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 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.
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。
The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.
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).
Every physical measurement ever conducted has yielded exactly one outcome: self-identical, non-contradictory, determinate. This paper argues that the Three Fundamental Laws of Logic (Identity, Non-Contradiction, and Excluded Middle) are not cognitive conventions but ontological constraints constitutive of physical distinguishability. From this thesis, quantum mechanics emerges as the unique stable structure at the interface between non-Boolean possibility and Boolean actuality. The framework derives rather than postulates: complex Hilbert space structure follows from interface constraints combined with the physical requirement of compositional consistency (local tomography); the Born rule follows from Gleason's theorem; unitary dynamics follows from information preservation requirements. One structural prediction, that quantum mechanics requires complex amplitudes, has been experimentally confirmed (Renou et al., 2021). The framework specifies twelve explicit conditions that would falsify it. Logic Realism Theory provides unified explanation for seventeen quantum phenomena that other interpretations treat as independent postulates, while maintaining realism, locality, and parsimony.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
This paper resolves the measurement problem and the problem of intersubjectivity—two foundational puzzles of quantum mechanics—within the Planck Core Framework, without requiring any additional postulates beyond the three axioms of the framework. I prove that measurement is not a mysterious non-unitary "collapse" of the wavefunction, but a unitary process of entanglement establishment between subsystems of the quantum bit network. The apparent collapse arises from the observer's limited access to the full entanglement structure: what we perceive as a definite measurement outcome is simply our location within a specific entanglement branch. The Born rule is derived, not postulated—the probability of an outcome is proportional to the number of entanglement threads in the corresponding branch. The Wigner's friend paradox is resolved by proving that different observers may occupy different branches, but their descriptions converge when they exchange information via entanglement. I further introduce the concept of the "information eye"—a hypothetical direct entanglement sensor—and prove that any macroscopic biological organism operating at biological temperatures will inevitably perceive a decohered, classicalised representation of reality, not the underlying quantum information network itself. The classical appearance of the macroscopic world is thus derived as a contingent consequence of thermodynamic constraints on biological evolution, not a fundamental feature of reality. This work completes the derivation of quantum mechanics from the Planck Core Framework, establishing that no special measurement postulate is required.