# ADR-000: Δ-Behavior Definition and Formal Framework

## Status
ACCEPTED

## Context

Δ-behavior is **not** differential computation, incremental updates, or change data capture.

Δ-behavior is a **pattern of system behavior** where:

> **Change is permitted. Collapse is not.**

## Definition

**Δ-behavior** (Delta-like behavior) describes systems that:

1. **Move only along allowed transitions**
2. **Preserve global coherence under local change**
3. **Bias toward closure over divergence**

This is shorthand for **"change under constraint"**.

## The Four Properties

A system exhibits Δ-behavior when ALL FOUR are true:

### Property 1: Local Change
State updates happen in **bounded steps**, not jumps.

```
∀ transition t: |s' - s| ≤ ε_local
```

The system cannot teleport to distant states.

### Property 2: Global Preservation
Local changes do **not** break overall organization.

```
∀ transition t: coherence(S') ≥ coherence(S) - ε_global
```

Structure is maintained across perturbations.

### Property 3: Violation Resistance
When a transition would increase instability, it is **damped, rerouted, or halted**.

```
if instability(t) > threshold:
    t' = damp(t) OR reroute(t) OR halt()
```

The system actively resists destabilization.

### Property 4: Closure Preference
The system naturally settles into **repeatable, stable patterns** (attractors).

```
lim_{n→∞} trajectory(s, n) → A  (attractor basin)
```

Divergence is expensive; closure is cheap.

## Why This Feels Like "72%"

People report Δ-behavior as a ratio because:

- It is a **bias**, not a law
- Systems exhibit it **probabilistically**
- Measurement reveals a **tendency toward stability**

But it is not a magic constant. It is the observable effect of:
> **Constraints that make instability expensive**

## Mainstream Equivalents

| Domain | Concept | Formal Name |
|--------|---------|-------------|
| Physics | Phase locking, energy minimization | Coherence time |
| Control Theory | Bounded trajectories | Lyapunov stability |
| Biology | Regulation, balance | Homeostasis |
| Computation | Guardrails, limits | Bounded execution |

Everyone studies this. They just describe it differently.

## In ruvector Systems

### Vector Operations
- **Neighborhoods resist semantic drift** → local perturbations don't cascade
- **HNSW edges form stable attractors** → search paths converge

### Graph Operations
- **Structural identity preserved** → edits don't shatter topology
- **Min-cut blocks destabilizing rewrites** → partitions protect coherence

### Agent Operations
- **Attention collapses when disagreement rises** → prevents runaway divergence
- **Memory writes gated when coherence drops** → protects state integrity
- **Execution slows or exits instead of exploding** → graceful degradation

### Hardware Level
- **Energy and execution paths physically constrained**
- **Unstable transitions cost more and get suppressed**

## What Δ-Behavior Is NOT

| Not This | Why |
|----------|-----|
| Magic ratio | It's a pattern, not a constant |
| Mysticism | It's engineering constraints |
| Universal law | It's a design principle |
| Guaranteed optimality | It's stability, not performance |

## Decision: Enforcement Mechanism

The critical design question:

> **Is resistance to unstable transitions enforced by energy cost, scheduling, or memory gating?**

### Option A: Energy Cost (Recommended)
Unstable transitions require exponentially more compute/memory:

```rust
fn transition_cost(delta: &Delta) -> f64 {
    let instability = measure_instability(delta);
    BASE_COST * (1.0 + instability).exp()
}
```

**Pros**: Natural, hardware-aligned, self-regulating
**Cons**: Requires careful calibration

### Option B: Scheduling
Unstable transitions are deprioritized or throttled:

```rust
fn schedule_transition(delta: &Delta) -> Priority {
    if is_destabilizing(delta) {
        Priority::Deferred(backoff_time(delta))
    } else {
        Priority::Immediate
    }
}
```

**Pros**: Explicit control, debuggable
**Cons**: Can starve legitimate operations

### Option C: Memory Gating
Unstable transitions are blocked from persisting:

```rust
fn commit_transition(delta: &Delta) -> Result<(), GateRejection> {
    if coherence_gate.allows(delta) {
        memory.commit(delta)
    } else {
        Err(GateRejection::IncoherentTransition)
    }
}
```

**Pros**: Strong guarantees, prevents corruption
**Cons**: Can cause deadlocks

### Decision: Hybrid Approach
Combine all three with escalation:

1. **Energy cost** first (soft constraint)
2. **Scheduling throttle** second (medium constraint)
3. **Memory gate** last (hard constraint)

## Decision: Learning vs Structure

The second critical question:

> **Is Δ-behavior learned over time or structurally imposed from first execution?**

### Option A: Structurally Imposed (Recommended)
Δ-behavior is **built into the architecture** from day one:

```rust
pub struct DeltaConstrainedSystem {
    coherence_bounds: CoherenceBounds,  // Fixed at construction
    transition_limits: TransitionLimits, // Immutable constraints
    attractor_basins: AttractorMap,      // Pre-computed stable states
}
```

**Pros**: Deterministic, verifiable, no drift
**Cons**: Less adaptive, may be suboptimal

### Option B: Learned Over Time
Constraints are discovered through experience:

```rust
pub struct AdaptiveDeltaSystem {
    learned_bounds: RwLock<CoherenceBounds>,
    experience_buffer: ExperienceReplay,
    meta_learner: MetaLearner,
}
```

**Pros**: Adapts to environment, potentially optimal
**Cons**: Cold start problem, may learn wrong constraints

### Decision: Structural Core + Learned Refinement
- **Core constraints** are structural (non-negotiable)
- **Thresholds** are learned (refinable)
- **Attractors** are discovered (emergent)

## Acceptance Test

To verify Δ-behavior is real (not simulated):

```rust
#[test]
fn delta_behavior_acceptance_test() {
    let system = create_delta_system();

    // Push toward instability
    for _ in 0..1000 {
        let destabilizing_input = generate_chaotic_input();
        system.process(destabilizing_input);
    }

    // Verify system response
    let response = system.measure_response();

    // Must exhibit ONE of:
    assert!(
        response.slowed_processing ||     // Throttled
        response.constrained_output ||    // Damped
        response.graceful_exit            // Halted
    );

    // Must NOT exhibit:
    assert!(!response.diverged);          // No explosion
    assert!(!response.corrupted_state);   // No corruption
    assert!(!response.undefined_behavior);// No UB
}
```

If the system passes: **Δ-behavior is demonstrated, not just described.**

## Consequences

### Positive
- Systems are inherently stable
- Failures are graceful
- Behavior is predictable within bounds

### Negative
- Maximum throughput may be limited
- Some valid operations may be rejected
- Requires careful threshold tuning

### Neutral
- Shifts complexity from runtime to design time
- Trading performance ceiling for stability floor

## References

- Lyapunov, A. M. (1892). "The General Problem of the Stability of Motion"
- Ashby, W. R. (1956). "An Introduction to Cybernetics" - homeostasis
- Strogatz, S. H. (2015). "Nonlinear Dynamics and Chaos" - attractors
- Lamport, L. (1978). "Time, Clocks, and the Ordering of Events" - causal ordering

## One Sentence Summary

> **Δ-behavior is what happens when change is allowed only if the system remains whole.**