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].
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 (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.
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.
Abstract Relativity describes the structure of a world once definite reality is in play. This paper proposes a complementary shift in perspective: that definite reality itself is constituted by physical encounter at the quantum boundary. Where Einstein showed that properties are relative to a frame of reference, this framework proposes that existence is relational—forged through interaction, not given in advance. The paper advances two claims of differing epistemic weight. 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. An empirical programme is proposed, centred on the Integration-Dissipation Ratio (RID = Sprod / CL), with explicit failure conditions. Independent computational validation on real human EEG (OpenNeuro ds005620, Bajwa et al. 2024, 21 subjects, propofol sedation) confirms the framework’s central prediction: RID is statistically invariant between wakefulness and light propofol sedation with preserved dreaming (p = 0.48, BF = 178.60), and collapses significantly under deep propofol where consciousness is abolished (p = 0.0065). The scalar proxy does not recover window-level linear coupling (r = 0.003); this is expected, as the proxy compresses spatially distributed, directed tensor structure into a single undirected scalar. Cross-species validation on rat neocortical LFP (DANDI 000041, Watson et al.) confirms magnitude consistency with human values (KR1–KR4 all pass). Human sleep deprivation EEG (OpenNeuro ds004902, 21 paired subjects) confirms elevated waking RID under homeostatic debt (p = 0.033). Overnight polysomnography (Sleep-EDF, 152 recordings) reveals a methodological boundary of the histogram-based entropy production proxy, suggesting that the thermodynamic signature of consciousness may require multi-dimensional operationalisation. Four datasets, two species, and three recording modalities are reported with full transparency. The scalar proxy analysis was not designed to test the full tensor equation; its role was to test whether an aggregate projection of the framework carried empirical signal. It did (p = 0.48, BF = 178.60). The framework’s fundamental equation, derived from a Nakajima-Zwanzig coarse-graining of quantum relative entropy (Appendix F), is the Relational Tensor: Σμν(t) = Π ∫ K(t−τ) (dCμν/dτ) dτ + Λμμδμν, where Σμν is the directed thermodynamic flux tensor, Cμν is the directed relational complexity tensor (identified as classical transfer entropy under decoherence), Π = αkBln 2 is the Landauer-bounded thermodynamic cost constant (α ≥ 1), K is a causal memory kernel, and Λμμδμν is the strictly diagonal local bath dissipation baseline. The empirical RID is formally recovered as the macroscopic scalar projection (global spatial summation) of this tensor. The proper test of the elementwise equation requires spatially resolved, directed, temporally filtered network analysis on high-density data. PC algorithm causal discovery confirms three robust causal edges (Sprod–CL, Sprod–Conscious, CL–Conscious), with the Sprod–CLcoupling persisting within conscious and unconscious states independently (r ≥ 0.995). A formal coarse-graining reduction establishes the mathematical chain from quantum relational entropy to macroscopic transfer entropy via the Nakajima-Zwanzig formalism, diagonal trapping of bath correlations, and decoherence-induced projection, with full dimensional consistency against the Landauer bound. The thermodynamic structure of the tensor equation is shown to belong to the same family as emergent gravity: when evaluated on a Bekenstein-saturating system at a local Rindler horizon, it reproduces both inputs to Jacobson’s (1995) thermodynamic derivation of Einstein’s field equations, establishing the framework as a specific instance of the information-geometry correspondence. The core theorems have been independently verified by two formally distinct proof kernels: Lean 4 (v4.14.0) and Rocq/Coq (v9.1.1), with zero placeholders in either verification. A matrix of sixteen falsifiable predictions defines the precise conditions under which the framework would be disconfirmed. Keywords: Relational Ontology, Quantum Decoherence, Consciousness, Ontological Resolution, Thermodynamic Irreversibility, Hard Problem of Consciousness, Integration-Dissipation Ratio, Relational Tensor, Nonequilibrium Brain Dynamics, Quantum Darwinism, Pixel Tax, Causal Discovery, Bayesian Model Comparison, Dual Formal Verification, EEG Validation
# Project Description: The Relational Closure Framework (RCF) ## 1. Project Purpose and Background (研究目的与背景) Modern fundamental physics faces deep structural crises, primarily manifested in the mathematical and conceptual incompatibilities between General Relativity (space-time geometry) and Quantum Mechanics (probability amplitudes and non-locality). Most contemporary attempts at unification (such as String Theory or Loop Quantum Gravity) retain the classical paradigm of "objects" (e.g., strings, loops, points) or pre-existing spacetime backgrounds as their ontological starting point. The purpose of this research project is to develop and formalize the **Relational Closure Framework (RCF)**, a radical, background-independent, and purely relational foundation for physics. RCF operates on the core philosophical and mathematical premise that **relations are ontologically prior to objects**. Objects, spaces, fields, and particles do not possess inherent existence; instead, they are emergent phenomena. The framework systematically models how stable physical entities and laws arise from self-consistent, discrete cyclic relations called **closures**. When discrete relations close upon themselves, they generate topological invariants (**holonomies**) that act as structural anchors. The ultimate goal of this project is to provide a rigorous mathematical derivation of the core pillars of physics—including conservation laws, the principle of least action, Born’s rule, the speed of light, and the emergence of continuous spacetime—directly from the discrete algebraic properties of these closure networks. ## 2. Methodology and Theoretical Architecture (方法论与理论架构) The project proceeds axiomatically by establishing four primary structural axioms and one fundamental derived theorem, utilizing tools from abstract algebra, higher category theory, operator algebras (von Neumann algebras), and quantum information theory: * **Axiom 1 (Primacy of Relations):** Pure distinction forms the precursor to all relations. There are no pre-existing entities or backgrounds. * **Derived Theorem 1 (Non-Commutativity):** Because relations must occur sequentially to establish distinction, non-commutativity is a mathematically derived necessity, providing the foundational algebra for quantum observables. * **Axiom 2 (Closure-Induced Invariance):** Stable structures exist if and only if relations form self-consistent cyclic configurations. These closures generate topological invariants (holonomies) across different topological scales. * **Axiom 3 (Chirality from Order):** Spatial and structural chirality (such as the \mathbb{Z}_3 minimal non-trivial chiral closure) emerges inherently from the algebraic ordering of relations. * **Axiom 4 (Bidirectional Group Exchange):** Observation and physical measurement are modeled not as passive recordings, but as bidirectional group-theoretic exchanges between overlapping relational systems forming a joint closure. ## 3. Expected Outcomes and Key Derivations (预期数学成果与推导) The primary expected outcome of this project is a self-consistent mathematical physics framework that derives standard physics as a macroscopic statistical limit of the discrete network. Specifically, the framework delivers the following five principal theoretical results: * **A Generalized Noether Theorem:** The project proves that physical conservation laws are equivalent to the non-trivial kernel of the relational exchange map. In the continuous limit, this reduces to standard Noether currents, but in the discrete regime, it predicts strictly discrete, quantized conserved numbers. * **Derivation of the Least Action Principle:** The action functional (S) is derived as a direct function of the network’s closure invariants (holonomies). The classical principle of least action (\delta S = 0) is shown to be equivalent to a non-Abelian Kirchhoff's law enforcing relational consistency across the network, recovering Maxwell and Yang-Mills fields in the continuous limit. * **Emergence of the Speed of Light (c) and Causality:** Utilizing the **Lieb-Robinson bound** on discrete networks, the project demonstrates that the propagation of structural constraints has a finite, maximum velocity (v_{LR}). This maximum velocity is identified as the physical speed of light (c = v_{LR}). Consequently, causal consistency (the absence of closed timelike curves) emerges as a mathematical corollary of network locality. * **Derivation of the Born Rule:** Quantum randomness is explained via the **Nakajima-Zwanzig projection method** acting on the information limits of localized observers. By applying the **Gleason-Busch theorem**, the quadratic probability rule (|\alpha|^2) is derived rigorously from finite additivity and non-contextuality within the closure space. * **Origin of Spacetime and Wave Functions:** Physical spacetime is reinterpreted as a "pseudo-continuity"—a statistical artifact arising from mutually constraining discrete closures. Similarly, wave functions and the Schrödinger equation are derived as the statistical envelopes of underlying discrete constraint-gap dynamics. ## 4. Empirical Predictions and Observational Significance (经验预测与实验检验) While primarily a foundational and theoretical physics project, RCF yields distinct, falsifiable empirical predictions that differentiate it from standard quantum mechanics and general relativity, providing a clear pathway for future experimental validation (e.g., via multi-ion traps or superconducting qubit benchmarks): * **Observer-Capacity Dependent Decoherence:** RCF predicts that a quantum system's phase dephasing rate (\gamma_{deph}) is explicitly bounded by, and scales non-linearly with, the total information storage/processing capacity (C) of the local physical observer. * **The Quantum "Locking Phase Transition":** The framework predicts that when the information required to specify a quantum state (I_{req}) approaches the maximum capacity of the environment/observer (C), the system will undergo a sharp, non-linear phase transition, dramatically freezing its lifetime within its pointer basis. * **Non-Linear S-Shaped Trigger Curves:** In quantum erasure or interferometry tracking experiments, the interference visibility (V) as a function of information saturation (\Delta = I_{req} - C) is predicted to follow a distinct Sigmoid (S-shaped) curve, rather than the linear or pure exponential decay predicted by standard environmental decoherence models. * **Cosmological Entropy Bounds:** The framework provides a localized, boundary-driven explanation for de Sitter space entropy, predicting the maximum cosmological observer complexity to be bounded near S_{dS} \approx 10^{122} based on the global finiteness of the constraint network.
## Title The Relational Closure Framework: From Relations to Matter via Causal Consistency ## Authors Mingdong He (Independent Researcher, ORCID: 0009-0009-5176-7571) ### Overview This deposit contains the complete source code and compiled PDF of the Relational Closure Framework (RCF) version 2.0C, a relational approach to the foundations of physics developed as a candidate contribution to Hilbert's sixth problem (the axiomatization of physics). The framework proposes that the universe is constituted by **relations** rather than objects. Stable relations form **closures** (self-consistent cycles); closures generate topological invariants (holonomies); and these invariants underpin all physical structures—conservation laws, the least-action principle, the Born rule, dimension, geometry, and matter—as emergent phenomena of the closure network at different scales. ### What is new in v2.0C Version 2.0C introduces a **causal consistency principle**: physical existence requires causal consistency as a precondition, not as a consequence of continuous spacetime. What appears as continuity is reinterpreted as **pseudo-continuity**—the statistical artifact of mutually constraining discrete closures. This reframing dissolves (rather than solves) the continuum-limit problem that has obstructed discrete approaches to physics for decades. Key new results in v2.0C: 1. **Causal consistency as physical precondition** — The speed of light is derived as the enforcement mechanism for causal consistency (not an arbitrary postulate). 2. **Randomness from constraint gaps** — Quantum probability arises naturally from the incompleteness of discrete constraints, not from an axiom. 3. **Wave function as constraint-gap envelope** — The wave function exists because the network is discrete; it "waves" because holonomy phases interfere; it is non-classical because it carries probability, not energy. 4. **Schrödinger equation from pseudo-continuity** — Via the Madelung transformation, the Schrödinger equation is shown to emerge from: (i) holonomy action stationarity (Hamilton–Jacobi), (ii) constraint-gap-induced probability density, and (iii) Noether conservation (continuity equation). The quantum potential is interpreted as the signature of pseudo-continuity failure. 5. **Fields as long-range constraint mechanisms** — All fundamental fields are intrinsically long-range because causal consistency is a global requirement. The de Sitter entropy (~10^122) is conjectured as the information-theoretic capacity bound on causal constraints. ### Core results (retained from v1.2) - A **generalized Noether theorem**: conservation is identified with the kernel of the exchange map, derived from axioms without assuming symmetry. - A **least-action principle**: the action is proved to be necessarily a function of closure holonomies; δS = 0 is equivalent to a non-abelian Kirchhoff law; the continuum limit recovers Maxwell and Yang–Mills equations. - The **Born rule**: stochasticity arises from the observer as an open dynamical system (Nakajima–Zwanzig); the square law |α|² is forced by the Gleason–Busch theorem under finite additivity. - The **Z₃–γ₅ algebraic correspondence**: a strict algebraic identity linking the minimal relational closure (3-cycle) to the Dirac chirality operator. - A **quantum error correction correspondence**: the Gauss-law operator is structurally identical to the toric-code stabilizer; vacuum stability is topologically protected. ### AI use disclosure This manuscript was prepared with the assistance of AI language models (Claude/ChatGPT/Gemini families) for typesetting, reference formatting, and consistency checking. All theoretical content—axioms, proofs, conjectures, and the causal consistency principle—is the sole intellectual contribution of the author. AI tools were used as a writing assistant, not as a source of scientific ideas.