Intent-Tensor-Theory

Drift-Lock Dynamics

Drift-Lock Dynamics

The Mechanism of Temporal Production

1. The Central Equation

The drift production scalar is defined as:

σ_θ(x, t) = 𝒟(x, t) · (1 - ℒ(x, t))

This multiplicative structure encodes the fundamental insight: Time emerges from the competition between change (Drift) and stability (Lock).

2. The Drift Magnitude 𝒟

2.1 Definition

𝒟(x, t) = α_𝓜 ‖∂𝓜_{ij}/∂t‖_F + α_Φ ‖∂(∇Φ)/∂t‖₂

2.2 Component Analysis

Memory Term: The Frobenius norm of the Memory Tensor time-derivative:

‖∂𝓜/∂t‖_F = √(Σ_{i,j} (∂𝓜_{ij}/∂t)²)

Intent Gradient Term: The Euclidean norm of the time-derivative of the gradient:

‖∂(∇Φ)/∂t‖₂ = √(Σ_i (∂²Φ/∂t∂x_i)²)

2.3 Properties

Proposition: 𝒟 ≥ 0 always.

Proof: Norms are non-negative, and α_𝓜, α_Φ > 0. ∎

Proposition: 𝒟 = 0 iff both ∂_t 𝓜_{ij} = 0 and ∂_t ∇Φ = 0 everywhere.

Physical Interpretation: 𝒟 = 0 means the glyph fields are static—no recursive evolution.

3. The Shell-Lock ℒ

3.1 Formal Definition

ℒ(x, t) = ⟨𝒞(x, t), 𝒞^{ref}(x)⟩ / (‖𝒞‖ · ‖𝒞^{ref}‖)

Where:

  • 𝒞 is the curvent vector field
  • 𝒞^{ref} is a reference configuration
  • ⟨·,·⟩ is the inner product

3.2 Domain and Range

Proposition: ℒ ∈ [-1, 1].

Proof: By Cauchy-Schwarz inequality: |⟨𝒞, 𝒞^{ref}⟩| ≤ ‖𝒞‖ · ‖𝒞^{ref}‖. ∎

For temporal purposes, we restrict to ℒ ∈ [0, 1].

3.3 Boundary Cases

ℒ ValueConfigurationMeaning
ℒ = 1𝒞 ∥ 𝒞^{ref}Perfect alignment
ℒ = 0𝒞 ⊥ 𝒞^{ref}Orthogonal (no lock)

4. The Unlock Factor (1 – ℒ)

Define the unlock factor:

U(x, t) := 1 - ℒ(x, t)

Properties:

  • U ∈ [0, 1]
  • U = 0 when ℒ = 1 (perfect lock)
  • U = 1 when ℒ = 0 (no lock)

Physical Interpretation: U measures how unlocked the system is:

  • High U: Free to evolve → Time flows
  • Low U: Locked in place → Time stalls

5. The Product σ_θ = 𝒟 · (1 – ℒ)

5.1 Algebraic Analysis

Expand:

σ_θ = 𝒟 - 𝒟 · ℒ

Two terms:

  1. 𝒟: Raw temporal production potential
  2. 𝒟 · ℒ: Temporal production suppressed by lock

5.2 Boundary Cases Table

Case𝒟ℒσ_θPhysical State
Static Locked010Frozen, no change
Static Unlocked000Frozen, but free
Dynamic Locked>010Active but suppressed
Dynamic Unlocked>00𝒟Maximum time flow
Partial Lock>0∈(0,1)𝒟(1-ℒ)Attenuated flow

6. Sensitivity Analysis

Sensitivity to Drift:

∂σ_θ/∂𝒟 = 1 - ℒ

High lock (ℒ ≈ 1) → Low sensitivity to drift changes.

Sensitivity to Lock:

∂σ_θ/∂ℒ = -𝒟

High drift → High (negative) sensitivity to lock changes.

7. Summary

The Drift-Lock mechanism:

σ_θ = 𝒟 · (1 - ℒ)

Key Results:

  1. Multiplicative structure: Both factors must be nonzero for Time to flow
  2. Lock suppression: (1 – ℒ) acts as a “gate” on temporal production
  3. Drift drive: 𝒟 provides the raw rate of change
  4. Boundary behavior: Perfect lock (ℒ = 1) or zero drift (𝒟 = 0) halts Time
  5. Linear response: Small perturbations yield linear changes in σ_θ

This mechanism is the engine of temporal emergence in ITT.

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