Abstract Causal set theory proposes that spacetime is a locally finite partial order (Bombelli et al., 1987; Sorkin, 2005). We extend this program by introducing remainder as a third primitive quantity. Each node of the causal graph carries three primitives—order, number, and remainder—and edges are discrete tick events in which remainder propagates between nodes. We define the tick-demand operator T̂ and the remainder operator R̂ with commutation relation [T̂, R̂] = iℏτ, and write the TCC Hamiltonian Ĥ = T̂ + V(R̂). This first-order Hamiltonian creates a structural tension with the second-order Schrödinger equation. We resolve this tension by conjecturing a two-component embedding of TCC dynamics—motivated by the physical interpretation that remainder propagates in two directions, accumulating and resolving—and applying the Foldy-Wouthuysen (FW) transformation. The Schrödinger equation emerges as the non-relativistic limit, with ℏτ = ℏ fixed by the correspondence principle. Three correction terms arise at next order: a kinetic energy correction (~∂⁴ρ), a Darwin-type contact term (~∂²V/∂ρ²), and a remainder-demand coupling (~(∂V/∂ρ)·∂ρ) with no analog in standard quantum mechanics. We discuss the measurement problem, causal irreversibility, the Born rule as an emergent property, testable predictions that distinguish TCC from existing programs, and catalog the open problems that must be resolved for TCC to make independent empirical predictions.
Abstract Causal set theory proposes that spacetime is a locally finite partial order (Bombelli et al., 1987; Sorkin, 2005). We extend this program by introducing remainder as a third primitive quantity. Each node of the causal graph carries three primitives—order, number, and remainder—and edges are discrete tick events in which remainder propagates between nodes. We define the tick-demand operator T̂ and the remainder operator R̂ with commutation relation [T̂, R̂] = iℏτ, and write the TCC Hamiltonian Ĥ = T̂ + V(R̂). This first-order Hamiltonian creates a structural tension with the second-order Schrödinger equation. We resolve this tension by conjecturing a two-component embedding of TCC dynamics—motivated by the physical interpretation that remainder propagates in two directions, accumulating and resolving—and applying the Foldy-Wouthuysen (FW) transformation. The Schrödinger equation emerges as the non-relativistic limit, with ℏτ = ℏ fixed by the correspondence principle. Three correction terms arise at next order: a kinetic energy correction (∂⁴ρ), a Darwin-type contact term (∂²V/∂ρ²), and a remainder-demand coupling (~(∂V/∂ρ)·∂ρ) with no analog in standard quantum mechanics. We discuss the measurement problem, causal irreversibility, the Born rule as an emergent property, testable predictions that distinguish TCC from existing programs, and catalog the open problems that must be resolved for TCC to make independent empirical predictions.
The Schrödinger equation iℏ ∂ₜψ = Ĥψ, with non‑relativistic Hamiltonian Ĥ = ℏ²|p̂|²/(2m) + V(x̂), has been quantum mechanics’s central dynamical statement for nearly a century. It governs atomic spectra, molecular bonding, semiconductors, neutron interferometry, and countless other empirical settings. The question is why it has the form it does. In standard QM, the Schrödinger equation is a postulate. The Hilbert space is given. The momentum operator p̂ = −iℏ∇ is given. The kinetic form |p̂|²/(2m) is inherited directly from classical mechanics — quantum theory simply adopts the classical T = p²/(2m) structure without deeper justification. The first‑order time derivative is chosen because empirically it works. Each piece is accepted because the resulting equation matches experiment. This walkthrough shows how the Event Density framework derives the Schrödinger equation, in its full standard form, from a smaller set of structural commitments about what reality is made of at the substrate level. Every element — the Hilbert space, the momentum operator, linearity, first‑order time structure, the kinetic‑plus‑potential form, the factor of 1/(2m) — emerges as a forced consequence rather than a separate postulate. The derivation runs through Stone’s theorem applied twice — once on spatial translation (forcing p̂ = −iℏ∇) and once on time translation (forcing existence and uniqueness of a self‑adjoint generator with automatic linearity and first‑order structure) — closed by Galilean Lie algebra integration that identifies the time‑translation generator with the kinetic‑plus‑potential operator. The factor of 1/(2m) emerges as the chain‑rule Jacobian of the Galilean commutator [Ĥ, K̂ᵢ] = −iℏ p̂ᵢ, not as a borrowing from classical mechanics. The first‑order time derivative is a consequence of operator exponentials, not a chosen feature.
We propose a minimal discrete model in which causality, time, spacetime geometry, and quantum measurement emerge from a single primitive concept: the node (boundary). A node is defined as an irreversible completely positive (CP) map representing interaction simultaneity—the fundamental unit where states undergo non-unitary transformation. Conventional causality is reconceptualized as a chain of such nodes connected by reversible propagation. Time is not an external parameter but the cumulative count of nodes; temporal flow velocity is proportional to node density. We demonstrate that Minkowski spacetime emerges as the continuum limit of a node network, with gravitational curvature arising from spatial gradients in node density. The quantum measurement problem dissolves naturally as nodes represent the only loci of irreversibility. We derive testable predictions for gravity-induced decoherence independent of gravitational phase shifts, offering clear falsifiability. This framework unifies gravity, quantum mechanics, thermodynamics, and temporal phenomenology through a single ontological primitive: the event itself.
This paper offers suggested improvements to the causal sets program in discrete gravity, which treats spacetime geometry as an emergent manifestation of causal structure at the fundamental scale. This viewpoint, which I refer to as the causal metric hypothesis, is summarized by Rafael Sorkin's phrase, "order plus number equals geometry." Proposed improvements include recognition of a generally nontransitive causal relation more fundamental than the causal order, an improved local picture of causal structure, development and use of relation space methods, and a new background-independent version of the histories approach to quantum theory. Besides causal set theory, à la Bombelli, Lee, Meyer, and Sorkin, this effort draws on Isham's topos-theoretic framework for physics, Sorkin's quantum measure theory, Finkelstein's causal nets, and Grothendieck's structural principles. This approach circumvents undesirable structural features in causal set theory, such as the permeability of maximal antichains, studied by Major, Rideout, and Surya, and the configuration space pathology arising from the asymptotic enumeration of Kleitman and Rothschild. The paper culminates in the theory of co-relative histories and kinematic schemes, combining the causal metric hypothesis, the histories approach to quantum theory, and Grothendieck's relative viewpoint. This leads to the derivation of causal Schrödinger-type equations as dynamical laws for discrete quantum spacetime.
This work develops a graph-theoretic and process-based foundation for indefinite causal order in quantum theory, demonstrating that causal indefiniteness is not an ad hoc or exotic phenomenon but an emergent structural property of quantum dynamics when the primitive unit of evolution is taken to be the П-такт (P-Tact) — a discrete quantum process composed of local unitary superposition and entangling unitary interaction. Building upon a sequence of previously published foundational works, this paper formulates indefinite causal order as a superposition of quantum process graphs, where each graph encodes a definite causal structure, and the quantum state corresponds to a coherent superposition over such graphs. Within this framework, causal order is not fundamental but arises only in coarse-grained or classical limits. We show that: Indefinite causal order naturally emerges from the superposition of process graphs, without invoking external control systems or higher-order operations. The quantum switch is reinterpreted as a special finite realization of a more general graph-superposition principle. Time and causality emerge as effective parameters in the infinite-graph (continuum) limit of discrete П-такт dynamics. The graph-theoretic formulation is fully compatible with — and structurally supports — the Everettian (Many-Worlds) interpretation, where branching corresponds to graph bifurcations rather than wavefunction collapse. Classical spacetime causality appears as a macroscopic ordering imposed on an underlying non-causal quantum substrate. This approach unifies quantum information, indefinite causal order, emergent temporality, and Everettian branching within a single formal framework based on discrete process units and their graph composition. The results close several open conceptual gaps in the interpretation of quantum switches, process matrices, and non-classical causal structures, offering a consistent ontology grounded in quantum dynamics itself. This paper directly builds upon and extends the following foundational works: Pavlov, M. (2026). Quantum Switch as an Emergent Process Structure from Discrete Process Tacts (П-такт). Zenodo.https://doi.org/10.5281/zenodo.18498181 Pavlov, M. (2026). Quantum Switch and Teleportation. Zenodo.https://doi.org/10.5281/zenodo.18509173 Pavlov, M. (2026). Emergence of Causality, Gravity, Time, and Three-Dimensional Space from Discrete P-Tact (П-такт) Dynamics. Zenodo.https://doi.org/10.5281/zenodo.18504244 Pavlov, M. (2026). P-Tact (П-такт) as the Primitive Unit of Quantum Theory. Zenodo.https://doi.org/10.5281/zenodo.18516711 Together, these works establish a coherent program aimed at demonstrating the universality of quantum mechanics, the absence of fundamental collapse, and the emergence of spacetime and causality from discrete quantum process dynamics.
Context: This manuscript serves as the foundational ontology paper for the Directed Acyclic Graph Interpretation (DAGI) research program at Whytics. Abstract: Quantum mechanics is remarkably successful in predicting experimental outcomes, yet it remains conceptually challenging: the status of the wavefunction, the meaning of "measurement", and the origin of the arrow of time are still debated. We propose an interpretation grounded in information theory and graph theory, representing quantum histories as a single Directed Acyclic Graph (DAG) of informational states. In this model, nodes represent quantum informational states (density operators) and directed edges represent physical interactions that update those states. Branching occurs precisely when an interaction creates a durable, distinguishable record that cannot be coherently erased at the relevant scale; decoherence corresponds to record proliferation into the environment; and quantum erasure corresponds to coherent record removal that can re-enable interference by merging previously distinguishable histories. Furthermore, we prove a Kolmogorov Consistency Theorem for finite-amplitude DAGs, guaranteeing that the framework yields a coherent probability measure on the decohered quotient graph and exactly recovers unitary Schrödinger evolution in the non-branching limit. We show how this ontology resolves standard quantum paradoxes without introducing ambiguous collapse postulates or an ontological multiplication of worlds, and we outline auxiliary operational predictions motivated by the model. In particular, DAGI motivates an "informational time dilation" program: an operational clock's inferred progress decreases with the rate of irreversible record creation under schedule-matched controls, and increases toward baseline when records are coherently erased. We additionally report initial IBM Quantum hardware results from the C1-LC program regarding light-cone transport, motivating a refined conformal hypothesis for how time and causal structure emerge from record density. Key Highlights: Novel Ontology: Replaces the ontological extravagance of the Many-Worlds Interpretation with a single, strictly causal DAG of informational records. Mathematical Rigor: Provides a mathematically rigorous Kolmogorov Consistency Theorem for finite-prefix cylinder sets, resolving standard measure-theoretic pathologies in history-based interpretations. Paradox Resolution: Resolves Wigner's Friend, Delayed Choice, and Schrödinger's Cat paradoxes through an explicit mathematical graph-refinement logic. Hardware Testability: Formulates the theoretical foundation for operational time dilation and causal-cone transport limits actively being benchmarked on IBM superconducting quantum processors.
We derive the deepest foundations of quantum mechanics and spacetime from the information-protection framework. The Lorentzian signature and constant speed of light emerge from the directed graph structure. The canonical commutation relation follows from the consensus delay, while an independent information-bandwidth bound constrains phase-space resolution. The wavefunction is identified as an information description held by an observer subnetwork, the measurement problem is resolved via the modified Schrodinger equation with a non-unitary relaxation term, and macroscopic commutativity emerges in the large-N limit. Bell nonlocality is resolved through the active-edge mechanism: entangled qubits are connected by dedicated graph edges whose consensus completes within a Planck time, providing an explicit realization of ER = EPR. Spin-1/2 emerges as a topological defect, and the spin-statistics theorem follows from braid group representations. The underlying network is deterministic; quantum probability is emergent for finite observers.