III. Mathematical Framework
A. Genotype--Phenotype Mapping: Epistasis and Pleiotropy Formalization
A central requirement for a CAS-based account of evolution is a rigorous description of how genetic variation translates into phenotypic traits that are subject to selection. Traditional population genetics often models loci as additive contributors to fitness, an assumption that simplifies analysis but obscures the reality of epistasis (nonlinear interactions among loci) and pleiotropy (a single locus influencing multiple traits). These two mechanisms are indispensable for explaining how complex, synchronized phenotypes emerge, because they create dependencies that tie multiple traits into integrated adaptive modules.
We formalize the genotype of an individual as a vector of allelic states:
g=(g1,g2,...,gL),gk{0,1} or gkR,\mathbf{g} = (g_1, g_2, \dots, g_L), \quad g_k \in \{0,1\} \ \text{or}\ g_k \in \mathbb{R},g=(g1,g2,...,gL),gk{0,1} or gkR,
where LLL denotes the number of loci, and gkg_kgk encodes either a discrete allelic state (biallelic approximation) or a quantitative effect size.
The phenotype is represented as a vector of traits:
T=(T1,T2,...,Tm),\mathbf{T} = (T_1, T_2, \dots, T_m),T=(T1,T2,...,Tm),
where each TT_\ellT corresponds to a functional dimension (e.g., wing aerodynamics, respiratory efficiency, visual acuity, neuromuscular coordination).
The mapping from genotype to phenotype combines additive, pleiotropic, and epistatic contributions:
T(g)=k=1LMkgk+1k<jLEkjgkgj+.T_\ell(\mathbf{g}) \;=\; \sum_{k=1}^{L} M_{k\ell} g_k \;+\; \sum_{1 \leq k < j \leq L} E_{kj\ell} g_k g_j \;+\; \eta_\ell.T(g)=k=1LMkgk+1k<jLEkjgkgj+.
