From Change to Computation. URFT begins not with particles or spacetime, but with change — encoded as a ripple. This lesson defines the computational field engine that powers URFT: the ripple field Φ(x, y, t), the curvature tensor Rᵢⱼ, and the fidelity layer that shapes system memory and emergence.

🔹 Section 1: The Ripple Field (Φ)

The ripple field is the dynamic core of URFT. It represents the system’s local state of change, not a particle or scalar property.
At every point (x, y) in space and time t, the field evolves according to ripple propagation and interaction laws.

Field notation:

Φ(x, y, t)

Think of Φ like an echo surface — not a fixed object, but an evolving displacement of potential.

🔹 Section 2: The Ripple Tensor (Rᵢⱼ)

Where classical physics uses scalar fields or vector gradients, URFT uses a second-order tensor to track local curvature and interaction geometry.

Tensor definition:

Rᵢⱼ = ∂²Φ / ∂xᵢ∂xⱼ

This allows URFT to model directional echo behavior, entanglement, and asymmetry.
It replaces the Laplacian in wave equations with a structure-aware propagation rule.

🔹 Section 3: Fidelity: The Resistance Layer

URFT introduces fidelity — a spatial field that resists ripple propagation.
It simulates entropy, memory decay, or containment, depending on the environment.

Fidelity function:

Λ(x, y) = resistance to change

High-fidelity zones absorb ripples (collapse wells).
Low-fidelity zones let ripples travel freely (vacuum).

🔹 Section 4: The Field Evolution Equation

With the components in place, URFT’s core simulation engine takes a numerically solvable form:

Φₜ₊₁ = 2Φₜ − Φₜ₋₁ + Δt² [ c² · ∇·(R · ∇Φ) − Λ(x, y)Φₜ ]

This captures:

Reversible motion
Local curvature effects
Directional propagation
Entropy via fidelity

🔹 Section 5: Why This Matters

This engine replaces the need for:

Heat equations
Wave equations
Scalar-based PDEs

It lets systems emerge, collapse, or rebound — not from parameter tuning, but from ripple structure itself.

This is the beating heart of URFT.

The Ripple Engine