The Triangular Root of the Genetic Code's Mathematical Substrate
Authors/Creators
Description
The origin of the genetic code's error-detecting properties remains a central question in molecular biology. We report the discovery of an Infinite Geometric Ladder generated by the digital roots of polygonal number sequences that uniquely identifies a family of circular codes. Specifically, the digital root cycles of triangular (3-sided), dodecagonal (12-sided), icosihenagonal (21-sided), tetracontaoctagonal (48-sided), and hexacontasexagonal (66-sided) numbers are mathematically identical. This shared identity arises because these side counts all satisfy the modulo-9 congruence condition (s ≡ 3 mod 9), generating a 'seed' motif {GCA, GAC, GTT} found in exactly 9 of the 216 maximal self-complementary C₃ circular codes. This geometric spine is further reinforced by a temporal resonance, where the division of 48 by 16 yields 3, mirroring the triangular root, and the 66-sided geometry provides an "over-completion" of the 64-codon alphabet. We further demonstrate a 216-24-9 Lock, where the entire 216-code universe is perfectly partitioned by the 24-sided harmonic into this unique 9-code family, exactly isolating its size within the 27 equivalence classes. Genomic analysis across three kingdoms of life (Escherichia coli, Saccharomyces cerevisiae, and Homo sapiens) reveals that members of this family exhibit significant frame-0 enrichment and frame-shift suppression. These findings suggest that the structural organization of the genetic code is grounded in a fundamental mathematical substrate, moving beyond stochastic models toward a deterministic geometric framework.
Abstract
Version 4 (V4) — Mathematical Integrity Update
This version supersedes V3 with two targeted mathematical corrections identified through a full computational integrity audit. First, the description of the geometric spine [3, 12, 21, 48, 66] has been made mathematically precise: these side counts are now correctly characterised by the modulo-9 congruence condition (s ≡ 3 mod 9), which is the provable, necessary and sufficient condition for the polygonal number formula to generate the identical digital root cycle. The previous recurrence relation stated in V3 was an approximation that held for only the first two terms. Second, the parenthetical description of the 24-sided polygonal cycle in the 216-24-9 Lock section has been corrected to reflect the computationally verified cycle (1,6,6,1,9,3,1,3,9). The 216-24-9 Lock arithmetic itself (216 ÷ 24 = 9) remains fully intact and unchanged. The abstract has also been updated to incorporate the corrected modulo-9 framing and to tighten the narrative flow. All core findings — the Infinite Geometric Ladder, the 9-code family, the 216-24-9 Lock, and the cross-kingdom genomic enrichment — are unchanged and fully verified.
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TriangularRoot_v4_FINAL (1).pdf
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