Collective Field Function

The Collective Field Function is the spatial field built by summing discrete pulses, each spread over space by a coupling kernel centered on its position. It is how the book moves from single binary pulse events to extended field behavior: individual computational pulses combine through Spatial Coupling Kernels to produce an emergent collective field on a discrete substrate, so the resulting field is quantized rather than a continuous approximation. The kernel is taken as a normalized Gaussian, G(x,σ) = (1/√(2πσ²)) exp(-x²/(2*σ²)), with widths σ_i = σ_0 × (i+1)^(−1/3). Its evolution in time is given by the Recursive Coupling Equation, which adds dependence on accumulated history. The formal statement is Ψ(x,t) = sum_i P_i(t) × G(x − x_i, σ_i), where Ψ(x,t) is the Collective Field Function, P_i(t) the single pulse state of pulse i, G the Spatial Coupling Kernel, x the spatial coordinate, x_i the position of the ith pulse, σ_i its characteristic interaction width, t time and i the pulse index. Dimensions from the book: Ψ [𝕃⁻³/²], P_i [∅], G [𝕃⁻¹], x, x_i, σ_i [𝕃], t [𝕋], i [∅]. It connects to the Spatial Coupling Kernel, Recursive Coupling Equation and Resonant Field of Becoming.