spw.quest • Semantic Gradients

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1 · Motivation

While the photonic model captures flow and interference in Spw parsing, a complementary framework is needed to quantify how meaning evolves across document space (position) and reader time (traversal). Semantic Gradients provide a scalar–vector field for local cohesion and global drift, facilitating adaptive highlighting, context-window optimization, link re-ranking, and entropy-aware reading paths.

2 · Gradient Definition

Symbol	Description	Units	Example
\(\phi(x)\)	Semantic potential at position \(x\)	bits / token	High in dense concepts
\(\nabla\phi\)	Semantic gradient (vector)	bits / token / char	Steepest change
\(\lVert\nabla\phi\rVert\)	Cohesion magnitude	bits / token / char	Low → cohesive
\(\Delta\phi\)	Drift along a path	bits / token	Traversal integral

^model[semantic.gradient]{
  potential φ : entropy-reduced meaning density,
  gradient ∇φ : ∂φ/∂xᵢ across document axes,
  cohesion   : ‖∇φ‖⁻¹  // high cohesion → low gradient
}

3 · Mathematical Formalism

3.1 Semantic Potential

\(\phi(x) = -\sum_{i} p_i(x)\log_2 p_i(x)\), where \(p_i(x)\) comes from local concept distributions (embeddings/co-occurrence windows).

3.2 Gradient Operator

∇φ = ⟨ ∂φ/∂x_sem , ∂φ/∂x_syn , ∂φ/∂x_prag ⟩
∂φ/∂x ≈ [φ(x+δ) - φ(x-δ)] / 2δ

3.3 Drift Integral

Δφ = ∫₍x₀→x₁₎ ∇φ · d𝓁   // shift along reader traversal

3.4 Divergence & Curl

Use \(\nabla\cdot\nabla\phi\) to find semantic sources/sinks; \(\nabla\times\nabla\phi\) to flag rotational inconsistencies.

4 · Integration with Parsing & Physics

n(x) = n₀ · [ 1 + κ · ‖∇φ(x)‖ ]  // turbulence → refractive index
// Couple to beam-weight lattice θ to bias precedence in high-drift zones.

5 · Experiments & Validation

^experiment[grad.validate]{
  input    : <doc_set>,
  compute  : ∇φ,
  correlate: <comprehension_scores>,
  output   : r²
}

6 · Future Directions

Validate cohesion–comprehension correlation (retention, eye-tracking).
Temporal decay: \(\phi(x,t)=\phi(x)\,e^{-\lambda t}\).
RL-guided navigation over low-drift paths.
Inter-document gradients for knowledge-graph expansion.
Hardware acceleration (FPGA/AR overlays).