Potential Field

Φ(j) = ∑_i=1¹⁰ x_i / √(j−i)²+ε²

Potential fields reveal influence and attraction.

Elastic-Chain Hamiltonian

H = ½k ∑(x_i−μ)² + ½κ ∑(x_i+1−x_i)²

Harmonic coherence measures structural synchronicity.

Resonance

R_ij⁽ⁿ⁾ = exp[−α|x_i−n x_j|]

Resonance reveals hidden harmonic relationships.

Phase Embedding

ψ_i = e^{2πi(x_i mod M)/M}

Phase embedding transforms discrete values into cyclic intelligence.