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.
We develop a conditionally closed reconstruction of quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime--vacuum substrate, and persistent causal source--response coupling. Wave functions, Hilbert spaces, complex amplitudes, density operators, the Born rule, commutation relations, Fock spaces, path integrals, completely positive maps, and Markovian dynamics are introduced only as derived mathematical or operational structures, not as additional physical axioms. The construction distinguishes the actual finite carrier and its causal history from the quantum state used to represent preparations, transformations, and measurement probabilities. It derives unitary dynamics from self-adjoint Hamiltonians, formulates spectral observables and probability currents, and states the explicit equilibrium, additivity, noncontextuality, and calibration conditions required for the Born representation. Measurement is described through instruments, pointer dynamics, stable records, weak values, continuous monitoring, diffusive and counting trajectories, and operational error--disturbance relations. Quantum trajectories remain record-conditioned predictive representations rather than automatically ontic paths. Composite systems, identical particles, spin, entanglement entropy, purification, steering, Bell correlations, dense coding, and no-broadcasting are developed on declared tensor-product and symmetry domains. Open-system dynamics are formulated through causal Volterra equations, memory kernels, initial-slip terms, auxiliary-mode realizations, and unitary dilations. Existence, positivity, complete positivity, trace preservation, no-signalling, and energy closure are treated as logically distinct properties. Semiclassical pseudodifferential calculus, controlled path-integral convergence, Ehrenfest-time bounds, semiclassical measures, decoherence, and macroscopic-record stability define the quantum--classical interface. Topological phases, defect-bound states, relativistic wave equations, infrared dressing, and effective quantum-field-theory domains are incorporated with explicit applicability and error conditions. Numerical spectral certification, statistical model comparison, twenty-five benchmark models, theorem-status ledgers, and a complete experimental statistical protocol provide falsification and replication criteria. The resulting framework preserves standard quantum mechanics on its validated domain while isolating finite-core and historical corrections as conditional, quantitatively bounded, and experimentally testable extensions. **Keywords** Quantum foundations; topological quantum mechanics; historical memory; three-axiom reconstruction; Born rule; quantum measurement; weak measurement; quantum trajectories; entanglement; spin and statistics; open quantum systems; Volterra equations; semiclassical limit; quantum topology; numerical certification; experimental falsification.
We reconstruct nonrelativistic quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by
20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.
This work continues the framework developed in Ref. [1], in which time is treated relationally as the +1 evolution parameter associated with the dimensional configuration space of a physical phenomenon, and quantum-mechanical structure is proposed to emerge from the information-preserving dimensional recording of a higher-dimensional wave field. Here we extend that construction beyond the single-system description and investigate whether additional central elements of quantum mechanics can arise without being introduced as independent quantum postulates. We first clarify the recording process in terms of physically distinguishable equivalence classes of parent configurations and show how changes of the parent field can produce effective measurement-induced state updating in the reduced description. For independently preparable systems, preservation of local linear superposition leads to bilinear composition and, subject to completeness of the accessible composite state space, to Hilbert-space tensor products. Joint interactions then generate nonfactorizable reduced states, providing a structural origin for entanglement before any probabilistic interpretation is introduced. We subsequently derive the Born probability rule from the geometry of orthogonal reduced alternatives together with unitary invariance, noncontextual consistency, continuity, normalization, and additive coarse-graining. The resulting probability structure is compared with the independent robustness-based reconstruction of De Raedt, Katsnelson, and Michielsen [2]. Using the derived composite structure and probability rule, Bell–CHSH correlations are obtained from generic interaction-generated pure entangled states without assuming a singlet state or Pauli spin operators, while operational no-signalling is preserved. Finally, we explore a possible geometric extension toward internal gauge symmetries: aligned three-dimensional, tilted three-dimensional, and fully four-dimensional parent-field deformations lead respectively to one, two, and three component complex internal representations, suggesting U(1), SU(2), and SU(3) type symmetry structures. Unlike approaches that take the quantum formalism as their starting point, or reconstruction programs based primarily on operational or informational axioms, the present framework seeks a concrete physical mechanism from which these structures emerge together. The fact that a single modification in the dimensional role of time, combined with information-preserving reduction, consistently generates multiple otherwise independent features of quantum theory makes this possibility physically significant even before genuinely new empirical predictions are established.
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 derives the complete mathematical structure of quantum mechanics—complex Hilbert spaces, unitary evolution, and Born-rule probabilities—from five information-theoretic axioms, without assuming quantum mechanics at the outset. Starting from an ontology of microhistories (fully specified microscopic trajectories), we define branches as equivalence classes of operationally indistinguishable microhistories. We then show that five physically motivated constraints on how information is stored, mixed, and transformed uniquely force the effective state space to be a complex projective space with unitary dynamics.