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Created January 20, 2026 15:38
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Trajectories, Not States

Trajectories, Not States

We propose a reframing of thermodynamics in which the fundamental object is not the state, but the trajectory.

Entropy does not count how many states exist. It measures how many futures remain executable from the present. Irreversibility arises because dynamics progressively eliminate paths. Once a constraint is enforced, the process cannot proceed otherwise. Time has a direction because options do not come back.

The second law is not about disorder. It is about constraint accumulation. Free energy measures the cost of continuing motion. Systems evolve toward trajectories that are easier to sustain and harder to escape.

Phase transitions occur when many possible trajectories collapse into a small set. After collapse, motion becomes stable and constrained. Learning transitions, including grokking, follow the same rule.

Understanding is not a property of answers, representations, or explanations. It is the loss of alternative executions.

Under this framing, learning dynamics and thermodynamics obey the same principle:

Time advances because possibilities disappear.
Entropy measures how few futures remain.
Irreversibility is the loss of options.


gaius@hert.org

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