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March 12, 2026Open Access

The Standard Genetic Code as a Dissipative Optimum: A No-Fit-to-the-SGC Framework from S³-MEP

Key Points

This research aims to derive a framework for the architecture and error robustness of the Standard Genetic Code (SGC) without fitting parameters to it.
Derived integers related to genetic code from spatial dimensions and gauge fiber dimensions of the S³-MEP framework.
Utilized permutation tests with physicochemical descriptors to assess statistical optimality.
Explored geometric interpretations of codon positions through Hopf fibration.
Confirmed statistical optimality of the SGC with significant p-values in permutation tests.
Established relationships between chiral tangent-space and coding-theoretic relations leading to the number of amino acids and codons.
Identified geometric interpretations of degeneracy and curvature at codon positions.

Abstract

This preprint derives a geometric framework for the architecture and error-robustness of the Standard Genetic Code (SGC) from the S³-MEP (Maximum Entropy Production) framework, with no parameters fitted to the SGC. Eight exact integers — 4 bases, codon length 3, 64 codons, 20 amino acids, 3 stop codons, 61 sense codons, mean degeneracy 3. 05, and the NLO bound ≤22 — are derived from the spatial dimension D=3 and the gauge fiber dimension K=7 of the S³-MEP framework (companion paper: DOI 10. 5281/zenodo. 18878097). Two independent routes yield NAA = 20: the chiral tangent-space relation 2 (D+K) = 20 and the coding-theoretic relation |Σ| (|Σ|+1) = 4×5 = 20. Their equality is equivalent to the algebraic identity K = (D²+D+2) /2 = 7, independently derived from Hurwitz–Cartan octonion completion. The wobble degeneracy at codon position 3 admits a geometric interpretation through the Hopf fibration S¹ → S³ → S². The fiber S¹ has zero intrinsic curvature (minimal information by the Čencov–Fisher–Rao theorem), while the base S² has positive curvature (maximal information). Positions 1–2 are associated with the base; position 3 with the fiber. The Klein four-group V₄ ⊂ Center (Isom (S³) ), acting on the four basis vectors of H₁ = VL₁/₂ ⊗ VR₁/₂, generates all three biological classifications of nucleotide bases: purine/pyrimidine (right isospin σR), amino/keto (left isospin σL), and Watson–Crick complementarity (antipodal map σL σR). Statistical optimality is confirmed by permutation tests on S₂₀ with 26 published physicochemical descriptors (none fitted to the SGC) and four parameter-free error channels. At the strongest explored PCA projection (5 principal components, 88% variance), Fisher's combined p-value is 5. 99 × 10⁻⁹ under a channel-independence approximation — exceeding the significance of the Freeland & Hurst (1998) benchmark. The formula NAA = |Σ| (|Σ|+1) retrodicts the ancestral code: |Σ|=2 → 6 AA (doublet phase), |Σ|=3 → 12 AA (early triplet), consistent with 10 independent phylogenetic reconstructions and the Trifonov (2004) chronological ranking, which exhibits phase boundaries at exactly the predicted positions (ranks 6 and 12). Keywords: genetic code, maximum entropy production, S³ geometry, Hopf fibration, wobble, amino acids, error minimization, permutation test, Fisher information, Klein group, dissipative structure, MaxCal

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