From Nearest-Neighbor Propagation to the Schr¨odinger Equation: A Rigorous Derivation Based on Step Counting and Fluctuation Corrections (Revised Version)

Since its introduction in 1926, the Schr¨odinger equation has been regarded as a
fundamental postulate of quantum mechanics, with its physical origin—especially
the appearance of the imaginary unit i—remaining unexplained. In this paper,
within the framework of Marker Ontology and starting from the nearest-neighbor
propagation mechanism of spatial primitives, we rigorously prove that the continuum
limit of nearest-neighbor propagation naturally converges to the Schr¨odinger
equation. The imaginary unit i is not an introduced mathematical assumption, but
rather an imaginary component that emerges naturally from the expansion of the
complex representation eiϕ of step counting in the continuum limit. This imaginary
component couples with the spatial diffusion term, endowing the effective diffusion
coefficient with an intrinsic imaginary part. The non-vanishing of the fluctuation
cross-term is numerically verified through Monte Carlo simulation (p < 10−6), confirming
that the magnitude of the second-order fluctuation term is sufficient to affect
the real part of the effective diffusion coefficient. We further provide a rigorous proof
of the cancellation forced by the conservation law: the random tessellation paper
has rigorously proved the invariance of the speed of light, and has argued that
the isotropy of the spatial primitive network is equivalent to the principle of relativity.
From these two postulates, the standard mathematical theorem of special
relativity directly yields the Lorentz transformation, whose algebraic corollary, the
normalization of the four-velocity, is reformulated in ZTD as the marker allocation
conservation relation. This conservation relation forces the real part of the effective
diffusion coefficient to vanish in the continuum limit, so that the fluctuation
correction must contribute exactly −D0 to cancel the bare real diffusion term. The
paper concludes with an appendix containing a purely real-number derivation, starting
from two purely real discrete states without presupposing any complex number
concept, in which the imaginary unit i and the complex wave function emerge naturally
in the continuum limit, completely eliminating the suspicion of “deriving the
imaginary from the imaginary.” We also provide an order-of-magnitude estimate
of the characteristic step number N0 (N0 = λ/ℓ0 ∼ 1028) from the properties of
the primitive network, and demonstrate the self-consistency relation between N0
and the quantum of action ℏ. This paper provides the Schr¨odinger equation with a
physical foundation that has been absent since its inception.
Keywords: Schr¨odinger equation; nearest-neighbor propagation; step counting;
complex wave function; characteristic step number; fluctuation correction; Monte
Carlo verification; conservation law forced cancellation; Marker Ontology