We present a discrete graph-based model in which quantum interference emerges from bidirectional update dynamics over a finite history space. An action functional is defined over graph evolution histories, and complex amplitudes are assigned via phase weighting. Summation over histories produces constructive and destructive interference consistent with the Born probability rule. Numerical simulations recover classical statistics when backward updates are disallowed and yield Bell-type correlations violating classical bounds up to the Tsirelson limit. These results indicate that quantum behavior can arise from minimal relational structure without assuming spacetime geometry, wavefunctions, or continuous fields.
We present a discrete graph-based model in which quantum interference emerges from bidirectional update dynamics over a finite history space. An action functional is defined over graph evolution histories, and complex amplitudes are assigned via phase weighting. Summation over histories produces constructive and destructive interference consistent with the Born probability rule. Numerical simulations recover classical statistics when backward updates are disallowed and yield Bell-type correlations violating classical bounds up to the Tsirelson limit. These results indicate that quantum behavior can arise from minimal relational structure without assuming spacetime geometry, wavefunctions, or continuous fields.
Introduction: We introduce a novel quantum-inspired simulator that approximates the Feynman path integral within a discrete, graph-theoretic framework, mapping logical qubits to graph vertices. Materials and methods: The entire pathfinding process is driven by gate operations, forming the core of our unique, three-tier search: (1) classical path probability boosting to identify primary routes, (2) Feynman-inspired vertex amplification to weight critical nodes, and (3) complex amplitude calculations, based on logical qubit interactions (edge-weighted), which are crucial for reproducing quantum interference. Results: While methods (1) and (2) identify the shortest classical paths, only the third tier successfully approximates superposition and interference effects through complex amplitude summation, where distinct phase contributions lead to either amplification or suppression. We show that interference patterns align with pairwise Pauli-Z correlations that effectively emulate entanglement phenomena. High gate dynamism, especially 4-qubit gate operations, are essential for the broad search of optimal routes, while boundary conditions may limit continuous exploration and refinement of pathways, emphasizing the need for fine parameter tuning. Simulations achieve convergence, with entropy plateaus confirming stability given a sufficient tolerance threshold for systems up to 8 vertices. The temporal evolution of vertex occurrence counts exhibits Gamma–Poisson (Cox process) statistics, with variance-to-mean ratios increasing over time, indicating non-Poissonian dynamics. A key finding is that quantum pathfinding, driven by destructive interference, effectively reveals dominant paths that may not be classically apparent. The observed destructive interference directly results from the summation of complex amplitudes where the phase differences equal approximately π radians (180), revealing complementary path and “anti-path” structures. Conclusions: The novelty of our simulator lies in its dual vertex-wave nature that provides new insights for quantum algorithm design and the study of quantum dynamics. Beyond the core framework, structural and mathematical analogies emerging from Simulation H—including a correlation structure consistent with a light-cone pattern—are reported as posthoc observations of the simulation output. These are not physical derivations, but suggest directions for further investigation at the intersection of graph-theoretic quantum simulation and relativistic quantum mechanics.
This preprint develops a discrete, fully relational framework in which spacetime structure, classical dynamics, and quantum-like behavior emerge from three minimal principles: relational existence without background manifold, local deterministic update, and finite per-element update capacity. The central organizing primitive is finite update capacity. At each update step, elements can realise only a bounded number of relational changes. Competition among admissible updates induces “blocked realisations,” which are defined micro-ontologically as energy. Proper time is identified with realised record growth. Spatial distance and effective dimension arise from intrinsic graph structure and shell scaling. Under broad locality conditions, marginal frontier stability constrains viable scaling regimes. Requiring compatibility between flux balance and the observed inverse–square law selects an effective three-dimensional scaling regime among integers. Classical force structure emerges from anisotropies in blocked realisation. A statistical layer appears through coarse-graining of deterministic microhistories. Composition and norm stability require unit-modulus complex weights; refinement invariance and classical coherence fix the phase rule θ=ℏ−1∫Edτ, with the same blocking-defined energy that generates intrinsic force. In decohered regimes, quadratic (Born-type) probability scaling follows from additive consistency under random-phase aggregation. The framework is programmatic: it does not yet derive general relativity or quantum field theory from explicit microscopic rule families. However, it establishes structural coherence linking geometry, inertia, inverse–square interaction, phase accumulation, and probabilistic outcomes under a single finite-capacity relational principle. The work also identifies structural empirical constraints, including compatibility between shell scaling and classical force behavior, stability of inertial response, and the possible appearance of capacity-induced high-energy dispersion within the intrinsic causal-cone bound. Explicit construction and simulation of proposal/selection rule families constitute the next stage of investigation.
We propose a discrete, node-based framework in which quantum mechanics emerges as a continuum approximation of underlying update dynamics. Fundamental entities are nodes related by causal update relations, with no probabilistic postulate assumed at the microscopic level. Transition amplitudes are assigned as minimal two-dimensional internal degrees of freedom, while observations correspond to coarse-grained node densities. We show that the Born rule is not a fundamental axiom but an inevitable quadratic statistical form arising from phase elimination, σ-additivity, and non-contextuality in the continuum limit. The Schrödinger equation appears as an effective generator of near-identity node updates, and probability measures are uniquely fixed by information‑geometric consistency. This framework reinterprets quantum theory as a shadow of discrete node dynamics rather than a primary description of reality.
This paper is Paper 003 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the Born Ledger: the probability scalarization of Binary Mechanics. Paper 002 derived finite polarized graph histories and the primitive finite amplitude law. In that construction, amplitudes are not assigned to instantaneous graph states, vertices, or hidden local phase registers. Instead, they arise as finite sums of Z4-weighted admissible histories grouped into coarse record classes. The present paper begins from that result and asks what probability ledger is forced once record-disjoint coarse alternatives are required to receive stable nonnegative weights. The central result is a no-fence scalarization theorem: the only admissible nonnegative scalarization of a finite complex record amplitude is its squared modulus. When normalized over an exhaustive record-disjoint partition, this yields the Born-type probability rule. The derivation does not assume Hilbert space, a wavefunction, a measurement postulate, a continuum path integral, primitive probability, or primitive branch selection. Interference appears before probability as cancellation inside finite history amplitudes, and probability appears only as the record-level scalarization of those amplitudes. No fine history is primitively deleted or selected; the ledger assigns weights to record-disjoint coarse alternatives generated from the full admissible finite history structure. The paper therefore supplies the Born-weighting step in the Binary Mechanics construction: finite graph histories first, finite amplitudes second, record ledgers third, normalized probabilities last.
We develop an information-theoretic reconstruction of quantum dynamics based on inference over action space. The fundamental object is a density of action states encoding the multiplicity of dynamical alternatives between configurations. Maximum-entropy inference introduces a finite resolution scale in action, implying that sufficiently close action contributions are operationally indistinguishable. We show that this indistinguishability, together with probability normalization and action additivity, selects complex amplitudes and unitary evolution as the minimal continuous representation compatible with action additivity, probability normalization, and inference under finite resolution. Quantum interference and unitarity therefore emerge as consequences of these assumptions rather than independent postulates. From the resulting propagator, the Lagrangian, Hilbert-space structure, and Schrödinger equation follow as derived consequences. In the infinitesimal-time limit, action differences universally fall below the resolution scale, making coherent summation the minimal consistent description at every step. The numerical value of the action scale is fixed empirically and identified with ℏ.
This paper shows how the L2 norm underlying the Born rule arises as a phase-invariant quadratic observable when linear spectral representations are applied to discrete systems governed by L1 structure. Rather than assuming Hilbert space geometry, the result demonstrates how quadratic probability emerges from a combination of discrete combinatorics, linear representation, and phase invariance. The central result establishes that, given an underlying L1 causal structure, a linear spectral representation of states, and invariance under phase rotations, the expected observable magnitude under uniform phase averaging reduces to a sum of squared amplitudes. This recovers the L2 norm and yields the Born rule after normalization. Within this framework: Discrete systems are modeled using L1-based combinatorial structure Observers introduce linear complex representations of these systems Phase variables act as unobservable degrees of freedom and are averaged using a uniform invariant measure Cross-terms cancel under phase averaging, leaving a purely quadratic observable This provides a structural explanation for why quantum probability takes a quadratic form. The result can be interpreted as a consequence of Parseval-type identities applied under phase invariance, rather than as an independent postulate. The paper includes: A formal derivation of the quadratic observable from phase averaging A proof that only the quadratic (p = 2) measure preserves underlying combinatorial volume Numerical validation demonstrating consistency with Parseval identity Simulations reproducing standard interference behavior and norm preservation A construction showing compatibility between defect-based dynamics and Schrödinger-type evolution under additional assumptions The work does not attempt a full derivation of quantum mechanics. Instead, it establishes how L2 structure and Born-rule probabilities arise naturally from discrete L1 systems when represented in a linear spectral form with phase invariance. This result connects discrete geometry, harmonic analysis, and quantum probability, and provides a bridge between combinatorial models and continuous Hilbert space representations.