Intra-Algorithm Freedom

Even with a fixed algorithm, branching and choice exist. The conditional action entropy $H(A_t \mid H_t)$ measures how many nontrivial options the system has at each step. Combined with empowerment—the mutual information $I(A_t ; S_{t+\Delta})$ between actions and future states—this captures meaningful choices rather than mere noise.

Inter-Algorithm Freedom

The policy manifold volume $V_\tau = \text{Vol}(\mathcal{R}_\tau)$ represents how many distinct algorithms (policies) the system can reach through learning, plasticity, or reconfiguration within timescale τ. A sorting algorithm has near-zero volume; a learning organism inhabits a vast, structured region of policy space.

Causal Emergence

When macro-level descriptions have more effective information than micro-level ones, we have causal emergence: $\text{EI}(\text{macro}) > \text{EI}(\text{micro})$. This is where high-level decisions become better levers on system behavior than raw micro-variables—the mathematical signature of genuine agency at scale.

Descriptive Regularity

The Bernshteyn–Rozhoň bridge connects local distributed algorithms to measurable colorings on infinite Borel graphs. High descriptive regularity means solutions are tame (Borel/Baire measurable) and locally implementable—not pathological axiom-of-choice constructions that cannot be realized by physical systems.

The Tree and Cloth Metaphor

The visualization shows algorithms as trees—computation unfolding through time—with a translucent cloth representing goal slack: the fiber bundle of micro-implementations that achieve the same macro-outcome. Thick cloth means many paths to the same goal; thin cloth means rigid, deterministic execution. This is freedom made geometric.

Composite Freedom Measure

The freedom score combines these dimensions:

Low scores indicate rigid automata with tiny policy manifolds. High scores correspond to systems that navigate policy space itself, exploiting slack between implementations and using macro-scale descriptions as powerful levers on their own dynamics.