Part I: Foundations

Measure-Theoretic Inevitability

Introduction
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Measure-Theoretic Inevitability

Consider a substrate-environment prior: a probability measure μ\mu over tuples (S,E,x0)(\mathcal{S}, \mathcal{E}, \mathbf{x}_0) representing physical substrates (degrees of freedom, interactions, constraints), environments (gradients, perturbations, resource availability), and initial conditions. Call μ\mu a broad prior if it assigns non-negligible measure to sustained gradients (nonzero flux for times ≫\gg relaxation times), sufficient dimensionality (nn large enough for complex attractors), locality (interactions falling off with distance), and bounded noise (stochasticity not overwhelming deterministic structure).

Under such a prior, self-modeling systems are typical. Define:

CT=(S,E,x0):system develops self-model by time T\mathcal{C}_T = {(\mathcal{S}, \mathcal{E}, \mathbf{x}_0) : \text{system develops self-model by time } T}

Then:

lim⁡T→∞μ(CT)=1−ϵ\lim_{T \to \infty} \mu(\mathcal{C}_T) = 1 - \epsilon

for some small ϵ\epsilon depending on the fraction of substrates that lack sufficient computational capacity.

Proof. [Proof sketch] Under the broad prior:
  1. Probability of structured attractors →1\to 1 as gradient strength increases (bifurcation theory)
  2. Given structured attractors, probability of boundary formation →1\to 1 as time increases (combinatorial exploration of configurations)
  3. Given boundaries, probability of effective regulation →1\to 1 for self-maintaining structures (by definition of “self-maintaining”)
  4. Given regulation, world model is implied (POMDP sufficiency)
  5. Given world model in self-effecting regime, self-model has positive selection pressure

The only obstruction is substrates lacking the computational capacity to support recursive modeling, which is measure-zero under sufficiently rich priors.

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Inevitability means typicality in the ensemble. The null hypothesis is not "nothing interesting happens" but "something finds a basin and stays there," because that's what driven nonlinear systems do. Self-modeling attractors are among the accessible basins wherever environments are complex enough that self-effects matter. Empirical validation is emerging: in protocell agent experiments (V20–V31), self-modeling develops in 100% of seeds from random initialization — self-models are indeed typical. High integration (Φ>0.10\Phi > 0.10) develops in approximately 30% of seeds, with the variance dominated by evolutionary trajectory, not initial conditions. The ensemble fraction for self-modeling is near unity; the fraction for rich integration is substantial but stochastic, consistent with the distinction between typicality (the structure will emerge) and universality (every trajectory reaches it).