LLMs and Carbon Chains

Title: A Formal Comparative Model of Finite-State Carbon Programs and Tokenized Sequences in Large Language Models (LLMs)

under construction 7/6/2025

Abstract: This paper presents a novel formal comparison between finite-state carbon chain programs, derived from constrained covalent bond rotations, and tokenized sequences in large language models (LLMs). Through structural, functional, and mathematical parallels, we propose that discrete molecular systems like programmable carbon chains can serve as analogs to computational token systems in artificial intelligence, opening new avenues for bio-inspired computation and molecular informatics.

1. Introduction Molecular systems, particularly programmable carbon chains, demonstrate discrete state transitions based on chemical rotations and bond conformations. Analogously, LLMs operate over sequences of discrete tokens governed by statistical and semantic transformations. Despite their different domains, both systems can be modeled with formal finite-state architectures, suggesting a powerful underlying computational similarity. This paper formalizes the comparison and explores its implications.

2. System Definitions

Concept Carbon Chain Program LLM Token Sequence Unit Covalent bond in discrete states Token (word, subword, character)Alphabet (\u03a3)Set of allowed rotational states (e.g., {0°, 180°})Vocabulary of tokensProgramSequence of bond states {s1, s2, ..., sn}Sequence of tokens {t1, t2, ..., tn}Transition Function (\u03b4)Physical rules governing bond-to-bond rotationsTransformer-based attention and embeddingsInitial StateFixed first bondBOS (Beginning of Sentence) tokenOutput3D spatial trajectory of terminal carbonText generation, embeddings, semantic vectors

3. Mathematical Formalism

3.1 Finite-State Carbon Chain (FSM Representation):Let \u03a3 = {0, 1, ..., m} denote the discrete rotational states. Let Q = set of molecular spatial configurations. \u03b4: Q \u00d7 \u03a3 \u2192 Q is the bond transition function. Starting from an initial state q0:

q0 --s1--> q1 --s2--> q2 --s3--> ... --sn--> qn

The final spatial configuration qn is a deterministic function of the sequence {s1, ..., sn}.

3.2 Tokenized LLM Sequence:Let T = token vocabulary, and H0 be the initial state (e.g., BOS). At each step i:

Hi = fi(Hi-1, ti)

Where fi represents the transformer function that computes the hidden state using previous context and current token.

4. Structural Correspondence

FeatureCarbon ProgramLLMInitial ConditionFixed orientationBOS tokenContextual LogicDownstream bonds affectedSelf-attention contextTransformationRotational transitionLearned embedding dynamicsOutput Interpretation3D structure or motifGenerated or decoded text

5. Encoding and EmergenceCarbon programs can encode structural motifs such as helices, sheets, or loops, derived from simple rotation rules. Similarly, LLMs generate syntactic or semantic constructs from token sequences. Both exhibit emergent complexity: protein-like folds in carbon chains and linguistic intelligence in LLMs.

6. Applications and Implications

• Molecular AI: Predict molecular behavior using LLM-style training on bond-state sequences.
• Bio-semantic Embedding: Learn molecular embeddings analogous to word embeddings.
• Shape-to-Text Translation: Map molecular conformation sequences to textual descriptions.
• Programmable Matter: Use molecular chains to compute logic operations.
• 
  

7. ConclusionFinite-state carbon chains and tokenized LLMs, though arising from different sciences, can be modeled with shared computational formalisms. This opens opportunities for bio-inspired computing, synthetic biology applications, and new hybrid AI-material systems where matter and machine converge through structure, symbol, and state transitions.

Keywords: carbon chains, finite-state machine, large language model, token sequences, modular rotation, molecular computing, biomatics, protein dynamics