Symbolic Genome Structures: A Recursive Information Framework for Molecular Pattern Encoding

The complexity and hierarchical organization of genomic information pose a fundamental challenge for both efficient data compression and the preservation of biologically meaningful patterns. Here we propose a symbolic, recursive information framework called ϕ ∞ that enables encoding of molecular sequence motifs and structures in DNA, RNA, and proteins in a unified algebraic form. In contrast to conventional sequence compression algorithms (e.g., Lempel-Zivbased methods) which maximize redundancy elimination but risk obscuring functional patterns, ϕ ∞ explicitly retains and represents genomic motifs and higher-order folding relationships during compression. The ϕ ∞ framework is defined as a recursive algebra over sequence alphabets, allowing repetitive or structurally conserved motifs to be represented by self-referential symbols across multiple scales. We demonstrate that ϕ ∞ can encode codon sequences and their amino acid translations, preserve RNA base-pairing interactions through augmented symbolic strings, and represent protein α-helix motifs via looped symbolic constructions. Results include a ϕ ∞ recursion tree derived for a segment of the human TP53 gene, illustrating how codons and regulatory repeats can be hierarchically parsed and compressed without loss of semantic information. Additionally, a symbolic folding map for an example RNA segment shows that ϕ ∞ encoding of a sequence together with its secondary structure (χ ⊕ ∆(χ)) is consistent with the direct encoding of the folded structure. Compared to probabilistic models (e.g., hidden Markov models) and recent deep learning embeddings, ϕ ∞ offers improved interpretability, structural consistency, and potential for recursive inference across molecular scales. This approach opens new avenues for integrative genomic data compression, pattern discovery, and knowledge-driven augmentation of computational protein folding tools.