This deposit contains two companion technical notes developing a finite-dimensional semantic state algebra for unresolved readings, classical decisions, and order-sensitive semantic operations. The first note separates semantic states from their response formats. A response format is defined as an alphabet together with a prescribed resolution map. For a two-reading construction, four selector-polarity states are distinguished explicitly. The polarity-only yes/no map is proven non-injective, whereas the full selector-polarity map is bijective. A separate rank-nullity theorem establishes the corresponding global obstruction for linear maps from higher-dimensional semantic spaces into lower-dimensional decision spaces. A real two-dimensional space is sufficient for storage; under the additional requirements of associative and commutative multiplication, a unit, and division by every nonzero element, the unique two-dimensional real extension is the complex field. The classical real sector is embedded conservatively. The second note extends this framework to semantic operators. Questions and contextual updates generate a unital complex operator algebra, while order dependence is represented exactly by commutators and the associated derived Lie algebra. State-dependent and global order-sensitivity measures are introduced. Complex dimension two is shown to be minimal for noncommuting complex-linear dynamics, while terminal binary decisions remain separate real-valued readouts. Operator complexification preserves sums, composition, identities, and commutators. The construction is structural rather than empirical. It assumes neither quantum ontology nor a Born rule and does not claim a complete formal semantics of natural language.
This deposit contains two companion technical notes developing a finite-dimensional semantic state algebra for unresolved readings, classical decisions, and order-sensitive semantic operations. The first note separates semantic states from their response formats. A response format is defined as an alphabet together with a prescribed resolution map. For a two-reading construction, four selector-polarity states are distinguished explicitly. The polarity-only yes/no map is proven non-injective, whereas the full selector-polarity map is bijective. A separate rank-nullity theorem establishes the corresponding global obstruction for linear maps from higher-dimensional semantic spaces into lower-dimensional decision spaces. A real two-dimensional space is sufficient for storage; under the additional requirements of associative and commutative multiplication, a unit, and division by every nonzero element, the unique two-dimensional real extension is the complex field. The classical real sector is embedded conservatively. The second note extends this framework to semantic operators. Questions and contextual updates generate a unital complex operator algebra, while order dependence is represented exactly by commutators and the associated derived Lie algebra. State-dependent and global order-sensitivity measures are introduced. Complex dimension two is shown to be minimal for noncommuting complex-linear dynamics, while terminal binary decisions remain separate real-valued readouts. Operator complexification preserves sums, composition, identities, and commutators. The construction is structural rather than empirical. It assumes neither quantum ontology nor a Born rule and does not claim a complete formal semantics of natural language.
Literary interpretation changes as readers encounter new passages, answer interpretive questions, and revisit earlier material in light of later context. Quantum cognition provides a probability calculus for judgments whose probabilities depend on context and probe order, but the use of Hilbert-space notation alone does not establish nonclassical behavior. This article develops Quantum Semantic Dynamics (QSD) as a constrained quantum-probabilistic model and distinguishes three claim levels: quantum-inspired representation, quantum-probabilistic structure, and intrinsic nonclassicality. The present paper advances only the second. The confirmatory implementation restricts the local semantic state to a two-dimensional space, fixes probe geometry in an independent pilot sample, reconstructs a population-level 2 × 2 density state from three linearly independent binary probe bases, and gives the off-diagonal state components an operational meaning through cross-basis judgments. Contextual evolution is restricted to a one-parameter completely positive trace-preserving contraction toward a preregistered context-target state rather than an arbitrary latent transition. Two probes form the randomized A → B/B → A sequence used for the quantum question-order (QQ) diagnostic; the third is measured in an independent arm for state identification. The paper derives the QQ equality, states its assumptions, and adds an uncertainty-style consistency constraint for the frozen noncommuting probe geometry. QSD is compared with explicitly defined classical comparator classes ranging from a static commutative model to matched sequential latent-state models. Compatibility with QQ or the Hilbert representation is not treated as proof of intrinsic nonclassicality; such a claim would require a separate contextuality analysis against an explicitly defined classical representability criterion. The resulting architecture makes the mathematical status, empirical burden, and rejection conditions of QSD explicit before data collection.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.
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.
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the pro-positional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality [symbols omitted] applicable to the PQM lattices; and truth-functionality [symbol omitted] applicable to both the lattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional [symbol omitted] semantics for three-or-higher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum state-induced expectation-function on a PQM truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.
We reconstruct complex quantum kinematics from a finite-dimensional real amplitude space equipped with projective measurement contexts and a positive mass geometry. Contextual completeness makes the generated algebra irreducible, so its neutral commutant has real, complex, or quaternionic scalar type. A positive-dimensional abelian group of neutral continuous phases selects exactly the complex branch. Normalizing its central generator yields a common operator J2 = −I, unique up to J ↦ −J, while the double commutant recovers the full algebra of complex-linear operators. The identity component of the orthogonal normalizer acts unitarily, and normalizer elements reversing the sign of J act antiunitarily. A commutator Laplacian identifies the neutral commutant and provides spectral-gap estimates for approximate neutrality and stable phase reconstruction. A three-context qubit example gives an explicit finite realization.
Complex Hilbert space is usually assumed at the start of quantum mechanics. This paper formulates a reverse construction for the one-particle and operator-algebraic setting. Starting from a real Hilbert space carrying an orthogonal representation, the admissible imaginary units are identified with the skew-adjoint square roots of − Id in the Real commutant of the complexified representation. The novelty is not the isolated use of complexification, spectral orientation, or polar decomposition, but their order of dependence: first the symmetry representation defines an intrinsic operator-valued phase space; then an oriented internal time flow selects a distinguished point of that phase space; only afterward are the usual complex one-particle, observable, probability, and Weyl–CCR structures invoked under their standard hypotheses. In the terminology used here, this oriented internal flow is the reversible or inner-generative side of a dual-time distinction. It is not a second observer clock and not an additional Lorentzian coordinate: mathematically it is the strongly continuous orthogonal group U ( t ) = exp ( t A ) on the real Hilbert space. Its Stone-dual variable is energy/frequency λ ; canonical conjugation pairs λ with − λ ; positive energy orients this pair and selects the canonical phase on the dynamic sector through A = J | A | = | A | J . Thus inner-generative time is the source of phase orientation, while irreversible record time, projection, memory, and decoherence enter only as later reduced-dynamics layers. No spectral gap at zero is required, and zero modes remain visible as residual phase data rather than being hidden by the construction. The selected phase is invariant under time-orientation-preserving symmetries and changes sign under time reversal, giving the usual antiunitary action after complex reconstruction. The paper also makes explicit the monoidal limitation: two selected local phases do not make the raw real tensor product the standard composite system; the standard composite is recovered only after imposing the J -balanced, equivalently complex, tensor product rule. Thus the result is a reconstruction of compatible complex kinematics on the selected one-particle/operator-algebraic sector, not a derivation of all quantum foundations from purely real data.