Digit-Wise Circular Representation

• Digit-wise circular representation is a framework that models each digit independently within a cyclic or modular structure, enabling distinct digit processing.
• It underpins advanced techniques in algebraic systems, digital convolutions, and topological data analysis, with practical applications from error detection to signal filtering.
• In neural computation, cyclic embeddings facilitate circuit specialization and localized error correction in large language models, enhancing interpretability and performance.

Digit-wise circular representation refers to a class of mathematical, algorithmic, and neural encoding strategies where every digit of a multi-digit object—whether a number, symbol sequence, or word—is assigned a role in a cyclic structure. Rather than processing numbers or sequences as monolithic values, digit-wise circular paradigms model each digit independently and frequently map them onto cyclic or modular coordinate systems, typically the unit circle. This framework has concrete implications in numeration theory, modular and digital signal processing, neural computation, and data encoding architectures, including contemporary LLMs.

1. Mathematical Foundations: Ring and Polynomial-Based Digit Systems

In algebraic settings, digit-wise circular representations arise in general digit systems over residue class rings $\mathcal{R} = \mathcal{E}[x]/(P)$, where $\mathcal{E}$ is a commutative ring and $P\in\mathcal{E}[x]$ is a polynomial (Scheicher et al., 2010). Each element $A \in \mathcal{R}$ admits a canonical expansion:

$A = e_0 + e_1 X + \cdots + e_h X^h,$

with digits $e_i$ from a finite digit set $\mathcal{N}$, chosen as a complete system of coset representatives modulo $(X)$. The division and digit extraction follows the backward division scheme:

$T(A) = \frac{A - D_\mathcal{N}(A)}{X}$

where $D_\mathcal{N}(A)$ is the unique coset representative of $A \mod X$. Successive application yields digit sequences $(D_\mathcal{N}(A), D_\mathcal{N}(T(A)), \ldots)$ which may be finite, periodic, or ultimately periodic.

The core sequence admits a “circular” characterization, especially in cases with non-monic $P$ or digit sets $\mathcal{N}$ not containing $0$, requiring consideration of zero cycles—finite expansions of $0$ using only “digits”—to pad or align digit decompositions. For example, in some CNS and finite field cases, digits may be selected from intervals not containing $0$, complicating the standard periodicity and padding strategies.

2. Circular Encoding in Signal, Convolution, and Modular Arithmetic

Digit-wise circular representation has foundational importance in digital convolutions, signal processing, and modular arithmetic. A notable construction is the exact, multiplication-free circular convolution transform where signals are encoded digit-wise using bit-shifts and modulo operations (Chandra, 2010). Here, cyclic integers are generated via powers of $2$ (as opposed to roots of unity), with the mapping

$2^N \equiv 1 \pmod{m}$

for chosen prime $m$, ensuring the digital analog of the unit circle is respected.

The digital Binomial Theorem (Nguyen, 2014) and related digital expansion techniques employ explicit digit-wise positionality. For example, expansion coefficients are determined by the sum-of-digits function $s(\cdot)$, restricting terms to carry-free additions (modulo base, often arranged circularly) and preserving the structural interplay between digitwise modular addition and global value updates.

This digital/circular structure is similarly pivotal in modular network coding and array codes, where byte-wise circular shifts yield cyclic convolution products over $Z_{2^m}$, and polynomial symbols are processed by shifting bytes and performing addition modulo $2^m$ (Shum et al., 2020). These operations are efficient in hardware and align inherently with circular digit representations.

3. Circular Coordinate Representations in Topological and Data Analysis

In topological data analysis (TDA) and nonlinear dimensionality reduction, circular coordinate frameworks extract circular structures in high-dimensional data using persistent cohomology (Luo et al., 2020). The classical circular coordinate seeks the smoothest representative of a 1-cocycle $\alpha$ via an $L_2$ norm minimization:

$$\bar{f} = \operatorname{argmin}_f \| \alpha + \delta_0 f \|_{L_2}$$

A recent generalization employs a penalty blending $L_1$ and $L_2$ norms:

$$\bar{f} = \operatorname{argmin}_f \{ (1-\lambda)\|\alpha + \delta_0 f\|_{L_1} + \lambda \| \alpha + \delta_0 f \|_{L_2} \}$$

($\lambda \in [0,1]$), which induces frequent “locally constant” regions punctuated by abrupt jumps. These abrupt transitions correspond to change-points or topological features and are encoded in a circular fashion—i.e., values wrap around as coordinates traverse the circle.

4. Algorithmic Representations: Circular Words, Permutations, and Digit Rotations

In combinatorics and theoretical computer science, digit-wise circular representations generalize the concept of circular words (necklaces) (Hegedüs et al., 2014). Three principal encoding methodologies are used:

• Tuples: Compactly represent circular words by $(u, n)$ where $u$ is a root word and $n$ is the length; $u^{n/|u|}$ then cyclically generates all conjugates.
• Trees (Tries): Graphically capture periodicity and all cyclic shifts within a word, highlighting border structures and common factors across rotations.
• Iterative Representations: Fractional power and cyclic shift operators construct circular words layer by layer, from primitive root to full structure.

For digit-wise encoding, such methodology can be modified for numeric alphabets (digits), supporting error detection, CRCs, and data compression based on cyclic permutations.

5. Neural Representations in LLMs

Recent empirical investigations in LLMs have shown that numbers are internally encoded and manipulated via digit-wise circular representations (Levy et al., 15 Oct 2024, Baeumel et al., 4 Aug 2025):

• Digit Embeddings: Each digit $t$ (in base $b$) is mapped into a circular embedding:

$$\text{circle}_b(t) = [\cos(2\pi t / b),\, \sin(2\pi t / b)]$$

and a probe recovers digit information from hidden states by training on supervised digit recovery tasks.

• Error Localization: Arithmetic errors produced by LLMs are fundamentally digit-local; mistakes are statistically distributed across digit positions rather than centered around numeric value. For instance, addition answers often fail in one digit (units, tens, or hundreds) independent of magnitude.
• Circuit Specialization: There exist modular neural subgroups—distinct MLP subnetworks or “circuits”—which exclusively process specific digit positions in arithmetic tasks. Activation swaps in such circuits causally and selectively alter predicted digits in model outputs without affecting others (Baeumel et al., 4 Aug 2025). Feature selection via Fisher Score identifies neurons contributing strictly to a digit’s subtask.

Aspect	Mathematics / Algorithms	Neural Computation
Encodes digits	Positionally (ring, polynomial, tuples)	Circular embedding per digit (unit circle)
Circular property	Explicit (modulo or cyclic shift/scheme)	Modular, digit-position-specific neural circuits
Error/periodicity	Periodic expansions, carry-free sums	Error localized to digits, not values

6. Implications and Applications

Digit-wise circular representation is central to diverse theoretical and applied domains:

• Numeration and Dynamical Systems: Provides a unified setting for digit systems over rings, enables complete periodic expansions (or establishes conditions for their periodicity and finiteness), and connects with shift radix systems governing admissibility properties (Scheicher et al., 2010, Dombek et al., 2011).
• Signal Processing: Enables fast convolution, robust filtering, and cryptographic primitives free of numerical overflow and roundoff errors (Chandra, 2010, Shum et al., 2020).
• Data Analysis and Visualization: Circular coordinates reveal topological structure, periodicity, and abrupt transitions in high-dimensional datasets (Luo et al., 2020).
• LLM Interpretability and Robustness: Explicit identification and manipulation of digit-wise neural circuits informs architectural refinement and error correction methodologies (Levy et al., 15 Oct 2024, Baeumel et al., 4 Aug 2025).

A plausible implication is that exploiting digit-wise circular representations may improve modular processing in both classical and neural architectures. In theoretical numeration, relaxing monic and zero-digit assumptions invites new phenomena—unstable periodic orbits and necessity for zero cycles—while in neural networks, understanding digit circuit specialization provides avenues for architectural and intervention-based advances.

7. Controversies and Open Challenges

• Uniqueness and Completeness: In generalized digit systems, not all elements admit finite representations unless supplemented by zero cycles or confined to critical submodules (Scheicher et al., 2010).
• Boundary and Symmetry Effects: In negative base systems, periodicity and symmetry of expansions can hinge on delicate choices of interval and digit set, especially in Pisot base and generalized transformation domains (Dombek et al., 2011).
• Numerical Reasoning in Neural Models: Despite high per-digit recovery accuracy, neural models still manifest non-negligible error rates—especially when probing digits outside natural positional boundaries—suggesting the digit-wise circular representation is a dominant but not exclusive mechanism (Levy et al., 15 Oct 2024).
• Field vs. Ring Arithmetic: In coding and convolution systems, circular arithmetic in rings (e.g., $Z_{2^m}$) introduces zero divisors and subfield constraints that may compromise theoretical guarantees (Shum et al., 2020).

Ongoing work continues to extend the principles of digit-wise circular representation to broader algebraic structures, more robust neural computing systems, and richer combinatorial frameworks, with active research focusing on the explicit characterization, causal manipulation, and algorithmic exploitation of digit-level cyclic structures.