The Coherosance Framework: Modular Symmetries in Figurate Number Sequences and Their Structural Resonance with the Genetic Code

AbstractWe present a systematic investigation of digital root cycles arising from triangular,Fibonacci, and tetrahedral figurate number sequences under modulo-9 reduction. Thetriangular sequence generates a unique period-9 digital root cycle [1, 3, 6, 1, 6, 3, 1, 9,9] shared exclusively by the dodecagonal (s = 12) polygonal family. From this cycle, afinite Triplet-Window Lattice is constructed exhibiting three provable algebraicproperties: stability under positional encoding, self-mirroring under digit reversal, anda 13-encoding property unique to its Bridge subfamily.Mapping the lattice digits {1, 3, 6, 9} bijectively to the nucleotide bases {G, A, C, T}defines Anchor (DR=1) and Bridge (DR=4) codon families. Exhaustive testing of all 24possible mappings identifies a canonical mapping (1→G, 3→A, 6→C, 9→T) thatuniquely recovers the seed motif {GCA, GAC, GTT} and yields Anchor/Bridge usageratios ranging from 1.10× to across nine divergent organisms (bacteria, yeast,human, archaea, virus, fly, plant, nematode). This mapping ranks third in overall signalstrength. A strict null model matching both GC content (~55.6%) and degeneracyprofile shows the preference exceeds random expectation in most lineages (p < 0.05 ineukaryotes, archaea, and virus).Analysis of human laminin genes reveals elevated representation of Anchor-encodedamino acids. While codon usage is shaped by multiple biochemical, translational, andevolutionary factors, these results indicate a non-trivial structural resonance betweenfigurate number symmetries and observed codon preferences under specific3 3. 3.06×mappings. The Coherosance Framework offers a novel mathematical lens for exploringbiological information architecture and warrants further investigation. More information: https://coherosance.com