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The Isomorphism of Complex Numbers and Latent Worlds 0; From Coding Stability to Modal Necessity

changzheng zhou, ZIQING zhou

Published Jun 16, 2026• 0 citations
Publisher

Abstract

This paper constructs a derivation framework for quantum mechanics starting from a physical encoder in a finite-temperature heat bath, without relying onprior quantum assumptions. By strictly distinguishing the signal space from thestate-label space, and adopting information conservation and computability as operational axioms, we prove that the steady-state power response of the encoder isuniquely locked by thermodynamic stability to a positive-definite quadratic formon the signal space. This quadratic form, via the parallelogram law, a posterioriinduces an inner product, which is then pulled back by the encoding map to a metric on the state space. Linearity arises from the composability of physical inputsand injectivity of the encoding map, rather than from causal inference of energy response. Under the requirement of algebraic closure, we analyse the thermodynamiccosts of octonions and quaternions: the non-associativity of octonions necessitatesthe introduction of exceptional constraint algebras for composite-system construction, whose maintenance generates irreversible entropy production exceeding theLandauer limit; the non-commutativity of quaternions leads to ambiguity in spectral decomposition and potential path-dependent entropy production, for which arigorous statistical mechanical derivation is yet to be completed. The complex fieldemerges as the unique natural choice, being the only scalar algebra that simultaneously satisfies finite-dimensionality, algebraic closure, associativity, commutativity,and normed divisibility. The Born rule is not derived from thermodynamic powerweights, but emerges as the unique self-consistent frequency assignment for thecontext-independent response of the encoder, supported by a Gleason-type uniqueness argument and the law of large numbers. Latent-world information is defined asthe difference between Shannon entropy and von Neumann entropy; its geometryis exactly the restriction of the Bures metric along off-diagonal generators, corresponding to a scaled version of the Killing form of the adjoint Lie algebra, and itsdynamics follows a decoherence-driven exponential decay equation, directly interfacing with the master equations of open quantum systems. Finally, we proposefalsifiable experimental predictions comparing the information-energy efficiency ofcomplex encoders with real-number quantum mechanics (RQM), and establish astatistical hypothesis-testing framework.

DOI
10.5281/zenodo.20713333

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This paper constructs a derivation framework for quantum mechanics starting from a physical encoder in a finite-temperature heat bath, without relying onprior quantum assumptions. By strictly distinguishing the signal space from thestate-label space, and adopting information conservation and computability as operational axioms, we prove that the steady-state power response of the encoder isuniquely locked by thermodynamic stability to a positive-definite quadratic formon the signal space. This quadratic form, via the parallelogram law, a posterioriinduces an inner product, which is then pulled back by the encoding map to a metric on the state space. Linearity arises from the composability of physical inputsand injectivity of the encoding map, rather than from causal inference of energy response. Under the requirement of algebraic closure, we analyse the thermodynamiccosts of octonions and quaternions: the non-associativity of octonions necessitatesthe introduction of exceptional constraint algebras for composite-system construction, whose maintenance generates irreversible entropy production exceeding theLandauer limit; the non-commutativity of quaternions leads to ambiguity in spectral decomposition and potential path-dependent entropy production, for which arigorous statistical mechanical derivation is yet to be completed. The complex fieldemerges as the unique natural choice, being the only scalar algebra that simultaneously satisfies finite-dimensionality, algebraic closure, associativity, commutativity,and normed divisibility. The Born rule is not derived from thermodynamic powerweights, but emerges as the unique self-consistent frequency assignment for thecontext-independent response of the encoder, supported by a Gleason-type uniqueness argument and the law of large numbers. Latent-world information is defined asthe difference between Shannon entropy and von Neumann entropy; its geometryis exactly the restriction of the Bures metric along off-diagonal generators, corresponding to a scaled version of the Killing form of the adjoint Lie algebra, and itsdynamics follows a decoherence-driven exponential decay equation, directly interfacing with the master equations of open quantum systems. Finally, we proposefalsifiable experimental predictions comparing the information-energy efficiency ofcomplex encoders with real-number quantum mechanics (RQM), and establish astatistical hypothesis-testing framework.
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.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but its algebraic origin—the fact that the complex number field is the only commutative, algebraically closed division algebra over the realnumbers—has not yet been rigorously linked to an interpretive framework of quantum reality. This paper proposes an axiomatic system for a latent world algebra,decomposing the descriptive structure of quantum states into a direct sum of anobservable subalgebra and a latent world subalgebra, and defining the multiplication rules and commutation relations between them. It is proved that this algebraicstructure is a concrete realization of a C*-algebra, where self-adjoint elements correspond to the algebraic representation of observables, and non-self-adjoint elementsencode the algebraic information of modal dimensions. Furthermore, the square-lawclassification of the latent world subalgebra and its correspondence with Cliffordalgebras are explored, interpreting the difference between J2 = −I (complex structure) and J2 = +I (real structure) as two distinct physical modalities. On thisbasis, the algebraic framework is extended to quaternionic and octonionic cases,arguing that the non-associativity of octonions may correspond to the non-localityof quantum entanglement. Finally, topological invariants such as Chern numbersand characteristic classes are represented as numerical values obtained from thepairing of central cohomology classes of the latent world algebra with K-theory,establishing an equivalence relation between topological transitions and algebraicprojection mappings. This study provides a rigorous mathematical foundation forunderstanding the algebraic necessity of quantum structures and opens a formalizedpath for exploring hypercomplex quantum theories.
This paper proposes a structuralist interpretational framework for the quantummeasurement problem based on the concept of a latent world algebra. The framework decomposes the algebra corresponding to a physical system into a direct sumof an observable quantum algebra and a modal Lie algebra: the former correspondsto the commutative structure of classical observables, while the latter encodes thequantum modal information represented by complex phases. The measurementprocess is rigorously defined as a projection mapping from the latent world algebraonto the observable quantum algebra; this mapping discards all modal informationand is not an algebraic homomorphism, thus precisely capturing, at the algebraiclevel, the information loss and non-unitary character of measurement. Through representation theory and the Gelfand–Naimark–Segal construction, the framework isshown to be fully compatible with the standard formalism of quantum mechanics and naturally yields the Born rule. From the perspective of quantum resourcetheory, the framework reveals the operational meaning of the imaginary part ofcomplex numbers: the imaginary-part coherence measure is positively correlatedwith the success probability advantage of complex measurement bases in quantumstate discrimination tasks, thereby providing an algebraic explanation for recentexperiments establishing the imaginary part as a quantum resource. On the dynamical side, the paper introduces the concept of information potential differenceand establishes its threshold condition, characterizing the emergence of the classical world as the continuous dynamics of irreversible loss of modal informationduring decoherence in open systems. The testability of the framework is embodiedin three quantitative predictions concerning the precision limits of complex phasemeasurements, the correlation between the imaginary-part resource measure andstate-discrimination advantage, and the tomographic reconstructability of modalinformation; each prediction comes with a clearly specified experimental platformand falsification criteria. This work transforms the quantum measurement problemfrom an interpretational debate into an analyzable algebraic-dynamical structure,offering a new theoretical pathway for understanding the fundamental status ofcomplex numbers in physics.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.