This paper extends the results of Hyperbolic Degree Bounds Signaling Dimension in Generalized Probabilistic Theories (Zenodo DOI: 10.5281/zenodo.22946778), which established that the signaling dimension of every canonical finite-dimensional Euclidean Jordan algebra (EJA) equals its Jordan rank. We investigate the consequences of this single-system identity under recursive composition. Combining the injectivity of local effect-space tensor products in standard non-signaling composites with the no-hypersignaling (NH) principle, we prove that the normalized linear-dimension ratio η_lin(S) = dim(S)/sign.dim(S)² is supermultiplicative. Consequently, any local excess above unity is amplified under repeated composition. For canonical EJA composites satisfying the structural axioms of Barnum, Graydon, and Wilce, product Jordan frames and NH enforce multiplicativity of composite rank. Together with established EJA classification results, this yields the conditional recursive constraint dim(J) ≤ rank(J)². Within this framework, quaternionic matrix Jordan systems and spin factors V_n with n ≥ 4 are excluded, while real matrix systems remain subcritical and complex matrix systems exactly saturate the bound. We also obtain finite-copy exclusion criteria and identify exact NH violations in specified EJA composition rules, including doubled complex and quaternionic composites. The analysis does not assume local tomography. Its conclusions are conditional on NH and the stated composition axioms; they do not establish unique selection of complex quantum theory. This work provides a compositional continuation of the preceding hyperbolic-signaling result, connecting classical simulation requirements to structural constraints on Jordan probabilistic theories.
This paper extends the results of Hyperbolic Degree Bounds Signaling Dimension in Generalized Probabilistic Theories (Zenodo DOI: 10.5281/zenodo.22946778), which established that the signaling dimension of every canonical finite-dimensional Euclidean Jordan algebra (EJA) equals its Jordan rank. We investigate the consequences of this single-system identity under recursive composition. Combining the injectivity of local effect-space tensor products in standard non-signaling composites with the no-hypersignaling (NH) principle, we prove that the normalized linear-dimension ratio η_lin(S) = dim(S)/sign.dim(S)² is supermultiplicative. Consequently, any local excess above unity is amplified under repeated composition. For canonical EJA composites satisfying the structural axioms of Barnum, Graydon, and Wilce, product Jordan frames and NH enforce multiplicativity of composite rank. Together with established EJA classification results, this yields the conditional recursive constraint dim(J) ≤ rank(J)². Within this framework, quaternionic matrix Jordan systems and spin factors V_n with n ≥ 4 are excluded, while real matrix systems remain subcritical and complex matrix systems exactly saturate the bound. We also obtain finite-copy exclusion criteria and identify exact NH violations in specified EJA composition rules, including doubled complex and quaternionic composites. The analysis does not assume local tomography. Its conclusions are conditional on NH and the stated composition axioms; they do not establish unique selection of complex quantum theory. This work provides a compositional continuation of the preceding hyperbolic-signaling result, connecting classical simulation requirements to structural constraints on Jordan probabilistic theories.
The signaling dimension of a probabilistic system is the minimum number of classical states required to reproduce, with shared randomness, every finite prepare-and-measure input-output correlation generated by the system. Frenkel and Weiner proved the complex-quantum simulation result that is equivalently expressed as signaling dimension n for an n-level system. Building on Frenkel’s later positive-multilinear-form/supply-demand formulation, we show that hyperbolic polarizations provide a general source of the multilinear structure required by that simulation mechanism, extending the resulting bound to arbitrary proper hyperbolicity cones. If the effect cone of an ordered probabilistic model is the hyperbolicity cone of a homogeneous polynomial p of degree d, then its signaling dimension is at most d. The proof uses the non-negativity of the polar form of p on the hyperbolicity cone and an order-theoretic identity relating the smallest hyperbolic eigenvalue of an effect to its minimum value over normalized dual states. As a corollary, every canonical Euclidean Jordan model has signaling dimension exactly equal to its Jordan rank. This simultaneously covers classical systems, real, complex and quaternionic matrix Jordan models, spin factors, the exceptional Albert algebra, and direct sums. For symmetric cones this yields a numerical identity between signaling dimension, Jordan rank, the cone Carathéodory number, and the optimal self-concordant barrier parameter. The general hyperbolic bound need not be tight; the ℓ1-cone/square-bit example of Ito and Lourenço provides a strict gap when viewed through signaling dimension. The result separates exact classical prepare-and-measure simulation dimension from internal state-space geometry at fixed signaling dimension and motivates the study of which operational invariants resolve the structure that signaling dimension alone does not capture.
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities.These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit.Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. {In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.}
Complex finite-dimensional quantum theory is distinguished within the Euclidean Jordan family by the matching between traceless observables and infinitesimal reversible dynamics. Existing characterizations typically impose this relation globally, or require a collision model capable of realizing all reversible dynamics. We ask how much of this dynamical structure is needed once the single-system state space is already assumed to be a simple Euclidean Jordan algebra and composition is available in the sense of Barnum–Graydon–Wilce. Let J be a nontrivial simple finite-dimensional Euclidean Jordan algebra admitting a BGW self-composite, and let Sₜ = exp(tΓ) be a unit-preserving reversible binary collision. Suppose equal product preparations are infinitesimally stationary and that the interaction induces a nonzero first-order reduced dynamics for at least one reference state. We prove that J ≅ Hₙ(C)ₛₐ for some n ≥ 2. The conclusion is rigid: after identifying J with Hₙ(C)ₛₐ, every productive stationary first-order footprint is a nonzero scalar multiple of the Hamiltonian commutator b ↦ i[b, ·], so the collision-induced map J₀ → Der(J) is automatically an isomorphism. On the standard complex matrix self-composite, the global stationary interaction itself is partial swap up to an overall rate. Conversely, partial swap supplies such a collision for every complex Hermitian EJA. The key step is that equal-state stationarity forces the global collision generator to commute with the connected diagonal automorphism action. The local infinitesimal footprint therefore determines a nonzero equivariant map J₀ → Der(J), where J₀ is the traceless subspace. Classification of simple EJAs then leaves only the complex Hermitian family, and one-dimensionality of the complex intertwiner space gives the rigidity statement. The result is a Jordan type-selection theorem, not a reconstruction from arbitrary operational theories: EJA structure and BGW composition are hypotheses. At this stage, however, neither local tomography, purification, collision universality, nor an a priori observable–generator correspondence is required. We also construct higher spin-factor collisions that are globally nontrivial and exactly stationary but have vanishing first-order local response, showing that the productivity hypothesis cannot simply be replaced by global nontriviality.
Complex finite-dimensional quantum theory is distinguished within the Euclidean Jordan family by the matching between traceless observables and infinitesimal reversible dynamics. Existing characterizations typically impose this relation globally, or require a collision model capable of realizing all reversible dynamics. We ask how much of this dynamical structure is needed once the single-system state space is already assumed to be a simple Euclidean Jordan algebra and composition is available in the sense of Barnum–Graydon–Wilce. Let J be a nontrivial simple finite-dimensional Euclidean Jordan algebra admitting a BGW self-composite, and let Sₜ = exp(tΓ) be a unit-preserving reversible binary collision. Suppose equal product preparations are infinitesimally stationary and that the interaction induces a nonzero first-order reduced dynamics for at least one reference state. We prove that J ≅ Hₙ(C)ₛₐ for some n ≥ 2. The conclusion is rigid: after identifying J with Hₙ(C)ₛₐ, every productive stationary first-order footprint is a nonzero scalar multiple of the Hamiltonian commutator b ↦ i[b, ·], so the collision-induced map J₀ → Der(J) is automatically an isomorphism. On the standard complex matrix self-composite, the global stationary interaction itself is partial swap up to an overall rate. Conversely, partial swap supplies such a collision for every complex Hermitian EJA. The key step is that equal-state stationarity forces the global collision generator to commute with the connected diagonal automorphism action. The local infinitesimal footprint therefore determines a nonzero equivariant map J₀ → Der(J), where J₀ is the traceless subspace. Classification of simple EJAs then leaves only the complex Hermitian family, and one-dimensionality of the complex intertwiner space gives the rigidity statement. The result is a Jordan type-selection theorem, not a reconstruction from arbitrary operational theories: EJA structure and BGW composition are hypotheses. At this stage, however, neither local tomography, purification, collision universality, nor an a priori observable–generator correspondence is required. We also construct higher spin-factor collisions that are globally nontrivial and exactly stationary but have vanishing first-order local response, showing that the productivity hypothesis cannot simply be replaced by global nontriviality.
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.