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

Yan Zhuang

Published Sep 3, 2026• 0 citations
Publisher

Abstract

Since its introduction in 1926, the Schr¨odinger equation has been regarded as afundamental postulate of quantum mechanics, with its physical origin—especiallythe appearance of the imaginary unit i—remaining unexplained. In this paper,within the framework of Marker Ontology and starting from the nearest-neighborpropagation mechanism of spatial primitives, we rigorously prove that the continuumlimit of nearest-neighbor propagation naturally converges to the Schr¨odingerequation. The imaginary unit i is not an introduced mathematical assumption, butrather an imaginary component that emerges naturally from the expansion of thecomplex representation eiϕ of step counting in the continuum limit. This imaginarycomponent couples with the spatial diffusion term, endowing the effective diffusioncoefficient with an intrinsic imaginary part. The non-vanishing of the fluctuationcross-term is numerically verified through Monte Carlo simulation (p < 10−6), confirmingthat the magnitude of the second-order fluctuation term is sufficient to affectthe real part of the effective diffusion coefficient. We further provide a rigorous proofof the cancellation forced by the conservation law: the random tessellation paperhas rigorously proved the invariance of the speed of light, and has argued thatthe isotropy of the spatial primitive network is equivalent to the principle of relativity.From these two postulates, the standard mathematical theorem of specialrelativity directly yields the Lorentz transformation, whose algebraic corollary, thenormalization of the four-velocity, is reformulated in ZTD as the marker allocationconservation relation. This conservation relation forces the real part of the effectivediffusion coefficient to vanish in the continuum limit, so that the fluctuationcorrection must contribute exactly −D0 to cancel the bare real diffusion term. Thepaper concludes with an appendix containing a purely real-number derivation, startingfrom two purely real discrete states without presupposing any complex numberconcept, in which the imaginary unit i and the complex wave function emerge naturallyin the continuum limit, completely eliminating the suspicion of “deriving theimaginary from the imaginary.” We also provide an order-of-magnitude estimateof the characteristic step number N0 (N0 = λ/ℓ0 ∼ 1028) from the properties ofthe primitive network, and demonstrate the self-consistency relation between N0and the quantum of action ℏ. This paper provides the Schr¨odinger equation with aphysical 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; MonteCarlo verification; conservation law forced cancellation; Marker Ontology

DOI
10.5281/zenodo.22274575

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Since its introduction in 1926, the Schr¨odinger equation has been regarded as afundamental postulate of quantum mechanics, with its physical origin—especiallythe appearance of the imaginary unit i—remaining unexplained. In this paper,within the framework of Marker Ontology and starting from the nearest-neighborpropagation mechanism of spatial primitives, we rigorously prove that the continuumlimit of nearest-neighbor propagation naturally converges to the Schr¨odingerequation. The imaginary unit i is not an introduced mathematical assumption, butrather an imaginary component that emerges naturally from the expansion of thecomplex representation eiϕ of step counting in the continuum limit. This imaginarycomponent couples with the spatial diffusion term, endowing the effective diffusioncoefficient with an intrinsic imaginary part. The non-vanishing of the fluctuationcross-term is numerically verified through Monte Carlo simulation (p < 10−6), confirmingthat the magnitude of the second-order fluctuation term is sufficient to affectthe real part of the effective diffusion coefficient. We further provide a rigorous proofof the cancellation forced by the conservation law: the random tessellation paperhas rigorously proved the invariance of the speed of light, and has argued thatthe isotropy of the spatial primitive network is equivalent to the principle of relativity.From these two postulates, the standard mathematical theorem of specialrelativity directly yields the Lorentz transformation, whose algebraic corollary, thenormalization of the four-velocity, is reformulated in ZTD as the marker allocationconservation relation. This conservation relation forces the real part of the effectivediffusion coefficient to vanish in the continuum limit, so that the fluctuationcorrection must contribute exactly −D0 to cancel the bare real diffusion term. Thepaper concludes with an appendix containing a purely real-number derivation, startingfrom two purely real discrete states without presupposing any complex numberconcept, in which the imaginary unit i and the complex wave function emerge naturallyin the continuum limit, completely eliminating the suspicion of “deriving theimaginary from the imaginary.” We also provide an order-of-magnitude estimateof the characteristic step number N0 (N0 = λ/ℓ0 ∼ 1028) from the properties ofthe primitive network, and demonstrate the self-consistency relation between N0and the quantum of action ℏ. This paper provides the Schr¨odinger equation with aphysical 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; MonteCarlo verification; conservation law forced cancellation; Marker Ontology
Abstract 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 contin uum 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, confirming that the magnitude of the second-order fluctuation term is sufficient to affect the real part of the effective diffusion coefficient, thereby providing numerical support for the cancellation of the real diffusion term and the transition to purely unitary evo lution. We further 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
Within the framework of ZTD (Marker Ontology), this paper provides a downgradederivation of the Schrödinger equation from two ontological axioms. The paper consists oftwo parts.Part I (Physics): From the finitude axiom, we rigorously derive that the change ofprimitive states requires time, excluding zeroth-order dynamics. Second-order dynamicsis chosen as the minimal nontrivial dynamics (ontological parsimony assumption). Fromground-state stability, elastic restoring force is derived; from nearest-neighbor coupling, thecoupling term is derived, yielding the coupled oscillator equation. Key clarification: spatialprimitives are the minimal distinguishable units of free fluctuation, not small balls with fixedsize. The refresh of primitives is asynchronous—synchronous refresh would requireinfinite information to coordinate, violating Axiom 2. Asynchronous refresh allows collectivefluctuation modes to persist. The offset ϕn is a measure of the collective fluctuationmode, not the size deviation of a single primitive. The oscillator equation naturally yieldstwo real degrees of freedom—displacement and normalized velocity. These two real degreesof freedom differ by 90◦ in phase in simple harmonic motion, naturally encoding a complexamplitude, from which the imaginary unit i emerges. Under the condition that the randomtessellation continuum limit holds, the oscillator equation converges to the Klein-Gordonequation, and then through a rigorous algebraic transformation to the Schrödinger equation.The old diffusion equation is shown to be a special case of the oscillator equation in theoverdamped limit.Part II (Mathematics): A conditional rigorous proof is given for the continuum limitof the Poisson-Voronoi coupled oscillator network. Rigorously proved are: energy conservation,uniform boundedness, H1loc boundedness of the piecewise linear interpolation, weakcompactness, the discrete edge sum identity, and the complete weak-form limit derivationfrom the unique core assumption (H’) to the Klein-Gordon equation. The unique core openproblem (H’) is precisely stated: does the vector graph Laplacian of the Poisson-Voronoiadjacency graph weakly converge to the continuous gradient operator with scaling O(l20)?This paper is a conditional rigorous proof: every step from (H’) to the conclusion is rigorous;the only remaining (H’) is an independent open mathematical problem.Keywords: Schrödinger equation; downgrade; coupled oscillator model; asynchronousrefresh; Poisson-Voronoi tessellation; graph Laplacian; spectral convergence; Klein-Gordonequation; ZTD theory
The Schr¨odinger equation is the central postulate of quantum mechanics. Standardquantum mechanics treats it as a fundamental assumption and does not deriveit from any deeper mechanism.In this paper, within the framework of ZTD (Marker Ontology), we prove astrong downgrade theorem: under the conditions of strict locality, unitarity,isotropy, and Markovianity, the continuum limit of the single-step evolution operatoron a discrete primitive network uniquely yields the Schr¨odinger equation.The core argument of the strong downgrade is:1. Unitarity + Markovianity → U = eiHtP ;2. Strict locality → H = αI + βA, higher-order terms are excluded;3. Isotropy → A → zI + l206 ∇2;4. Continuum limit → Schr¨odinger equation.This paper contains two appendices:Appendix 1 (Purely Real-Number Derivation): Decompose the unitaryoperator into real and imaginary parts, proving that the imaginary unit i emergesnaturally from real orthogonal evolution, eliminating the criticism of “deriving theimaginary from the imaginary.”Appendix 2 (Generalization to the Potential Term): Under the conditionthat “external constraints only modify the self-energy,” generalize the free-particleequation to the complete Schr¨odinger equation.Keywords: Schr¨odinger equation; strong downgrade; uniqueness theorem; discrete primitivenetwork; locality; unitary evolution; purely real-number derivation; potential term;ZTD theory
Quantum mechanics has achieved unparalleled mathematical success, but has left a seriesof black boxes at the level of physical pictures: What does it mean that state vectors livein a complex Hilbert space? Why does the imaginary unit i appear? Where does thecommutation relation [ ˆX , ˆp] = iℏ come from? Why does measurement cause collapse? Whydoes the Born rule hold?In this paper, within the framework of ZTD (Marker Ontology), we provide physicalontological interpretations for every mathematical object of quantum mechanics.The positioning of this paper must first be made clear: this is a conceptual frameworkpaper, not a complete mathematical derivation. The goal of this paper is to establisha mapping relationship between the ZTD physical picture and the mathematical structure ofquantum mechanics. For each item in the list, rigorous mathematical argumentation needsto be gradually developed in subsequent work. Among the mathematical derivations givenin this paper, some are already rigorous (such as the discrete derivation of the commutationrelation), some are still conceptual (such as the derivation of the Born rule), and some carrycircularity risks (such as the borrowing of the Fermi golden rule for the Born rule). Theseare honestly marked in the text.The core mappings include:1. Complex Hilbert space corresponds to the complex amplitude space on the primitivenetwork;2. State vectors are the mathematical generalization of diffuse patterns;3. The wave function is the physical distribution of the diffuse pattern on the primitivenetwork;4. The imaginary unit i is the phase encoding of step counting;5. The commutation relation is the continuum limit of the discrete adjacency matrix;6. Unitary evolution is the pattern maintaining itself;7. Measurement is the capture of the diffuse pattern by an atom;8. The Born rule is the statistical result of capture;9. Quantum entanglement is the sharing of the same birth event;10. The uncertainty principle is a necessary consequence of spatial discreteness.This paper does not invent new mathematics, nor does it propose new testable predictions.Its goal is: to turn quantum mechanics from “mathematical mechanics” backinto “physical mechanics.”Keywords: quantum mechanics; mathematical structure; physical mapping; ontology; diffusepattern; unitary evolution; ZTD theory
The Schrödinger equation is derived classically assuming that particles present local \nrandom spatial fluctuations compatible with the presence of the zero-point field. With- \nout specifying the forces arising from this permanent matter-field interaction but ex- \nploring its fundamental properties (homogeneity, isotropy and random aspect) to justify \nthe emergence of the continuity equation in one-particle context, these fluctuations are \ndescribed in terms of the probability density. Specifically, the starting point is the as- \nsumption that the local activities, which turn the path followed by the particle totally \nunpredictable, must be associated with an energy proportional to \n@ \nP \n=@ \nt \n. The polar form \nof the wave function, which connects the obtained classical equations with the corre- \nsponding quantum equation, emerges as a by-product of the approach.
The Schrödinger equation need not be postulated. This paper demonstrates that the imaginary unit iii in quantum mechanics and the Schrödinger equation itself are both derived consequences of the Dirac equation when formulated in the real Clifford algebra Cl(1,3)\mathrm{Cl}(1,3)Cl(1,3) - the Spacetime Algebra introduced by Hestenes (1966). No new physics is proposed; the result collects and organises existing mathematical results into a single self-contained derivation chain: Cl(1,3)  →  Hestenes-Dirac  →  γ0-split  →  Pauli  →  Schroedinger\mathrm{Cl}(1,3) \;\to\; \text{Hestenes-Dirac} \;\to\; \gamma_0\text{-split} \;\to\; \text{Pauli} \;\to\; \text{Schroedinger}Cl(1,3)→Hestenes-Dirac→γ0​-split→Pauli→Schroedinger The role of iii is played by the spin-plane bivector γ21=γ2γ1\gamma_{21} = \gamma_2\gamma_1γ21​=γ2​γ1​, a real geometric object satisfying γ21  2=−1\gamma_{21}^{\;2} = -1γ212​=−1 as a consequence of the Minkowski metric. Complex-valued wavefunctions are thus projection artifacts: the residue of real four-dimensional geometry viewed from a three-dimensional observer frame. Key Results - The imaginary unit iii in the Schrödinger equation is identified as the spin-plane bivector γ21\gamma_{21}γ21​ - a directed plane in spacetime, not an abstract algebraic postulate.- The Schrödinger equation is derived as the non-relativistic limit of the Hestenes--Dirac equation via the γ0\gamma_0γ0​-split (observer frame choice), with iii inherited from the spacetime bivector structure.- Two of the five standard axioms of quantum mechanics (complex Hilbert space and the Schrödinger equation) are demoted from postulates to derived theorems. The measurement postulate and tensor product rule are retained.- Full compatibility with the Renou et al. (2021) no-go theorem and the Chen et al. (2022) experimental confirmation: the formulation is isomorphic to complex quantum mechanics, not a real restriction of it. Cl(1,3)\mathrm{Cl}(1,3)Cl(1,3) contains the required complex structure via γ21\gamma_{21}γ21​.- Convergence with Penrose's twistor programme: both approaches conclude that complex structure in quantum mechanics and Lorentzian spacetime geometry have a common origin. Document Contents - complex_wavefunctions.pdf - Main paper (13 pages). Self-contained derivation from Cl(1,3)\mathrm{Cl}(1,3)Cl(1,3) to Schrödinger, with compatibility analysis (Renou/Chen/Penrose), axiom restructuring, and pedagogical appendix.- complex_wavefunctions.tex - LaTeX source. Standalone, no external dependencies. Scope and Originality This paper presents no new mathematical results. The identification i↔γ21i \leftrightarrow \gamma_{21}i↔γ21​ is due to Hestenes (1967); the non-relativistic reduction in geometric algebra is given in Doran & Lasenby (2003). The contribution is expository: collecting the derivation chain into a single path with the geometric identity of iii tracked at each step, and situating the result in the context of the Renou et al. no-go theorem and its experimental confirmation.