Reproducible Geometric Alignment in Protein Conformational Space: Internal Validation of Golden-Ratio Proximity and π-Scale Toroidal Enrichment

Abstract "This study investigates whether protein backbone conformations exhibit reproducible alignment to specific mathematical targets, specifically the Golden Ratio (ϕ) and π. Using a systematic panel of 2,400 protein structures (comprising over 534,000 ϕ/ψ angle pairs), we tested two preregistered geometric heuristics: golden-ratio proximity and toroidal distance enrichment near π. Applying a 'protein-equal weighting' methodology to prevent large structures from dominating the results, the data was split into discovery and validation subsets. Both heuristics replicated across splits with high statistical significance (p<0.0001). The results suggest that protein folding is not merely a stochastic physicochemical process but follows a highly optimized, non-random geometric framework, potentially functioning as a biological form of 'data compression' for structural stability." Description This goes in the "Notes" or "Description" box to explain your methodology. "Key Findings: Golden Ratio (ϕ) Proximity: Observed mean deviation of ~0.779 vs a null mean of ~0.838, indicating proteins 'tune' their angles toward the golden ratio for stability. π-Scale Enrichment: A consistent ~12.6% hit rate for π-correlated geometry, significantly higher than random null models (p<0.0001). Scale: This analysis represents a high-power validation using a non-redundant dataset of 2,400 proteins. Methodological Perspective: The identification and testing of these specific mathematical constants were driven by a neurodivergent research framework (ASD/ADHD). This approach prioritized cross-disciplinary pattern recognition—bridging number theory and molecular biology—to identify structural 'signatures' that are often overlooked in traditional biochemical analysis. Files Included: protein_geometry_manuscript_draft_v6.pdf: Full technical report and discussion. per_protein_summary_v6.csv: Raw data for all 2,400 proteins analyzed. paper_ready_validation_figure_v6.png: Visual distribution of the angular clustering." protein_geometry_analysis_v6_validation_fixed.py: Code for testing DISCLAIMER I was not aware that I had to disclose whether I used AI as a resource. With that being said: Generative AI was used to assist with [literature screening / coding support / draft language revision]. All AI-assisted outputs were independently checked by the author, and the author takes full responsibility for the final analysis and text. This is encompassing all the work that has been done and will be done.