Entangled String–Number Representations

• Entangled string–number representations are hybrid encodings that simultaneously capture syntactic structure and numerical value, ensuring reversible arithmetic manipulations.
• They employ methods like Smullyan and Markov codings where operations such as concatenation mirror arithmetic addition or matrix multiplication.
• These representations underpin advances in formal logic, combinatorics, and AI, facilitating optimized data encoding, Gödel numbering, and transformer model analysis.

Entangled string–number representations encode the intricate duality between strings and numbers, where a single mathematical object simultaneously carries both a syntactic (string-like) and a semantic (number-like) interpretation. This dual view underlies foundational constructions in logic, formal arithmetic, combinatorics, quantum information theory, and empirical studies of learned representations in artificial intelligence. Such representations arise in settings as diverse as weak arithmetical theories, algebraic structures, optimally encoded data, quantum states with fixed entanglement, and transformer-based LLMs, each revealing distinct facets of string–number entanglement.

1. Foundational Concepts and Definitions

A string–number representation involves mapping numbers (typically nonnegative integers) onto strings (finite sequences over an alphabet) in such a way that the number is recoverable from its string and, frequently, vice versa. The entanglement occurs when operations carried out in one domain (e.g., concatenation of strings) correspond naturally to operations in the other (e.g., arithmetic addition), and the coding is robust enough for mathematical manipulation.

A further refinement appears in logic and formal arithmetic, where "ur-strings" denote primitive concatenative objects that may lack explicit projection and length functions but can be effectively manipulated via coding constructs. In computational and algebraic contexts, these representations must be injective and preserve structural properties under concatenation or matrix product.

2. Smullyan Coding and its Arithmetic Realization

Smullyan coding, inspired by foundational studies in formal systems, implements binary strings via arithmetization suitable for weak arithmetic bases. Sequences are encoded by indirect binary string coding, often leveraging pairing functions such as

$\operatorname{pair}(x, y) := (x+y)^2 + x$

effective in base theories like PA₀ or PA₁. Extraction of sequence segments uses a β-function, e.g.,

$$B(x,i,w) := \exists u,v,q\; [w = (u,v) \,\wedge\, u = q(1+(i+1)v) + x \,\wedge\, x < 1+(i+1)v ]$$

where the parameters $u, v, q$ are tailored to encode and decode finite sequences.

Smullyan coding eschews exponential growth by employing a length-first (also: dyadic, bijective base-2) numeral system. Here, a function $A(x)$ replaces each letter in a string with "a," retaining the length profile—e.g.,

$A(\operatorname{sm}(n)) = l(n) - 1$

with length determined by the largest power of 2 dividing $n+1$. Concatenation and pairing exploit reserved separators $b$, ensuring codes are unambiguous: $(x, y) = A(x)\, b\, x\, y$ Selection of the separator is nontrivial; strategies include constant separator, growing separator, and extension to larger alphabets. These methods enable reversible, low-strength arithmetic coding for sequences (or ur-strings) under resource-constrained arithmetic logics.

3. Markov Coding via Special Linear Monoids

Markov coding translates binary strings into sequences of $2 \times 2$ matrices with nonnegative integer entries and determinant 1: $$SL_2(\mathbb{N}) = \{ M \in \mathbb{N}^{2\times2} \mid \det(M)=1 \}$$ Defining letters as generators,

$$A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \ 1 & 1 \end{pmatrix}$$

any binary string maps to a unique matrix by left-to-right multiplication of letter matrices.

Notably, while tally-based codings grow exponentially, Markov's method yields linear growth, as illustrated by

$$(BA)^n = \begin{pmatrix} F_{n-1} & F_n \ F_n & F_{n+1} \end{pmatrix}$$

where $F_n$ is the nth Fibonacci number—highlighting the direct connection between matrix products and combinatorial sequences. Canonical normal forms, Euclidean division, and (sometimes) additional axioms (e.g., left cancellation) are needed to ensure uniqueness and proper decomposition, especially in weak arithmetical theories. Diophantine definability of relevant functions confirms their utility in metamathematics and computability.

4. Entanglement: Syntactic–Semantic Duality and Formal Applications

Both Smullyan and Markov codings exemplify the entanglement between number and string representations. In Smullyan's approach, arithmetic operations encode functional properties—pairing guarantees order retrieval; β-functions act as projectors. In Markov's approach, concatenation corresponds to matrix multiplication, with group-theoretic structure enforcing unique string interpretation.

These entangled representations enable full arithmetization of syntax, e.g., Gödel numbering for proofs or programs, and facilitate the construction of partial satisfaction predicates, models of weak theories, and computability constructs. The interaction between arithmetic operations and formal language operations deepens the understanding of the boundaries between arithmetic and syntax, crucial for proofs of incompleteness and the metamathematical paper of model strength.

5. Conditions for Effective and Robust Coding

Successful entangled string–number coding requires arithmetic frameworks with sufficient expressivity but limited growth. Smullyan coding is suited for base theories supporting recursive pairing but not requiring exponentiation (length-first encoding sidesteps this issue). Markov coding mandates that the ambient arithmetic (discretely ordered commutative rings or similar) supports unique matrix normal forms; Euclidean division principles or Bézout conditions may be necessary.

Certain coding strategies rely on additional axioms (e.g., cancellation, Editor Axiom tc8) to capture concatenation faithfully. Attention to these conditions ensures that codings are injective, that operations are reversible, and that representations can be manipulated in logical and algebraic settings.

6. Broader Significance and Practical Impact

Entangled string–number representations serve as theoretical bridges across logic, combinatorics, computability, and formal language theory. They illuminate the logical strength of weak theories by demonstrating that sophisticated syntactic operations are representable within limited arithmetic. The ability to recover sequences, sets, or syntactic objects from numbers underpins not only foundational results (Gödel’s theorems, MRDP theorem) but also applications in algorithmic information theory, programming language theory, and combinatorial constructions.

These representations also provide templates for real-world coding problems, such as prefix-free data encoding, probabilistic code assignment (via distribution functions and Catalan encodings), and optimized encoding for partial recursive functions (see (Ogden, 2012) for relevant constructs).

In artificial intelligence contexts, as shown in (Marjieh et al., 3 Feb 2025), entangled string–number representations arise naturally within the representational geometry of transformer models trained on natural language. Here, the blending of string and number features—such as Levenshtein and log-linear distances—reflects the dual demands of text and numerical reasoning, with significant implications for model performance in realistic decision scenarios.

7. Mathematical Formulas and Structural Summary

Key mathematical expressions highlight the structural properties of entangled string–number representations:

• Smullyan pairing: $\operatorname{pair}(x, y) = (x + y)^2 + x$
• β-function for projections: $B(x, i, w)$ as defined above
• Markov matrix product: $(BA)^n$ expresses Fibonacci sequence entries in matrix form
• Length extraction in length-first encoding: $l(x) = y \iff y$ is the maximal power of 2 with $y \le x+1 < 2y$
• General coding of sequences as matrix products: $B\, A(n_0)\, B\, A(n_1)\, \cdots\, B\, A(n_{k-1})$

These codings achieve injectivity, reversibility, and robust growth control, essential for rigorous mathematical applications in logic and theoretical computer science.

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The paper and application of entangled string–number representations reveal a deep unity of arithmetic and syntax, fundamental for modern mathematics, logic, and computational theory. Whether in the context of weak arithmetic, algebraic groups, data encoding, or machine-learned representation, the entanglement between string and number remains a central organizing principle.