Four-Spatial-Dimensional Quantum Scattering: A Controlled Toy-Model Study of Barrier-Shape Coupling and Extra-Coordinate Propagation

Abstract

This non-peer-reviewed exploratory preprint tests whether a coupled quantum Hamiltonian with one additional spatial coordinate can produce barrier transmission distinct from ordinary one-dimensional quantum mechanics when both calculations share exactly the same observable slice V(x,0). A prescribed self-similar Gaussian barrier is compared using three calculations: fixed-width 1D, shrinking identical-slice 1D, and full coupled (x,w) scattering with harmonic confinement. The shrinking 1D slice is itself non-monotonic, so an upward turn is not uniquely extra-dimensional. The full model nevertheless gives a quantitatively different curve, with a sampled minimum near V0=18 and T=9.87e-3 at V0=64 versus 6.09e-9 for the identical slice. This is a reproducible coupled-channel result within a mathematical toy Hamiltonian, not evidence that nature has an accessible fourth spatial dimension. The barrier family is prescribed rather than derived from a constitutive field law. The record includes the manuscript and a supplement containing code, complete numerical results, metadata, and convergence checks. ChatGPT and Sarvam AI assisted with brainstorming and language development. ChatGPT also assisted with literature triage, control design, numerical implementation and auditing, and drafting. AI systems are not authors. Chaman Prakash Kanth remains responsible for the manuscript, assumptions, metadata, and publication decision.