The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic

Michael T. M. Emmerich

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The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic

Michael T. M. Emmerich

arXiv:2609.03896v1Sep 3, 2026math.HOcs.DMmath.DSmath.NTnlin.CG

TL;DR

The paper asks how a different representation can make modular and multiplicative arithmetic structures visible. It develops a recursively growing array of autonomous prime clocks with locally generated valuation coordinates. The resulting representation uniquely captures positive integers and rationals, reaches some positive algebraic irrationals, and has a stated boundary at general algebraic numbers.

Problem
The paper asks how familiar arithmetic structures can be organized in one prime-indexed representation that makes them easier to see.
Method
The paper builds a recursively growing array of autonomous two-hand clocks whose common +1 evolution generates primes and local valuation coordinates.
Results
The valuation representation uniquely captures positive integers and rationals, while rational valuation levels include certain positive algebraic irrationals but not all algebraic numbers.
Takeaways & Limitations
The coordinates expose divisibility and multiplicative operations while connecting clock-based discrete dynamics with standard number-theoretic structures.
Takeaways & Limitations
The rational valuation extension does not include all algebraic numbers and requires additional structure for richer algebraic number fields.

Abstract

from arXiv · show

The way numbers are represented strongly influences which arithmetic structures are easy to see. The \emph{prime clockwork} is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit \texttt{mod} or \texttt{div} operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=ν_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this \emph{grand cycle} is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

1 Introduction

The prime clockwork reorganizes familiar number-theoretic structures into a prime-indexed dynamical representation generated from an elementary common-tick recursion. Its clocks, valuation readouts, and coordinate operations are designed to support visual intuition while remaining connected to standard notation.

The construction asks whether divisibility, congruences, factorization, arithmetic functions, and valuations can be organized in one prime-indexed representation.
A common +1 signal generates prime periods recursively, while each clock independently updates seconds and minutes readings without explicit mod or div operations.
Valuations are derived from ring events, minutes addresses, and local history, distinguishing the maintained autonomous clock hands from their multiplicative observation.
The representation is motivated as a visual and intuitive route through modular and multiplicative number theory, with possible alternative proofs using cycles, counting, and state transitions.
The contribution connects discrete dynamical systems to standard number-theoretic notation without replacing the latter.

2 The basic clockwork

The basic clockwork is an autonomously updated array of prime-period clocks whose seconds and minutes readings generate valuation coordinates and recursively reveal the primes. Its unit-step recursion installs a clock exactly when no existing clock rings.

2 The basic clockwork: For period p, the seconds reading cycles through p states while the minutes reading counts completed p-cycles, yielding n = pM_p(n) + R_p(n).
2 The basic clockwork: The valuation readout is zero away from rings and, at a ring, equals one plus the earlier valuation addressed by the current minutes value.
2.1 The unit-step recursion: Each active clock advances its seconds counter, carries completed turns into its minutes counter, grows the array when no seconds counter is zero, and then generates valuations from stored history.
2 The basic clockwork: The maintained dynamical state consists of autonomous clock hands, while valuation histories provide synchronized multiplicative observations.
2.1 The unit-step recursion: The figures present the recursively generated trajectory, distinguishing finite seconds readings from derived valuation readings and marking newly installed prime clocks.
2.1 The unit-step recursion: At time n > 1, the recursion installs a new clock if and only if n is prime.

3 Integers on the clockwork

The clockwork organizes finite prime-clock states into a single grand-cycle orbit, then uses locally generated valuation readouts to recover arithmetic structure.

3.1 Grand cycles from Cartesian products: A nonzero rotation of a prime clock visits all positions exactly once, supplying the local cyclic fact behind the product-orbit construction.The same cyclic fact is later reused in the unique-factorization argument.
3.1 Grand cycles from Cartesian products: The grand cycle visits every joint state in the Cartesian product exactly once before returning, with length p1p2 ··· pk.Its construction uses clock rotations and counting before CRT provides an algebraic interpretation.
3.1 Grand cycles from Cartesian products: The common tick orders the Cartesian product into one reversible orbit, while CRT recognizes that product as a residue system modulo p1p2 ··· pk.The dynamical proof and CRT identify the same finite state space from different perspectives.
3.2 The valuation readout and factorization depth: The valuation readout is generated from a ring event, a minutes address, and the clock’s earlier valuation history rather than from the minutes counter alone.This readout supplies prime multiplicity while the two hands remain the autonomous dynamical state.
3.2 The valuation readout and factorization depth: Valuation rows do not repeat, and coordinatewise additivity gives V(ab) = V(a) + V(b), enabling reconstruction of integers and unique factorization.For example, V(72) = (3, 2, 0, 0, . . .).

3.3 Non-repetition and unique factorization from prime clocks

The paper proves that valuation rows never repeat, then uses those intrinsic coordinates to recover unique prime factorization and recast several arithmetic operations geometrically.

Non-repetition: Valuation rows never repeat: every time after 1 has either a positive existing-prime coordinate or a newly installed prime coordinate.A repeated nonzero row would descend through a ringing clock’s minutes address to an earlier repeated row.
Unique factorization: Every integer n > 1 has a unique prime factorization, recovered directly from its valuation row without assuming prior factorization.The synthesis map from finite-support nonnegative exponent vectors reconstructs the original integer time.
Intrinsic coordinates: The valuation vector is a bijective static representation of positive integers by finite-support nonnegative prime-exponent vectors.The two clock hands remain dynamically relevant when advancing integers additively, even though they are unnecessary for static reconstruction.
Divisibility, gcd, and lcm: Divisibility becomes weak componentwise order on valuation coordinates, while gcd and lcm correspond to coordinatewise minimum and maximum.In a two-prime slice, these operations appear as the lower-left and upper-right corners of the exponent vectors.
Perfect powers: An integer is a k-th power exactly when every prime exponent is divisible by k, so perfect powers form exponent sublattices.Perfect squares are the even sublattice, and the largest perfect-power exponent is the gcd of the nonzero exponents.
Euler’s totient: Euler’s totient counts grand-cycle states avoiding every zero alarm; for the 2-, 3-, and 5-clocks, eight states give φ(30) = 8.The same clockwork count yields the prime-power formula φ(p^a) = p^(a−1)(p−1).

4 From integers to rational numbers

The clockwork extends from positive integers to positive rationals by turning valuation coordinates into signed integers while retaining finite residue states where denominators are invertible. Rational multiplication becomes coordinate addition, and rational uniqueness follows from additive prime-vector reasoning.

4.1 Signed valuation vectors from integer pairs: Rational equivalence is preserved because equivalent integer pairs have equal cross-products and therefore equal valuation vectors.
4.1 Signed valuation vectors from integer pairs: Rational uniqueness follows additively: equal signed vectors imply equal cross-products, establishing equality of the represented pair classes without product cancellation.
4.3 Valuation vectors for rational numbers: Multiplication of positive rationals corresponds exactly to addition of signed valuation vectors, with each prime coordinate changing independently.
Signed valuation coordinates uniquely represent positive rationals, extending integer valuations by subtraction of denominator exponents.
4.2 Backward and rational-step motion of the seconds counters: Finite seconds counters remain reversible and support admissible rational steps, but a denominator divisible by p has no residue on the p-clock.
4.3 Valuation vectors for rational numbers: Valuation coordinates and elapsed-cycle minutes must remain distinct: valuations encode multiplicative depth, whereas minutes count completed p-cycles.

5 Algebraic numbers

Rational exponent levels extend the valuation representation beyond positive rationals to a multiplicatively rich class of algebraic numbers. The extension includes certain positive algebraic irrationalities but has a precise boundary because it is not closed under addition.

5.1 Rational valuation levels: The representation consists of finite-support rational prime-exponent vectors whose positive real products define the represented values.
5.1 Rational valuation levels: Rational exponent vectors are unique, and powers scale every coordinate while roots divide every coordinate.
Every positive rational root of a positive rational belongs to the rational-exponent representation, which therefore includes certain positive irrational algebraic numbers.
5.1 Rational valuation levels: Adjacent half-levels refine the exponent coordinate by multiplication by 2^1/2, while finite residue clocks remain unsubdivided.
5.2 A natural boundary: The class is not closed under addition, so the rational-exponent representation does not include all algebraic numbers.

6 What the clockwork reveals

The clockwork organizes recursive growth, finite-state dynamics, modular phase, and multiplicative coordinates in one prime-indexed representation. Its main value is expository: standard arithmetic structures become adjacent and can be traversed through a common dynamical language.

The representation places recursion, finite-state dynamics, modular phase, multiplicative depth, and counting observables in one prime-indexed organization.
The common unit tick generates a grand cycle through every state in the Cartesian product of the finite seconds spaces.
The valuation vector is the multiplicative layer: it is unique for integers and rationals, and multiplication evolves each coordinate independently.
Maintaining clock hands makes local unit-step evolution immediate, avoiding reconstruction and refactorization at every additive step.
Its proposed contribution is expository and pedagogical rather than a claim of new theorems, connecting elementary number theory, visual reasoning, and discrete dynamical systems.

7 Further perspectives

The paper presents the prime clockwork as a representational synthesis of recursive prime generation, autonomous clock dynamics, valuation coordinates, and classical number theory. It positions the construction as an expository and proof-oriented perspective rather than a source of new number-theoretic theorems.

7 Further perspectives: The paper relates its recursive clock construction to prior work on prime vectors, grand cycles, clock-based congruences, cyclic sieves, and alternative proofs of arithmetic theorems.It explicitly distinguishes the present paired residue/valuation representation from closer sieve and gap-structure constructions.
7 Further perspectives: The prime clockwork’s intended distinction is its synthesis of recursive prime generation, paired clock states, valuation history, and ordered dynamical presentation.Its contribution is organizational and representational rather than a collection of new number-theoretic theorems.
7 Further perspectives: Prime-vector coordinates support descriptions of divisibility, division, signed and rational coordinates, and algebraic irrationality within the model’s stated boundary.The representation reaches certain positive algebraic irrationalities, while ordinary addition remains outside the model.
7 Further perspectives: Continuous counterparts and richer algebraic-number settings remain deliberately undeveloped directions beyond the paper’s elementary core.The author specifically leaves open useful continuous analogues of grand cycles and related dynamical structures.
7 Further perspectives: The construction is intended for visual and intuitive number theory, alternative proofs and explanations, and an entry point from modular arithmetic into dynamical-systems language.The paper frames the synthesis as a useful perspective on modular structure rather than a reinvention of established theory.

Tool and computational resource disclosure

The paper discloses using ChatGPT for language editing and auxiliary checks while retaining human responsibility for the mathematics and references. It also states the manuscript’s intended CC BY 4.0 licensing.

Tool and computational resource disclosure: ChatGPT was used for language editing and as an auxiliary check for possible gaps or inconsistencies in mathematical arguments.The disclosure follows the Leiden Declaration.
Tool and computational resource disclosure: The author did not treat ChatGPT as an author or mathematical authority and verified the arguments and references.The author assumes responsibility for the manuscript and remaining errors.
Tool and computational resource disclosure: The manuscript is intended for release under the Creative Commons Attribution 4.0 International license.The stated license is CC BY 4.0.

A A compact executable form of the prime clockwork

The appendix supplies compact executable views of the prime clockwork, with division-free two-hand updates, online prime installation, and local valuation-history readouts. The implementations expose the same recursive process in Python and Common Lisp.

A A compact executable form of the prime clockwork: Each loop performs one unit step, advancing every installed seconds hand and carrying one unit to minutes when a hand returns to zero.The update uses increment, comparison, reset, and carry rather than remainder or division operations.
A A compact executable form of the prime clockwork: When no installed clock rings, the current integer is installed as a new period in state (0, 1), generating the next prime.The Common Lisp view isolates this online prime-discovery process.
A A compact executable form of the prime clockwork: At a ring, the minutes reading addresses the clock’s own valuation history and adds one, while non-ring states have zero valuation.The program output includes primes, active seconds and minutes readings, and nonzero factorization coordinates.
A A compact executable form of the prime clockwork: The Python implementation advances both clock hands and generates valuation readouts from each clock’s minutes address and local history.The executable views separate online prime discovery from the full clock-and-valuation trajectory.

B Complex-exponential picture of the seconds counters

The appendix embeds the seconds counters in continuous complex exponentials while preserving the discrete event logic of prime discovery. The visualization represents cyclic seconds motion, not minutes-based valuation depth.

B Complex-exponential picture of the seconds counters: The unit clock’s positive-real crossings occur exactly at integer times, supplying the events when the recursive process inspects installed prime clocks.This makes integer sampling explicit in the continuous construction.
B Complex-exponential picture of the seconds counters: For each discovered prime, the complex embedding preserves the seconds state, with positive-real alignment equivalent to divisibility by that prime.The paper states z_p(n) = p iff p divides n.
B Complex-exponential picture of the seconds counters: The discrete rule “grow when no clock rings” becomes an absence-of-synchronization test among continuous rotations sampled at unit-clock events.A newly discovered period is installed at its own positive-real reference point.
B Complex-exponential picture of the seconds counters: The continuous cone and ribbon visualize the maintained cyclic seconds layer, while completed turns belong to minutes and valuation readouts remain separate.The embedding does not display prime-exponent depth directly.

C A portrait run for integers 1 through 30

Tables 1 and 2 show the run’s factorized outputs and simultaneous prime-residue states through n=30. Gray rows mark internally generated primes, while active columns expose the evolving clock array.

Table 1 records each integer’s prime-exponent output, with gray rows marking newly encountered primes generated by the recursion.
Table 2 displays simultaneous residue-clock states through prime 29, although only columns with p ≤ n are active during the dynamic run.
A zero in an active residue column marks that prime clock’s ring, while reading across rows gives joint states and down columns follows individual clocks.