The Crystallographic Restriction and the Six-Fold Boundary: Why Six Is Forced, Why Five Is the Baseline, and Why Quasicrystals Are Rendered

The foundational architecture, geometric intuitions, and motivating philosophy of this framework originated independently with the author prior to and separate from any AI involvement. Subsequent mathematical derivation, connection to established physics literature, computational verification, and error-checking were conducted through extended technical work with AI systems (Anthropic’s Claude and Google’s Gemini). The author directed this process, evaluated and selected among proposed derivations, and takes full responsibility for the accuracy and originality of the final work.