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.
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.
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.
Dirac called the imaginary unit in the Schrödinger equation the deepest thing in quantum mechanics, and admitted he did not understand it. This note offers a candidate: i is the complex place of a single number — Q, the real root of x⁴ − x − 1 — and the strangeness of quantum mechanics is the strangeness of that one number's arithmetic, seen from the inside. For a century quantum theory has been taught as a list of postulates — complex amplitudes, a Born rule, unitary evolution, a dagger — accepted because they work and never because they are understood. Grant a single identification, that the state space of a quantum system is the complex place of Q's field, and those postulates stop being assumptions. They become theorems. The imaginary unit is the field's complex structure; the dagger is Galois conjugation; observables are real because they are Galois-fixed; unitary evolution is the field's unit circle; and the Born rule — the squared modulus, long the least-explained law in physics — is the field's own norm, made unique by Gleason's theorem. Time reversal and charge conjugation appear as named elements of the Galois group S₄, the Weyl group of su(4). And a proved theorem locates the classical/quantum boundary precisely: in the family xⁿ − x − 1 the root is Pisot — its powers settle, its world definite — only through three dimensions, and at the fourth that settling fails. That, on the reading proposed here, is where the classical world ends and the quantum world begins. The dozen postulates of the physics textbooks are thus compressed to one here, named plainly as a posit, with every claim labeled at its evidential tier and the composition of many systems left as the open edge. It is not a metaphor and not a fit. Quantum Mechanics is the arithmetic of a single number — and, at last, a single thing to be right or wrong about.
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.
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.