Determinism

Determinism: The Math of Areloria

In Areloria, "random" is a forbidden word in the core simulation. We operate on Absolute Causality.

1. Kappa Invariant (Fixed-Point Math)

To avoid floating-point drift across different browsers and hardware, we use the Kappa Standard:

• Kappa = 1000
• All simulation values are scaled by Kappa and treated as 64-bit integers.
• Example: A move speed of 1.25 units/tick is represented as 1250.

2. Psi Evolution Formula

The state transition of the world is governed by the Trinitarian Psi Evolution:

Psi[n + 1] = kappa * (Psi[n] odot A_ARE)^3 + 3 * P(Psi[n])

• Psi[n]: Current state vector at tick n.
• A_ARE: Axiomatic coupling operator.
• P(Psi[n]): Plexity function (the weight of simulation entities).

3. Replayability

Every meaningful world state must be reproducible from:

1. Initial Seed
2. Tick Number
3. Input Stream
4. Deterministic Engine Version

Replay is our proof layer. If a combat result cannot be replayed, it did not happen.

4. ARE Determinism Gate

Our CI/CD pipeline enforces strict determinism rules for Level-A paths:

• Forbidden: Math.random(), Date.now(), new Date(), randomUUID().
• Required: Sorted Map/Set iterations.
• Requirement: All simulation decisions must be traceable to the tick-derived seed.