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.
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.
For a century, the postulates of quantum mechanics have been taken as irreducible axioms. Bohr, von Neumann, and Feynman each concluded that asking for physical mechanisms underlying the Born rule, wavefunction collapse, or the Pauli exclusion principle constitutes a category error. This consensus was never proved — it was an inference from the failure ofknown attempts.This paper challenges that consensus by exhibiting a candidate framework — Modular Entropic Gravity (MEG) — built on four geometric axioms, in which the six foundational postulates of quantum mechanics are derived consequences and the five standard mysteries of quantum theory have precise physical explanations reducible to classical mechanisms. We make a precise logical claim, not an empirical one. We do not assert that MEG is the correct final theory. We assert that it is internally consistent: its derivations are valid within the framework, no quantum postulate appears as a hidden input, and the programme is non-circular in a precise sense — it derives the probabilistic and interpretive postulates of textbook quantum mechanics from a deeper operator-algebraic level plus the MEG axioms.Several derivations have open steps, all identified explicitly. These concern the technical completion of MEG as a specific theory and do not affect the existence argument: a frame-work of this kind demonstrably can be constructed, as MEG shows. Whether MEG is the correct such framework is an empirical question. That a coherent framework of this kind exists is established here.
The Born rule — the identification of measurement probability with the squared modulus of quantum amplitudes — has resisted derivation from deeper principles for a century. Every attempt within the Hilbert space framework faces the same structural obstacle: complex amplitudes and real probabilities are different kinds of mathematical objects, and connecting them requires either a new postulate or a covert assumption that smuggles in the probability structure being derived.We argue that in Modular Entropic Gravity (MEG), this obstacle does not arise. The MEG state-to-field map (Axiom 1) sends any quantum mixture ρ = |α|2ρA + |β|2ρB with orthogonal branches to the entropy field configuration Sρ = |α|2SA + |β|2SB . The Born weights |α|2 and |β|2 appear directly as convex coefficients of a real field decomposition — not as complex amplitudes requiring conversion. The factorisation of the measurement map through the coarse-graining channel (established in prior work) guarantees that measurement outcomes respect this convex structure, and we prove a readout faithfulness propositionestablishing that an unbiased branch-resolving apparatus produces outcome frequencies equal to the convex coefficients.The traditional Born rule problem — how do complex amplitudes become real probabilities? — is not answered in MEG. It is rendered unnecessary. In a framework where the physical description is a real entropy field whose branch decomposition carries the Born weights as mixture fractions, the amplitude-to-probability conversion is not a measurement postulate but a feature of the representation change from Hilbert space to entropy field. We directly confront the objection that this merely relocates the Born rule into the density matrix, and show that the relocation changes its logical status: from a postulate connecting two categorically different mathematical objects to an inherited structural property of the physical state representation. The information-geometric results of the MEG programme — the Petz recovery fidelity F 2 = |α|2 and the PLES–Petz equilibrium coincidence — are reinterpreted as independent confirmations of the consistency of this convex structure.
The complete mathematical structure of quantum mechanics is derived from first principles, tracing all fundamental results to the constraint axiom via octonionic algebra. Without postulates, empirical fitting, or measurement assumptions, the following results are established: (1) Quantum states must reside in a complex Hilbert space over the complex numbers. (2) The canonical commutation relation between position and momentum follows from octonionic non-commutativity together with holographic information bounds. (3) The Heisenberg uncertainty principle arises as a mathematical theorem from the Cauchy–Schwarz inequality. (4) The Schrödinger equation emerges from time translation symmetry. (5) Spin one-half arises from the structure of the relevant Clifford algebra. (6) The Pauli matrices originate from the quaternion subalgebra contained within the octonions. (7) The Born rule appears as the unique probability measure on complex projective space. Related resourcesAdditional preprints, theoretical frameworks, and ongoing work by the author are available at:https://murad-ahmadov.github.io/
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
The complete mathematical structure of non-relativistic quantum mechanics — Hilbert space, canonical commutation relations, the uncertainty principle, the de Broglie relation, the Schrödinger equation, the Born rule, spin-statistics, and quantum entanglement — is shown to be derivable from three geometric axioms: the spatial universe is an S³ hypersurface propagating at velocity c through the quaternionic vacuum H ≅ ℝ⁴, subject to an isoclinic rotation enforced by the extremality condition. Phase space and quantisation of ℏ. The null constraint |v₄| = c forces the phase space to be T*S³. The Hopf fibration S¹ ↪ S³ → S² with first Chern class c₁ = 1 quantises the symplectic flux, determining Planck's constant ℏ as both the minimum symplectic area and the conserved Noether charge of the contact-conformal symmetry. The universality of ℏ — its identical value across all particles and interactions — is proved to be a consequence of the isoclinic condition: only isoclinic rotation preserves the Hopf fibration structure required for symplectic flux quantisation. Five principal results. (i) Born rule. The probability measure |ψ|² arises from marginalisation over the inaccessible Hopf fibre coordinate; it is not postulated. (ii) Quantum entanglement. Entanglement is topological, forced by the universal Hopf linking number lk = 1. (iii) Bell correlations. The correlator E(a,b) = −a·b is a consequence of SO(4) rotational invariance on S³, without hidden-variable supplementation. (iv) Classical–quantum decomposition. Classical physics is the radial projection of the membrane dynamics; quantum mechanics is its rotational projection. The two regimes are not separate postulates but orthogonal projections of a single geometric structure. (v) Quantisation obstruction. The Groenewold–van Hove quantisation obstruction does not apply to S³ because C^∞(S³) = U(su(2)) by the Peter–Weyl theorem; the algebra of observables is globally well-defined. Falsifiable predictions. Two predictions are stated with explicit observational protocols: — No gravitational decoherence: the geometric origin of ℏ implies that gravitational interaction cannot degrade entanglement coherence. Testable with the MAQRO space mission via matter-wave interferometry at mass scales m ~ 10⁻¹⁷ kg over baseline ~ 10³ km. — Perfect Bell correlations at arbitrary separation: the topological origin of entanglement (lk = 1) predicts no geometric suppression of Bell correlations with increasing spatial separation, in contrast to models coupling entanglement to background curvature.
Building on the evolutionary time field theory, this paper proves that the core mathematical structures of quantum mechanics can be derived from the constrained geometry of the evolutionary time field. Starting from the symplectic structure of the constrained geometry of the evolutionary time density and the order parameter, we establish effective canonical commutation relations and derive the time-energy uncertainty relation. The quantum excited states form a Fock space, whose linear structure yields the superposition principle, and the effective dynamics reduces to the standard Schrödinger equation in the flat-space limit. The measurement postulate corresponds to the coupling of the order parameter to an external dissipative channel—the contrapositive of the information bottleneck theorem applies directly, and the Born rule is reduced to an application of the Principle of Minimum Irreversible Cost. This paper further reveals the topological origin of the imaginary unit i, the geometric nonlocality mechanism of many-body entanglement, and the collapse threshold dissipation effect. The conclusion is that the core structures of quantum mechanics—Hilbert space, operator algebra, superposition principle, uncertainty relation, Schrödinger equation, measurement postulate, and Born rule—can all be derived from the constrained geometry of the evolutionary time field. This paper is the tenth in a series. The evolutionary time field theory is established in a companion paper (DOI: 10.5281/zenodo.21436542), the gauge field origin in a second (DOI: 10.5281/zenodo.21438149), the Hawking radiation quantum effects in a third (DOI: 10.5281/zenodo.21436715), and the axiomatic framework in a fourth (DOI: 10.5281/zenodo.21325460).