Geometric framework for biological evolution

Biological evolution sustains complex dynamics without any fitness function, yet virtually all evolutionary algorithms depend on one. Genesis is an open-source platform designed to test, empirically, what an artificial system needs to sustain evolutionary dynamics after complete fitness removal. Evolution in Genesis is governed by physical constraints, relational dominance, and adaptive regulation - no scalar fitness, no designer-specified objectives. Across experiments totalling over one million evolutionary generations, Genesis has: (i) shown that constraint-driven selection can sustain evolutionary activity after complete fitness removal (7/12 runs; Wilson 95% CI [30.2%, 82.5%]; p<0.01, Cohen's d=1.47 vs. baselines); (ii) produced a sham-controlled negative result demonstrating that niche construction alone does not break the complexity plateau; and (iii) provided preliminary evidence that speciation-protected niche construction initiates structural diversification that unprotected secretion cannot. These findings establish empirical boundaries for fitness-free evolution and open a new direction: meta-evolution of physics, in which the laws governing an evolutionary system are themselves evolved.

Evolutionary computation has long promised to deliver both high-performance optimization tools as well as rigorous scientific simulations of Darwinian evolution. However, modern algorithms frequently abandon evolutionary fidelity for physics-inspired heuristics or superficial biological metaphors. This paper derives a suite of advanced gradient-based optimization algorithms directly from evolutionary first principles. We introduce Darwinian Lineage Simulations (DLS) to prove that, in an asexual context, Fisher's and Wright's historically opposed views of evolution are actually formally equivalent; One can partition Fisher's deterministically-evolving total population into Wright's randomly-drifting sub-populations. We prove that proper bookkeeping requires introducing a specific kind of structured noise (the DLS noise relation). Crucially, any bookkeeping choices which satisfy this relation will yield a faithful simulation of evolution. Using this vast representational freedom, we prove that a broad family of battle-tested optimization algorithms are already perfectly compatible with evolutionary dynamics. These include: Stochastic Gradient Descent as well as many regularizations/approximations of Newton's method and Natural Gradient Descent. By simply adding DLS noise (i.e., evolutionarily faithful genetic drift), these algorithms become scientifically valid in silico simulations of Darwinian evolution. Finally, we demonstrate that even the state-of-the-art Adam optimizer can be brought into evolutionary compliance through a minor mathematical surgery.

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ of an outcome on contemporaneous covariates can be regarded as a field on covariate space. We study how this field evolves when the covariates of individual units move as a diffusion and the outcome attached to each unit changes according to its own dynamics. Under mild regularity we show that $m_t$ satisfies a second-order evolution equation whose diffusion term is $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, the trace of the Hessian of the regression function weighted by the instantaneous covariance $Σ$ of the covariate increments, whose transport term is the drift of the time-reversed covariate process, and whose source term is the conditional mean rate of change of the outcome. The equation is a regression-theoretic reading of a classical transport law, and it yields a Feynman-Kac representation of $m_t$ along reversed covariate paths. We connect the Hessian appearing in the equation to the Hessian estimated by quadratic regression (under Gaussian design the best quadratic approximation has Hessian $E[\nabla^2 m(X)]$ exactly) and to the leading bias of kernel smoothers. For binary outcomes governed by bistable replicator dynamics with imitation among units with similar covariates, the closed equation is of Nagumo type, its planar travelling fronts are exactly logistic in $(x,t)$, and a single logistic regression of the outcome on covariates and time identifies the ratio of diffusion to reaction and the direction of invasion. We discuss identification under panel versus repeated cross-sectional designs and a two-stage estimation strategy, and we illustrate the results with simulations.