decomposition yields unique additive digit vectors, while multiplicative and “twist” fibres give logarithmic and primary-component coordinate systems on ${\mathbb{Z}}_q^{\times }$. These digit spaces form natural arenas for finite Fourier analysis, providing basis functions that already respect the Seifert fibration and the zero-divisor seams.
• Stability and profinite coherence. Not only do the constants $i_q,{\pi }_q,e_q$ stabilise; whole classes of mixed-radix vector units are shown to be prefix-stable. Fixed “low” digits therefore behave as conserved infrared data when the modulus is enlarged, a property expected to be crucial for multiscale simulations of finite-precision physics.:contentReference[oaicite:7]index=7

Outlook: With the composite-cardinality foundations now in place, the programme is poised to move from construction to application. The profinite stability of the canonical constants, the explicit Seifert-fibred geometry of $S_q$, and the mixed-radix harmonic toolkit together constitute a self-contained calculus on finite, relational spacetime. We are therefore ready to deploy this framework to concrete physical settings—from finite-precision lattice gauge theories and discrete quantum systems to number-theoretic models of cosmology—and to test how faithfully it can capture, unify and predict observable phenomena within a single, fully finite universe.