Speed and Velocity
on the Ring
Formal definitions for ring trajectory, velocity, speed, orientation, and conservation — standalone foundations for MPRC motion analysis on Z₂₅₆ · v1.1
Closed Ring, Two Arcs
Standard real-line mechanics defines velocity as a signed difference on an open interval. On a closed ring, this breaks: between any two positions there are exactly two arcs, and the "direction" of a step is not given by sign alone. This chapter gives the formal definitions that replace real-line intuitions on Z256 — the ring used throughout the MPRC framework.
Direction is resolved by choosing the shorter arc. This is not a convention — it is the intrinsic metric of a closed ring. Speed and orientation are then fully decoupled quantities.
These definitions underpin all motion analysis in the MPRC framework: acoustic ring trajectories, LQC clock-drift detection, and the oloid two-circle decomposition. Every chapter that uses velocity, speed, or orientation implicitly relies on this foundation.
| Quantity | Symbol | AGC Effect | Notes |
|---|---|---|---|
| Speed | s[n] | Destroyed | |v[n]| compressed toward zero |
| Velocity magnitude | |v[n]| | Destroyed | Same as speed |
| Velocity direction | sign(v[n]) | Preserved | Positive rescaling leaves sign invariant |
| Orientation bit | b[n] | Preserved | Depends only on sign(v[n]) |
The Ring as Closed System
A ring trajectory is a sequence of positions on a closed ring of finite circumference. Unlike a real-line trajectory, the ring has no boundary and no preferred cut-point. Every trajectory on the ring eventually returns to its starting position.
The closure has one immediate consequence: between any two positions on ℜ there are exactly two arcs. Direction is resolved by choosing the shorter one. This is not a convention — it is the intrinsic metric of a closed ring.
Signed Shortest Arc — Velocity on Z₂₅₆
The magnitude |v[n]| is the arc length of the step. The sign is the direction. Both are contained in a single quantity — no separate direction flag is needed.
AGC-Invariant Direction Signal
Proposition 6.2 — AGC invariance. The orientation bit b[n] is invariant under any strictly positive rescaling of φ.
Proof. AGC replaces φ[n] with α·φ[n] for α > 0. The difference scales by α, preserving sign. Hence b[n] is unchanged. □
This invariance is non-trivial. Automatic Gain Control is applied to virtually all real-world audio and sensor signals before analysis. Speed and velocity magnitude are both destroyed by AGC — the ring compresses toward zero. The orientation bit survives because it depends only on the sign of the step, which is preserved under positive rescaling.
Zero Net Angular Momentum — Closed Trajectories
Remark 7.3. This is a consequence of closure, not an independent physical law. The postulate with real content is that physical signals form closed trajectories on the ring. That claim must be stated explicitly wherever the theorem is applied. The theorem itself is purely algebraic — it holds for any closed sequence on any ring.
Edge Cases on Z₂₅₆
| Case | Condition | Resolution |
|---|---|---|
| Antipodal step | Both arcs equal length (step = 128 on Z₂₅₆) | Direction undefined. Flag as degenerate. Do not emit. |
| Stationary step | v[n] = 0 | Orientation bit b[n] = 1 by convention. No contribution to net angular momentum. |
| Transient AGC | Short gain-settling window | Proposition 6.2 holds under steady-state rescaling only. Transient AGC is explicitly excluded. |
The antipodal degenerate case (step = 128) corresponds to the half-ring boundary on Z₂₅₆. This is structurally distinct from the vacuum positions {0, 64, 128, 192} — the antipodal step is a transition crossing the diameter, not a forbidden state.