We develop Restricted Closure Theory (RCT), a time-symmetric completion of quantum cosmology that resolves the set-selection problem in decoherent histories while preserving strictly unitary dynamics. Working within the Two-State Vector Formalism (TSVF), we decompose the final-time Hilbert space into a Coherent subspace—pointer-stable macrostates—and a complementary Knot subspace encoding macroscopic inconsistencies. Imposing a Null-Knot Boundary Condition (⟨Φ|PK = 0) eliminates all macroscopically defective histories via Sorkin’s quantum-measure preclusion theorem. This yields global macroscopic closure with probability one while preserving Born-rule statistics up to O(ε) corrections determined by tail weights in the classical correspondence limit. RCT therefore provides a minimal, no-signaling, TSVF-consistent mechanism for selecting a unique globally coherent quasiclassical history without altering Schrödinger dynamics. Connections to decoherence theory, TSVF, boundary-condition cosmology, absorber proposals, and the Frauchiger–Renner family of paradoxes are examined.
We show that the operational quantum formalism -- the Born rule, Lüders state updating, quantum interference, non-Markovian effective dynamics, and Bell inequality violation at the Tsirelson bound 22 -- arises within Accessibility Theory (AT) from the Aperture construction together with explicit coherence and locality assumptions stated in the paper. AT is a framework built on real graded spectral triples and a single algebraic selection principle. The Principle of Universal Accessibility Balance requires three independent measures of the complexity of a spectral triple -- its algebraic, gauge-theoretic, and geometric content -- to be exactly equal and minimized, uniquely selecting the algebra C⊕H⊕M3(C) and with it the Standard Model gauge group, particle content, four-dimensional Lorentzian spacetime, three generations, and gravitational dynamics. Restriction to a codimension-one geometric boundary reduces this algebra to its commutative center C⊕C⊕C -- the Aperture -- which defines a permanent information bottleneck for any embedded observer. Coherence conditions on inference through this bottleneck, together with Gleason's theorem on the 48-dimensional internal Hilbert space, uniquely determine the Born rule; the remaining operational features follow from the same observer-level framework under the stated assumptions. At the ontological level the theory is deterministic and state-realist, while the operational quantum formalism appears at the observer level as a consequence of structurally limited access to the underlying algebraic reality.
We investigate the probability sector of PT-symmetric quaternionic spacetime (PTQ) from a structural and geometric perspective. The aim of this work is not to claim a universal derivation of the Born rule or a solution of the measurement problem, but to identify a constrained and internally coherent route by which the Born-rule form becomes admissible within the projected pseudo-Hermitian sector of PTQ.The analysis proceeds in three steps. First, we show that the probability sector must be defined only after restriction to a real, PT-even, projectively invariant observable scalar sector. Second, we demonstrate that consistency of the projected dynamics selects a pseudo-Hermitian metric structure, leading to a physically distinguished G-inner product rather than a naive kinematic norm. Third, we show that the same G-metric induces a conserved probability current, whose density is j⁰ = ψ†Gψ and whose spatial integral is preserved under the projected evolution.These ingredients jointly imply a constrained route: projected observability → physical G-inner product → conserved probability current → Pᵢ = |⟨ψ | φᵢ⟩ᴳ|². Within this route, the Born-rule form is not introduced as an independent postulate, but appears as the probability assignment compatible with observability, metric compatibility, and norm-preserving dynamics in the PT-even sector. We clarify the scope of the result: the construction is restricted to the unbroken PT-even projected sector and does not address collapse, branch selection, or fully general measurement theory. Within these boundaries, however, PTQ provides a disciplined and auditable framework in which the probability sector is structurally constrained.