# **Coherence Gradient Cases — RTT/1**  
### *Case Studies for the Coherence Tensor Engine (CTE)*

Coherence gradients describe **directional changes in coherence** across conceptual, computational, physical, and dimensional regimes.  
These case studies illustrate how the Coherence Tensor Engine (CTE) evaluates:

- coherence gradient magnitude  
- gradient direction  
- coherence curvature  
- drift sensitivity  
- collapse‑point formation  
- stability envelopes  

Each case demonstrates one or more CTE operators:

- **CTE‑Gradient**  
- **CTE‑Compute**  
- **CTE‑Field**  
- **CTE‑Collapse**  
- **CTE‑Stabilize**

---

## **1. Structural Gradient Cases**

### **Case 1 — Structural Invariant Gradient (R1 → R2)**  
**Scenario**  
A conceptual invariant (symmetry) propagates into computational structures, forming a stable coherence gradient.

**CTE Output**  
```json
{
  "regime": "R1-R2",
  "gradient_magnitude": 0.72,
  "gradient_direction": "R1→R2",
  "coherence_curvature": 0.33,
  "drift_sensitivity": 0.12,
  "stability_envelope": 0.81
}
```

---

### **Case 2 — Constraint‑Driven Gradient (R2 → R3)**  
**Scenario**  
A computational constraint enforces coherence across physical calibration.

**CTE Output**  
```json
{
  "regime": "R2-R3",
  "gradient_magnitude": 0.68,
  "gradient_direction": "R2→R3",
  "coherence_curvature": 0.41,
  "drift_sensitivity": 0.27,
  "stability_envelope": 0.74
}
```

---

## **2. Coherence Gradient Cases**

### **Case 3 — Coherence Ridge Alignment (R1 ↔ R4)**  
**Scenario**  
Conceptual and dimensional coherence gradients align, forming a coherence ridge.

**CTE Output**  
```json
{
  "regime": "R1-R4",
  "gradient_magnitude": 0.83,
  "gradient_direction": "R1↔R4",
  "coherence_curvature": 0.52,
  "drift_sensitivity": 0.18,
  "stability_envelope": 0.79
}
```

---

### **Case 4 — Drift‑Sensitive Coherence Gradient (R2 ↔ R3)**  
**Scenario**  
Computational drift influences physical coherence gradients.

**CTE Output**  
```json
{
  "regime": "R2-R3",
  "gradient_magnitude": 0.81,
  "gradient_direction": "R3→R2",
  "coherence_curvature": 0.57,
  "drift_sensitivity": 0.44,
  "stability_envelope": 0.63
}
```

---

## **3. Boundary Gradient Cases**

### **Case 5 — Abstraction‑Measurement Gradient (R1 → R3)**  
**Scenario**  
Coherence forms at the boundary between conceptual abstraction and physical measurement.

**CTE Output**  
```json
{
  "regime": "R1-R3",
  "gradient_magnitude": 0.69,
  "gradient_direction": "R1→R3",
  "coherence_curvature": 0.38,
  "drift_sensitivity": 0.22,
  "stability_envelope": 0.71
}
```

---

### **Case 6 — Gradient‑Boundary Alignment (R2 ↔ R4)**  
**Scenario**  
Aligned gradients across computational and dimensional regimes produce boundary coherence.

**CTE Output**  
```json
{
  "regime": "R2-R4",
  "gradient_magnitude": 0.88,
  "gradient_direction": "R2↔R4",
  "coherence_curvature": 0.47,
  "drift_sensitivity": 0.33,
  "stability_envelope": 0.68
}
```

---

## **4. Tensor‑Field Gradient Cases**

### **Case 7 — Multi‑Regime Gradient Tensor (R1 ↔ R2 ↔ R3)**  
**Scenario**  
A multi‑regime coherence tensor binds conceptual, computational, and physical coherence gradients.

**CTE Output**  
```json
{
  "regime": "R1-R2-R3",
  "gradient_magnitude": 0.94,
  "gradient_direction": "tensor",
  "coherence_curvature": 0.63,
  "drift_sensitivity": 0.29,
  "stability_envelope": 0.84
}
```

---

### **Case 8 — Dimensional Tensor Gradient (R2 ↔ R4)**  
**Scenario**  
Dimensional tensors constrain computational coherence gradients.

**CTE Output**  
```json
{
  "regime": "R2-R4",
  "gradient_magnitude": 0.88,
  "gradient_direction": "R4→R2",
  "coherence_curvature": 0.55,
  "drift_sensitivity": 0.37,
  "stability_envelope": 0.73
}
```

---

## **5. Collapse‑Point Gradient Cases**

### **Case 9 — Collapse Basin Gradient (R3 → R4)**  
**Scenario**  
Physical drift amplifies dimensional coherence curvature, forming a collapse basin.

**CTE Output**  
```json
{
  "regime": "R3-R4",
  "gradient_magnitude": 0.91,
  "gradient_direction": "R3→R4",
  "coherence_curvature": 0.71,
  "drift_sensitivity": 0.52,
  "stability_envelope": 0.44,
  "collapse_point": "R4:0.82"
}
```

---

### **Case 10 — Collapse Ridge Gradient (R2 ↔ R3)**  
**Scenario**  
Computational drift reduces coherence while physical drift increases coherence sensitivity.

**CTE Output**  
```json
{
  "regime": "R2-R3",
  "gradient_magnitude": 0.86,
  "gradient_direction": "R2↔R3",
  "coherence_curvature": 0.62,
  "drift_sensitivity": 0.49,
  "stability_envelope": 0.48,
  "collapse_point": "R3:0.77"
}
```

---

## **6. Canonical CTE Gradient Snippet**

```json
{
  "regime": "R1-R4",
  "gradient_magnitude": 0.83,
  "gradient_direction": "R1↔R4",
  "coherence_curvature": 0.52,
  "drift_sensitivity": 0.18,
  "stability_envelope": 0.79
}
```

---

## **Status**

- **Version:** 1.0  
- **Status:** canon‑stable  
- **Category:** rtt‑structural  
- **Module Path:** `/docs/rtt/Coherence_Tensor_Engine/`