Digit-wise Circular Representations

• Digit-wise circular representations are constructs that treat digits or symbols as cyclic elements, providing a framework for modular arithmetic and combinatorial analysis.
• They employ tuple, trie, and iterative methods to encode circular sequences compactly, facilitating efficient transformations like digital convolution and re-radix operations.
• Applications span digital signal processing, cryptographic schemes, and neural network computations, offering robust and precise error-resilient techniques.

Digit-wise circular representations are mathematical and algorithmic constructs in which the elements (typically digits, but sometimes bits or symbols) of a string or number are treated with explicit reference to their positions and cyclic structure. Such representations arise in diverse domains: digital convolution, combinatorial and geometric analysis of permutation spaces, symbolic representations of words and numbers, modular arithmetic algorithms, and the internal mechanisms of large neural networks handling arithmetic and symbol manipulation. Common to all is the fact that elements are processed locally as parts of a circle or cyclic group, whether for efficient computation, representation compression, analysis of transformation distance, or interpretability of numerical processing.

1. Mathematical Foundations: Cyclic Groups and Circular Structures

The algebraic foundation of digit-wise circular representations is rooted in cyclic groups and modular arithmetic. In digital signal processing, a canonical example is the number-theoretic transform presented in "Fast Digital Convolutions using Bit-Shifts" (Chandra, 2010). Instead of complex exponentials, digit-wise circular representations employ powers of 2 modulo a carefully chosen prime $m$ such that $2^N \equiv 1 \pmod{m}$ for the transform length $N$ (a power of two). The resulting sequence $\{1,2,2^2,\ldots,2^{N-1}\}$ wraps around the digital circle:

$$X(u) = \sum_{t=0}^{N-1} f(t)\,\cdot\,2^{ut} \pmod{m}$$

$$x(t) = \frac{1}{N} \sum_{u=0}^{N-1} \langle X(u)\cdot 2^{-ut} \rangle_{m}$$

Unlike classical DFT, all operations are additions, bit-shifts, and modular reduction. The modulus $m$ is selected (perhaps via Carmichael's Theorem generalizing Euler's Theorem) to ensure a cyclic group of requisite order, guaranteeing exactness, invertibility, and immunity to quantization and overflow errors.

In combinatorics, the paper of circular sequences and multipermutohedra (Aravamuthan, 2010) models permutations as paths on the vertices of a geometric polytope, transitioning between arrangements via adjacent transpositions ("switches"). The bound

$x([n]) \leq s(y) \leq x([n]) + l\cdot r$

captures the minimal number of digit-wise operations needed to cycle between configurations, with each position treated as a "gate" marking potential operations.

2. Representational Schemes: Tuples, Trees, and Iterative Methods for Circular Words

Symbolic representation of circular words and digit-strings admits several compact and algorithmically tractable forms (Hegedüs et al., 2014):

• Tuple Representation: A circular word (necklace) is encoded as $(u,n)$ where $u$ is a (possibly primitive) root and $n$ is the length. The cyclic structure is captured compactly, with $u^{n/|u|}$ furnishing all conjugates.
• Tree (Trie) Representation: The set of all cyclic shifts is represented graphically; branches mark positions of divergence, and repeated structure is evident in shared paths. Each digit-wise rotation corresponds to a path in the tree.
• Iterative Representation: Successive application of fractional power and cyclic shift operators rebuilds a circular word from a minimal seed:

$(u, \ell_1, k_1, \ell_2, k_2, \ldots, \ell_m)$

with $u_0 = u,$ $u_i = u_{i-1}^{\ell_i/|u_{i-1}|}$, and shifts as needed, compactly encoding complex periodicity.

These representations are extensible to digit-wise contexts. For cyclic digit sequences, tuple and iterative formulations allow both compression and rapid enumeration of conjugates, facilitating efficient encoding, pattern recognition, and transformation.

3. Algorithms and Transformations: Bitwise, Modular, and Combinatorial Methods

Digit-wise circular representations underpin exact, efficient computation in several settings:

Digital Convolution

Bit-shift based transforms (Chandra, 2010) realize circular convolution over finite integer fields. Multiplication by $2^k$ is a bit-shift, and all arithmetic is modulo $m$. The Carmichael construction extends applicability to arbitrary power-of-two lengths, and bit-level manipulations achieve precise, efficient computation without floating-point roundoff.

Transpositions and Distance in Circular Strings

For circular binary strings, the minimal sequence of transpositions needed to transition between strings is determined by analyzing partitions induced by the digits (Nakanishi, 2017). Key functions:

$$f_1(S, r) = \max\left\{ \sum_{j=1}^r s_{i_j} \right\}, \quad f_2(S, r) = \min\left\{ \sum_{j=1}^r s_{i_j} \right\}$$

Bounds relating the weights of partitions yield sharp criteria for the transposition diameter:

$$d(S, T) = k-1 \iff f_1(S,1) > f_1(T,1) \geq f_2(T,1) > f_2(S,k-1)$$

Digit-wise circular structure enables efficient combinatorial optimization in sequence alignment, genomic rearrangement, and error-tolerant digital systems.

Arithmetic Digit Manipulation

Closed-form digit-extraction and "circular radix" algorithms are realized entirely in primitive arithmetic operations (Boylan, 2021):

• Trailing-Digit Truncation: $\overline{T}_b^n(x) = \lfloor x/b^n \rfloor$
• Digit Selection: $$D_b^n(x) = \overline{T}_b^n(x) - b\cdot \overline{T}_b^{n+1}(x)$$
• Re-Radix Function: Rearranges digits circularly around a marker.

Such formulations enable modular digit-by-digit manipulation required in data encoding, cryptography, base conversion, and analysis of pathological mathematical functions.

4. Neural Computation: Circular Digit Representations in LLMs

Recent probing and causal intervention studies of LLMs (Levy et al., 15 Oct 2024, Baeumel et al., 4 Aug 2025) demonstrate that digits in numbers are encoded and manipulated via circular representations:

• Circular Mapping for Digits: Each digit $t$ in base $b$ is represented as a vector

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

with distinct circuits (groups of MLP neurons) dedicated to each digit position. Evidence from feature probing shows that digits are reconstructed independently from per-layer activations using $\text{atan2}$ on the probe output.

• Digit-wise Error Patterns: Arithmetic mistakes occur as isolated digit errors, not value-based noise. For example, the addition $132+238+324+139 = 833$, but models may output $633$ or $823$. This reflects independent processing of each digit within its circular group.
• Causal Interventions: Targeted manipulation of the circular representation of a particular digit (e.g., adding $+5\ \bmod\ 10$ to the tens digit) yields predictable, digit-local changes in the output, confirming the modular, compositional nature of these internal representations.
• Fisher Score-Based Circuit Identification: The circuits responsible for each digit's computation are isolated using cross-class mean activations and Fisher scores:

$$F_{i,d} = \mathbb{E}_{c \in C_d}\left[ (H_{i,c,d} - H_{i,d})^2 \right] / \mathbb{E}_{c \in C_d}\left[ \text{Var}(a_i(x)|c) \right]$$

These properties are robust across model size, tokenization, and persist for arithmetic operations including addition, subtraction, and multi-digit tasks.

5. Applications in Digital Systems, Cryptography, and Algorithm Design

Digit-wise circular representations have practical consequences across several technological domains:

• Exact, Efficient Digital Convolutions: Eliminating quantization and overflow errors in digital signal processing with modular, bit-shift-based transforms (Chandra, 2010).
• Combinatorial Optimization on Cyclic Structures: Modeling worst-case operation counts for cyclic digit rearrangements and automated hardware transitions (Aravamuthan, 2010).
• Efficient Representation and Analysis of Repetitive and Cyclic Patterns: Compact encoding and rapid enumeration of cyclic repeats in numerical, symbolic, and biological sequences (Hegedüs et al., 2014).
• Robustness and Interpretability in Deep Learning: Direct theoretical and empirical linkage between digit-wise cyclic processing and error patterns in neural arithmetic and symbolic computation (Levy et al., 15 Oct 2024, Baeumel et al., 4 Aug 2025).
• Cryptographic and Coding Schemes: Controlled, invertible digit rearrangement for secure data manipulation (Boylan, 2021).

6. Limitations, Generalizations, and Open Problems

Digit-wise circular representations are not universally optimal and face limits in generalization:

• Alphabet Size and Structure: Some combinatorial and structural results hold only for binary alphabets and require extension for larger digit sets (Hegedüs et al., 2014).
• Carry and Inter-Digit Dependencies: While most representations treat digits independently, real-world arithmetic includes carries and correlations that may require more complex (still modular) circuits (Baeumel et al., 4 Aug 2025).
• Multiple Representation Coexistence: Neural systems may maintain redundant, overlapping mechanisms alongside digit-wise circular representations (Levy et al., 15 Oct 2024), presenting interpretability challenges.
• Symbolic vs Numeric Encoding: The transition between word-form numbers and digit-wise representations remains an active area (Levy et al., 15 Oct 2024).

7. Historical Context and Theoretical Implications

The formalization and exploitation of digit-wise circular representations reflects an evolution from group-theoretic and combinatorial constructs to algorithmic and neural implementation:

• Number-Theoretic Foundations: Rader Transforms, Carmichael's generalization, and permutohedral geometry provide mathematical rigor and generality in cyclic structures (Chandra, 2010, Aravamuthan, 2010).
• Symbolic Representation Theory: Advances in tuple, trie, and iterative methods unify symbolic and digit-based approaches (Hegedüs et al., 2014).
• Algorithmic Arithmetic: Closed-form manipulation in base conversion and pathological functions, such as the Conway Base-13, reveal circular digit processing as both an analytic tool and practical algorithm (Boylan, 2021).
• Neural Computation: The discovery and quantification of digit-wise modular circuits in LLMs opens avenues for both improved algorithmic design and deeper understanding of compositional reasoning in neural systems (Levy et al., 15 Oct 2024, Baeumel et al., 4 Aug 2025).

A plausible implication is continued convergence between algebraic, combinatorial, and neural paradigms in the representation and manipulation of cyclic (including digit-wise) data structures, with open research in generalization, optimization, and application domains.