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From Nearest-Neighbor Propagation to the Schrödinger Equation: A Rigorous Derivation Based on Step Counting and Fluctuation Corrections

This paper derives the Schrödinger equation as the continuum limit of nearest-neighbor propagation within Marker Ontology, demonstrating that the imaginary unit emerges naturally from step-counting fluctuations and that Monte Carlo simulations support the transition to unitary evolution by canceling real diffusion terms.

Original authors: YanZhuang

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: YanZhuang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Turning a "Guess" into a "Story"

For nearly 100 years, the Schrödinger equation has been the most important rule in quantum mechanics. It tells us how tiny particles (like electrons) move and behave. However, since it was first written in 1926, physicists have treated it like a magic spell: it works perfectly, but no one knew why it worked or where the strange imaginary number $i$ came from. It was just assumed to be there.

This paper claims to finally solve that mystery. The author, Yan Zhuang, argues that the Schrödinger equation isn't a magical guess. Instead, it is the natural result of space and time being made of tiny, discrete "pixels" (like a digital image) rather than being smooth and continuous.

The Core Metaphor: The Pixelated Universe

Imagine the universe isn't a smooth, continuous stage, but a giant, invisible grid made of tiny dots called "spatial primitives." Think of these like the pixels on a high-definition screen, but for reality itself.

The Walk: A particle (or a wave of energy) doesn't glide smoothly through space. Instead, it hops from one pixel to its immediate neighbors, step by step.
The Clock: Time also ticks in tiny steps. Every time the particle hops to a new pixel, one "time step" passes.
The Count: As the particle hops, it keeps a mental tally of how many steps it has taken. This is called "Step Counting."

Where Does the Magic Number " $i$ " Come From?

In math, the number $i$ is the "imaginary unit." In the Schrödinger equation, it's the reason particles wave and oscillate instead of just diffusing (spreading out like ink in water).

The paper explains that $i$ isn't a magic assumption. It emerges naturally from the Step Counting:

Imagine you are walking in a circle. Every step you take turns you slightly.
If you take a huge number of tiny steps to complete a full circle (360 degrees), your path looks smooth.
The author shows that if you count these steps mathematically, the "turning" of the step count creates a complex number (a number with a real part and an imaginary part).
The Analogy: Think of a digital clock. If you look at it from far away, the numbers seem to change smoothly. But if you zoom in, you see it jumps from one number to the next. The paper argues that the "jump" (the discrete step) creates a tiny rotation. When you zoom out to the human scale, all those tiny rotations add up to create the imaginary number $i$ .

So, the imaginary unit $i$ is just the mathematical signature of a particle taking tiny, discrete steps through a pixelated universe.

The "Diffusion" vs. "Oscillation" Problem

Usually, when things move randomly (like a drop of ink in water), they spread out and flatten. This is called diffusion.

Diffusion Equation: "The ink spreads out until it's flat."
Schrödinger Equation: "The particle waves and keeps its shape, never flattening out."

The paper asks: How do we get from spreading ink to a waving particle?

The answer lies in Fluctuations (tiny random errors).

In a perfect grid, the particle would just spread out (diffuse).
But in the author's model, the grid is slightly irregular (like a Voronoi pattern, which looks like a honeycomb where the cells aren't all the same size).
Because the grid is irregular, the "steps" vary slightly in length. The paper uses computer simulations (Monte Carlo) to show that these tiny irregularities create a "cross-term" (a side effect) that perfectly cancels out the spreading effect.
The Result: The spreading stops, and the wave starts to rotate (oscillate) instead. This rotation is what we see as the Schrödinger equation.

The "Step Number" ( $N_0$ )

The paper calculates a specific number called $N_0$ .

Imagine a wave of light. It has a wavelength (the distance between two peaks).
$N_0$ is the number of tiny "pixel steps" it takes to cover that one wavelength.
The author estimates this number is huge: about $10^{28}$ .
Why this matters: Because this number is so massive, the individual steps are so tiny that our eyes (and our math) can't see them. We only see the smooth, continuous wave. This explains why the universe looks smooth to us, even if it's actually made of pixels.

What the Paper Does NOT Claim

It is important to stick to what the paper actually says:

It does not claim to have built a new computer or a new medical device.
It does not claim to have solved the mystery of gravity or black holes yet.
It does not claim to have proven that "hidden variables" (the idea that quantum randomness is just a lack of knowledge) are definitely true, though it suggests this framework fits that idea.
It focuses strictly on free particles (particles not being pushed or pulled by forces) and explains how the Schrödinger equation arises from the basic structure of space and time.

Summary

This paper tells a story:

Space is made of tiny pixels.
Particles hop from pixel to pixel, counting their steps.
This counting creates a natural "turning" effect, which mathematically becomes the imaginary number $i$ .
Tiny irregularities in the pixel grid cancel out the natural spreading of the particle, forcing it to wave instead.
The result is the Schrödinger equation, which is no longer a mysterious rule, but a logical consequence of a pixelated universe.

The author concludes that the "magic" of quantum mechanics is actually just the physics of counting steps on a very fine grid.

Technical Summary: From Nearest-Neighbor Propagation to the Schrödinger Equation

Problem Statement
Since its formulation in 1926, the Schrödinger equation has served as a fundamental postulate of quantum mechanics, yet its physical origin—specifically the emergence of the imaginary unit $i$ —remains unexplained. Standard interpretations often treat the wave function as a mathematical tool for probability amplitudes without addressing its physical reality or the necessity of complex numbers. This paper addresses the gap in deriving the Schrödinger equation from more fundamental physical principles, aiming to explain the origin of the imaginary unit and the transition from diffusive to unitary evolution without relying on ad hoc mathematical assumptions like Wick rotation.

Methodology
The derivation is conducted within the framework of Marker Ontology, which posits that space and time are discrete, composed of "spatial primitives" (minimal pixels with scale $\ell_0$ ) and "temporal primitives" ( $t_P$ ). The core mechanism is nearest-neighbor propagation, where a wavefront advances from a primitive to all its nearest neighbors in each time step.

The methodology proceeds through the following steps:

Discrete Formulation: The state of the wavefront is defined by a discrete complex amplitude $\Psi(n) = |\psi(n)| e^{i\phi(n)}$ , where the phase $\phi$ is derived from step counting ( $N$ ), the accumulated number of propagation steps. The phase is defined as $\phi = 2\pi N / N_0$ , where $N_0$ is a characteristic step number corresponding to a full $2\pi$ cycle.
Discrete Evolution Equation: A discrete evolution equation is established where the amplitude at the next time step is the average of neighbor amplitudes multiplied by a phase factor $e^{i\Delta\phi_0}$ .
Continuum Limit: The authors perform a Taylor expansion of the discrete equation to derive a continuous partial differential equation. This involves expanding the spatial neighbors and the complex phase factor $e^{i\Delta\phi_0} \approx 1 + i\Delta\phi_0$ .
Fluctuation Analysis: To address the cancellation of the real diffusion term, the paper introduces fluctuation corrections arising from the non-uniformity of the nearest-neighbor distances in a random Poisson-Voronoi tessellation. The authors analyze cross-terms between spatial derivatives and phase fluctuations.
Numerical Verification: A Monte Carlo simulation is performed on a 3D Poisson-Voronoi tessellation to verify the statistical properties of the fluctuation cross-terms.

Key Contributions and Results

Natural Emergence of the Imaginary Unit $i$ : The paper demonstrates that the imaginary unit is not an external mathematical assumption but emerges naturally from the first-order expansion of the complex representation of step counting ( $e^{i\Delta\phi_0}$ ) in the continuum limit. The term $i\Delta\phi_0$ couples with the spatial diffusion term, endowing the effective diffusion coefficient with an intrinsic imaginary part.
Derivation of the Schrödinger Equation: By combining the time derivative with the spatial diffusion term (modified by the imaginary component), the authors derive an equation of the form $\partial_\tau \Psi = i D_0 \nabla^2 \Psi$ . By defining the quantum of action $\hbar = \ell_0^2/t_P$ and effective mass $m = \hbar/(2D_0)$ , this recovers the standard free-particle Schrödinger equation: $i\hbar \partial_\tau \Psi = -(\hbar^2/2m)\nabla^2 \Psi$ .
Fluctuation Cancellation Mechanism: The derivation identifies that the standard Schrödinger equation corresponds to a limit where the real part of the diffusion coefficient is canceled by second-order fluctuation terms. Monte Carlo simulations ( $M=10^5$ samples) verify that the fluctuation cross-term $\langle \delta\phi_j \cdot \delta_j \rangle$ is non-zero (statistical significance $p < 10^{-6}$ ) and that the magnitude of the second-order fluctuation term is comparable to the real diffusion coefficient $D_0$ . This provides numerical support for the transition from diffusive to purely unitary evolution.
Characteristic Step Number ( $N_0$ ): The paper estimates the characteristic step number $N_0 = \lambda/\ell_0$ to be on the order of $10^{28}$ for visible light, based on a primitive scale $\ell_0$ comparable to the Planck length. This large number justifies the continuum limit approximation, as the phase increment per step is infinitesimal ( $\sim 10^{-28}$ rad).
Self-Consistency of $N_0$ and $\hbar$ : The paper establishes a self-consistency relation showing that $N_0$ (characterizing phase discreteness) and $\hbar$ (characterizing action discreteness) are different macroscopic manifestations of the same underlying discrete spatiotemporal structure.

Significance and Claims
The paper claims to provide the first rigorous physical derivation of the Schrödinger equation from a discrete, nearest-neighbor propagation mechanism, a foundation absent since 1926.

Physical Reality of the Wave Function: It reinterprets the wave function not as a mere probability amplitude but as a description of a "diffuse, indivisible physical event" (a collective synchronized fluctuation) propagating on a primitive network.
Resolution of the "Wick Rotation" Mystery: The paper argues that the "Wick rotation" (the appearance of $i$ ) is not an artificial analytic continuation but a necessary macroscopic manifestation of the $U(1)$ rotation structure inherent in the discrete step-counting mechanism.
Unitary Evolution: The unitary nature of quantum evolution is traced back to the stepwise phase accumulation and the specific cancellation of real diffusion terms by geometric fluctuations in the primitive network.

Theoretical Boundaries and Limitations
The authors explicitly acknowledge several limitations and areas for future work:

Exact Cancellation Condition: While the Monte Carlo simulation confirms the magnitude of fluctuation terms is sufficient to affect the real diffusion coefficient, the precise analytic condition for the exact cancellation (how fluctuations are self-consistently determined by the instantaneous wave function state) is not yet derived.
Massive Particles: The current derivation applies to the $p=1$ limit (photon-like propagation). Generalizing to $p < 1$ to derive the specific masses of massive fermions (like electrons) and bound-state structures is a goal for subsequent work.
Lorentz Invariance: The discrete primitive network does not possess continuous Lorentz symmetry at the microscopic scale; the recovery of Lorentz invariance in the macroscopic limit is not addressed.
Bell's Inequality: The fluctuation mechanism is framed within statistical hidden variables; whether this framework can reproduce nonlocal correlations revealed by Bell's inequality requires future rigorous examination.
Photon Treatment: The photon is used as an example of the propagation mechanism, but the paper notes that the zero-mass limit requires a separate treatment (starting from Dirac or Maxwell equations) which is deferred.