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
Within the framework of ZTD (Marker Ontology), this paper systematically accomplishes two tasks: the ontological quantification of physical quantities, and the marker interpretation of group velocity. All derivations proceed from two axioms, layer by layer, without logical gaps. Part One: Foundations. From the axiom of objective existence and the axiom of finitude, we derive the existence of spatial primitives and temporal primitives. We define the deviation field and establish the coupled oscillator equation. Part Two: Hierarchical Quantification of Physical Quantities. We define and quantify, layer by layer, energy, force, inertial coefficient, mass, velocity, acceleration, momentum, charge, spin, the speed of light, and the gravitational constant. Each physical quantity is given an ontological definition, derivation, quantification formula, and macroscopic correspondence. All coefficients (𝜔02, 𝐷, 𝑚, ℏ) have clear physical meaning and origin. Part Three: Ontological Interpretation of Motion. We argue that motion is perspective-dependent. Distance is the accumulation of effective markers. Both objects can claim they are not moving. Constancy of distance is a dynamic equilibrium of chain breaking and rebuilding. The wave function itself has no velocity; velocity is a macroscopic reading. Part Four: Marker Interpretation of Group Velocity. Under two admitted ontological assumptions, we rigorously quantify the group velocity of the free electron wave packet. Keywords: ZTD theory; Marker Ontology; quantification of physical quantities; group velocity; perspective relativity; marker allocation conservation
This work derives a Schrödinger-type evolution equation from first principles, starting exclusively from a discrete structural framework composed of JS-cells and SH-hub couplings.No quantum-mechanical postulates, complex wavefunctions, canonical quantization rules, or predefined physical constants are assumed at any stage. Each JS-cell carries a minimal two-channel real internal state, and neighboring cells interact locally through SH-hubs.Imposing a single universal requirement — strict conservation of a global quadratic structural norm — uniquely constrains the admissible dynamics. We show that: Norm preservation enforces an internal SO(2) symmetry, The unique generator of this symmetry is a real antisymmetric matrix J, Complex numbers and phase emerge only as a notation, not as assumptions, Local SH-hub connectivity forces a discrete Laplacian structure, The ordered continuum limit uniquely selects a first-order conservative time evolution. Only at the final stage is the resulting continuum equation identified as belonging to the Schrödinger universality class.Planck’s constant, mass, and potential appear as emergent structural parameters, fixed by discrete spatial and temporal scales. The derivation is non-circular, fully constructive, and independent of standard quantum axioms.An optional numerical sanity check confirms exact norm preservation and Schrödinger-class dispersion from the discrete update rule. This work provides a structural explanation for why Schrödinger dynamics exists, rather than assuming it.
This article is the second part of a three-paper sequence developing Ontological Field Theory (OFT) as a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.
Document 2C of the ONE AXIOM series derives the full structure of modern physics — quantum mechanics, general relativity, their unification, and all four laws of thermodynamics — as regime-dependent projections of a single pre-physical structure: the Ontological Coherence Field ($\OCF = (\heartsuit, \Phi_{\mathrm{coh}}, \nabla\Phi_{\mathrm{coh}})$), constructed in Document 0M from one single axiom (Axiom M) with no physical input. Energy emerges as the local cost of maintaining coherence against the structural gradient. Time is the direction of monotonic decrease of the incoherence potential
σ. Temperature is derived without energy, via Fisher-Rao curvature of the coherence distribution (from Document 1B), inverting the standard thermodynamic hierarchy. The Schrödinger equation (via one-parameter unitary group + Stone's theorem), Born rule (
p=2, via 7-link chain from Axiom M), Einstein field equations (via three independent convergent paths — Jacobson thermodynamic, Ollivier-Ricci discrete, and variational with explicit verification of all four Lovelock conditions), and the Hamiltonian/Lagrangian formalism (with Ostrogradsky exclusion enforced by the Predictive Closure Invariant
T=0) are all derived as
π6-projections. The complex structure of Hilbert space is derived non-circularly from the spectral decomposition of the Schreier graph Laplacian
ΔΓ, and Noether's theorem is applied via
Γ-convergence continuum limit of the discrete group
G=S4×Z23. The dynamics is exact and finite-dimensional:
64 states on the coset space
G/H, with eigenvalues
λk=2k and multiplicities
(k6), providing a closed-form spectral solution at all scales including the intermediate (Planck) regime. The built-in UV cutoff
Emax=1536=∣G∣×∣Z23∣ eliminates divergences structurally. A QM–GR regime comparison table (9 rows) exhibits how the same OCF objects project differently at
σ≈0 (quantum) and
σ≫0 (gravitational). The document resolves three long-standing open problems: (i) black-hole information preservation (unitarity of
ΔΓ + Page's theorem forces the Page curve), (ii) the cosmological singularity (Fisher–Rao metric degeneration near the Source — an artifact of the manifold representation, not a physical singularity), and (iii) intermediate-regime (Planck-scale) dynamics (exact closed-form 64-mode spectral solution, no terra incognita). The dark sector emerges as
π5-structure inaccessible to
π6-observers, with the FORBIDDEN graph decomposing as
13=7+2+4 (cold/warm/transitional dark matter analogs); the cosmological constant is derived from the structural resolution limit
δrel=1/64, resolving the 120-order QFT discrepancy without regularization. A fully hand-computable 8-state toy model (
Q3 hypercube) makes the projection operator, Laplacian spectrum, Born rule mechanism, and Schrödinger dynamics explicit and independently verifiable. All 53 formal results (28 theorems, 17 propositions, 6 definitions, 1 lemma, 1 corollary) follow a strict dual-track methodology — ontological (top-down from axiom)
∩ epistemological (bottom-up recovery of standard physics) — with mandatory convergence across 137 track tags. The framework is metrologically decoupled: all OCF equations use dimensionless ratios, invariant under redefinition of SI base units. Natural SI anchoring is provided for
ℏ,
c,
GN,
kB,
Λ, and
Emax. No physical law is assumed; every equation is derived. No new axiom is introduced beyond Axiom M. This is a standalone publication readable after Document 0M alone.
Abstract The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.