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.
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.
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.
Chirality in active and passive fluids gives rise to odd transport properties, most notably the emergence of robust edge currents that defy standard dissipative dynamics. While these phenomena are well-described by continuum hydrodynamics, a microscopic framework connecting them to their topological origins has remained elusive. Here, we present a lattice model for an isotropic chiral random walk that bridges the gap between classical stochastic diffusion and unitary quantum evolution. By equipping the walker with an internal degree of freedom and a tunable chirality parameter, p, we interpolate between a standard diffusive random walk and a deterministic, topologically non-trivial quantum walk. We show that the topological protection characteristic of the unitary limit (p=1) remarkably persists into the dissipative regime (p<1). This correspondence allows us to theoretically ground the robustness of edge flows in classical chiral systems using the bulk-boundary correspondence of Floquet topological insulators. Our results provide a discrete microscopic description for odd diffusion, offering a powerful toolkit to predict transport in confined geometries and disordered chiral media.
We derive the continuous spacetime limit of the one dimensional lazy discrete time quantum walk, obtaining explicit macroscopic evolution equations for a three state model in the presence of decoherence. While continuum limits of two state quantum walks are well established, an explicit continuous spacetime formulation for the lazy three state walk, particularly including noise, has not previously been constructed. Using an SU(3) representation of a Grover type coin together with a Lindblad formulation of decoherence acting either on the coin or the spatial subspace, we systematically expand the discrete dynamics in both space and time to obtain continuum master equations governing the coarse grained evolution. The resulting generators yield a genuine partial differential equation description of the walk, going beyond purely probabilistic or spectral correspondences. We show that the unitary limit is governed by a Dirac-type SU(3) Hamiltonian describing ballistic advection of left and right moving modes coupled by local symmetric mixing, with the rest state acting as an additional internal degree of freedom. Coin dephasing selectively damps internal coherences while preserving coherent spatial transport, whereas spatial dephasing suppresses long range spatial interference and rapidly drives the dynamics toward classical behaviour. This continuum framework clarifies how internal symmetry, rest state coupling, and distinct decoherence channels shape large scale transport in lazy open quantum walks, and provides a foundation for future extensions toward multichannel quantum transport models and quantum-inspired algorithms.
Abstract We derive the continuous spacetime limit of the one dimensional lazy discrete time quantum walk, obtaining explicit macroscopic evolution equations for a three state model in the presence of decoherence. While continuum limits of two state quantum walks are well established, an explicit continuous spacetime formulation for the lazy three state walk, particularly including noise, has not previously been constructed. Using an SU (3) representation of a Grover type coin together with a Lindblad formulation of decoherence acting either on the coin or the spatial subspace, we systematically expand the discrete dynamics in both space and time to obtain continuum master equations governing the coarse grained evolution. The resulting generators yield a genuine partial differential equation description of the walk, going beyond purely probabilistic or spectral correspondences. We show that the unitary limit is governed by a Dirac-type SU ( 3 ) Hamiltonian describing ballistic advection of left and right moving modes coupled by local symmetric mixing, with the rest state acting as an additional internal degree of freedom. Coin dephasing selectively damps internal coherences while preserving coherent spatial transport, whereas spatial dephasing suppresses long range spatial interference and rapidly drives the dynamics toward classical behaviour. This continuum framework clarifies how internal symmetry, rest state coupling, and distinct decoherence channels shape large scale transport in lazy open quantum walks, and provides a foundation for future extensions toward multichannel quantum transport models and quantum inspired algorithms.
A discrete-time quantum walk (DTQW) is a lattice model of a single particle whose two-component internal state is repeatedly mixed by a unitary "coin" and shifted in space, chirality by chirality. In this work, we present a framework for coupling such a walk to a background electromagnetic (U(1)) gauge field. Instead of introducing traditional link variables common to the existing literature, we embed the gauge phase directly within the coin operator already required by the free walk, thereby eliminating the need for extra local degrees of freedom. We demonstrate this construction in two stages: first, using only coin phases to generate a purely spatial potential (A_1) while structurally forcing the temporal potential (A_0) to vanish (the temporal, or Weyl, gauge); and second, incorporating a single post-shift phase to recover a fully general potential (A_mu = (A_0, A_1). In both regimes, we show that the gauge transformation law for the coin phases is rigorously forced by the consistency requirement between the left- and right-moving sectors rather than assumed. We verify the construction non-perturbatively through the exact lattice dispersion relation, the walk’s convergence rate to the continuum Dirac equation, and its intrinsic avoidance of the fermion doubling problem.
This paper develops a discrete, information-theoretic framework in which physical reality is modeled as a lattice or relational graph of interacting “space cells,” each evolving by local update rules. The framework treats space as an active information-processing substrate rather than as a passive background. Time is modeled as an ordering of state transitions rather than as a fundamental substance. Energy is provisionally associated with event density, update rate, action flow, or structured information-processing intensity, with Planck’s constant interpreted as a conversion scale between discrete transitions, phase, and action. The paper combines formal mathematical development with a summary of toy-model probes used to test whether the framework can support the kinds of structures required of a physical theory. Mathematically, it distinguishes theorem-level consequences of local/unitary dynamics from continuum-limit derivations and from simulation-supported but still speculative model-selection branches. Finite-radius local updates imply causal cones; unitary updates preserve norm; local trace-preserving operations preserve no-signaling at the reduced-state level; split-step quantum-walk updates support Dirac-like low-momentum continuum behavior; and Schrödinger-like dynamics arise as a nonrelativistic positive-energy envelope approximation rather than as the most immediate continuum limit. The space-cell framework is organized around two candidate geometric realizations. A distortable-cell model assigns deformation, strain, connection, or topology-bearing degrees of freedom to cells and links. An alternative extrinsic χ-compression model assigns an additional internal scalar coordinate , interpreted as compression, event-density elevation, hidden displacement, or update-rate modulation. The χ option is not initially treated as a literal fourth spatial dimension; it is a parsimonious internal state variable to be compared against deformation-based models under causal, weak-field, Lorentz-like, source-specific, and false-source controls. The toy-model program summarized here tests whether local update rules can generate stable, bounded, identity-preserving informational structures that survive perturbation, scattering, false-carrier pressure, and geometry-like coupling. These simulations do not prove that nature uses space cells, but they have produced mechanism-level evidence for relational identity preservation, local reconstruction, false-carrier rejection, conservation-ledger requirements, photon-like interaction bookkeeping, and χ/deformation model-selection criteria. This paper does not claim to have derived the Standard Model, full quantum field theory, or the Einstein field equations. It presents a staged mathematical and computational research framework. The strongest current claim is that local information-processing rules can be formalized in ways that recover several necessary structural features—locality, conservation-like invariants, causal propagation, no-signaling compatibility, Dirac/Schrödinger continuum regimes, and candidate weak-field geometry equations—while toy-model branches provide constrained tests of whether those structures can support physically relevant informational excitations.