Quantum structure in this framework arises from a single analytic kernel defined on the analytic exterior. Boundary wavefunctions are not physical fields but coordinate charts of this kernel, and the familiar canonical operators—Hamiltonian, momentum, position, and angular momentum—appear as boundary projections of one intrinsic analytic generator. Schrödinger evolution becomes the time‑chart expression of analytic flow, while measurement amplitudes correspond to residue components selected by boundary projection. Probability is not an added postulate but the squared residue norm of the projected component, supplied by the invariant residue pairing. A further refinement comes from recognizing that each admissible boundary chart carries its own symmetry group, and each symmetry group forces a unique invariant pairing. The historical operator formalism mixes Hilbert symmetry, Lorentz symmetry, and analytic symmetry, creating the appearance of incompatible dynamical regimes. In this framework, measurement is not merely chart selection but symmetry selection: a boundary projection is well defined only when the symmetry group of the chart aligns with the symmetry of the underlying analytic flow. Symmetry‑compatible charts yield separable boundary expressions, while incompatible charts produce analytic cross‑terms—the geometric counterpart of entanglement. The analytic interval unifies these regimes. When the interval is nonzero, interior analytic structure persists and the boundary receives unitary evolution. When the interval collapses to a null boundary, interior analytic structure is lost and the boundary state becomes static. Because both evolution and collapse are boundary limits of the same analytic flow, the measurement problem dissolves: the apparent duality between unitarity and collapse reflects symmetry misalignment rather than a physical conflict. In this view, quantum mechanics becomes boundary geometry, and its operational content is the shadow cast by the interior analytic kernel. Dynamics, measurement, and probability emerge as coordinated manifestations of a single geometric structure. This is version 1.0 of the manuscript.
Quantum structure in this framework arises from a single analytic kernel defined on the analytic exterior. Boundary wavefunctions are not physical fields but coordinate charts of this kernel, and the familiar canonical operators—Hamiltonian, momentum, position, and angular momentum—appear as boundary projections of one intrinsic analytic generator. Schrödinger evolution becomes the time‑chart expression of analytic flow, while measurement amplitudes correspond to residue components selected by boundary projection. Probability is not an added postulate but the squared residue norm of the projected component, supplied by the invariant residue pairing. A further refinement comes from recognizing that each admissible boundary chart carries its own symmetry group, and each symmetry group forces a unique invariant pairing. The historical operator formalism mixes Hilbert symmetry, Lorentz symmetry, and analytic symmetry, creating the appearance of incompatible dynamical regimes. In this framework, measurement is not merely chart selection but symmetry selection: a boundary projection is well defined only when the symmetry group of the chart aligns with the symmetry of the underlying analytic flow. Symmetry‑compatible charts yield separable boundary expressions, while incompatible charts produce analytic cross‑terms—the geometric counterpart of entanglement. The analytic interval unifies these regimes. When the interval is nonzero, interior analytic structure persists and the boundary receives unitary evolution. When the interval collapses to a null boundary, interior analytic structure is lost and the boundary state becomes static. Because both evolution and collapse are boundary limits of the same analytic flow, the measurement problem dissolves: the apparent duality between unitarity and collapse reflects symmetry misalignment rather than a physical conflict. In this view, quantum mechanics becomes boundary geometry, and its operational content is the shadow cast by the interior analytic kernel. Dynamics, measurement, and probability emerge as coordinated manifestations of a single geometric structure. This is version 1.0 of the manuscript.
Standard quantum descriptions treat interference, entanglement, measurement, and no signaling as separate operational rules, yet their shared structural features point to a deeper geometric origin. In this work we identify that origin by developing the analytic exterior: a non‑spatiotemporal analytic domain specified intrinsically by a single analytic direction, a compatibility condition, and a singularity–residue geometry. A relational bundle then supplies the curvature and boundary projections that generate the operational structure of the amplitude calculus. This intrinsic specification requires neither explicit analytic functions nor global coordinates. Within this analytic framework, familiar quantum structures emerge naturally as boundary effects. The Hilbert‑space inner product appears as the boundary pairing induced by residue geometry; analytic flow along the distinguished direction reveals the Schrödinger‑type evolution already encoded in analytic displacement; and boundary evaluation combined with normalization invariance yields the Born rule as the unique probability assignment compatible with the analytic exterior. Interference, entanglement, measurement, and no signaling thus arise not as independent postulates but as geometric consequences of singularities, curvature, and contextual transport. The relational object introduced in [6] is recovered as the boundary value of an analytic‑exterior amplitude kernel, exposing the analytic origin of its interference and measurement properties. The analytic exterior therefore provides a unified, non‑spatiotemporal foundation in which amplitudes, probabilities, and correlations are boundary manifestations of a single underlying analytic geometry. Quantum mechanics emerges as the boundary theory of this analytic structure, offering a unified and conceptually coherent reformulation parallel in spirit to the relational architecture developed in our earlier work. Version 1.1. This revision adds a new subsection clarifying the ontological divergence between the analytic–exterior framework and the standard Lorentzian manifold ontology. The update explains why the analytic exterior cannot arise within the classical spacetime model and introduces the notion of mathematical ontology as a representational constraint. No mathematical results or definitions have been changed.
The persistent challenge of unifying quantum mechanics with general relativity is often traced to a series of apparent structural tensions: complex versus real ontology, probabilistic versus deterministic dynamics, linear versus nonlinear evolution, and spectral discreteness versus geometric continuity. We argue that these tensions do not signal a fundamental incompatibility between the theories, but rather originate from surplus structure in the conventional formulation of quantum mechanics. The term unification is used in a structural sense, referring to compatibility of quantum dynamics with classical field theory rather than a modification of general relativity. Beginning from the physically indispensable requirement of matter stability, we prove that the Hamiltonian spectrum can always be shifted to become non-negative. The consequent positivity of energy enforces one-sided temporal frequency support for all physical unitary orbits. Equivalently, all scalar temporal components of physical states arise as boundary values of functions analytic in the lower half of the complex time plane, selecting the Hardy space—or, more generally, analytic tempered distributions—as the admissible solution space. In these spaces, the real and imaginary parts of the wave function are not independent; they are linked uniquely via the temporal Hilbert transform. Hence, the complex structure of quantum mechanics emerges as a representational artifact of an underlying real operator algebra. Building on our earlier work on Unified Quantum Mechanics and the Unified Real Wave Equation, we establish a rigorous proof of exact equivalence between the real-field evolution and the Schr\"odinger equation on the physical subspace. Probabilistic quantities arise naturally as quadratic invariants of analytic signals, rather than as independent postulates. We further show that quantum spectral discreteness arises from operator-theoretic confinement and boundary conditions on a smooth spacetime manifold, and does not imply any fundamental discreteness of spacetime. The resulting framework provides a real-field formulation of quantum mechanics and gravitation, with stability and causality implemented as foundational principles, and demonstrates that the Einstein field equations are compatible with a unified field-theoretic description of spacetime and matter. Extending the same analyticity principle to spacetime geometry, we propose an analytic formulation of the Einstein field equations, from which classical general relativity is recovered exactly as the real projection.
This paper presents Dimensional Evolution — Quantum Measurement and Identity Formation (DE-QM), a geometric framework addressing the quantum measurement problem while preserving the full predictive structure of standard quantum mechanics. Within DE-QM, the wavefunction is treated as a resonance-geometry field rather than a probability amplitude. Superposition arises from deterministic traversal across multiple resonance-compatible configurations supported by an inverse domain, rather than from the coexistence of mutually realized states. Measurement collapse is described as a geometric stability transition governed by anchoring thresholds imposed by environmental and detector constraints. Probability emerges as a derived measure of geometric overlap between scan traversal and detector geometry, recovering the Born rule without introducing intrinsic randomness or observer-dependent postulates. The framework integrates decoherence, entanglement, and relativistic structure into a unified geometric description that remains fully compatible with nonrelativistic and relativistic quantum mechanics, including the Schrödinger and Dirac formalisms. Entanglement is treated as a shared inverse-domain resonance geometry, enabling coordinated collapse without superluminal signaling. Measurement outcomes are shown to be observer-independent and physically grounded in macroscopic anchoring geometry, consistent with established experimental results. This manuscript serves as the foundational quantum-mechanical component of the broader Dimensional Evolution framework and builds directly on prior canonical works, including the DE Core paper, companion papers CP1–CP5, and specialized studies addressing the Mott problem and Dirac structure. DE-QM does not modify quantum mechanics’ mathematical formalism; rather, it provides a coherent geometric ontology underlying wavefunction structure, superposition, collapse, and probability. --- Project website: https://de-theory.com This work is part of the Dimensional Evolution research program. Additional materials, updates, and contact information will be made available via the project website as the work expands.
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
The persistent challenge of unifying quantum mechanics with general relativity is often traced to a series of apparent structural tensions: complex versus real ontology, probabilistic versus deterministic dynamics, linear versus nonlinear evolution, and spectral discreteness versus geometric continuity. We argue that these tensions do not signal a fundamental incompatibility between the theories, but rather originate from surplus structure in the conventional formulation of quantum mechanics. Beginning from the physically indispensable requirement of matter stability, we prove that the Hamiltonian spectrum can always be shifted to become non-negative. The consequent positivity of energy enforces one-sided temporal frequency support for all physical states, selecting the Hardy space of analytic signals as the admissible solution space. In this space, the real and imaginary parts of the wave function are not independent; they are linked uniquely via the temporal Hilbert transform. Hence, the complex structure of quantum mechanics emerges as a representational artifact of an underlying real operator algebra. Building on our earlier work on Unified Quantum Mechanics and the Unified Real Wave Equation, we establish a rigorous proof of exact equivalence between the real-field evolution and the Schrödinger equation on the physical subspace. Probabilistic quantities arise naturally as quadratic invariants of analytic signals, rather than as independent postulates. The resulting framework provides a realfield formulation of quantum mechanics and gravitation, with stability and causality implemented as foundational principles, and demonstrates that the Einstein field equations are compatible with a unified field-theoretic description of spacetime and matter. Extending the same analyticity principle to spacetime geometry, we propose an analytic formulation of the Einstein field equations, from which classical general relativity is recovered exactly as the real projection.