This paper presents a structural reconstruction of the formal core of quantum mechanics fromthe properties of a bounded, linear, distinguishability-reducing projection acting on a generalfunctional space. Effective quantum states are identified with equivalence classes under theprojection ΠQM : H∗ → HQM and the Hilbert-space structure of HQM is induced by a positiveoperator K on the underlying space.Within this framework, the characteristic elements of quantum theory emerge as direct conse-quences of projection geometry and operator structure. Linearity yields superposition, quadraticnorms on equivalence classes give rise to the Born rule, non-factorizability leads to entangle-ment and commuting local algebras ensure no-signalling. Schrödinger-type unitary dynamicsarises from projection of symmetric generators, Dirac-type operators follow from algebraic com-patibility with Clifford structures and violations of classical Bell inequalities together with theTsirelson bound are enforced by operator-norm constraints in the projected C∗-algebra.No hidden variables are introduced and no modification of standard empirical quantummechanics is proposed. Rather than postulating the quantum formalism as axiomatic, theanalysis shows that its defining structures form a highly constrained and stable outcome oflinear projection, equivalence-class geometry, and norm-preserving dynamics.Rather than introducing new physical dynamics, the purpose of this work is to clarify why theestablished quantum formalism admits so little structural freedom once a small set of minimalrequirements is imposed.
This paper presents a structural reconstruction of the formal core of quantum mechanics fromthe properties of a bounded, linear, distinguishability-reducing projection acting on a generalfunctional space. Effective quantum states are identified with equivalence classes under theprojection ΠQM : H∗ → HQM and the Hilbert-space structure of HQM is induced by a positiveoperator K on the underlying space.Within this framework, the characteristic elements of quantum theory emerge as direct conse-quences of projection geometry and operator structure. Linearity yields superposition, quadraticnorms on equivalence classes give rise to the Born rule, non-factorizability leads to entangle-ment and commuting local algebras ensure no-signalling. Schrödinger-type unitary dynamicsarises from projection of symmetric generators, Dirac-type operators follow from algebraic com-patibility with Clifford structures and violations of classical Bell inequalities together with theTsirelson bound are enforced by operator-norm constraints in the projected C∗-algebra.No hidden variables are introduced and no modification of standard empirical quantummechanics is proposed. Rather than postulating the quantum formalism as axiomatic, theanalysis shows that its defining structures form a highly constrained and stable outcome oflinear projection, equivalence-class geometry, and norm-preserving dynamics.Rather than introducing new physical dynamics, the purpose of this work is to clarify why theestablished quantum formalism admits so little structural freedom once a small set of minimalrequirements is imposed.
This work presents an axiomatic and structural reconstruction of core elements of quantum mechanics. Starting from minimal principles — symmetry under vanishing preference, normalization, consistency of probability assignments, and coherence-preserving dynamics — we derive several central features of the quantum formalism rather than postulating them. Key results include: A geometric derivation of the constants 1/21/\sqrt{2}1/2 and 1/21/21/2 from symmetry and normalization. A theorem-level derivation of the Born rule (including the qubit case) from noncontextuality and additivity. A reconstruction of Schrödinger dynamics from reversible norm-preserving structure. A derivation of the tensor-product structure for composite systems from composition axioms and local tomography, with entanglement emerging as a necessary consequence. Formal treatment of Bell–CHSH nonlocality, Tsirelson’s bound, and no-signalling. A conditional reconstruction of complex Hilbert space structure from intrinsic phase dynamics and compositional consistency. The manuscript is self-contained and framed in standard mathematical-physics language. It does not propose modifications to quantum mechanics, but instead clarifies why many of its defining structures arise inevitably from deeper structural constraints. Prior computational and model-based work by the authors (including diagnostic toy models exploring phase locking, coherence stabilization, and recursive-time dynamics) is acknowledged but intentionally not reproduced here, in order to keep the present work focused on axiomatic structure rather than model-specific architecture. This upload is intended as a foundations and reconstruction contribution, suitable for readers interested in the mathematical and conceptual structure of quantum theory.
This volume explains why the standard formalism of quantum mechanics is not merely successful, but structurally inevitable. Rather than proposing a new interpretation or modifying existing equations, this work derives quantum mechanics as the only stable descriptive framework compatible with the FSPS (Foundational Structure / Projected Structure) model. Within FSPS, observable physics (PS) arises as a projection from a non-geometric foundational layer (FS), and this projection necessarily introduces non-commutativity, probabilistic structure, and unitary evolution. The volume combines the axiomatic foundation of FSPS–00 v2.2 with a focused derivation showing that classical mechanics is structurally unstable under projection, while the Hilbert-space formalism of quantum mechanics represents the minimal information-preserving description. Measurement, wavefunction collapse, and probabilistic outcomes are treated not as additional physical processes but as re-projection and stabilization events within PS. No changes are made to Schrödinger dynamics, operator algebra, or experimental predictions. This work serves as a unifying explanation for why quantum mechanics has its present form, consolidating earlier FSPS-based analyses into a single, coherent account aligned with modern quantum theory.
Motion-First Reconstruction of Quantum Kinematics establishes the algebraic conditions under which a distinguishability theory admits Hilbert-space representation, without invoking probabilities, measurement postulates, or dynamical assumptions. The reconstruction begins from a primitive motion-history domain and a distinguishability functional defined on finite-support multiplicity bundles of histories. From explicitly stated structural constraints—most centrally the third-order obstruction principle and signed residual nonnegativity—the distinguishability functional is shown to admit a symmetric quadratic kernel representation. This kernel is proven to be positive semidefinite, allowing application of the Moore–Aronszajn construction to obtain a canonical real Hilbert space in which distinguishability corresponds to squared norm differences. The representation is canonical up to isometric equivalence and preserves the distinguishability structure exactly. Distinguishability-preserving transformations are shown to correspond to orthogonal operators, and tensor-product factorization follows under an explicit subsystem independence condition. Under an additional cyclic distinguishability symmetry, a compatible complex structure emerges, yielding the standard complex Hilbert-space framework used in quantum theory. The results isolate the representational kinematic core of Hilbert-space structure as a consequence of algebraic distinguishability constraints rather than as an independent postulate. The reconstruction clarifies the precise structural boundary separating distinguishability theories that admit Hilbert-space representation from those requiring more general representational geometries. This work is representational in scope. It does not assume or derive measurement postulates, probability rules, observable algebras, or dynamical evolution laws. Instead, it establishes the minimal algebraic conditions under which Hilbert-space structure becomes representationally inevitable. These results provide a structural foundation for subsequent work investigating how physical compositional principles on motion histories may give rise to the distinguishability constraints identified here.
This paper reconstructs the operational content of finite-dimensional quantum mechanics from deterministic Hamiltonian dynamics defined on a compact Kähler manifold Σ. An ontic microstate evolves under a Liouville-preserving flow generated by a Hamiltonian vector field, while observable statistics arise only after applying a smooth many-to-one projection π:Σ→CPⁿ⁻¹ to epistemic coordinates. The induced epistemic measure is taken to be absolutely continuous with respect to the Fubini–Study measure, which is selected as the canonical invariant density under the symmetry and regularity assumptions developed in companion work. For a class of Hamiltonians compatible with the projection structure, the projected dynamics on CPⁿ⁻¹ defines a holomorphic vector field. By the Ashtekar–Schilling correspondence, this reproduces the standard Schrödinger form of unitary evolution on the operational state space. Measurement interactions are modelled as Hamiltonian evolution on a joint system–apparatus surface, with outcomes represented by context-dependent regions on CPⁿ⁻¹ defined relative to the measurement setting. When evaluated with the Fubini–Study measure, the resulting region weights reproduce Born-form outcome statistics for the measurement constructions considered. Composite systems are represented by joint ontic structures that need not factorise, while the operational projection recovers the usual reduced-state formalism and subsystem dynamics. Overall, the analysis shows how unitary evolution, measurement statistics, mixed-state structure, and contextual dependence in finite-dimensional quantum mechanics can be realised within a deterministic geometric framework, without introducing fundamental stochastic processes or additional collapse postulates. Programme background: https://www.constraintsurfacedynamics.com — Glossary: https://glossary.constraintsurfacedynamics.com
This paper argues that quantum mechanics is not a separate postulate system to be reconciled with spacetime physics. It is the boundary algebra of a non-invertible projection: the algebraic structure that a globally coherent bulk field is forced to produce when it becomes observable. The bulk field is a Majorana spinor, Ξ = CΞ̄, on M^5 = R^4 × S^1(τ), governed by a single conservation law. The projection Π onto the observable boundary is not an external measurement device added afterward: it is the form the field must take in order to admit an observable image at all. What survives Π defines the observable sector; what enters ker Π is lost structurally, not thermodynamically. From this single mechanism, four boundary structures emerge: the Hilbert space of quantum mechanics, physical time, matter stability, and the Born rule. These are four boundary readings of one conservation law, K_τ = ξη = α^−1, resolved by the same tangential dynamics on the invariant surface Σ. Observable states are equivalence classes [Ξ] = Ξ + ker Π. The Born rule is the canonical quadratic probability measure induced by the projection geometry. Measurement irreversibility is the non-injectivity of Π, with non-vanishing protected by the transcendence of π. Physical time is not assumed in advance: it emerges as the output of chiral projection through a conformal map from the compact cyclic fibre to the irreversible real line. The Wheeler–DeWitt equation is not a paradox but the correct timeless description of the bulk. Spacetime emerges after quantum mechanics, not before it. The fine-structure constant is not a free parameter. It is the unique observable residue forced by the self-coherent projection structure. The double-slit interference pattern, the Born rule, and measurement irreversibility are derived as structural corollaries of this paper in the companion work doi:10.5281/zenodo.19654630
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.