We introduce a discrete angular lattice model in which reality is generated by a four-component state vector S = (S1, S2, S3, S4) with the sum of all components equal to 1, evolving under strict plus or minus 90 degree orthogonal rotations on an elastic triadic lattice. The dynamics are governed by a single master state-transition equation derived from a nonlinear elastic potential that enforces exact 90 degree symmetry, together with dual feedback loops and a central attractor. A reproducible Python simulator confirms the existence of stable non-trivial attractors with dominant S3 and S4 phases and an emergent Fermat-spiral phase accumulation. Through mesoscopic averaging we also exhibit a suggestive formal correspondence between the discrete dynamics and the structure of the Schrödinger (and Dirac) equation in the S3-dominant regime. This continuum correspondence is presented as a candidate emergence pathway rather than a fully rigorous derivation; a multi-site spatial implementation and detailed continuum analysis are left for future work. This work explores one discrete angular model that produces interesting dynamical attractors and a formal link to continuum quantum mechanics.
We introduce a discrete angular lattice model in which reality is generated by a four-component state vector S = (S1, S2, S3, S4) with the sum of all components equal to 1, evolving under strict plus or minus 90 degree orthogonal rotations on an elastic triadic lattice. The dynamics are governed by a single master state-transition equation derived from a nonlinear elastic potential that enforces exact 90 degree symmetry, together with dual feedback loops and a central attractor. A reproducible Python simulator confirms the existence of stable non-trivial attractors with dominant S3 and S4 phases and an emergent Fermat-spiral phase accumulation. Through mesoscopic averaging we also exhibit a suggestive formal correspondence between the discrete dynamics and the structure of the Schrödinger (and Dirac) equation in the S3-dominant regime. This continuum correspondence is presented as a candidate emergence pathway rather than a fully rigorous derivation; a multi-site spatial implementation and detailed continuum analysis are left for future work. This work explores one discrete angular model that produces interesting dynamical attractors and a formal link to continuum quantum mechanics.
We introduce the discrete angular tetravalent lattice as a minimal classical geometric model. The system consists of a four-component density evolving under strict +90° cyclic permutations and conditional spatial shifts. We construct a minimal non-unitary quantum walk with complex amplitudes and prove that, under a full quantum dephasing channel, its intensity evolution exactly recovers the classical tetravalent process. By analysing the continuum limit of this decohered process we demonstrate a duality: in the passive regime the system yields a dissipative complex wave equation, while in the active regime the continuum dynamics acquire the algebraic skeleton of the two-dimensional Weyl operator ∂x + i∂y. Quantitative scaling tests confirm a wave-speed duality: the absolute information edge propagates at the hyperbolic light-cone speed c = 1, while the macroscopic density ridge propagates at c = 1/2. Furthermore, a spatial domain wall between opposite chiral cycles (A and Aᵀ) traps a robust one-dimensional chiral edge current driven by classical skipping orbits; the current remains stable under strong spatial disorder. We discuss the scope of this construction relative to phenomenological active quantum walks, clarifying the boundary between classical geometric mechanisms and fully coherent multi-sector quantum dynamics.
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 paper develops the quantum dynamical layer of the Quantum Lattice Model (QLM) by showing how the standard wave equations of quantum mechanics arise as continuum descriptions of coherent phase-action transport on a discrete Planck-scale lattice. The framework is constructed from the per-radian Planck primitives {ℏ, ℓP, tP}, where each lattice tick advances physical phase by one radian and transports one quantum of reduced action. The corresponding phase-action and phase-flow relations are dS = ℏ dθ and E = ℏ dθ/dτ, while the invariant lattice transport identity c = ℓP/tP fixes the maximal causal phase-transport speed of the lattice. Starting from symmetric nearest-neighbor phase transport between Planck cells, the continuum limit yields the relativistic wave operator (1/c²)∂t² − ∇². Introducing intrinsic rest-phase rotation with frequency ω0 = mc²/ℏ produces the Klein-Gordon equation as the continuum description of synchronized spatial phase transport and local rest-phase rotation. A first-order relativistic factorization yields the Dirac equation, while extraction of the rapidly rotating rest-phase carrier produces the nonrelativistic Schrödinger equation governing slow envelope dynamics. Within this framework, the quantum wavefunction is interpreted as a coherent phase-envelope amplitude describing phase-action transport across the lattice. The conserved density ρ = |φ|² arises from the continuity equation of Schrödinger evolution, and the associated current takes the phase-flow form J = (ℏ/m)ρ∇θ. The paper also shows how interaction sectors enter the same transport framework as modular coupling layers. Scalar potentials contribute additional local phase rotation to the slow envelope, while routing-weighted transport modifies the effective capacity of lattice links. Coulomb coupling and weak-field gravitational coupling are presented as examples: the Coulomb potential appears as a scalar phase-rotation term, and gravitational routing modulation of the rest-phase carrier yields the weak-field potential V = mΦ. These results organize the Schrödinger, Klein-Gordon, and Dirac equations as successive continuum limits of deterministic phase-action transport on the Planck lattice, providing the quantum dynamical layer of the Quantum Lattice Model.
Overview In the centennial year of the Schrödinger Equation (1926–2026), this work introduces a novel conceptual framework where quantum dynamics emerges from the intrinsic topological properties of a discrete, non-orientable space-time lattice. The core of the model is the "Möbius Pixel": a fundamental 0D/0+1D quantum unit defined on a non-orientable manifold. By employing the formalism of Pseudo-Hermitian Quantum Mechanics, we demonstrate how the antilinear symmetries inherent in surfaces like the Möbius strip and the Klein bottle naturally generate real energy spectra and complex dynamical structures. Key Scientific Contributions Emergent Dynamics: Derivation of a modified Schrödinger equation from a discrete pixel lattice (pi, pi*), featuring emergent Dirac and Majorana-type coupling terms. Topological Foundation: A rigorous link between non-orientable geometry and the Nambu-like doubled representation of quantum states. Pseudo-Hermiticity: Use of the metric operator η and the involutory operator C to ensure spectral stability and positive-definite inner products in the presence of torsional couplings. Unified Vision: A "symphonic" perspective where fundamental interactions and mass are interpreted as rotovibrational resonances and topological inertia of the lattice. Technical Abstract We construct a toy-model framework in which quantum dynamics emerges from the topological structure of non-orientable surfaces. Using a discrete pixel lattice, we derive a pseudo-Hermitian Hamiltonian whose continuum limit reproduces a Schrödinger equation with additional Dirac and Majorana-type terms. The reality of the spectrum is guaranteed by the pseudo-Hermiticity condition. In this framework, the observable wavefunction emerges as an energy-resolved sector of a higher-dimensional pseudo-Hermitian system. Technical Focus: Pseudo-Hermitian Stability and Lagrangian Spectrum This framework is centered on the formalization of a stable, real-valued spectrum for non-orientable pixel dynamics. The core mathematical challenge addressed is the preservation of a positive-definite inner product in the presence of torsional (antilinear) terms. 1. The Metric Operator η and Involutory C To ensure spectral stability, we define the Hamiltonian H as pseudo-Hermitian with respect to the metric operator η: H† = η H η⁻¹ We introduce a linear, involutory operator C (C² = I) that commutes with the Hamiltonian [H, C] = 0. This operator C is essential for constructing a strictly positive-definite inner product: ⟨ψ | φ⟩_C = ⟨ψ | η C | φ⟩ This construction guarantees that the "Möbius Pixel" states reside in a physical Hilbert space with a consistent probabilistic interpretation, neutralizing the instabilities typically associated with non-Hermitian torsional couplings. 2. Lagrangian Extension and Hamiltonian H The internal dynamics is derived from a 0+1D Lagrangian density accounting for vibrational (n) and topological (k) degrees of freedom. The extension to the full Hamiltonian H includes: Mass-topological Term: m_top c² Internal Excitation Spectrum: Defined by the functional f(n,k), representing the energy levels of the discrete pixel. Torsional Coupling: An antilinear term λ₄ Φ T inducing mixing between the pi and pi* sectors. The resulting spectrum E_n,k remains real under the stability condition: (m_top c² + λ₃ f(n,k))² ≥ (λ₄ Φ_k)² This inequality provides a natural threshold for the lattice, where Planck-scale incompressibility acts as a physical cut-off. Call for Formalization and Collaboration I have established the foundational structure for η and C within the context of the C-formalization. I am looking for collaborators (Theoretical Physicists, Mathematicians) to: Generalize the C operator for an N-pixel lattice interaction. Refine the Lagrangian to include full Lorentz invariance in the continuum limit. Analyze Torsional Modes as symmetry-protected sectors in a 4D pseudo-Hermitian field theory.
This paper introduces a discrete interaction lattice derived from the directed geometry of tukutuku weaving and formulates it as a two-component unitary dynamical system. It is shown that the lattice recovers the 1+1 dimensional Dirac equation in the continuum limit, establishing a rigorous connection to relativistic quantum dynamics. At the discrete level, however, a structural obstruction emerges: a sign-flip interface fails to support a localized zero-mode due to broken chiral symmetry. This failure is identified as a fundamental limitation of the single-step model, leading to the necessity of a period-2 staggered lattice and a symmetrized split-step formulation. The work provides a mathematically grounded framework while clearly distinguishing proven results from heuristic structural correspondence.
This work introduces the Phase–Lattice Atomic Model, a fundamentally geometric and dynamical reinterpretation of atomic structure. Instead of a point-like nucleus and probabilistic orbital clouds, the model proposes that mass–energy is distributed over a spherical surface through discrete, phase-coherent micro-spheres arranged along quantized spiral pathways. Each micro-sphere carries a triple-phase structure separated by 120°, and the requirement of global phase compatibility over successive Planck-scale steps enforces a natural “phase-selection rule” for propagation. This phase-coherent dynamics generates a quantized surface lattice composed of finite spiral windings, with the total number of phase-carrying nodes bounded by the experimentally observed proton–electron mass ratio (≈1836). The interaction between phase delay accumulation and phase-compatibility constraints produces amplitude envelope modulation, giving rise to a natural mechanism for electronic fine-structure splitting—without invoking continuous orbital wavefunctions. The framework provides:• a geometric foundation for spin, exclusion, and handedness;• a discrete mechanism for fine-structure variation;• a surface-dynamical alternative to point-charge quantum orbitals;• a unifying interpretation of classical and quantum behavior based on minimal phase disturbance and least-action pathways. This preprint consolidates the theoretical structure, graphical representations, derivations, and physical implications of the Phase–Lattice model and is intended to make the framework publicly accessible while formal journal submission is in progress.