The saccade project

Darwinian Evolution

Mapping population genetics to Developmental Constraint Theory

The System

The system is a reproducing biological population evolving over generational time.

S (system): A population of organisms
X (configuration space): Genetic configuration space (genotype–phenotype mappings)
E (environment): Selection pressures (resource availability, predation, climate, competition)

At any time $t$ t, the population occupies a distribution across X, defined by genotype frequencies $p_{i} (t)$ , where: $\sum p_{i} (t) = 1$

Viability is determined by a fitness function: $f (x_{i}, E)$

The admissible set is defined as: $A (E) = {x \in X ∣ f (x, E) \geq θ}$

where $θ$ is the viability threshold.

Governing Structure

Evolutionary dynamics are governed by replicator dynamics:

$\frac{d p_{i}}{d t} = p_{i} (f (x_{i}, E) - \overset{ˉ}{f})$ where:

$\overset{ˉ}{f} = \sum p_{j} f (x_{j}, E)$

This defines how genotype frequencies change under differential reproductive success.

Inheritance across generations follows:

$p_{i} (t + 1) = \frac{p_{i} (t) \cdot f (x_{i}, E)}{\overset{ˉ}{f}}$

These equations describe redistribution of population states under environmental selection.

Constraint Formation

Constraint occurs through environmental shift:

$E_{0} \to E_{1}$

which produces: $A (E_{1}) \subset A (E_{0})$

This reduces admissible genotype–phenotype configurations.

Previously viable configurations fall below threshold
Survival probability decreases for $x \notin A (E_{1})$
Viability limits are defined by $f (x, E) \geq θ$

Constraint is therefore:

reduction of admissible state-space under environmental change

Reorganization

When population states fall outside admissible space:

Genotypes with $f (x_{i}, E_{1}) < θ$ decrease in frequency
Replicator dynamics redistribute probability mass

Population reorganizes toward: $x \in A (E_{1})$

This produces:

allele frequency shifts
phenotype redistribution
structural population change

Over repeated constraint cycles:

divergence accumulates
subpopulations separate
speciation occurs when gene flow approaches zero

Structural Correspondence (SACCADE)

Evolution satisfies the DCT sequence:

Signal — Environmental gradients produce differential fitness ( $\partial f / \partial E \neq 0$ )
Arrival — Genetic variation distributes population across X
Context — Environment E defines admissible set $A (E)$
Constraint — $A (E_{1}) \subset A (E_{0})$
Adaptation — Replicator dynamics redistribute $p_{i}$
Distribution — Inheritance propagates viable configurations
Evolution — Iterated constraint cycles reshape population structure

These transitions occur without foresight or teleology.

Constraint Regime Outcome

What persists:

Genotype distributions within $A (E)$
Population structures aligned with viability constraints
Stable reproductive pathways

What causes failure:

Environmental shift beyond adaptive capacity
Loss of viable configurations
inability to reorganize within new admissible bounds

Scope and Limits

This mapping does not introduce new mechanisms or modify evolutionary theory.

Darwinian evolution remains fully governed by established biological principles.

This formulation expresses those dynamics as:

admissible state-space reduction
constraint-induced reorganization
distribution under replicator dynamics

Structural Conclusion

Darwinian evolution satisfies Developmental Constraint Theory as a population-level instantiation of ordered constraint formation within genetic configuration space.