Scattering amplitudes from geometry of kernel-free functionals on piecewise random null paths with memory

We introduce a simple functional of algebra-valued fields with definite Lorentz properties that is kernel-free yet it encodes the structure of tree-level scattering amplitudes. The fields are evaluated at the points of a piece-wise straight paths in the anti-self-dual plane of complexified Minkowski space, with the directions of consecutive segments correlated so that the path has a velocity memory. We show that momentum-space reduction of the functional generates the cyclic Parke--Taylor factor, off shell in terms of light-front bilinears and on shell in terms of spinor brackets. The same reduction, organized differently, can be naturally promoted to twistor space without any twistor-space input: a closed cyclic chain of Penrose transforms on a twistor line, in which the Parke--Taylor denominators arise as boundary terms of the integrals along the paths. Finally we show that derivatives with respect to one of the path parameter generate numerators that together with the Parke-Taylor denominator build scattering amplitudes. Lorentz invariance fixes the spins of the inserted fields and the homogeneities of the corresponding twistor functions without any use of supersymmetry. The three-point outputs reproduce the kinematic forms of the light-cone cubic vertices of chiral higher-spin theory, and the derivative monomials of order four generate the components of the $\overline{\rm MHV}$ supermultiplet of $\mathcal N=4$ super Yang--Mills theory, off shell and on shell.