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

{
  "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

{
  "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

{
  "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

{
  "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

{
  "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

{
  "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

{
  "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

{
  "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

{
  "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

{
  "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#

{
  "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/

Updated Jul 8, 2026