The Schrödinger equation is a core postulate of quantum mechanics. Standard quantummechanics treats it as a fundamental assumption and does not ask for its origin. Within theZTD (Zhongji Tongyi Dugui, Ultimate Unification Rule) framework, this paper provides aconditional derivation starting from two ontological axioms.We introduce the deviation field ϕn as the fundamental field and the statistical body asa physical interpretation. The mathematical definition of the deviation field is the relativedeviation of the effective size of a statistical body from the ground-state size; the deviation ofa single primitive is discrete and bounded below, while the deviation of a statistical body is astatistical average and can be arbitrarily small. The mathematical derivation deals with thedeviation-field equation on a Poisson–Voronoi graph, and the core mathematical problem isthe PVVL conjecture (Poisson–Voronoi Vector Laplacian Convergence Conjecture) formallyproposed in this paper.The derivation chain is: Axiom II → spatial primitives, time primitives → asynchronousrefresh → deviation field ϕn → cost function S → Lagrangian L → variational principle →equation of motion → conserved quantity S → identified as energy → Klein–Gordon equation→ Schrödinger equation.The contributions of this paper are fourfold:1. Conditional theorem: Under the assumption that the PVVL conjecture holds, thecontinuum limit of the discrete deviation-field equation converges rigorously to theKlein–Gordon equation, and then, via a standard physical limit (time averaging +low-frequency limit), to the free-particle Schrödinger equation.2. Independent mathematical conjecture: We formally propose the PVVL conjecture,state its relation to existing results of Armstrong–Venkatraman, García Trillos et al.,explain that it is an independent open mathematical problem, and provide a discussionof special cases and a numerical verification scheme.3. Ontological clarification: Energy emerges from the equation of motion rather thanbeing predefined; the imaginary unit i emerges from the rotational coupling of two realdegrees of freedom; first-order dynamics cannot be fundamental dynamics, as inferredfrom ontology; the total cost is a sum of three terms, as inferred from local additivity,lowest-order expansion, and symmetry.4. Unification with the electromagnetic reconstruction: The first-order diffusionequation in the electromagnetic reconstruction is the effective equation of the secondorderoscillator equation of this paper in the sourced, overdamped, quasi-static limit.The oscillator equation is the fundamental dynamics; the diffusion equation is its effectivelimit.Statement of theoretical boundaries: In addition to the two axioms, the derivationchain relies on physical assumptions such as locality, translation invariance, the principle ofleast action, and the form of the Lagrangian; the core continuum limit relies on the PVVLconjecture proposed in this paper, and the geometric constant c0 remains to be determined.The goal of this paper is to derive the form of the equation, not to derive the mass spectrum.1Keywords: ZTD theory; Schrödinger equation; conditional theorem; PVVL conjecture;Poisson–Voronoi tessellation; graph Laplacian; emergence of energy; electromagnetic diffusionequation
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