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\section{Mathematical Foundations and Biological Correspondences}
\label{sec:math}
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\paragraph{Learnon (\Lrn).} Empirical risk minimization:
\begin{equation}
\theta^\star \in \arg\min_{\theta}\ \mathbb{E}_{(x,y)\sim \mathcal{D}}[ \ell(f_\theta(x),y) ] + \lambda \Omega(\theta),
\end{equation}
with gradient updates $\theta_{t+1}=\theta_t-\eta_t\nabla\widehat{\mathcal{L}}(\theta_t)$; RL maximizes $J(\pi)=\mathbb{E}_\pi[\sum_t \gamma^t r_t]$ in MDPs. \emph{Biology:}~ Hebbian/Oja \citep{Hebb1949,Oja1982}, reward-modulated prediction errors \citep{SuttonBarto2018}.

\paragraph{Evolon (\Evo).} Population pipeline $P_{t+1}=\mathcal{R}(\mathcal{M}(\mathcal{C}(P_t)))$ with fitness-driven selection. \emph{Biology:}~ Price equation $\Delta \bar{z}=\frac{\mathrm{Cov}(w,z)}{\bar{w}}+\frac{\mathbb{E}[w\Delta z]}{\bar{w}}$; replicator $\dot{p}_i=p_i(f_i-\bar{f})$ \citep{Price1970,TaylorJonker1978}.

\paragraph{Symbion (\Sym).} Resolution/unification; soundness and refutation completeness \citep{Robinson1965Resolution}.

\paragraph{Probion (\Prb).} Bayes $p(z|x)\propto p(x|z)p(z)$; VI via ELBO $\mathcal{L}(q)=\mathbb{E}_q[\log p(x,z)]-\mathbb{E}_q[\log q(z)]$; \emph{Biology:}~ Bayesian brain \citep{KnillPouget2004}.

\paragraph{Scholon (\Sch).} A* with admissible $h$ is optimally efficient; DP/Bellman updates $V_{k+1}(s)=\max_a[r(s,a)+\gamma\sum_{s'}P(s'|s,a)V_k(s')]$.

\paragraph{Controlon (\Ctl).} LQR minimizes quadratic cost in linear systems; Kalman filter provides MMSE state estimates in LQG \citep{Kalman1960,Pontryagin1962,TodorovJordan2002}.

\paragraph{Swarmon (\Swm).} PSO updates $v_i(t+1)=\omega v_i(t)+c_1 r_1(p_i-x_i)+c_2 r_2(g-x_i)$; ACO pheromone $\tau\leftarrow (1-\rho)\tau+\sum_k \Delta\tau^{(k)}$.

