Quantum pyramidal cosmology: emergent universe from ternary quantum chaos

Abstract

Comprehensive Edition v9 of Quantum Pyramidal Cosmology (QPC) — a self-consistent model in which the observable Universe emerges from ternary quantum chaos. This edition adds rigorous information-theoretic foundations for the Aura Functional via quantum relative entropy, establishes a Hierarchical Noether Theorem as a consequence of the SHIFT functor structure, provides a complete Russian translation of the work, and resolves several key mathematical ambiguities identified in previous versions. Major enhancements in this edition: 1. Constructive Definition of the Aura Functional The Aura Functional, previously introduced axiomatically, is now rigorously defined as the quantum relative entropy between a state and its LOCK projection: A(ρ) = S(ρ || Π∞(ρ)) = Tr[ρ(log ρ - log(Π∞[ρ]))]. This definition grounds the Aura Functional in standard quantum information theory, establishes its positivity and monotonicity as theorems (rather than postulates), and allows the variational principle to be derived as a consequence rather than an independent conjecture. 2. Hierarchical Noether Theorem Building on the categorical structure of the SHIFT functor, we establish a hierarchical analogue of Noether's theorem. Every continuous hierarchical symmetry at level n lifts via SHIFT to a symmetry at level n+1, with corresponding conserved hierarchical currents J_n that form a tower under coarse-graining. The theorem is explicitly illustrated with hierarchical energy conservation, where the effective Hamiltonians H_n = S^n(H_0) form a tower of conserved generators, providing a mathematical foundation for the renormalization group flow of Hamiltonians. 3. Complete Russian Translation The full manuscript is now available in Russian, expanding accessibility to the Russian-speaking theoretical physics community. The translation includes all new mathematical developments and maintains terminological consistency with the established English version. 4. Clarification of Hierarchical Dynamics The text now explicitly distinguishes the roles of the qutrit substrate (static configurational space of logical potentialities), the Aura Functional (which evolves according to CPTP dynamics), and effective laws (which emerge as stable attractors of this evolution). This resolves ambiguities present in earlier formulations regarding the nature of the fundamental substrate and its relation to emergent structures. 5. Enhanced Projector Formulation The treatment of hierarchical projection operators now explicitly acknowledges that alternative projection scenarios generate manifolds that are unstable with respect to a given hierarchical basis, but may generate their own nested hierarchies upon a change of projection basis — remaining unobservable from our stable branch or manifesting as anomalous intersections of spectral sectors. This replaces the previously binary "dead end" formulation with a richer ontological landscape. Mathematical contributions established in this edition:- Constructive definition of the Aura Functional- Positivity and monotonicity theorems for the Aura- Aura critical points as LOCK fixed points- Hierarchical Noether Theorem with lifted generators and conserved currents- Functorial lifting of symmetries via SHIFT- Spontaneous breaking of hierarchical symmetries at LOCK fixed points- Hierarchical energy tower H_n = S^n(H_0) The full mathematical apparatus now comprises:- Axiomatic foundations (Sections 2-4)- Aura Functional: axiomatic (Section 6) and constructive (Section 7)- Hierarchical Projection Towers (Section 8)- SHIFT Functor (Section 9)- Hierarchical Noether Theorem (Section 10)- Variational Principle (Section 11)- Local Hierarchical Dynamics (Section 12)- Renormalization as Hierarchical Flow (Section 13)- Cosmological Interpretation (Section 14)- Hierarchy of Effective Laws (Section 15) All cross-references have been updated to reflect the new section numbering. Relationship to previous versions: v1–v3: Initial formulation, Zenodo doi:10.5281/zenodo.21423122v4: Added Zurek correspondence and experimental proposalsv5: Companion paper on operational semantics, Zenodo doi:10.5281/zenodo.21527673v6 (July 2026): Added categorical foundations, resonant unpacking, noise as resource, lay summary; revised to clarify role of Decoherence vs. Chaos in resonant unpackingv7: Added expanded Lay Summary with hierarchical symmetries sectionv8: Added new Axiom on Emergence of Effective Laws; corrected projector terminologyv9 (current): Added constructive definition of Aura via relative entropy; added Hierarchical Noether Theorem with full proof and energy conservation example; complete Russian translation; revised hierarchical projector formulation; updated all cross-references Key features of the QPC framework (all editions): Microscopic Hamiltonian H_QCA dynamically generating asymmetryHierarchy of emergent levels via quantum phase transitions and RG flowEinstein's equations from entanglement entropy; Λ as a dynamical attractorPrecise correspondence with Zurek's decoherence theoryResonant unpacking: Decoherence selects from Chaos by resonance with existing contextNoise as a resource: physical noise as a window into the multiverseAura Principle: structure formation at decoherence boundaries, with implications for astrobiology and the genetic codeCategorical and topos-theoretic foundations: QPC operations as functors between topoiTen falsifiable predictions (CMB, collider, cold-atom, neural, noise spectroscopy)Experimental proposal for H_cross detection using dual-BEC cold-atom platformsPreliminary computational confirmation in the "Next Step" software platform (v5.3.0)Comprehensive lay summary in accessible languageFull Russian translation now available Note on parallel evolution: The convergence between QPC and topos-theoretic physics (Döring–Isham program, Caramello's bridges) is not a case of intellectual borrowing but of parallel evolution in theory space — independent approaches converging on compatible mathematical structures for describing emergence and contextuality. This edition represents a significant maturation of the mathematical foundations of QPC. By grounding the Aura Functional in standard quantum information theory and establishing the Hierarchical Noether Theorem as a consequence of the categorical structure, the framework now possesses both greater rigor and broader applicability. The constructive definition of Aura via relative entropy eliminates the previously axiomatic character of the variational principle, while the Noether theorem demonstrates the genuine physical content of the SHIFT functor beyond formal elegance. Future directions identified in this edition include operator-algebraic realizations, continuous hierarchical dynamics, numerical investigations of hierarchical fixed points, representation-independent formulations, and higher-categorical formulations of effective theories. Keywords: Quantum cosmology, Open quantum systems, Decoherence, Emergence, Quantum information, Category theory, Renormalization, Complex systems, Noether theorem, Relative entropy, Hierarchical structures