A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.
A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.
The Admissibility of Natural Physics, Part I begins where an admissibility-first reconstruction of single-system quantum mechanics ends. Its central question is not whether the familiar tensor-product formalism can be assumed, but whether any physical composition law can be derived without importing composite systems, global state spaces, entanglement, locality, or signalling as primitives. The book develops a disciplined reconstruction of binary quantum composition from operational and structural requirements. It first records the exact content and limitations of the inherited single-system closure, establishes dependency and non-circularity statutes, and introduces the possible object of natural physics only as a provisional target ledger rather than as an assumed ontology. It then examines joint preparation, product states and effects, marginal consistency, operational no-signalling, minimal and maximal probabilistic composites, local tomography, and the classification of admissible tensor-product representations. Particular attention is given to the distinction between a mathematical representation theorem and the uniqueness of a physical composition rule. Real, complex, quaternionic, Jordan-algebraic, minimal, maximal, restricted, and non-tomographic alternatives are compared under explicitly declared admissibility conditions. The complex Hilbert-space tensor product is recovered only on branches satisfying the required factorization, coherence, scalar, and operational closure conditions. Where several inequivalent composites survive, the result is a classified family rather than a falsely asserted uniqueness theorem. Entanglement is introduced only after composition has been commissioned. It is defined as the failure of an admitted joint state to belong to the convex closure of admitted product preparations. The analysis proves, within the resulting predictive grammar, that entanglement is a property of composite-state geometry, not an additional force, substance, interaction mechanism, or signalling channel. Throughout, the argument enforces quotient primacy, refinement stability, finite diagnostic closure, regeneration stability, explicit dependency receipts, local termination, and the possibility of Undefined as a legitimate scientific outcome. Part I therefore supplies neither a speculative ontology nor a presupposed many-world formalism. It provides a conditional classification theorem for binary physical composition and identifies precisely what remains to be established before interaction, repeated composition, particle identity, locality, fields, symmetries, and particle content may be addressed.
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply PrL(fail)≤C∣ΩL∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.
How a finite real cycle acquires a complex structure and a rule for composition Quantum theory uses a common imaginary unit when individual systems become a composite. This framework constructs that compatibility from finite real source histories. On the complete contrast sector of a three-cycle, the forward and backward comparisons give J=3P−P2,J2=−I The complex structure appears after the real comparison is specified. At a qualified unmarked boundary, opposite reference shifts select QAB=2I−JA⊗JB The range is the balanced complex tensor product. Its many-source composition is associative. A finite controlled contact supplies a nonfactorizable complex-linear operation, and exact residual bounds quantify departure from the selected sector. A source model that produces the contact A reversible controller compares operation orders on 24 proper signed frames. Its ordered, arrow-insensitive axis detector produces the retained cyclic response. The contact algebra gives [[G,G],[G,G]][G,G]≅C3. The construction prints the controller, scratch cleanup, full contrast probes, admissible operations and refusal states. A triaxial real response tensor realizes the detector and provides an explicit map between material contrasts and event-count contrasts. A record must survive the test of future operations A present tensor can hide a material marker that a later operation reveals. The framework derives independent force-gradient calibration and a delayed odd-field release. With a nonzero release field, energy and vector torque reconstruct the entire marker change: DU=−Δp⋅E,Dτ=Δp×E.Δp=∣E∣2E×Dτ−DUE The test includes shared calibration errors, stray fields, finite release pulses, marker relaxation and a complete operation inventory. It distinguishes detection of a retained record from agreement with its frozen prediction and from discrimination between physical theories. From proved structure to existing experiments The paper adds a conditional probability-correspondence theorem for the balanced composite, examines independent-source quantum-network experiments, and reproduces all eight measured rows of a public torque dataset. These supply composition benchmarks and demonstrated apparatus capabilities. Their published significances are kept attached to the hypotheses they actually test. The calibrated material release law is also a conventional electrostatic prediction; it is not presented as a QTT-exclusive signal. One material coefficient, two readouts A new archival calculation transfers a published same-particle libration calibration to a separate precession measurement. In the qualified rotor regime, the cross-readout relation is Ξ=ωlib2EpωsΩElcosϕl=alibaprec. An unchanged dipole-to-inertia coefficient gives unity. The calculation yields a slope of 2.909 from the libration calibration, compared with the published precession slope of 2.950, a central difference of 1.4135%. This shared electrostatic consistency relation eliminates the separate dipole and inertia; unequal field gains, dynamical corrections and shared uncertainty remain explicit. Reversal and long-term dipole-stability observations identify complementary laboratory capabilities. The complete matched-tensor delayed-release test remains an experimental target. When do two descriptions differ in any future experiment? The finite-reference extension follows a controlled source contact through shared references, fresh references, inverse operations, intermediate measurements and feedback. Its complete order-three response is gn=∑k=02pkζkn,ζ=e2πi/3,g3=1,g2=g1. A finite observable-span algorithm tests two fully specified models across every permitted future word. It either certifies agreement or returns the preparation, operation sequence, terminal outcome and exact probability difference. In the populated compatible-instrument benchmark, the complete quantum representation gives the same records without an additional target fit. A source-selected physical coupling or preparation remains a distinct route to predictive value. The framework includes nine vector figures, the retained source proofs and countermodels, complete reference-history benchmarks, calibration and vector-reconstruction theorems, runnable exact and numerical checks, published-input receipts, a data assessment and an experimental packet specification. The introduction defines the QTT terms needed to follow the construction without prior familiarity with the corpus. The probability-interface theorem, finite-memory bounds and covariance-weighted witness specify which further laboratory comparisons can distinguish physical models. Source anchors QTT Main Book, especially pp. 56, 248-250, 352 and 827-829; Artian's Completed-Event Four-Capacity Theorem; Frame-cycle derivation map, completed-event lexicon, and Artian's Universe and laboratory readout. Stable citation: 10.5281/zenodo.22882084.
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.
Complementarity can organize a relational starting point for quantum reconstruction, but a completion relation does not yet supply preparations, effects, probabilities, continuous trans- formations, composites, or a scalar field. We formulate the missing passage as an assumption-control problem. Beginning from the typed primitive C= [a⊣b], we separate incidence, operationalization, continuity, scalar coherence, transition probability, effects, instruments, composition, purification, and process identification. Every negative reconstruction bridge is tied to an exact premise package and a typed target; geometric or pre-operational witnesses are not promoted into countermod- els of a larger theory they do not instantiate. At exact complex-closure resolution, let AOPT denote the finite-dimensional causal operational framework of an external reconstruction theo- rem, L local equivalence, and E equivalent-system purification. Finite classical theory witnesses AOPT + L without E, while finite-dimensional real quantum theory witnesses AOPT + E without L; the external theorem closes only AOPT + L+ E. Neither selector is derived from primitive complementarity here. The real-quantum stress test is sharpened by the two-rebit decomposition Sym(4) = Sym(2) ⊗Sym(2) ⊕R(J ⊗J): product effects see a nine-dimensional shadow of a ten- dimensional carrier. The identity setting together with one additional algebraically transverse joint-interaction setting gives complete terminal-product state exposure. This later observability remains distinct from original local tomography, universal process tomography, and complex-field selection. The contribution is a source-generated dependency architecture, exact finite witnesses, and a precise selector-origin problem: complex closure is formally conditional, while the physical origin of local equivalence and equivalent-system purification remains open.