The Mathematical Core
The biquadratic kernel, the four phase states, the phase rail, and the compose operation — the algebraic foundation of the Golden Phase Runtime.
The Biquadratic Kernel
Everything starts with one equation. φ (phi) is the golden ratio, (1 + √5)/2 ≈ 1.618. α (alpha) is a small constant, ≈ 0.00730. Substituting u = y² turns the quartic into an ordinary quadratic u² − φu + α = 0, which has two roots (the two sheets).
The Four Phase States
| State | y value | Rail position | Role |
|---|---|---|---|
| Φ0 | −√z₊ = −1.2702 | 0.000000 | sleeping / identity |
| Φ1 | −√z₋ = −0.0673 | 0.919408 | ready |
| Φ2 | +√z₋ = +0.0673 | 1.022203 | running |
| Φ3 | +√z₊ = +1.2702 | 1.941611 | kernel |
The Compose Operation
The compose operation (∘) combines two states into a third. It works in three steps: (1) Add the two states' rail positions. (2) Fold: if the sum exceeds the rail end (RAIL ≈ 1.9416), reflect it back (2·RAIL − x). (3) Snap: round to the nearest state using midpoints as boundaries.
The algebra is non-associative: for some triples, (a∘b)∘c ≠ a∘(b∘c). This is a feature — it means the algebra itself carries ordering information.
The Associator–Reflection Theorem
A triple (a, b, c) fails associativity if and only if the fold step activates in exactly one of the two groupings (a∘b)∘c and a∘(b∘c).
This theorem characterises exactly which triples are non-associative and proves that the failure is caused by the reflection step in compose. It is the mathematical foundation for Golden Phase OS's ability to detect reordering without timestamps or sequence numbers.