The incompatibility of quantum mechanics and general relativity arises from ashared foundational error: both frameworks treat time as a background parameterrather than a physical process. Relational Actualism (RA) dissolves this tensionby replacing both the Schr\"odinger equation and the Einstein field equationswith a single **Covariant Stochastic Graph Rewriting System**—the Engineof Becoming. The algorithm operates on the growing causal Directed Acyclic Graph(DAG) of actualization events at three levels: (1) a graph-theoretic primitivelayer describing candidate extensions of the causal boundary; (2) a quantummeasure layer assigning complex amplitudes to causal histories in afoliation-independent way, using Sorkin's quantum measure formalism; and (3) anemergent layer in which the Feynman path integral and the Einstein field equationsare the macroscopic descriptions: the quantum measure is exact at Level 2 andthe Feynman integral is its large-μ approximation; the Einstein fieldequations are the unique consistent Level 3 description, proved in the companionpaper RACL via Lovelock's theorem,the Lean-verified LLC, and the Pacₜ conservation theorem. The foliation-independence of the quantum measure(causal invariance) is the RA statement of Lorentz covariance at the discretelevel, and its combinatorial core has now been formally verified in the Lean 4theorem prover. Furthermore, by defining actualization via an irreversibleincrease in quantum relative entropy and leveraging the Continuous FunctionalCalculus, we prove the Stationarity Criterion for the Rindler thermal state,resolving the Unruh paradox without breaking Lorentz covariance. The algorithmmakes a specific, near-term falsifiable prediction: a strict null result forgravity-mediated entanglement experiments (the Bose-Marletto-Vedral protocol),because the spacetime metric is updated strictly from actualized vertices andcannot be placed in quantum superposition. Substantial progress on thegenerative measure: the Benincasa-Dowker sign criterion(SBDG(v) > 0) provides the discrete actualization filter,yielding a Poisson-CSG formula cₖRA(x) ∝ e⁻μμkwith μ = λ Vcoh pₜh that is local andenvironment-dependent, recovering dₛ = 3 in dense environments anddₛ = 2 (Durhuus-Jonsson-Wheater) in sparse ones.
The incompatibility of quantum mechanics and general relativity arises from ashared foundational error: both frameworks treat time as a background parameterrather than a physical process. Relational Actualism (RA) dissolves this tensionby replacing both the Schr"odinger equation and the Einstein field equationswith a single Covariant Stochastic Graph Rewriting System—the Engineof Becoming. The algorithm operates on the growing causal Directed Acyclic Graph(DAG) of actualization events at three levels: (1) a graph-theoretic primitivelayer describing candidate extensions of the causal boundary; (2) a quantummeasure layer assigning complex amplitudes to causal histories in afoliation-independent way, using Sorkin's quantum measure formalism; and (3) anemergent layer in which the Feynman path integral and the Einstein field equationsare the macroscopic descriptions: the quantum measure is exact at Level 2 andthe Feynman integral is its large-μ approximation; the Einstein fieldequations are the unique consistent Level 3 description, proved in the companionpaper RACL via Lovelock's theorem,the Lean-verified LLC, and the Pacₜ conservation theorem. The foliation-independence of the quantum measure(causal invariance) is the RA statement of Lorentz covariance at the discretelevel, and its combinatorial core has now been formally verified in the Lean 4theorem prover. Furthermore, by defining actualization via an irreversibleincrease in quantum relative entropy and leveraging the Continuous FunctionalCalculus, we prove the Stationarity Criterion for the Rindler thermal state,resolving the Unruh paradox without breaking Lorentz covariance. The algorithmmakes a specific, near-term falsifiable prediction: a strict null result forgravity-mediated entanglement experiments (the Bose-Marletto-Vedral protocol),because the spacetime metric is updated strictly from actualized vertices andcannot be placed in quantum superposition. Substantial progress on thegenerative measure: the Benincasa-Dowker sign criterion(SBDG(v) > 0) provides the discrete actualization filter,yielding a Poisson-CSG formula cₖRA(x) ∝ e⁻μμkwith μ = λ Vcoh pₜh that is local andenvironment-dependent, recovering dₛ = 3 in dense environments anddₛ = 2 (Durhuus-Jonsson-Wheater) in sparse ones.
Relational Actualism (RA) is a foundational framework in which on-shell actualization events — irreversible physical interactions satisfying ΔS(ρ‖σ₀) > 0 that write permanent vertices into a growing causal Directed Acyclic Graph — are taken as the primitive constituents of physical reality. From this single ontological commitment, with no new fields or free parameters, the framework derives the results of Papers 1–3 and unifies quantum mechanics, general relativity, and emergent complexity under one algorithm. The Engine of Becoming is a five-step Covariant Stochastic Graph Rewriting System. Step 1 identifies the causal frontier ∂G. Step 2 assigns quantum amplitudes to all possible extensions via Sorkin's foliation-independent quantum measure. Step 3 filters candidates by the entropy criterion ΔS > 0 — a Lorentz scalar, formally verified as frame-independent in Lean 4. Step 4 selects actualizing events with probability given by the quantum measure, yielding the Born rule without postulate. Step 5 recomputes the effective metric from the actualization density λ(x) via Gᵤᵥ = 8πG(Tᵤᵥ^(A) + Θᵤᵥ^(λ)), where Θᵤᵥ^(λ) = ξ[∇ᵤ∇ᵥ(ln λ) − gᵤᵥ□(ln λ)]. Quantum mechanics is the Level-2 (slowly varying λ) limit. General relativity is the Level-3 (dense graph) limit. The Einstein equations with Λ = 0 are the unique macroscopic output, established by the Local Ledger Condition (Lean-verified), the BDG action as the unique discrete Lorentz-invariant action in d = 4 (Glaser 2014), and Lovelock's uniqueness theorem. Time in RA is not a background parameter. Time is the process of the causal DAG growing. Proper time along a worldline is the count of actualization events τ = N_act/λ₀. The arrow of time is not derived from entropy gradients: it is encoded in the directed edges of the DAG, which are irreversible by construction. Ontic randomness at Step 4 is irreducible: the outcome of an actualization event is not determined by any prior fact in V. This is not epistemic uncertainty but the universe's genuine openness. The framework generates an emergent complexity hierarchy through iterated Step-5 coarse-graining. Biological assembly corresponds directly to actualization events (each JOIN operation in Assembly Theory is an exothermic on-shell energy transfer, confirmed by B3LYP/6-311+G* DFT across all five primary biological bond types). The Causal Firewall is the μ = 1 Erdős-Rényi percolation transition: the same critical actualization density governs QCD confinement–deconfinement, flat galactic rotation curves, fault-tolerance thresholds in quantum error correction, and the onset of stable biological complexity. Biosignatures should correlate with thermodynamic environments sustaining μ ≥ 1. The complete falsifiable prediction set of the program comprises 18 specific predictions spanning particle physics, quantum optics, cosmology, quantum computing, and astrobiology, all tabulated here. Hard walls are stated with precision: Bianchi compatibility, amplitude locality (discrete intrinsic proof), continuum limit convergence, and the covariant □(ln λ) extension. Collaboration targets are identified for each.
Relational Actualism (RA) proposes that the transition from quantum potentia to classical actuality is grounded in a precise physical criterion: an irreversible increase in quantum relative entropy ΔS(ρ∥σ0) > 0 with respect to the vacuum. The companion paper Relational Actualism: The Topology of Matter (RATM) has now established, via 67 Lean-verified theorems, that the BDG-stable causal topology types in 4D correspond exactly to the Standard Model particle spectrum—and that electrons and photons, the particles from which qubits are built, propagate at the minimum BDG actualization robustness score of 1. Qubit fragility is structurally grounded: no stable particle with lower BDG score exists. In the perturbative regime this coincides with the on-shell dispersion relation pμ pμ = m2c2, at which a mediating boson crosses the kinematic threshold from virtual to real and an irreversible vertex is inscribed into the growing causal directed acyclic graph (DAG) of the universe. This paper develops the implications of this framework—established in the companion papers RAQM, RAGC, RACI, and RAHC—for two fields that have lacked a precise physical account of irreversibility at the quantum level: quantum computing and quantum thermodynamics. For quantum computing, the Kinematic Snap provides a principled physical floor on decoherence, distinguishes genuinely off-shell (decoherence-immune) interaction regimes from on-shell ones, and gives the threshold theorem a physical rather than purely operational interpretation. For quantum thermodynamics, RA grounds Landauer’s principle in the causal DAG, resolves the Maxwell’s demon paradox without informationtheoretic postulates, gives the fluctuation theorems a precise actualization-event interpretation, and establishes that the thermodynamic arrow of time is structural rather than emergent. A key technical result: the quantum relative entropy—the central quantity in both quantum information theory and RA’s actualization criterion—is frame-independent by a compiled Lean 4 theorem (frame_independence in RA_AQFT_Proofs_v10.lean), proved using the continuous functional calculus unitary-conjugation lemma from the Lean-QuantumInfo library. The RA suite now has 101 Lean-verified results (zero sorry tags) across RA_AQFT_Proofs_v2.lean and the new RA_D1_Proofs.lean, which proves the BDG particle classification including colour confinement and the particle universe closure theorem. The RA suite now has 93 Lean-verified results across two proof files (RA_AQFT_Proofs_v2.lean: 26 results; RA_D1_Proofs.lean: 67 results), all with zero sorry tags and no axioms beyond the Mathlib standard library. A key quantitative connection: the Kinematic Coherence Bound threshold pth = αEM ≈ 1/137 ≈ 7.3°ø10−3 is shown to be consistent with the galactic-scale coupling constant ξ of the companion paper RADM to within ∼2% using the full baryonic disk mass, at the Planck scale, via the thermodynamic field equations established in RAGC. The Kinematic Coherence Bound threshold pth = αEM ≈ 1/137 ≈ 7.3°ø10−3 also enters the Poisson-CSG generative measure of RAGC (equation for cRA k ): it is the per-link actualization probability that determines whether the causal DAG is locally 3D or 2D, connecting quantum error correction thresholds to galactic-scale dimensional reduction and the Tully-Fisher relation. The fault-tolerance threshold, the speed of light c = lP/tP, and the biological Causal Firewall are all the same Erd˝os-R.nyi percolation transition at different scales of the causal graph. The Einstein field equations are shown to be the unique consistent macroscopic description of the RA-CSG by Lovelock’s theorem, with GR exact at μ ≫1 and RA’s departures from GR (rotation curves, Hubble tension, Planck corrections) arising precisely where μ ∼ 1. Throughout, implications are stated conditionally on the correctness of the RA framework, which is currently under peer review.
We argue that quantum randomness is a logical necessity arising from the nature of intrinsically relational properties—properties lacking definite states outside specific contexts of causal interaction. This motivates Relational Actualism (RAQM), an ontological framework classifying physical properties as invariant, classical, or intrinsically relational. RAQMgrounds wave function actualization in physical interaction rather than observer-relative facts. The actualization criterion is physically grounded and mathematically precise: an interaction constitutes an actualization event when it produces an irreversible increase in the entanglement entropy of the system with its asymptotic environment, generating a robust, redundant causal record. In weakly coupled QFT, a sufficient condition is the exchange of a real, on-shell excitation (pμ pμ = m2c2). This criterion is basis-independent, nonperturbative, and requires no new fundamental constants. To reconcile a globally consistent wave function with special relativity, the Tomonaga-Schwinger equation establishes a foliation-independent global update mechanism. We derive a consistency condition demonstrating that RAQM’s actualization map preserves identical observable statistics at spacelike-separated points regardless of hypersurface choice—a derived result, not a postulate—strictly forbidding superluminal signaling via algebraic microcausality. The no-signaling result applies rigorously to the non-selective (outcome-averaged) actualization map; conditioned branches are treated explicitly. Three specific, discriminating empirical predictions are stated: a finite actualization propagation velocity bounded by the Lieb-Robinson speed, an environment-dependent transition duration scaling as ℏ/ΔEenv, and a sharp reversibility threshold at the actualization criterion. RAQM is distinguished from Relational Quantum Mechanics (which relativizes facts), dynamical collapse models (which ground collapse in stochastic gravitational thresholds rather than causal interaction), decoherence-only models (which leave the outcome problem open), and Everettian Many-Worlds (which denies definite events). Classical behavior emerges dynamically when continuous actualization events eliminate all unconstrained degrees of freedom. The companion papers RAGC and RACI extend these results to gravity, cosmology, and biological complexity; RAHC [Sandeman, 2026e] extends the framework further to the emergence of agentic intelligence, formalizing the recursive coarse-graining hierarchy that generates successive levels of causal closure from the microscopic DAG; key theorems in those papers have been formally verified in Lean 4/Mathlib. A second Lean 4 file (RA_AQFT_Proofs_v10.lean) formalises the finite-dimensional AQFT proof program and proves the frame-independence and Rindler stationarity theorems; see the note on formal verification in the companion paper RAGC [Sandeman, 2026b].
Relational Actualism I introduces a discrete framework for fundamental physics in which spacetime is modeled as a finite causal graph evolving through irreversible event formation. Building on causal set theory, the framework defines a sequential growth process governed by the Benincasa–Dowker–Glaser (BDG) action and a local conservation law, the Local Ledger Condition. Unlike standard formulations of quantum theory and relativity, the theory treats actualization itself as primitive and does not assume a continuum spacetime, wavefunction, or collapse rule. The paper establishes the core kernel of the theory—its axioms, growth rule, acceptance criterion, and active support surface—and identifies open targets including quantum-statistical frequency laws, covariance closure, and macroscopic spacetime structure.
The Problem of Time — the apparent incompatibility between quantum mechanics' time parameter t and general relativity's proper time τ — has been treated as a fundamental obstacle to quantum gravity. I argue that this is a level-confusion: both theories define time within their own entity-theoretic frameworks, generating an apparent opposition that disappears when time is relocated to a meta-structural layer. Drawing on the meta-level temporal structure T_SDT := (V_G, ≺) (Zeng 2026), the paper formalizes QM and GR time as two projections of a single Universal Time Formula, parameterized by entity-theoretic tokens. Four token specializations recover the corresponding time concepts. A Bridge Theorem establishes mutual compatibility of all token projections. A bidirectional QM-GR channel is constructed (forward: GR geometry → QM phase; reverse: discrete chain-pair → Lorentzian spacetime), composing into a self-consistent closed loop. Three further results move beyond dissolution toward unification. First, a Madelung-bridge construction derives the Schrödinger equation — Hilbert space, unitarity, Hamiltonian, quantum potential, Born rule — from the determination-graph axioms together with the continuum embedding (CL1), with no independent QM input; the derivation extends to Klein-Gordon. Second, a Planck-scale token μ = t_P / |A_{v'}| is constructed on the discrete 𝒢_D, where |A_{v'}| is the admissibility cardinality; under CL1, this single step-measure recovers QM (|A| ≫ 1) and GR (|A| = 1) as its two continuum limits, with Einstein's equation proved in the classical regime (|A| = 1) via a two-stage argument (Benincasa-Dowker action convergence + standard scalar-tensor variation). The exact quantum-gravitational correction is derived as a scalar-tensor equation with φ = 1/|A|. Third, the discrete determination graph without continuum embedding constitutes a self-contained Planck-scale quantum gravity theory: it has a well-defined path integral, a finite-dimensional S-matrix (unitary in the κ = 0 sector, isometric in the κ > 0 sector; no UV divergences), preserves information across black hole horizons, resolves singularities, derives the Past Hypothesis, and provides structural origins for gauge symmetry, Hawking radiation, spectral dimensionality, and the spin-statistics connection. The Problem of Time is not merely dissolved — it is replaced by a constructive framework in which QM and GR are unified as two limits of a single discrete structure.
Quantum mechanics describes the potential future; general relativity describes the actualized past; the present is the process of actualization that turns the one into the other. That single move is Time–Actualization (TA), the foundational ontology set out here. TA assumes the quantum formalism and the relativistic causal structure, and derives neither: it adds no new physics, makes no new prediction, and asks to be judged on coherence and unifying power rather than on a measurement. It proposes a common ontological reading of measurement, EPR and the structural arrow of time. D5 locates the einselecting environment in the event's causal past; the couplings within that past remain imported, and the system/environment split is relocated rather than removed. In detail. The ontology is a minimal triplet (ρ, ν, ⟨P,≺⟩): a positive normalized state ρ on a local net of algebras (the conditional future potential), a primitive tempo ν (the rate of becoming), and a growing causal set ⟨P,≺⟩ (the past). What is not derived, and is named as such throughout, is the Born rule, Einstein's equations, and the constants. Eight core hypotheses yield seven consequences. A structural arrow of time, prior to thermodynamics (D1). A partial dissolution of the measurement problem (D2), with definiteness posited and the trigger open. Conditional dissolutions of EPR and the signalling worry (D3–D4), requiring a propagation law. A preferred basis anchored in the couplings carried by the causal past (D5). A conditional thermodynamic reading of the structural arrow (D6), requiring a posited bridge to physical reset, and further thermal assumptions for its value. And a conditional coherence lemma (D7): the actualization dynamics is generally covariant and no-signalling once the substrate carries the standard locality of physics. D7 claims less than the Rideout–Sorkin Bell causality condition, which is a screening-off analogue of Bell's local causality and is not inherited. We then develop the proposed physics of the tempo ν: a correspondence with a local quantum speed scale and a saturation choice, both posited. Quantum speed limits alone do not bound an ontic event count. The missing bridge must specify event resolution, additive counting and an applicable composable speed bound. The substrate cadence that reconstructs proper time is kept distinct from a system's tempo; their operational connection remains to be supplied. TA aims at empirical equivalence to quantum mechanics on a causal-set substrate. We label the admissibility requirement P5; its verification, including instrumented multi-time statistics, remains open. No distinctive prediction is claimed. The causal-set "swerve" is inherited from discreteness, generic and not specific to actualization. We close with the open dynamics, its system/substrate interface, the conditional constitution of systems, and the inherited limits of the programme.