Starting from the arithmetic structure of the Bost-Connes (BC) system, we systematically investigate the arithmetic origins of each of the five von Neumann axioms of quantum mechanics. The main results are as follows: (I) The Hilbert space structure is uniquely constructed from C*-positivity; (II) Self-adjointness is an intrinsic feature of the C*-algebraic framework; (III) The numerical form of the Born rule P = |c_n|^2 is uniquely determined by the overlap relation between KMS states; (IV) Unitary evolution is uniquely determined by the strict monotonicity of zeta; (V) The tensor product decomposition is a direct consequence of the fundamental theorem of arithmetic. The derivation introduces no measurement, observer, or wave function collapse. Probability emerges naturally within the BC framework as an objective relational property between two KMS states.
Starting from the arithmetic structure of the Bost-Connes (BC) system, we systematically investigate the arithmetic origins of each of the five von Neumann axioms of quantum mechanics. The main results are as follows: (I) The Hilbert space structure is uniquely constructed from C*-positivity; (II) Self-adjointness is an intrinsic feature of the C*-algebraic framework; (III) The numerical form of the Born rule P = |c_n|^2 is uniquely determined by the overlap relation between KMS states; (IV) Unitary evolution is uniquely determined by the strict monotonicity of zeta; (V) The tensor product decomposition is a direct consequence of the fundamental theorem of arithmetic. The derivation introduces no measurement, observer, or wave function collapse. Probability emerges naturally within the BC framework as an objective relational property between two KMS states.
Starting from the finite p-adic reduction of the Bost–Connes system and the established dynamics of matter, we give a systematic account of five axioms: quantum states, composite systems, unitary evolution, observables, and Born probabilities. Complex amplitude patterns on a finite arithmetic space, together with its normalized inner product, support pure-state and density-operator descriptions. Joint configurations of the participating degrees of freedom realize the tensor structure, while self-adjoint exchanges that balance energy and group charge generate joint unitary evolution. The full joint state retains coherence among matter, background, and auxiliaries; local mixtures arise by reducing that same state. Spectral couplings correlate the components distinguished by an observable with orthogonal records. Statistical consistency determines their Born weights, while the same instrument specifies conditional states and successive event histories. Character-based matter patterns, persistent phases, and arithmetic spectral responses give these structures their physical content. The five axioms thus connect within a common process of states, interactions, and records, providing a unified description of the evolution of matter and its observable manifestations in this framework.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
We construct an arithmetic model space ℓ²(Z>0) carrying additive shifts and multiplicative scalings coupled by the distributive law, and show that close analogues of all five standard quantum axioms arise from the interplay of additive and multiplicative structures, together with explicitly stated probabilistic and dynamical assumptions. The Beckner-Babenko theorem singles out ℓ² as the unique Fourier-isometric norm; Gleason's theorem yields Born's rule; Wigner-Stone gives unitary dynamics; unique factorization gives the tensor product. The Riemann Hypothesis is equivalent to positive semi-definiteness of a spectral quadratic form analogous to Born-type inner-product positivity.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
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.