Wave-Particle Duality and the Quantum Measurement Problem Derived from a Prime Number Lattice: Interference, Born Rule, and Observer Effect as Consequences of Z3 Symmetry Breaking on the Vortex Circle
This paper demonstrates that wave-particle duality and the quantum measurement problem — as exemplified by the double-slit experiment — emerge directly from the variational CTF‑PLCT framework built on the 2a×3b2a×3b prime number lattice. The lattice potential V(θ)=sin2(3θ/2)V(θ)=sin2(3θ/2) on the vortex circle S1S1 reduces to a generalized Mathieu equation whose eigenstates form a spine singlet (classical rest mass) and a degenerate Hard Wall / Temporal doublet (E1=E2E1=E2, V=3/4V=3/4). The doublet superposition ∣ψ1+ψ2∣2=4cos2(θ/2)∣ψ1+ψ2∣2=4cos2(θ/2) reproduces Young’s interference formula exactly. Measurement is identified with spontaneous Z3Z3 symmetry breaking via selection of the gradient sign dV/dθdV/dθ — the same mechanism already shown to explain why cc is invariant (Hard Wall, dV/dθ>0dV/dθ>0) while the fine‑structure constant αα runs (Temporal, dV/dθ<0dV/dθ<0). The Born rule for symmetric superpositions follows from equal gradient magnitudes ∣dV/dθ∣=33/4∣dV/dθ∣=33/4, which is a pure Tier‑1 ratio 33/2433/24. Which‑path information is shown to be equivalent to the mod‑9 zone label, and the Z3Z3 symmetry predicts a falsifiable three‑slit asymmetry. All claims are verified numerically in a reproducible Colab notebook. Two open problems are stated explicitly.
This paper demonstrates that wave-particle duality and the quantum measurement problem — as exemplified by the double-slit experiment — emerge directly from the variational CTF‑PLCT framework built on the 2a×3b2a×3b prime number lattice. The lattice potential V(θ)=sin2(3θ/2)V(θ)=sin2(3θ/2) on the vortex circle S1S1 reduces to a generalized Mathieu equation whose eigenstates form a spine singlet (classical rest mass) and a degenerate Hard Wall / Temporal doublet (E1=E2E1=E2, V=3/4V=3/4). The doublet superposition ∣ψ1+ψ2∣2=4cos2(θ/2)∣ψ1+ψ2∣2=4cos2(θ/2) reproduces Young’s interference formula exactly. Measurement is identified with spontaneous Z3Z3 symmetry breaking via selection of the gradient sign dV/dθdV/dθ — the same mechanism already shown to explain why cc is invariant (Hard Wall, dV/dθ>0dV/dθ>0) while the fine‑structure constant αα runs (Temporal, dV/dθ<0dV/dθ<0). The Born rule for symmetric superpositions follows from equal gradient magnitudes ∣dV/dθ∣=33/4∣dV/dθ∣=33/4, which is a pure Tier‑1 ratio 33/2433/24. Which‑path information is shown to be equivalent to the mod‑9 zone label, and the Z3Z3 symmetry predicts a falsifiable three‑slit asymmetry. All claims are verified numerically in a reproducible Colab notebook. Two open problems are stated explicitly.
A new `wave-particle non-dualistic interpretation at a single-quantum level' is presented by showing the physical nature of Schrodinger's wave function as an ‘instantaneous resonant spatial mode’ to which a particle's motion is confined. The initial phase associated with a state vector is identified to be related to a particular position eigenstate of the particle and hence, the equality of quantum mechanical time to classical time is obtained; this equality automatically explains the emergence of classical world from the underlying quantum world. Derivation of the Born rule as a limiting case of the relative frequency of detection is provided for the first time, which automatically resolves the measurement problem. Also, the Born rule derivation is supplemented with a geometrical interpretation. `What’s really going on' in Young's double-slit and Wheeler's delayed-choice experiments is explained at a single-quantum level. Also, an interference experiment is proposed to verify the correctness of the non-dualistic interpretation.
A new `wave-particle non-dualistic interpretation at a single-quantum level' is presented by showing physical nature of Schrodinger's wave function as an ‘instantaneous resonant spatial mode’ to which a particle's motion is confined. The initial phase associated with a state vector is identified to be related to a particular position eigenstate of particle and hence, equality of quantum mechanical time to classical time is obtained; this equality automatically explains emergence of classical world from underlying quantum world. Derivation of Born rule as a limiting case of the relative frequency of detection is provided for first time, which automatically resolves measurement problem. Also, Born rule derivation is supplemented with a geometrical interpretation. It's shown that non-dualistic interpretation statistically yields Copenhagen interpretation. ``What’s really going on? in Young's double-slit experiment is explained at a single-quantum level. Also, an interference experiment is proposed to verify correctness of non-dualistic interpretation.
Abstract: We present a computational demonstration of the Zeno Threshold (Zα) on the paradigmatic double-slit experiment. A Gaussian wave packet is propagated on a 256×128 tight-binding lattice using only two ingredients: graph topology (nearest-neighbor hopping) and unitarity (U = e-iHΔt, with the imaginary unit i forced by information conservation). No wave equation is assumed; interference fringes emerge as a strict consequence of lattice unitarity. We then introduce a which-path detector modeled as a tensor-product qubit that entangles with the particle at the slits. The entanglement entropy of the reduced detector state grows from zero toward Smax = ln 2 ≈ 0.693 nats. At the moment this entropy exceeds the observer's finite Zeno Threshold Zα, the observer's rendering engine is forced to perform Born Rule truncation—algorithmic garbage collection—erasing the off-diagonal coherence terms. The interference pattern is destroyed, leaving two classical bumps. Computational Verification Highlights: Emergent Interference: Fringes emerge strictly from discrete graph topology and unitarity, without assuming continuous wave equations. Forced Complex Phases: The complex unit i is mathematically forced by probability conservation (unitarity). The Z-Scale Breach: Which-path detection creates real entanglement entropy that breaches the observer's finite processing limit. Born Rule Garbage Collection: Truncation at the Z-Scale breach produces a detection pattern identical to the classical, incoherent sum |ψ1|2 + |ψ2|2. The full Python verification script and simulation code are available at github.com/srdrymn/atr-double-slit-simulation.
The double-slit interference experiment serves as a fundamental experiment in quantum mechanics. Core theories including wave-particle duality, quantum superposition and measurement collapse are established on the basis of this phenomenon. The mainstream Copenhagen interpretation and Bohmian pilot-wave theory rely heavily on mathematical statistical fitting to interpret experimental outcomes, yet they fail to reveal the real physical evolution mechanism of photons at the microscale. Moreover, they inherently suffer from symmetry breaking, causal discontinuity and logical inconsistency. The symmetry-breaking analogy widely adopted in academic circles contains dimensional mismatch and conceptual substitution, representing a cognitive bias that places mathematical tools above physical mechanisms. Based on the symmetry principle of classical physics and the rule of continuous causal evolution, this paper systematically identifies the core defects of mainstream quantum interpretations and points out the paradigm limitation of quantum mechanics that prioritizes mathematical fitting over physical ontology. A spatial particle resonance propagation model is proposed. This model explains that photons are essentially the resonance manifestation of space particles. When resonance passes through double slits, the resonance body splits symmetrically, and then recombines after passing through the slits to form interference. Measurement will break the symmetry of resonance, resulting in the disappearance of interference fringes. This model maintains continuous causality and geometric symmetry, and provides a new physical mechanism explanation for quantum uncertainty and measurement effects in double-slit experiments.
The double-slit experiment combines three observational facts: interference fringes in the coherent regime, position-resolved detections, and continuous fringe suppression under which-path resolution. Standard quantum mechanics accommodates all three through postulates without deriving them from deeper structure. This paper shows that the standard far-field interference law and the coherence-to-which-path transition follow as structural corollaries of results established in Quantum Mechanics as the Boundary Algebra of a Non-Invertible Projection (doi:10.5281/zenodo.19535588) and its companion (doi:10.5281/zenodo.19571321). Two results suffice for the algebraic derivation: the Born rule as the canonical quadratic measure on Im(Π), and the kernel spinor as the Hilbert quadrature of the boundary field. A third result — measurement irreversibility as a kinematic consequence of the projection geometry — belongs to the physical interpretation of measurement and temporal ordering within the same framework. The paper derives the Fraunhofer intensity law I(x) ∝ |Ψ₁ + Ψ₂|² and the detector-overlap formula with fringe visibility V = |γ|, within the standard Fraunhofer approximation and without additional postulates. It does not claim a first-principles derivation of the full boundary propagator or of single-event arrival statistics. A companion numerical simulation implements all three coherence regimes. A self-contained appendix summarises the three imported results in compact form.
This paper presents a finite, regulator-based resolution of wave–particle duality, showing that quantum duality is not a fundamental paradox but a natural consequence of domain separation. Within the Dual Regulator Framework, wave behavior is shown to arise exclusively in the virtual domain (momentum space, off-shell propagation), regulated by the finite invariant Λφ=ln(φ2),\Lambda_\varphi = \ln(\varphi^2),Λφ=ln(φ2), while particle behavior arises exclusively in the real domain (position space, on-shell detection), regulated by the Resolution Gap RG=ln(φ)/ln(π).\mathrm{RG} = \ln(\varphi)/\ln(\pi).RG=ln(φ)/ln(π). The double-slit experiment is reinterpreted as a virtual-domain interference process followed by a real-domain localization event. Which-path detection is shown to force a premature virtual-to-real transition, preventing interference from forming. Partial which-path information produces a quantitative reduction in visibility governed by a finite exponential law. The framework yields explicit, testable predictions with zero adjustable parameters, including: a φ-harmonic correction to interference fringe spacing, a sharp detector-coupling threshold for which-path information, an exact functional form for visibility–distinguishability relations, and scale-dependent corrections for matter-wave interferometry (electrons, neutrons, molecules). All results derive solely from the Trinity constants {π, φ, e}. Complementarity is shown to be a consequence of domain incompatibility, not a primitive principle. Wave–particle duality dissolves into a physically interpretable, observer-independent mechanism. This paper forms Paper II in a planned sequence addressing foundational inconsistencies in quantum field theory, following the finite resolution of the measurement problem.