NumeroLogic: Structured Number Reasoning

• NumeroLogic is a family of frameworks that explicitly encodes numerical data using symbolic, neural, and combinatorial methods for improved numeral reasoning.
• It introduces innovations such as augmented tokenization, hierarchical numeral modeling, and combinatorial digit analysis to enhance arithmetic task performance.
• The approach integrates efficient indexing, alternative numeration systems, and unified number-theoretic transforms to offer precise, actionable computations in modern algorithms.

NumeroLogic refers to a family of theoretical, algorithmic, and representational frameworks for encoding, reasoning, and operating with numbers—spanning symbolic, structural, neural, and combinatorial approaches. It encompasses innovations in number representation for machine learning models, arithmetic logic in alternative numeration systems, combinatorial constructions for digit-based properties, and formal methods for the efficient enumeration of number classes. Across these perspectives, a unifying theme is the precise control or explicit logic applied to the structure, semantics, or processing of numerical data, often with the objective of improving computational, learning, or representational properties.

1. Explicit Number Encodings for Enhanced LLM Numeracy

Numerical reasoning remains a distinct weakness for LLMs trained on standard textual corpora, primarily due to the limitations of plain text number representation. In standard practice, a number such as “42” is given as a sequence of ASCII characters, leaving a decoder-only model unaware of where higher place-values begin until the entire sequence is processed. NumeroLogic (Schwartz et al., 30 Mar 2024) proposes an augmentation in which every number is prefixed with its digit count (and, for floating-point numbers, a tuple indicating integer and fractional digit counts), demarcated by specialized tokens:

• Example for integer: 42 → <sn\>2<mn\>42<en>
• Example for float: 3.14 → <sn\>1.2<mn\>3.14<en>

This encoding informs the model about the structural scope of each number before emitting/examining digits, acting as a form of chain-of-thought (CoT) induction and providing immediate place value awareness. Experimental results demonstrate substantial accuracy gains in arithmetic reasoning (e.g., addition accuracy from 88.37% to 99.96% and subtraction from 73.76% to 97.20% in NanoGPT), along with consistent improvements across multiplication, trigonometric, and square root tasks. In general NLP tasks (e.g., MMLU), even modest overall gains are observed, with numerically dense sub-tasks benefiting most (e.g., STEM: +0.79% absolute improvement). The encoding is implemented via preprocessing (with regular expressions) and postprocessing routines, leaving the base model architecture unchanged.

2. Hierarchical and Continuous Approaches to Numeral Modeling in LLMs

Traditional LLMs treat numbers as discrete tokens, conflating numerals with words in a fixed vocabulary; this obscures numerical continuity and can cause large out-of-vocabulary (OOV) rates for numerals. “Numeracy for LLMs” (Spithourakis et al., 2018) introduces two core advances:

Hierarchical Softmax for Word–Numeral Separation

The probability of the next symbol is decomposed as

$$p(s_t\mid h_t) = \sum_{c_t\in\{\text{word},\,\text{numeral}\}}p(c_t\mid h_t)\,p(s_t\mid c_t,\,h_t)$$

allowing separate numerical and lexical branches with distinct embeddings. Empirical results show that, on numeral subsets, perplexity is improved by two and up to four orders of magnitude compared to non-hierarchical models.

Continuous Density Modeling with Mixture of Gaussians

Numerals are modeled directly as draws from a probability density function $$q(v) = \sum_{k=1}^K\pi_k\mathcal{N}_k(v; \mu_k, \sigma_k^2)$$ (with mixture weights computed from the model’s hidden state). Probabilities over numerals are approximated as:

$$\tilde{Q}(v\mid r) = \int_{v-\epsilon_r}^{v+\epsilon_r}q(u)\,du = F(v+\epsilon_r) - F(v-\epsilon_r)$$

with $\epsilon_r = 0.5 \times 10^{-r}$ for r decimal digits of precision. This approach enables open-vocabulary number modeling. Regression metrics show that modeling numerals with this continuous MoG reduces mean absolute percentage error (MAPE) by 18% and 54% relative to the best alternative on clinical and scientific data sets, respectively.

3. Digit-Based Logic and Combinatorics: Divisibility Properties

A canonical example of digit-centric Number Logic is the paper of numbers divisible by the product of their nonzero digits in a specified base (Sanna, 2018). For base $b\geq 3$, define

$p_{b,0}(n) = \prod_{\substack{j \ d_j \neq 0}} d_j$

and

$$\mathcal{N}_{b,0}(x) = \{n \leq x : p_{b,0}(n)\text{ divides } n\}$$

Using combinatorial and analytic optimization (over digit frequency distributions and entropy-type expressions), Sanna proved that the count of such numbers satisfies

$$x^{\rho_{b,0} + o(1)} < \#\mathcal{N}_{b,0}(x) < x^{\eta_{b,0} + o(1)}$$

with explicit constants $0 < \rho_{b,0} < \eta_{b,0} < 1$ (optimized via constraints involving expressions such as $S_{b,0}(s) = \sum_{d=1}^{b-1}d^s$). In base 10, the improved bounds are

$x^{0.526} < \#\mathcal{N}_{10,0}(x) < x^{0.787}$

for large x, narrowing previous results. This demonstrates that digit-based divisibility, despite being restrictive, yields a set of natural density polynomial in x, providing structural insight into the arithmetical logic operated at the digit level.

4. Alternative Numeration Systems and Automata-Theoretic Arithmetic

NumeroLogic also encompasses numeration systems that generalize or depart from standard positional notation. Notably:

Fibonacci-Equivalent Two’s Complement

The Fibonacci complement system (Labbé et al., 2022) defines the value of a binary word $w = w_{k-1}\dots w_0$ as

$(w) = -w_{k-1} F_k + \sum_{i=0}^{k-1} w_i F_i$

with $F_i$ Fibonacci numbers and the most significant bit assigned negative weight. Operations such as addition are carried out as in classical two’s complement, using digitwise addition followed by normalization through a finite-state transducer (the Berstel adder, extended with new transitions to handle negative numbers). The system is shown, with a constructive proof, to define an increasing bijection between $\mathbb{Z}$ and an admissible language $D\subseteq\{0,1\}^*$ avoiding forbidden patterns (e.g., consecutive 1’s, improper padding).

This approach generalizes the hardware-friendly properties of two’s complement to Fibonacci-based numeration and demonstrates the compatibility of arithmetic logic with automata-theoretic constructions.

5. Canonical and Efficient Indexing: Bijections for Finite-Decimal Numbers

Efficient number indexing is addressed in (Rithvik, 14 Aug 2025), which constructs an explicit bijection between all real numbers having terminating decimal expansions and the naturals $\mathbb{N}$, using a 4-tuple parametrization $(\text{sign}, N_1, N_2, N_3)$:

• sign: +1 or –1
• $N_1$: integer part (absolute value)
• $N_2$: number of leading zeroes after the decimal for absolute numbers <1
• $N_3$: significant digits in the fractional part (with trailing zeros suppressed).

A canonical mapping is defined, and every such number is assigned a unique index via closed-form, lexicographic formulas. Both the forward (number→index) and inverse (index→number) functions run in O(1) time, with no iteration or searching, e.g.,

$$C(K) = \begin{cases} 1 & K = 0 \ K(K+1) + 2 & K > 0 \ \end{cases}$$

for complexity level K. Lexicographic position is calculated by summing counts for tuples with lesser components, enabling rapid and exact enumeration. This systematic enumeration provides a theoretical foundation for finite decimal arithmetic in computing, constructive number theory, and data indexing, demonstrating that the subset of computable reals (constructively meaningful numbers) is precisely and efficiently manageable via NumeroLogic methods.

6. Morphic and Combinatorial Dynamics in Non-Standard Representations

Sum-of-digits functions and associated combinatorial dynamics are key in the analysis of Zeckendorf and base-φ (golden ratio) expansions (Dekking, 2020). These expansions yield unique representations of integers as sums of non-consecutive Fibonacci numbers or powers of φ, respectively, with associated sum-of-digits sequences that are “morphic”; that is, they are projections of fixed points of morphisms on infinite alphabets. First-difference sequences (discrete analogs of derivatives) of these sum-of-digits functions are characterized by generalized Beatty sequences (of the form $V(n) = p\lfloor n\alpha \rfloor + qn + r$), providing an explicit combinatorial description of where the sum increases, decreases, or remains constant. The morphic and Beatty structures highlight deep interplay between arithmetic, symbolic dynamics, and combinatorics central to contemporary NumeroLogic.

7. Unified Number-Theoretic Transforms via Modular Exponential Theorems

Generalized number-theoretic transforms (Semotiuk, 2020) leverage an “exponential function theorem”:

$(x \bmod p) \equiv( s^x \bmod (s^p-1) )$

and define direct and inverse transforms:

$$X(k) = \sum_{i=0}^{N-1} x(i)s^{-(ki\bmod N)} \pmod{M}$$

$$x(i) = \frac{1}{N} \sum_{k=0}^{N-1} X(k)s^{(ki\bmod N)} \pmod{M}$$

for arbitrary base s (possibly complex), unifying classical transforms (Mersenne, Gauss, Fourier) as special cases. These frameworks formalize the “duality” of arithmetic operations in transform space (shift, sum, linearity, convolution), controlled precisely by the modular and exponential structure. Dual forms, periodicity, and orthogonality properties are formalized; these yield a highly generalizable logic for exact, algebraic computation on residue rings, further broadening the scope of NumeroLogic.

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NumeroLogic, across these domains, designates both the specifically named encoding method and the wider principle of explicit, structurally-informed arithmetic, combinatorial, and representational logic applied to numbers and numerals in symbolic, neural, or modular-algebraic regimes. Central to these developments are formal abstractions and precise transformations that enable both deep learning architectures and classical algorithms to reason effectively about number, structure, and semantics.