The SMI Portal Algebra

Oscillation, Collapse, and Resonance in Timeline Dynamics

Structured Multiversal Interactions (SMI) requires a precise language for how timelines cycle, collapse, and stabilize. The SMI Portal Algebra provides that language: a unified operator framework describing oscillatory, amplifying, and stabilizing portals, and the measurement windows that convert quantum resonance into classical timeline selection.

1. Portals as Operators

A portal is a transformation acting on timeline space. In the simplest case, a portal is represented by an operator U or a set of Kraus operators Kᵢ:

• U ρ U† (unitary evolution)
• Σ Kᵢ ρ Kᵢ† (general quantum channel)

2. Three Portal Types

Oscillatory portals cycle states with periodic structure.
Stabilizing portals drive all states toward a single attractor branch.
Amplifying portals bias probability toward a preferred branch without eliminating others.

3. Oscillatory Portals

Example: the shift operator X on a 3‑level system:
X|0⟩ = |1⟩, X|1⟩ = |2⟩, X|2⟩ = |0⟩.
This produces a 3‑cycle resonance pattern.

4. Measurement Windows

A measurement window of length m consists of:
1. Apply the oscillatory portal m times.
2. Measure in the basis |k⟩.
3. Collapse to one branch.
4. Repeat.

Measurement windows convert continuous oscillation into discrete timeline selection.

5. Stabilizing Portals

A stabilizing portal makes one branch a fixed point. Example: a portal that drains probability from |1⟩ and |2⟩ into |0⟩ until the system converges to |0⟩.

6. Amplifying Portals

An amplifying portal biases the system toward a preferred branch but preserves a mixed distribution. The attractor is soft: |0⟩ dominates but |1⟩ and |2⟩ remain nonzero.

7. The SMI Portal Loop

The full SMI loop is:

Oscillate → Measure → Amplify/Stabilize → Re‑enter oscillation

Probability Flow Across Branches

Animated evolution of (p₀, p₁, p₂) through the SMI loop.

Blue: branch 0 · Orange: branch 1 · Red: branch 2