We present a complete derivation of all core postulates of quantum mechanics from the single primordial axiom of Idempotent Emergent Ontology (IEO): existence is confirmation, and confirmation is idempotent (E^2 = E). No prior assumptions—Hilbert space, wave function, probability amplitude, Schrödinger equation, or measurement postulate—are introduced as external inputs. The derivation proceeds through a systematic 19-theorem architecture: Peaks 1-3 establish the mathematical foundation: the GNS construction uniquely generates the Hilbert space from the confirmation algebra; the imaginary unit i emerges as the algebraic necessity of the confirmation closed loop; the Born rule is derived as the unique solution to trace-quota minimization under idempotent projection. Peaks 4-6 derive the dynamics: the Schrödinger equation emerges from the modular automorphism group restricted to the observable domain; measurement collapse is the re-execution of the idempotent projection; quantum entanglement is the split-window projection of a shared confirmation projector. Peaks 7-19 extend the framework to the complete quantum formalism: decoherence, path integrals, quantum tunneling, identical particle statistics, quantum logic, the Dirac equation, quantum field theory, geometric phases, S-matrix algebra, and the zero-redundancy closure theorem. All 19 standard postulates of quantum mechanics are downgraded from axioms to derived theorems. The framework contains zero free parameters—the effective Planck constant \hbar_{\text{eff}} = 1/\chi is rigidly determined by the Jones index \chi = 4. Quantum mechanics is thus revealed not as the fundamental language of the universe, but as the inevitable algebraic structure of any finite observer executing confirmation operations on the full-dimensional modal space. A detailed "Boundary of Honesty" distinguishes rigorously proven theorems from directional inferences requiring further mathematical hardening.本文从自幂涌现本体论(IEO)的唯一原初公理——存在即确认,确认即幂等(E^2 = E)——出发,完整导出了量子力学的全部核心公设。全推导不预设希尔伯特空间、波函数、概率幅、薛定谔方程或测量公设。论证通过系统的19峰定理架构展开: 第一至三峰建立数学基础:GNS构造从确认代数中唯一生成希尔伯特空间;虚数单位i作为确认闭合环的代数必然而涌现;玻恩规则作为幂等投影下迹配额极小化的唯一解被导出。 第四至六峰导出动力学:薛定谔方程从模自同构群在显在域上的限制中涌现;测量坍缩是幂等投影的再度执行;量子纠缠是共享确认投影的分窗口表观分裂。 第七至十九峰将框架扩展至完整量子形式体系:退相干、路径积分、量子隧穿、全同粒子统计、量子逻辑、狄拉克方程、量子场论、几何相位、S矩阵代数、零冗余性闭合定理。 量子力学的全部19条标准公设全部从公理降格为导出定理。框架零自由参数——有效普朗克常数 \hbar_{\text{eff}} = 1/\chi 由Jones指标 \chi = 4 刚性锁定。量子力学因此被揭示为不是宇宙的基本语言,而是任何有限观测者对全维模态空间执行确认操作时必然涌现的代数结构。 详细的“诚实边界”标注严格区分了已完成严格证明的定理与待进一步数学硬化的方向性推断。
We present a complete derivation of all core postulates of quantum mechanics from the single primordial axiom of Idempotent Emergent Ontology (IEO): existence is confirmation, and confirmation is idempotent (E^2 = E). No prior assumptions—Hilbert space, wave function, probability amplitude, Schrödinger equation, or measurement postulate—are introduced as external inputs. The derivation proceeds through a systematic 19-theorem architecture: Peaks 1-3 establish the mathematical foundation: the GNS construction uniquely generates the Hilbert space from the confirmation algebra; the imaginary unit i emerges as the algebraic necessity of the confirmation closed loop; the Born rule is derived as the unique solution to trace-quota minimization under idempotent projection. Peaks 4-6 derive the dynamics: the Schrödinger equation emerges from the modular automorphism group restricted to the observable domain; measurement collapse is the re-execution of the idempotent projection; quantum entanglement is the split-window projection of a shared confirmation projector. Peaks 7-19 extend the framework to the complete quantum formalism: decoherence, path integrals, quantum tunneling, identical particle statistics, quantum logic, the Dirac equation, quantum field theory, geometric phases, S-matrix algebra, and the zero-redundancy closure theorem. All 19 standard postulates of quantum mechanics are downgraded from axioms to derived theorems. The framework contains zero free parameters—the effective Planck constant \hbar_{\text{eff}} = 1/\chi is rigidly determined by the Jones index \chi = 4. Quantum mechanics is thus revealed not as the fundamental language of the universe, but as the inevitable algebraic structure of any finite observer executing confirmation operations on the full-dimensional modal space. A detailed "Boundary of Honesty" distinguishes rigorously proven theorems from directional inferences requiring further mathematical hardening.本文从自幂涌现本体论(IEO)的唯一原初公理——存在即确认,确认即幂等(E^2 = E)——出发,完整导出了量子力学的全部核心公设。全推导不预设希尔伯特空间、波函数、概率幅、薛定谔方程或测量公设。论证通过系统的19峰定理架构展开: 第一至三峰建立数学基础:GNS构造从确认代数中唯一生成希尔伯特空间;虚数单位i作为确认闭合环的代数必然而涌现;玻恩规则作为幂等投影下迹配额极小化的唯一解被导出。 第四至六峰导出动力学:薛定谔方程从模自同构群在显在域上的限制中涌现;测量坍缩是幂等投影的再度执行;量子纠缠是共享确认投影的分窗口表观分裂。 第七至十九峰将框架扩展至完整量子形式体系:退相干、路径积分、量子隧穿、全同粒子统计、量子逻辑、狄拉克方程、量子场论、几何相位、S矩阵代数、零冗余性闭合定理。 量子力学的全部19条标准公设全部从公理降格为导出定理。框架零自由参数——有效普朗克常数 \hbar_{\text{eff}} = 1/\chi 由Jones指标 \chi = 4 刚性锁定。量子力学因此被揭示为不是宇宙的基本语言,而是任何有限观测者对全维模态空间执行确认操作时必然涌现的代数结构。 详细的“诚实边界”标注严格区分了已完成严格证明的定理与待进一步数学硬化的方向性推断。
This paper provides a rigorous derivation of all core postulates of quantum mechanics from a single axiom: the existence of binary distinction. No prior assumptions—Hilbert space, wave function, probability amplitude, or unitary evolution—are introduced as external inputs. The derivation proceeds in six chapters: Chapter 1 establishes the axiom of binary distinction and proves that the imaginary unit i is the algebraic necessity of iterative distinction operations: Δ applied twice equals a reversal (-1), therefore one distinction corresponds to multiplication by i. Chapter 2 demonstrates that complex probability amplitudes are the unique algebraic carriers of the binary split between confirmed information (real part) and concealed information (imaginary part). Chapter 3 proves the Born rule as the unique solution to information deviation minimization under truncation constraints—the squared modulus is the sole functional form compatible with positivity, normalization, and classical-degenerate compatibility. Chapter 4 derives unitary evolution as the traversal of all indistinguishable paths during the suspension interval between confirmation operations. Schr\"{o}dinger's equation emerges as the differential limit of this traversal under modular flow. Chapter 5 establishes that measurement collapse is not a mysterious nonunitary process but the re-execution of the confirmation operation, projecting the suspended superposition onto a single confirmed eigenstate. The collapse probability is identical to the Born rule derived in Chapter 3. Chapter 6 demonstrates that quantum entanglement is the split-window projection of a shared confirmation projector—two entangled particles share the same confirmation projection in the full-dimensional information field, and spatial separation is merely the appearance of local windows in the observable domain. Bell inequality violation follows directly from the non-homomorphic property of the truncation projection. The entire formalism emerges without fitting parameters, without interpretation debates, and without external physical assumptions. Quantum mechanics is thus revealed not as the fundamental language of the universe, but as the inevitable cognitive structure of any finite observer executing information confirmation and truncation on the full-dimensional information field. All conclusions are strictly demarcated by a "boundary of honesty," distinguishing logically necessary deductions from directional inferences requiring further mathematical hardening.
This work presents a rigorous derivation of quantum mechanics from purely a priori ontological principles, a result not previously achieved within a single self-contained framework. Beginning with the apodictically certain proposition "I am"—immune to all doubt through performative contradiction—we establish through necessary deduction the Principle of Ontological Continuity (POC): Being is continuous and cannot cease or arise ex nihilo. From this single ontological foundation, the complete mathematical structure of quantum mechanics emerges as necessary consequence. We demonstrate that any mathematical representation of ontological continuity must have Hilbert space structure, that ontological conservation necessarily implies unitary evolution, and that the Schrödinger equation follows via Stone's theorem as mathematical necessity, not empirical postulate. The Born rule is derived through three independent converging arguments (Gleason's theorem, direct conservation requirements, and Zurek's envariance), establishing quantum probability as ontological necessity rather than interpretative supplement. Classical foundational problems—wave function collapse, the measurement problem, Bell's non-locality—are resolved without additional assumptions. Collapse emerges as effective description of decoherence; the observer is reconceptualized as self-aware subsystem without ontological dualism; Bell violations confirm ontological unity rather than presenting paradox. Quantum phenomena—superposition as ontological potentiality, tunneling as manifestation of continuity, Pauli exclusion as consequence of ontological structure—are shown to be necessary rather than mysterious. The work establishes quantum mechanics with apodictic certainty mediated through deductive necessity, demonstrating that quantum reality could not have been otherwise. This monograph integrates foundational ontology, mathematical formalism, and resolution of interpretative problems within unified self-contained framework. It deliberately closes at non-relativistic quantum mechanics, establishing stable foundation for future extensions (quantum field theory, general relativity, quantum gravity, normative ethics).
This paper presents a rigorous demonstration that the fundamental origin of the natural affinity between information ontology and quantum mechanics lies in the eternal asymmetry between global information (the full information algebra 𝔄) and local information (the observable algebra 𝒩). Starting from the core axioms of the Pan-Information Unified Field Theory (PIUFT)—idempotency of information confirmation and trace-preserving information conservation—the paper strictly proves that the kernel space ker(E) necessarily exists and that any observer is necessarily a local information structure, permanently unable to access the full information algebra. The conditional expectation E: 𝔄 → 𝒩 acts as the projection bridge, transforming the global structure of completeness, determinacy, commutativity, real numbers, and classicality into the local structure of incompleteness, indeterminacy, non-commutativity, complex numbers, and quantumness. All core features of quantum mechanics—the emergence of Hilbert space via the GNS construction, operator non-commutativity, complex probability amplitudes, the uniqueness of the Born rule, and quantum entanglement—are rigorously derived as inevitable algebraic consequences of local information incompleteness. The universal incompleteness of physical laws—singularity breakdown in general relativity, the quantum measurement problem, the redundancy of free parameters in the Standard Model, and the second law of thermodynamics—is unified under a single algebraic root: the asymmetric allocation of information quotas between the global and the local. The trace conservation law τ(𝔄) = τ(𝒩) + τ(ker(E)) guarantees that global and local information are not two independent worlds but two perspectives on the same hyperfinite type III₁ von Neumann factor, constituting a strict dialectical unity. The paper further derives a series of testable observational predictions, including the precise proportions of dark energy, dark matter, and ordinary matter, all algebraically determined by the Jones index χ = 4 of the A₁₁ subfactor with zero free parameters. This work establishes the first complete philosophical and mathematical framework that unifies quantum foundations, cosmological dark components, the incompleteness of physical laws, and the cognitive boundaries of observers under a single principle of information asymmetry.
ONE AXIOM BRIDGES: Possibility Selection Principle — Formalization, Collapse, and Born Rule Description: This document provides the formal mathematical bridge between the pure algebraic foundations of the ONE AXIOM project and its physical applications. It presents a rigorous derivation of the fundamental laws of physics starting from a single axiomatic symmetry group: G = S4 x Z2^3. Key Scientific Contributions: Derivation of the Born Rule (p=2): Proves that quantum probability is not a postulate but a theorem. The exponent p=2 is shown to be the unique solution emerging from the OCR-OER adjunction structure. Quantum Collapse as Domain Update: Formalizes "collapse" as a relational update of the Schreier graph domain (Gamma to Gamma_A) rather than a physical process, resolving long-standing paradoxes of "spooky action at a distance". Emergence of Spacetime and Gravity: Demonstrates the construction of 4D spacetime (M4) from group representation theory. Einstein's field equations are derived via two independent, proven paths: Jacobson's thermodynamic approach and Ollivier-Ricci curvature on the Schreier graph. Resolution of the Cosmological Constant Problem: Provides a natural derivation of the cosmological constant Lambda ~ 10^-122 from the "coherence floor" of the universe, eliminating the need for fine-tuning. Bekenstein-Hawking Entropy: Derives the coefficient lambda = 1/4 through triple-verified categorical and structural paths. Methodology: The work employs a "Dual-Track Derivation" methodology, requiring convergence between Ontological [O] (structural) and Epistemological [E] (computational) paths before a result is considered proven. Status: This version marks the completion of the foundational triad of the ONE AXIOM framework (0C, 0A and 0B documents)
This work derives the fundamental framework of quantum mechanics intrinsically from the first-principle axioms of UD duality theory. Starting from the inherent binary expansive–condensative attributes of the UD ontology, four fundamental physical fields are naturally decomposed by spontaneous symmetry breaking across spatial and material existential domains. A two-component complex spinor structure is uniquely constructed, and the imaginary unit i is proven to be not merely a mathematical convention but a structurally necessary ingredient required by both the compact SU(2) symmetry and the orthogonal inter-containment of U and D attributes. Within the flat spacetime limit of the UD gravitational action, a system of coupled Klein–Gordon field equations is established, governed by a single self-consistent fundamental mass scale m₀=1/(2πe^π) with no adjustable free parameters. The coupling constant is determined analytically from the UD vacuum equilibrium background values and the U↔D duality symmetry. Through explicit construction of four-component Dirac spinors and auxiliary field decomposition, the on-shell equivalence between the UD coupled field equations and the free Dirac equation is rigorously demonstrated. The non-relativistic limit further recovers the Pauli equation and the Schrödinger equation sequentially, reconstructing the core dynamical equations of quantum mechanics from a unified foundational origin. Global U(1) phase invariance of the UD action is verified via total-derivative decomposition, yielding the standard Noether conserved current. The Born probability interpretation is deduced directly from the global normalization axiom by isolating localized laboratory-scale excitations from the cosmic background, eliminating the need for independent quantum postulates. Based on the intrinsic properties of the UD field components, a self-consistent ontological interpretation is provided for wave–particle duality, double-slit interference, and the quantum measurement problem. In this framework, the measurement problem is dissolved rather than solved: no collapse postulate is required, and determinism is restored at the ontological level. The theory naturally implies that the continuous D_U field permits quantum superposition, while condensed D_D intrinsic entities possess definite classicality without physical particle-state superposition. Three testable theoretical predictions are presented, establishing falsifiable physical consequences of the UD quantum formulation. This paper completes the intrinsic derivation of the foundational structure of quantum mechanics from UD theory. Together with the previously derived general relativity framework, it realizes the homologous unification of gravity and quantum mechanics within a single self-contained axiomatic system.
The postulates of quantum mechanics — complex Hilbert space structure, unitary evolution, the Born rule, definite measurement outcomes, the canonical commutation relations, and the Pauli exclusion principle — have traditionally been taken as axioms. In this work we show that, within the framework of Modular Entropic Gravity (MEG), these structures admit a unified derivation from four geometric axioms governing the informational structure of the quantum vacuum.The framework is non-circular: its inputs are operator-algebraic structures of quantum field theory — including modular Hamiltonians, the JLMS identity, and the Bisognano–Wichmann theorem — which are defined independently of any probability interpretation. The probabilistic and dynamical rules of textbook quantum mechanics then arise as an infrared effective description of this underlying structure.The six results are obtained in logical order: (Q1) the emergence of complex structure from the entropion mass gap; (Q2) the L2 Hilbert space and Schrödinger equation from the long-wavelength limit of the vacuum propagator; (Q3) the Born rule from a relative entropy identity combined with the PLES–Petz correspondence; (Q4) definite measurement outcomes from PLES boundary conditions imposed by classical apparatus; (Q5) the canonical commutation relations from modular flow via the Bisognano–Wichmann and Araki reconstruction; and (Q6) the Pauli exclusion principle from the Z3 topological structure of the MEG vacuum.Three of these results (Q1, Q2, Q5) are established as complete theorems within the framework. The remaining three (Q3, Q4, Q6) are structurally complete derivations in which the underlying mechanism is fully identified, and any remaining conditions concern either formal infinite-dimensional extension or quantitative refinement.The paper therefore presents a coherent and non-circular programme in which the postulates of quantum mechanics are reconstructed from a smaller set of underlying principles, together with an explicit account of the remaining technical steps required for complete formalisation.