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The Complex Numbers and Unitarity of the Self-Referential Universe 1; The Foundational Role of Complex Structures in Self-Referential Systems

changzheng zhou, ZIQING zhou

Published Mar 11, 2026• 0 citations
Publisher

Abstract

Starting from two fundamental assumptions, information conservation and logical closure, this paper demonstrates that the complex number structure is theinevitable mathematical foundation for self-referential systems to achieve infinitesemantic expansion. Firstly, it proves that information conservation requires theevolution operator to preserve the inner product, thus belonging to the unitarygroup; logical closure requires the system to be able to generate new propositionsinfinitely, which forces the evolution operator to have a continuous eigenvalue spectrum. The joint constraint of the two uniquely leads to the complex number fieldas the algebraic framework for self-referential systems. It further elucidates thesemantic loading of the complex phase: the squared modulus encodes propositionalcertainty, and the argument encodes potential logical associations; unitary evolutiongenerates new states through phase reorganization without increasing informationentropy, thus avoiding logical freezing. By examining quantum mechanics and theblack hole information paradox, the core role of complex structures in physical theories is demonstrated. Starting from three constructive principles (countability,reversibility, continuity), the complex number field is independently derived as thesimplest algebraic solution, and it is rigorously proven that the real number fieldand quaternions cannot meet the requirements. Self-referential systems, semanticexpansion, and logical freezing are formally defined, and the uniqueness of complex numbers is rigorously proven using Schur’s lemma, the Frobenius theorem,and recent no-go theorems (Życzkowski et al. [9], Masanes [11]). A self-referentialautomaton model is constructed for numerical simulation, visually showing thatin the complex number case, the phase continues to evolve and semantic entropyfluctuates, while the real number case quickly falls into logical freezing. Finally, thetheoretical significance is summarized and future research directions are prospected.

DOI
10.5281/zenodo.18946987

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Starting from two fundamental assumptions, information conservation and logical closure, this paper demonstrates that the complex number structure is theinevitable mathematical foundation for self-referential systems to achieve infinitesemantic expansion. Firstly, it proves that information conservation requires theevolution operator to preserve the inner product, thus belonging to the unitarygroup; logical closure requires the system to be able to generate new propositionsinfinitely, which forces the evolution operator to have a continuous eigenvalue spectrum. The joint constraint of the two uniquely leads to the complex number fieldas the algebraic framework for self-referential systems. It further elucidates thesemantic loading of the complex phase: the squared modulus encodes propositionalcertainty, and the argument encodes potential logical associations; unitary evolutiongenerates new states through phase reorganization without increasing informationentropy, thus avoiding logical freezing. By examining quantum mechanics and theblack hole information paradox, the core role of complex structures in physical theories is demonstrated. Starting from three constructive principles (countability,reversibility, continuity), the complex number field is independently derived as thesimplest algebraic solution, and it is rigorously proven that the real number fieldand quaternions cannot meet the requirements. Self-referential systems, semanticexpansion, and logical freezing are formally defined, and the uniqueness of complex numbers is rigorously proven using Schur’s lemma, the Frobenius theorem,and recent no-go theorems (Życzkowski et al. [9], Masanes [11]). A self-referentialautomaton model is constructed for numerical simulation, visually showing thatin the complex number case, the phase continues to evolve and semantic entropyfluctuates, while the real number case quickly falls into logical freezing. Finally, thetheoretical significance is summarized and future research directions are prospected.
This paper establishes the fundamental framework of self-generative dynamics.Starting from two minimalist axioms—information conservation and computability—we rigorously derive the complex numbers as the inevitable mathematical foundation for describing physical reality. By defining a self-generative system, we couplethe recursive evolution of the system with the information potential difference, proving that unitarity and logical closure jointly compel the eigenvalues of the evolutionoperator to lie on the complex unit circle, thereby excluding the real numbers andquaternions as a global number field. We further introduce the self-referential operator and an infinite-layer Hilbert space structure, revealing the critical thresholdln2 of the information potential difference as a universal criterion for the quantumclassical phase transition. A correspondence is established between the spectralgap and Jones’ discrete spectrum, laying the groundwork for subsequent studieson geometric emergence and algebraic stability. This paper provides first-principlesupport for the unified program of the self-referential universe and generative dynamics.
Previous work has deepened the mathematical foundations of self-referential systems from the two directions of information geometry and category theory. Information geometry reconstructs self-referential systems as geodesic motion on a manifoldof quantum states, introducing quantum Fisher information and the Bures metric tocharacterize their dynamics and critical behavior; category theory reconstructs selfreferential systems as self-referential monads on tensor categories, revealing theiralgebraic and entanglement essence through sheaf categories and monad structures.This paper proves that these two descriptions are profoundly equivalent: the Fisherinformation flow corresponds to the derivative of the self-referential monad multiplication, and the Bures metric corresponds to the Riemannian metric on the sheafcategory. We construct an explicit functor from the sheaf category to the information manifold and prove that this functor constitutes a categorical equivalence;under this equivalence, the multiplication natural transformation of the monad induces the generator of the Fisher information flow, while the inner product structureof the morphism space in the sheaf category precisely yields the Bures metric. Thisunification provides a dual perspective for understanding the information processing and algebraic structure of self-referential systems, lays a unified mathematicallanguage for physical predictions such as the black hole information paradox andgravitational wave echoes, and reveals deep connections among quantum gravity,information theory, and category theory.
The presence of the imaginary unit iii in quantum mechanics is frequently treated as a computational postulate. We demonstrate that the complex structure of Hilbert space is not a decorative choice but a structural necessity derived from the first principles of modern physics: continuous symmetry, the requirement of unitarity to preserve probability, and the correspondence between generators and observables. By examining the Lie algebra of the unitary group, we prove that a real-valued theory fails to map self-adjoint observables to generators of continuous evolution without sacrificing local tomography or the existence of non-trivial phases. We conclude that iii is the unique logical bridge between symmetry and observability.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, yet its ontological status—whether complex numbers aremerely a mathematical tool or a constitutive element of physical reality—has longremained undecided. Starting from mathematical necessity, this paper argues thatthe complex structure is not only the descriptive language of quantum mechanicsbut also an ontological constraint condition of physical reality. Based on Frobenius’ theorem on the classification of division algebras over the real numbers, wereveal: if the descriptive framework is required to be both commutative and algebraically closed, then the complex structure is the unique logically necessary choice.This mathematical constraint, when mapped onto physical reality, implies that the”imaginary part” corresponds to a modal dimension that is not directly observablebut constrains observable phenomena. The measurement process is thus reconceptualized as a projection from the complex structure onto a real subspace, whereinformation loss is an inherent feature of the mathematical structure rather thanan additional dynamical assumption. We demonstrate that this framework is compatible with research on imaginarity as a resource in quantum resource theory andwith analyses of Fisher information limits in quantum metrology, and we proposethree testable predictions with quantum metrological precision. This interpretationadvances the quantum measurement problem from a matter of interpretational debate to a level of structural analysis, providing a rigorous mathematical foundationfor understanding the quantum-classical transition.
A complete scientific theory encompasses not only mathematical formalism andphysical predictions but also a reflection on its own status. The ”MathematicalUniverse of Self-Reference” series has established, in previous works, a comprehensive framework ranging from the necessity of complex numbers to experimentaltests. However, where does this framework itself stand within the philosophy ofscience? How does it relate to mainstream interpretations of quantum mechanics(Copenhagen, Many-Worlds, QBism, etc.)? How does it respond to the century-olddebate between realism and anti-realism? This paper does not aim to ”defend” theself-referential universe but rather to provide a meta-theoretical positioning: wedemonstrate that the self-referential universe is essentially a form of modal structural realism. It inherits the core insights of structural realism but transcends thestatic nature of traditional structural realism through the concept of ”modal dimensions” (the imaginary part of complex numbers). On this basis, we analyze thecompatibility and divergences between the self-referential universe and other interpretive frameworks, revealing its unique position within the genealogy of scientificphilosophy. Finally, we discuss the implications of the self-referential universe forunderstanding core concepts such as ”reality,” ”information,” and ”measurement,”offering a philosophical ”self-reflexive” reflection for the entire series.
This paper constructs an interpretative framework of quantum mechanics thattreats the complex structure as a modal dimension, reconstructing quantum measurement as an information-geometric phase transition rather than an instantaneouswavefunction collapse. By introducing a rigorous decomposition of Jordan algebrasand Lie algebras, the latent-world subspace is proven to form a bi-module structureunder the Jordan product and the projected Lie bracket, thereby being compatible with finite-dimensional algebraic frameworks. In the pure-state limit, the sectional curvature of the Bures metric and the quantum Fisher information satisfy anexact differential-geometric identity; the environmental corrections for the mixedstate case are honestly labeled as scaling conjectures constrained by renormalizationgroup. The information potential difference is rigorously defined via the relative entropy of coherence, and the Landauer principle yields the minimal thermodynamiccost of erasing coherent information in the quasi-static limit, with the ln 2 threshold re‑positioned as a calibratable benchmark in non‑equilibrium, finite‑temperatureenvironments. The physical inadequacy of the real-number framework under thesame Hilbert-space dimension is established by the exclusion principle of Bell-typeexperiments, proving that imaginarity is the physical origin of quantum interferencerather than a notational convenience. By introducing the Frobenius theorem as aselection constraint and Bell-type experiments as empirical constraints, the ternarycircular argument is broken into a unidirectional implication chain, establishingthe epistemological foundation of constraint structural realism. Four experimental predictions with explicit falsification conditions provide an operable interfacefor experiments, completing the proof closure from heuristic program to axiomaticsystem.