From ZTD Axioms to the Schr\"odinger Equation:\\ Downgrade Derivation of the Coupled Oscillator Model and Conditional Rigorous Proof of the Continuum Limit (First Version)

Abstract

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