Nonergodic Development: How High-Dimensional Systems with Low-Dimensional Anchors Generate Phenotypes

Biological development unfolds in a high-dimensional state space. Gene regulatory networks, signaling cascades, and metabolic pathways create a system with vastly more degrees of freedom than the genome that parameterizes it. This dimensionality has a fundamental consequence: the system is nonergodic—it cannot explore its state space in biological time.

This nonergodicity is not a limitation to be overcome but a feature that enables stable phenotypes. The genome acts as a low-dimensional anchor that constrains which regions of state space are reachable, while environmental history determines which specific attractor the system occupies within those constraints.

We quantify the relationship between anchor dimensionality and state space dimensionality:

Let $L$ be the dimension of the genotype (anchor) space and $k$ the dimension of measured phenotypic traits. Let the developmental system evolve on a manifold $\mathcal{M}$ with effective dimension $m_{\text{eff}}$. The dimensional gap is:

$\Delta_D = m_{\text{eff}} - (L + k)$

When $\Delta_D \gg 0$, the developmental system has far more degrees of freedom than can be specified by the genome or captured by phenotypic measurement. This creates a fundamental ambiguity:

• The same genotype can produce different phenotypes (depending on which trajectory/attractor)
• The same phenotype can arise from different mechanisms (genotype-determined vs. trajectory-determined)

Sierra et al. (2025) demonstrated that cooperative mammalian species exhibit lower cancer prevalence than competitive species. The same pattern admits both interpretations:

• Allele interpretation: Cooperative lineages have accumulated cancer-suppressing alleles through selection.
• Trajectory interpretation: Cooperative environmental cues enable slower, more coordinated development with fewer attractor bifurcations, yielding lower cancer mortality as an emergent property.

These interpretations are non-identifiable from cross-species comparative data. They diverge only in predictions for intervention: the allele model predicts that changing an organism's environment will not change its cancer risk; the trajectory model predicts substantial phenotypic shifts.

We create two "worlds" with identical genotype distributions but different environmental regimes. The developmental trajectories in each world converge to different attractor basins, producing dramatically different phenotype distributions.

The genetic distance between worlds is $F_{ST} \approx 0$ (by construction), yet phenotypic distance is $P_{ST} \gg 0$. A GWAS would find no significant variants and conclude "missing heritability"—but the heritability is not missing, it is trajectory-based.

Coherence is self-stabilizing: organisms that maintain high coherence maintain high coherence. The property that prevents cellular bifurcation (cancer) is the same property that stabilizes group dynamics (cooperation). Nonergodicity doesn't just trap individual trajectories—it creates nested attractors at every scale where the same coherence parameter operates.

The "fractal" nature of the architecture means that solving the social dilemma (stabilizing the group) automatically solves the cellular dilemma (suppressing cancer). This is not coincidence but consequence: the same dynamical structure operates at both scales.

Biological development is nonergodic. The state space is too vast to explore; trajectories are trapped; phenotypes are attractor states, not algorithmic outputs.

The genome is a low-dimensional anchor that constrains which attractor basins exist, but environmental history determines which basin is entered. This anchor-trajectory duality offers one lens on the genotype-phenotype relationship: the mapping is many-to-one not only because of noise or missing variants, but also because trajectory information that distinguishes outcomes is projected out by genotype-phenotype analysis.

Genotype does not algorithmically determine phenotype. Genotype anchors a nonergodic developmental system; environmental history traps it in an attractor; the phenotype is the trapped state.