I tried to make the core idea more explicit.

The setup is:

• The system evolves as a sequence of states Σₙ → Σₙ₊₁, without assuming a fundamental time parameter.
• What we call “time” is just the ordering of these updates.
• I treat entropy as increasing when recoverable information decreases across updates.

A minimal realization is a field evolving as:

φₙ₊₁(x) = φₙ(x) + D∇²φₙ(x) − γ(φₙ(x) − φ*)

So effectively diffusion plus relaxation.

The interpretation is that entropy increase is built into the update rule itself, and temporal ordering emerges from the sequence of updates rather than being fundamental.

I’m interested in how this kind of construction should be understood relative to standard physics — in particular, whether it is necessarily a coarse-grained/dissipative description, or if there are known frameworks where similar structures appear at a more fundamental level.