Spatial Coupling Kernel

The Spatial Coupling Kernel is the normalized Gaussian that spreads each discrete pulse over space, so that pulses at different positions can overlap and combine into a collective field. In the Collective Field Function each pulse state is multiplied by a kernel centred on that pulse's position, and the field is the sum over all pulses. The book derives the kernel for binary substrate interactions as a Gaussian whose integral over space is 1, which it says ensures probability conservation. Its width shrinks slowly with pulse index, σ_i = σ_0 × (i+1)^(−1/3), a law the book attributes to substrate correlation scaling. With history-dependent evolution given by the Recursive Coupling Equation, the kernel is how discrete computational events become collective field dynamics. The formal statement is G(x,σ) = (1/√(2πσ²)) exp(-x²/(2*σ²)), with ∫G(x,σ)dx = 1 and σ_i = σ_0 × (i+1)^(−1/3), where G is the Spatial Coupling Kernel, x the spatial coordinate (measured from the pulse position x_i), σ the characteristic interaction width, σ_0 a reference width and i the pulse index. Dimensions from the book's Where-list: G [𝕃⁻¹], x, x_i, σ_i [𝕃], i [∅]. It connects to the Collective Field Function, Recursive Coupling Equation and Resonant Field of Becoming.