This paper establishes a carrier-classification theorem for finite-dimensional pure-state ontology in quantum mechanics. The starting point is carrier identification. Physical pure-state predictions are invariant under nonzero complex rescaling of Hilbert-space representatives. Consequently, the empirical pure-state carrier is projective Hilbert space (equivalently, the rank-one density operators) rather than the unquotiented Hilbert-space vectors themselves. The central question is therefore not which ontology should be preferred, but which ontological assignments are defined on the empirical carrier. The main result is an exact projective descent and factorization theorem. A representative-level ontology assignment is well-defined on the empirical pure-state carrier if and only if it is invariant under all nonzero complex rescalings, equivalently if and only if it factors uniquely through the projective quotient. Admissible pure-state ontology is therefore characterized by descent and unique factorization through the projective carrier. Assignments that distinguish vector representatives lying over the same ray are not invariants of the standard pure-state datum and require an explicitly extended carrier. The theorem yields a positive carrier classification. Rays, rank-one density operators, projective subspaces, histories, records, and explicitly extended structures carrying gauge, branch, measure, pointing, or selection data are all admissible carrier classes when their defining data are supplied. A nontrivial branch decomposition is not selected by the bare Hilbert-space datum alone. Once a branch or measurement context is fixed, the associated probabilities and branch weights are projective quantities: they depend on the ray rather than on a distinguished vector representative. As an application, representative-level universal-state formulations that employ branch weights, typicality measures, observer-location claims, or “our branch” claims cannot remain simultaneously projectively well-defined and bare-Hilbert minimal. Such formulations must either factor through projective or density carriers, supply branch-selection structure (for example through factorization, decoherence, histories, records, or measurement context), or supply additional pointing, conditioning, indexical, gauge, or selection data. More broadly, the paper develops a carrier-first foundation for pure-state ontology. Rather than beginning with an ontological proposal and asking whether it reproduces quantum predictions, the theorem begins with the empirical carrier and classifies the ontology classes naturally associated with it through descent, factorization, and explicit carrier extension. License note: Distributed under CC BY-NC-ND 4.0.
This paper establishes a carrier-classification theorem for finite-dimensional pure-state ontology in quantum mechanics. The starting point is carrier identification. Physical pure-state predictions are invariant under nonzero complex rescaling of Hilbert-space representatives. Consequently, the empirical pure-state carrier is projective Hilbert space (equivalently, the rank-one density operators) rather than the unquotiented Hilbert-space vectors themselves. The central question is therefore not which ontology should be preferred, but which ontological assignments are defined on the empirical carrier. The main result is an exact projective descent and factorization theorem. A representative-level ontology assignment is well-defined on the empirical pure-state carrier if and only if it is invariant under all nonzero complex rescalings, equivalently if and only if it factors uniquely through the projective quotient. Admissible pure-state ontology is therefore characterized by descent and unique factorization through the projective carrier. Assignments that distinguish vector representatives lying over the same ray are not invariants of the standard pure-state datum and require an explicitly extended carrier. The theorem yields a positive carrier classification. Rays, rank-one density operators, projective subspaces, histories, records, and explicitly extended structures carrying gauge, branch, measure, pointing, or selection data are all admissible carrier classes when their defining data are supplied. A nontrivial branch decomposition is not selected by the bare Hilbert-space datum alone. Once a branch or measurement context is fixed, the associated probabilities and branch weights are projective quantities: they depend on the ray rather than on a distinguished vector representative. As an application, representative-level universal-state formulations that employ branch weights, typicality measures, observer-location claims, or “our branch” claims cannot remain simultaneously projectively well-defined and bare-Hilbert minimal. Such formulations must either factor through projective or density carriers, supply branch-selection structure (for example through factorization, decoherence, histories, records, or measurement context), or supply additional pointing, conditioning, indexical, gauge, or selection data. More broadly, the paper develops a carrier-first foundation for pure-state ontology. Rather than beginning with an ontological proposal and asking whether it reproduces quantum predictions, the theorem begins with the empirical carrier and classifies the ontology classes naturally associated with it through descent, factorization, and explicit carrier extension. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional carrier-descent theorem for canonical projective quantum measurement channels. Under explicit universal-carrier, Ward-admissibility, and finite projective-sector hypotheses, each nonzero sector carries a compact U(1) phase carrier, and a finite family of orthogonal sectors determines a central sectorwise phase torus. Haar averaging over this compact carrier gives the conditional expectation E_P(X) = ∑ⱼ Πⱼ X Πⱼ, whose trace-dual state map is the Lüders nonselective measurement channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The fixed observable algebra is the block algebra B_fix = {X : [X, Πⱼ] = 0 for all j}, while the classical outcome algebra is its center Z_out = span{Π₁, …, Πₘ}. The Born weights pⱼ(ρ) = Tr(Πⱼρ) are the unique coefficients representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except in the trivial one-sector case, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; hence the associated boundary-channel datum is a boundary-channel projection rather than full reversible Heisenberg–Schrödinger collapse, i.e. full-algebra unitary equivalence. The paper proves that canonical finite projective measurement is the endpoint of a carrier-to-phase-to-Haar boundary-channel descent: phase selection determines Haar projection, Haar projection determines the Lüders channel, central restriction determines the Born weights, and the collapse criterion certifies the obstruction to full reversible Heisenberg–Schrödinger collapse in every nontrivial projective context. License note: Distributed under CC BY-NC-ND 4.0.
Quantum mechanics represents physical states through complex amplitudes, state vectors, and density operators, yet the ontological status of these objects remains interpretation-dependent. This paper develops a closure-based formulation in which a quantum state is interpreted as a representation of contextually partial relational closure: the state possesses definite mathematical and relational structure while remaining unresolved relative to one or more observational contexts. Within this language, decoherence differentiates closure channels and redistributes locally accessible coherence into relational structure; environmental redundancy stabilizes records; and classical facts emerge as approximately fixed, effectively commuting, redundantly recorded structures. Measurement is separated into three problems—context, weight, and outcome—so that decoherence, Born weighting, and unique outcome selection are not conflated. The paper then distinguishes two stronger research programs from the exact reformulation. First, it asks whether Hilbert-space quantum mechanics can be reconstructed from primitive closure, incompatibility, composition, and consistency principles. Second, it formulates a falsifiable closure-history extension in which physically relevant information might exist beyond the present density operator. Quantum Partial Closure Theory is therefore presented as a relational interpretation and information calculus, a reconstruction program, and a prospective experimental framework rather than as an established replacement for quantum mechanics. Keywords: quantum foundations; closure; wavefunction ontology; coherence; entanglement; decoherence; measurement; classical emergence; quantum information; Hilbert-space reconstruction The wavefunction question Quantum theory is operationally clear about what a wavefunction does. For a normalized pure-state vector ψ∈H and a projective measurement with alternatives {Π_i}, the Born weights are p_i=Tr(ρ_ψ Π_i ), ρ_ψ=ψψ^†. For a rank-one outcome Π_i=e_i e_i^†, this reduces to p_i=|e_i^† ψ|^2. Yet the mathematical clarity of the formalism does not by itself settle what the state vector is ontologically. Interpretations of quantum theory differ over whether the state should be regarded as physical, epistemic, relational, branch-relative, dispositional, or otherwise. The present paper proposes a specific interpretive and reconstructive language: partial relational closure. The central claim is not that the wavefunction is literally a new material substance called closure. Rather, ▭(ψ "is interpreted as a Hilbert-space representation of coherent partial relational closure" .) The word partial is crucial. It does not mean that a state is a scalar fraction of being real or complete. It means that information and constraint are distributed across local, coherent, and relational sectors, and that closure is always specified relative to a context. A Bell state, for example, can be locally indefinite while being relationally exact. A decohered state can be closed with respect to one effective macroscopic context while retaining a larger quantum embedding. A selective outcome can be locally definite without implying that every larger-scale relation in the universe is globally closed. This formulation is motivated by three observations. First, projective dephasing is naturally closure-like because it is idempotent. Second, modern quantum information already supplies exact measures for local coherence, correlations, and record redundancy. Third, the quantum-to-classical transition is naturally relational: local coherence can disappear from a subsystem while information migrates into system-environment correlations. The paper has three epistemic layers. Layer A uses established quantum mathematics. Layer B derives exact information identities and interprets them in closure language. Layer C states reconstruction conjectures and an explicitly speculative beyond-QM closure-history hypothesis. These layers are kept separate throughout. A quantum system is not structureless before measurement. Its amplitudes, phases, conservation relations, and correlations are tightly organized. What remains unresolved is how that structure will become locally definite relative to a chosen measurement context. The partial-closure view therefore distinguishes three broad regimes: Coherent partial closure. Alternatives remain phase-related and can interfere. Contextual differentiation. Interaction or dephasing suppresses coherence between alternatives relative to a context without necessarily choosing one. Selective local closure. A particular outcome is conditioned or realized relative to the measurement relation. In this language, wave-like behavior corresponds to spatially or otherwise contextually unresolved coherent closure, while particle-like detection corresponds to localized closure. Entanglement is closure stored in relations rather than in isolated subsystems. Decoherence is not simply destruction: in a globally pure description it can be a transfer of local coherence into relational entanglement. Classical objectivity appears when relevant records become stable and redundantly accessible. The framework does not assign consciousness a privileged dynamical role. An observer is a partial closure process for constraining possibilities. Explainer video Watch on YouTube (Roots of Reality): https://www.youtube.com/watch?v=UYx_omk3h3A The video, its thumbnail and description are included in this record.
Physics may have mistaken a quotient for an origin. Quantum theory represents a preparation by a trace-one positive operator ρ. This is exact for relative outcome weights, but it is produced by the many-to-one map N(σ)=σ/Tr(σ), for which N(λσ)=N(σ) for every λ>0. If total response scale is inert under every admissible continuation, the identification loses nothing relevant. If one declared continuation can distinguish σ from λσ, however, normalisation has erased a future-relevant distinction. The trace-one state remains an exact terminal quantum reading, but it cannot by itself be a faithful future-sufficient carrier. The reason this order is easy to miss is structural: once the quotient is taken as the complete input, every downstream construction that factors through it is necessarily blind to the positive-ray scale already identified away. The mathematical ingredients used below are largely established and are not presented as isolated novelties. Jordan–von Neumann, Frobenius, and finite-dimensional Gleason–Busch-type results serve instead as consecutive links in a conditional dependency-order theorem. Equality of complete future-response profiles first fixes the minimal predictive quotient. Under explicit locks, positive refinement fixes square-root access; quadratic recomposition fixes a real inner-product carrier; and the stated return-algebra, positive-involution, and central exhaustive-phase locks then select its complex Hilbert representation. Positive additive effect weights acquire the unique pre-normalised trace form Wω(E)=Tr(σωE); and only after the non-zero total sω=Tr(σω) is closed does one obtain ρω=σω/sω and the Born reading. The familiar finite-dimensional Hilbert–trace–Born structure therefore reappears, but in a different dependency position. Two no-go results make this dependency claim exact within their stated domains: canonical ray normalisation cannot by itself faithfully carry active scale, and, in dimension at least two, a normalised invertible congruence is affine on the full state space only when its transport is scale-uniform. A common-profile theorem states when independently reconstructed readings are uniquely profile-equivalent over one fixed common future-profile image, rather than merely compatible descriptions. Under locks L0–L9 on one typed finite-dimensional assembly datum, these results form a single conditional genealogy; the paper neither derives quantum theory from NIR alone nor asserts that nature realises the locks. Its central result is not a new quantum rule. If two realised or admissibly generated representatives σ and λσ on the same positive ray, with σ≠0 and λ≠1, have different complete declared future-response profiles, no faithful representation of those profiles can factor solely through canonical ray normalisation N. The trace-one state is not wrong: it is the exact answer to a later, conditional question.
The Admissibility of Natural Physics, Part I begins where an admissibility-first reconstruction of single-system quantum mechanics ends. Its central question is not whether the familiar tensor-product formalism can be assumed, but whether any physical composition law can be derived without importing composite systems, global state spaces, entanglement, locality, or signalling as primitives. The book develops a disciplined reconstruction of binary quantum composition from operational and structural requirements. It first records the exact content and limitations of the inherited single-system closure, establishes dependency and non-circularity statutes, and introduces the possible object of natural physics only as a provisional target ledger rather than as an assumed ontology. It then examines joint preparation, product states and effects, marginal consistency, operational no-signalling, minimal and maximal probabilistic composites, local tomography, and the classification of admissible tensor-product representations. Particular attention is given to the distinction between a mathematical representation theorem and the uniqueness of a physical composition rule. Real, complex, quaternionic, Jordan-algebraic, minimal, maximal, restricted, and non-tomographic alternatives are compared under explicitly declared admissibility conditions. The complex Hilbert-space tensor product is recovered only on branches satisfying the required factorization, coherence, scalar, and operational closure conditions. Where several inequivalent composites survive, the result is a classified family rather than a falsely asserted uniqueness theorem. Entanglement is introduced only after composition has been commissioned. It is defined as the failure of an admitted joint state to belong to the convex closure of admitted product preparations. The analysis proves, within the resulting predictive grammar, that entanglement is a property of composite-state geometry, not an additional force, substance, interaction mechanism, or signalling channel. Throughout, the argument enforces quotient primacy, refinement stability, finite diagnostic closure, regeneration stability, explicit dependency receipts, local termination, and the possibility of Undefined as a legitimate scientific outcome. Part I therefore supplies neither a speculative ontology nor a presupposed many-world formalism. It provides a conditional classification theorem for binary physical composition and identifies precisely what remains to be established before interaction, repeated composition, particle identity, locality, fields, symmetries, and particle content may be addressed.
Title Ontological Uniqueness of Quantum Mechanics: A Pre-Quantum Classification Theorem Description This work addresses one of the most fundamental questions in theoretical physics: Why does nature employ quantum mechanical structure rather than alternative mathematical frameworks? We prove a rigorous classification theorem showing that quantum mechanics is not one possible theory among many, but the unique mathematical structure compatible with ontological continuity—the principle that "being cannot cease to be." Unlike previous approaches that either interpret quantum mechanics within its existing formalism (many-worlds, Bohmian mechanics) or derive it from operational measurement axioms (Gleason, Hardy, Chiribella-D'Ariano-Perinotti), this work operates at a pre-quantum logical level: we classify which geometric structures are ontologically admissible before introducing any quantum formalism. Main Result: Any theory satisfying minimal ontological requirements—continuity of being, compositional independence of subsystems, and reversibility of closed-system evolution—must describe the space of identity-classes as complex projective space ℂℙⁿ equipped with the Fubini–Study metric. Complex Hilbert space, unitary dynamics, the Schrödinger equation, and the Born rule emerge as necessary geometric consequences, not as postulates. Alternative structures (real or quaternionic projective spaces, non-homogeneous manifolds) are proven ontologically inadmissible through rigorous exclusion arguments. Key Contributions: Rigorous derivation showing that ontological composition forces a symplectic structure, which combined with homogeneity and isotropy uniquely determines Kähler geometry, selecting ℂℙⁿ as the only admissible space. Complete defense against circularity concerns, with detailed appendices showing that the axioms assume no quantum structure—they are purely ontological requirements for coherent continuous being. Systematic distinction from Gleason/Kochen-Specker theorems and operational axiomatizations, clarifying that this work classifies geometries before quantum mechanics, not constraints within it. Testable predictions in cosmology (Hubble tension resolution, matter fluctuation scaling) and quantum foundations (decoherence rates in molecular interferometry), while rigorously separating structural necessity from empirical determination. Significance: This work transforms quantum mechanics from an empirically successful but conceptually mysterious framework into an ontologically inevitable structure—comparable to Einstein's geometrization of gravity. It shows that quantum structure has no alternative: it is the only mathematical form compatible with coherent, continuous, compositional being. The manuscript includes comprehensive technical appendices with full geometric proofs, axiom justifications, and comparative analysis with existing foundational approaches. Suitable for researchers in quantum foundations, mathematical physics, philosophy of physics, and differential geometry. Document Status: Preprint submitted to Foundations of Physics (January 2026). Related Work: This work is part of the Law of Continuity of Being (LCS) research program, which derives physical and ethical structures from fundamental ontological principles. See author's ORCID profile for related publications. Open Access: This preprint is made available under CC BY 4.0 license to facilitate discussion and feedback from the research community prior to peer review.
This paper is Paper 004 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the Hilbert-space representation layer of Binary Mechanics from the mature finite graph-history formulation established in Paper 002 and the probability scalarization developed in Paper 003. The central claim is that Hilbert space is not a primitive arena inserted at the foundation. Instead, it arises as the minimal effective representation of finite Binary Mechanics amplitude data. A realized Binary Mechanics state is a resolved finite polarized graph. From that graph, the theory generates admissible finite future-history families. These histories carry phase structure through their update sequence, and coarse-grained alternatives receive finite complex amplitudes by summing the contributions of all admissible histories in the corresponding class. Once Paper 003 supplies the no-fence scalarization of finite complex record amplitude into squared-modulus record weights, the standard finite-dimensional complex Hilbert-space structure becomes the natural ledger for the resulting amplitude distribution. Basis states represent distinguishable coarse completion channels, linear superposition records unresolved admissible history content, the inner product records the squared-modulus weight structure, and norm-preserving transformations represent closed changes of effective description. The paper emphasizes that Hilbert space is therefore a reconstruction layer, not a substrate ontology. The underlying ontology remains the finite resolved graph and its admissible history fan. Hilbert space appears because it is the compact and faithful representation of the amplitude ledger generated by Binary Mechanics itself. The paper also clarifies how tensor products, entanglement, unitary structure, and continuum quantum descriptions are to be understood as later effective layers built from finite graph-history mechanics rather than assumed at the start. This work continues the staged reconstruction program of Anaxiomatic Mechanics and pq-spacetime by showing how ordinary quantum representation theory begins to emerge from finite graph histories, phase-bearing admissible futures, and no-fence probability scalarization.