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.
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.
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.
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].
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.
This paper proposes that definite reality is constituted through physical encounter at the quantum boundary, not given in advance. Where Einstein showed that properties are relative to a frame of reference, this framework proposes that existence is relational: forged through interaction. Two claims of differing epistemic weight are advanced. The first is ontological: that relatedness is fundamental, meaning definiteness arises through encounter, not prior to it. The second is a metaphysical proposal: that consciousness, as known in integrated biological organisms, is best understood as the intrinsic interior of recursively organised relational actualisation. All results have been independently reproduced three times using separately authored analysis code, and are fully reproducible from public datasets using standard tools. The Coin invariant, 4ħ ln 2 · G · ρ_rev = c³, ties gravitational coupling, information density, and entropy bookkeeping into one exact constraint. Four headline measurements confirm the constitutive coupling across regimes: R_ID is invariant across conscious states (Bayes Factor 178.60; collapse p = 0.0065 under deep anaesthesia); α·k_B ln 2 is uniform across 26 gate voltage settings in a single-electron quantum dot at 10 mK, where the Onsager coefficient M = p(1−p) holds to machine precision and sech²(A/2) is an exact thermodynamic law with zero free parameters (ΔBIC = 1,508 over Gaussian); the constitutive coupling signature is confirmed in solar coronal plasma with a 1.21× active/quiet R_ID ratio (N = 444,889 pixels, p ≈ 0); and a closure-proximity correction field yields 19.5% LOOCV improvement in nuclear binding at 77σ from null, with a mechanistic shell-closure gradient (doubly magic 58%, singly magic 31%, non-magic 10%) that pinpoints the structural origin. The nuclear result is measured against a minimal four-term SEMF with no shell or pairing corrections: the closure field is the sole non-additive element. A Regime C constitutive law, mc² = α_C·E_conf·ln(N_raw/N_phys), yields the nucleon mass to 0.03% (938.6 vs 938.9 MeV), the proton spin fraction ΔΣ = 0.321 (measured 0.28–0.33; testable at EIC), the Madelung orbital filling rule (19/19 orbitals), and all seven nuclear magic numbers — all with zero adjustable parameters. The nucleon mass, quark spin fraction, and cosmological visible matter fraction are three ratios of the same N_raw = 3 × 8 × 27 = 648 information partition — they are not fitted independently; and the LIGO merger-efficiency floor α_C is exactly linked to the proton spin fraction ΔΣ and visible-matter fraction f_vis by the algebraic corollary α_C = (ΔΣ − f_vis)/(2(1 − ΔΣ − f_vis)), so these quantities are mutually determined faces of the single confinement partition. The Boundary-Addressability Principle derives the Planck/QCD hierarchy from N_raw = 648 alone: ln(E_P/E_conf) = 2 ln(N_colour)/α_C + log₂(N_f) − log₂(N_raw)·α_C⁴ = 45.8798, yielding G to 0.016% from CODATA with zero free parameters and no G as input — a derivation of Newton’s constant from the confinement information partition, now axiomatically complete: the quartic correction −log₂(648)·α𝐶⁴ is closed by the Bandwidth Saturation Axiom (Axiom 7, Appendix Q), the seventh foundational element of TRF. The LIGO merger efficiency floor is derived as a cross-regime theorem: α_C = ln(N_spin/N_F)/(2 ln(N_colour)) = 0.148798, residual 1.79×10⁻⁶. Applied to the de Sitter horizon, the Coin recovers ρ_Coin = ρ_critical algebraically without QFT vacuum-energy additivity; the 648 partition fixes a 17.0% flavour-visibility fraction within confined matter; its cosmological projection to the visible-matter abundance is a zero-parameter structural extension, consistent with Planck’s 15.8% value and independently supported by the electromagnetically weighted quark momentum fraction ≈ 17–19% from global PDF fits. The fine-structure constant is derived to 0.004% accuracy (1/α = 137.031 vs CODATA 137.036) via the Relational Closure Principle: the photon IS the relational encounter (ds² = 0) and cannot occupy a free state in its own information budget, so its self-occupancy fraction f_F/N_gauge corrects the bare EM information cost. This is a conditional one-loop derivation; the sole remaining closure is the formal derivation of the partition factor Z = 1 + f_F/N_gauge from the Coin entropy bookkeeping. A matrix of falsifiable predictions defines the conditions under which the framework would be disconfirmed. The Koide formula K = (m_e + m_μ + m_τ)/(√m_e + √m_μ + √m_τ)² = 2/3 is derived as a structural theorem of TRF’s Z₃ generation symmetry (Appendix P, Sections P.12–P.13): the Flavour Equipartition Principle fixes r = √2 in the Z₃ parameterisation, and the Lepton Encounter Principle — lepton mass is the square of the irreversible relational projection onto the vacuum reference — closes the derivation with zero free parameters entering the ratio. The lepton mass ratios are recovered to m_μ/m_e = 206.875 (0.052%) and m_τ/m_μ = 16.8173 (0.0016%); the 0.0006% empirical departure from exact K = 2/3 is the expected O(x⁴) correction in the Coin entropy expansion — a prediction, not a discrepancy. The absolute lepton mass scale and the 1.68% m_e offset remain open and are orthogonal to the ratio derivation. Keywords: Relational Ontology, Quantum Decoherence, Consciousness, Ontological Resolution, Thermodynamic Irreversibility, Hard Problem of Consciousness, Integration-Dissipation Ratio, Relational Tensor, Thermodynamic Coin, Quantum Darwinism, Pixel Tax, Causal Discovery, Bayesian Model Comparison, Dual Formal Verification, EEG Validation, Solar Coronal Validation, Cross-Domain Invariance, Onsager Uniformity, Master Constitutive Law, Nucleon Mass, Proton Spin Crisis, QCD Bag Constant, Cosmological Constant, Nuclear Magic Numbers, Yang-Mills Mass Gap, Regime C, Madelung Rule, Atomic Ionisation Energies, Nuclear Magic Numbers Derivation, Iterative Relational Actualisation, Pauli Exclusion, Dark Matter Prediction, EM-Weighted Momentum Fraction
We present an observer-independent, basis-independent, non-perturbative criterion for wave function collapse: an interaction constitutes a genuine actualization event — a wave function collapse in the ontologically real sense — if and only if it produces an irreversible increase ΔS(ρ‖σ₀) > 0 in the quantum relative entropy of the system state ρ with respect to the Minkowski vacuum state σ₀. The criterion requires no observer, no Heisenberg cut, and no preferred foliation. In weakly coupled quantum field theory it coincides exactly with the on-shell condition pᵘpᵤ = m²c², recovering the perturbative picture as a special case. Off-shell (virtual) processes satisfy ΔS = 0 and are not actualization events. This criterion dissolves — rather than solves — the measurement problem: the question "why does a definite outcome occur?" becomes "why did this interaction produce an irreversible environmental record?", which is a question about interaction physics, not quantum foundations. The Schrödinger's cat paradox resolves cleanly: a real cat is a densely self-coupled actualization network, never in superposition; the isotope alone is the locus of genuine quantum indeterminacy. Three experimental predictions discriminate this framework from all current interpretations: (1) a finite actualization propagation velocity bounded by the Lieb-Robinson speed v_LR ~ Jℓ/ℏ, producing a measurable time delay Δt ≥ |r_A − r_B|/v_LR before decoherence propagates between sites; (2) an environment-dependent transition duration τ_act ~ ℏ/ΔE_env with universal slope dτ_act/d(1/ΔE_env) = ℏ, testable by tuning environmental coupling in superconducting circuits; and (3) a sharp reversibility threshold ΔS* above which quantum transitions cannot be undone regardless of intervention, in contrast to dynamical collapse models (no sharp threshold), Everett (no genuine threshold), and decoherence-only accounts (no irreversibility). The Rindler consistency check is formally verified in Lean 4: the Rindler thermal state satisfies ΔS(ρ_β‖σ₀) = 0 under modular flow (theorem rindler_stationarity in RA_AQFT_Proofs_v10.lean), confirming that no actualization event occurs in the Minkowski vacuum for either an inertial or uniformly accelerating observer. Frame independence — S(UρU†‖σ₀) = S(ρ‖σ₀) for any unitary U — is also formally verified (theorem frame_independence).
Abstract For over a century, theoretical physics has treated General Relativity and Quantum Mechanics as fundamentally incompatible: one posits a continuous, deterministic spacetime manifold; the other, a discrete, probabilistic Hilbert space. The Algorithmic Reality Model (ARM) resolves this schism by identifying a single computational mechanism—the Zeno Threshold (Zα)—as the universal bound governing both quantum collapse and spacetime geometry. ARM treats physical reality as the macroscopic projection of a finite-bandwidth computational substrate operating on a timeless, pre-geometric state (the Singleton). By enforcing Landauer's principle of irreversible erasure, ARM derives the entirety of the gravitational sector from seven pre-geometric, quantum informational axioms. Key Derivations & Unifications Emergent Spacetime: Spatial geometry emerges from the Quantum Fisher Information Metric (QFIM) of the observer's state space, while time emerges from the Tomita-Takesaki modular flow. The Lorentzian signature (-,+,+,+) is algebraically forced by the compact/non-compact structure of their isometry generators. The Holographic Scalar (The 3→1 Unification): Applying the Nyquist-Shannon sampling theorem to the holographic boundary yields the exact Bekenstein-Hawking factor of 1/4 as a universal 2D anti-aliasing limit. This simultaneously fixes the area-law coefficient and structurally derives Newton's constant as G = c3ξmin2 / ℏ. Systematic Boundary Deviations: The framework quantitatively resolves macroscopic anomalies as computational resolution limits. Dark energy emerges as the minimum Landauer maintenance cost of the cosmological event horizon, and the MOND acceleration scale (a0) emerges as a thermodynamic noise floor, algebraically locked by the structural identity a02 = (8πG/3)ρΛ. Thermodynamic State Truncation: Wavefunction collapse, the black hole information paradox, and EPR non-locality are rigorously unified as manifestations of the computational substrate's finite Landauer memory budget. Computational Computability Because ARM is built on discrete informational axioms rather than continuous manifolds, the entire framework is natively computable. The structural derivations in this paper—including emergent geodesic lensing, the Bekenstein-Hawking threshold, and the MOND noise floor—have been explicitly verified via sparse Hermitian lattice simulations. The complete open-source computational validation suite is available at: github.com/srdrymn/algorithmic-reality-model-validations This manuscript represents the master synthesis and culmination of the Algorithmic Theory of Reality (ATR) foundational framework.