Phase Evolution
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Lesson 01
Stage 2

Lesson 01

What Is Phase?

Phase describes where a repeating process is within its cycle. A complete cycle returns the system to its starting position.

Phase can represent many kinds of repeating processes: a rotating wheel, a wave on the sea, an electrical cycle, a swinging pendulum, a heartbeat. In each case, the phase is a single number that tells you where in the cycle the system currently is.

Use the controls to play the cycle, change how quickly it evolves, reverse its direction, or move through it by hand.

Learning objectives

  • Describe phase as a position within a repeating cycle.
  • Recognise that 0° and 360° are the same point in the cycle.
  • Predict how speed and direction change the evolution of phase.
  • Read phase both as degrees and as a fraction of one full cycle.
Phase circleA circular track with a point at 0 degrees, moving clockwise.0°90°180°270°360° = 0°
Current phase
0.0°

0.00 rad

Fraction of cycle
0.00

0% of one full cycle

Before time exists

Before any reference process exists, there is no measured time — only an ordered succession of relational states. We label that succession with an ordering parameter, λ.

Φ(λ) = Φ₀ + Ω λ
  • Φ(λ)phase at ordering step λ
  • Φ₀starting phase
  • Ωintrinsic rate of phase change
  • λordering parameter (not time)

λ is not physical time. It simply labels the succession of relational states independent of any clock.

ORDERINGλ

No clock · no timeline

From relational ordering to measured time

Phase Differential Theory does not begin with physical time. Instead, it begins with an ordered succession of relational phase states.

This ordering exists independently of clocks and provides the fundamental progression of the system.

Ordering is fundamental. Measurement comes later.

The role of Phase-Snap

Phase-Snap creates persistent physical structures that maintain identity through successive relational states.

Persistence allows systems to compare one state with another and makes stable reference processes possible.

Relational ordering (continues uninterrupted)

Persistent structures emerge (Phase-Snap)

Structure 1
Structure 2
Structure 3
Structure 4
Structure 5
Structure 6

Introducing measured time

Stable repeating physical processes provide reference standards that allow observers to assign numerical values to the ordering of events.

Clocks therefore measure relational ordering. They do not create it.

  1. 1
    Relational ordering
    Ordered succession of phase states (λ).
  2. 2
    Persistent structures
    Phase-Snap produces objects with identity through change.
  3. 3
    Reference processes
    Stable, repeating structures act as measurement standards.
  4. 4
    Measured time (t)
    Numerical values assigned to the underlying ordering.

Two starting points

Standard physics and Phase Differential Theory differ on what comes first.

Fundamental PDT

Phase first

  • Relational ordering is fundamental.
  • λ labels the ordered succession of states.
  • Clocks measure that ordering; they do not create time.

Standard physics

Time first

  • Time is fundamental.
  • Phase evolves with respect to time.
  • Clocks are assumed to already exist.

What to notice

  • Phase repeats after one complete cycle.
  • 0° and 360° represent the same position in the cycle.
  • Speed changes how quickly phase evolves.
  • Direction changes the order in which phase positions are visited.
  • Phase is not the physical object itself; it describes its position within a repeating process.

Key idea

  • PDT begins with relational ordering, not pre-existing time.
  • Ordered succession is fundamental.
  • Phase-Snap creates persistent structures.
  • Persistent structures make reliable reference processes possible.
  • Clocks measure the ordering of events.
  • The familiar variable t is the measurable representation of this deeper relational ordering.

Once measured time has been defined using physical reference processes, intrinsic phase evolution can be expressed in the familiar form:

Φ(t) = Φ₀ + ω t

Effective post-Phase-Snap description

A note on this lesson

This lesson demonstrates the standard mathematical idea of phase in a repeating cycle, reframed within the ontology of Phase Differential Theory: relational ordering is fundamental, Phase-Snap produces persistent structures, and measured time is the numerical representation of that ordering obtained via stable reference processes. Clocks measure the ordering of events — they do not create it. Later lessons will explore how multiple phase systems interact.

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Next: Coupled Systems