$G \text{ - } \psi \text{ - } D$

Master Equations

The central object is a Boltzmann-like probability measure over histories $\gamma$ compatible with a structure $O$:

$$ \mathbb{P}(\gamma \mid O) = \frac{1}{Z_O}\exp!\bigl(-\mathcal{C}_O[\gamma]\bigr), \qquad \gamma \in \Gamma_O, $$

where $\mathcal{C}_O[\gamma]$ is the total description cost of history $\gamma$ given structure $O$.

The cost functional decomposes into three regimes:

$$ \textbf{G} - \boldsymbol{\psi} - \textbf{D} $$

where $\textbf{D}$ is discrete, $\boldsymbol{\psi}$ spectral and $\textbf{G}$ geometric description of structure $O$.

The predictable physics emerges:

$$ \text{law (short } \psi\text{)} ;\Rightarrow; \text{predictable } D ;\Rightarrow; \text{small } \mathcal{C}_D ;\Rightarrow; \text{large } \mathbb{P}(\gamma\mid O) $$

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