Quantum theory and Lorentzian geometry both make essential use of ordered cones, but a mathematical cone isomorphism does not show that the corresponding orders have the same physical origin. This paper asks whether quantum-statistical positivity and an independently reconstructed one-sided transfer order can be represented as two aspects of a single physical transfer sector. A finite-dimensional local transfer semigroup defines a proper cone CT . A statistical state–effect duality, pulled back through a dynamically faithful process–state representation, defines a typed map ♯T : VT → VT ∗ . Witness pullback completeness fixes the dual order operationally, while causal-sharp compatibility (CSC) requires ♯T (CT ) = CT ∗ . An explicit counterexample shows that CSC does not follow from statistical self-duality, witness completeness, or cone homogeneity alone. If admissible invertible order filters act transitively on int CT , the cone is symmetric and therefore Jordan-representable. Intersecting an independent complex-matrix branch JT ≃ Hermn (C) with a quadratic causal-transfer algebraic boundary forces n = 2, giving CT ≃ Herm2(C)+≃ L4. Familiar qubit–Lorentz identities then become cross-pipeline consistency tests between independently reconstructed quantities. The construction is local, finite-dimensional, and conditional; it does not derive spacetime dynamics or gravity.
Quantum theory and Lorentzian geometry both make essential use of ordered cones, but a mathematical cone isomorphism does not show that the corresponding orders have the same physical origin. This paper asks whether quantum-statistical positivity and an independently reconstructed one-sided transfer order can be represented as two aspects of a single physical transfer sector. A finite-dimensional local transfer semigroup defines a proper cone CT . A statistical state–effect duality, pulled back through a dynamically faithful process–state representation, defines a typed map ♯T : VT → VT ∗ . Witness pullback completeness fixes the dual order operationally, while causal-sharp compatibility (CSC) requires ♯T (CT ) = CT ∗ . An explicit counterexample shows that CSC does not follow from statistical self-duality, witness completeness, or cone homogeneity alone. If admissible invertible order filters act transitively on int CT , the cone is symmetric and therefore Jordan-representable. Intersecting an independent complex-matrix branch JT ≃ Hermn (C) with a quadratic causal-transfer algebraic boundary forces n = 2, giving CT ≃ Herm2(C)+≃ L4. Familiar qubit–Lorentz identities then become cross-pipeline consistency tests between independently reconstructed quantities. The construction is local, finite-dimensional, and conditional; it does not derive spacetime dynamics or gravity.
Quantum theory satisfies the mathematically powerful property of self-duality, meaning that its state and effect spaces are made isomorphic by an inner product. This is why both quantum states and effects can be represented by positive Hermitian operators. The seminal Koecher-Vinberg theorem shows that self-duality is a rather special property: any system that is self-dual and also homogeneous, meaning that the group of symmetries of its cone of unnormalised states acts transitively on the strictly positive states, must be isomorphic to a Euclidean Jordan algebra. EJAs have been classified and are known to be just a slight generalisation of quantum systems. This is why the Koecher-Vinberg theorem, and hence self-duality and homogeneity, have been used in several reconstructions of quantum theory from first principles. While homogeneity has an operational derivation from the ability to steer states, self-duality currently lacks such a clear operational motivation. In this paper we prove an alternative to the Koecher-Vinberg theorem, substituting for self-duality the more operational property of pure transitivity: symmetries of the normalised state space act transitively on the pure states. We show that any system that is homogeneous and satisfies pure transitivity must be self-dual, and hence isomorphic to an EJA. Together with various ways of singling out the complex matrix algebras from among EJAs, and known ways of ruling out classicality in these, this yields several new and concise reconstructions of quantum theory. For example, quantum systems, classical systems, and composites of these are the only ones that are homogeneous, satisfy pure transitivity, and allow locally tomographic composites; fully quantum systems are the only ones that are homogeneous, have continuous pure transitivity, and allow a correspondence between observables and generators of reversible transformations.
We develop a spacetime-free reconstruction from a faithful state on a structured family of observable algebras. Compatible half-sided modular inclusions generate positive translations and, after finite homogeneous assembly, a pointed Lorentz cone. Each faithful standard split inclusion determines a canonical Petz--Gram transfer, and we isolate a faithful modular boundary-code condition with three clauses: compact discarded response, a uniquely dominant displacement sector, and complete-order recovery of product defects. Under this condition, overlap transport, and the assembly hypotheses, we prove strict tail contraction, completely bounded two-boundary factorization, exact coherently aligned M2(C) embeddings, selection of three response dimensions, and complex multiplication up to the opposite algebra. Exact finite-dimensional and type-III counterexamples show that none of the three local code clauses follows from faithfulness, split or modular nuclearity, positivity, KMS symmetry, or covariance alone. A genuinely interacting spin-1 AKLT/Haldane chain realizes the logical edge algebra without microscopic qubits, with closed-form transfer bands and completely bounded encoding distortion 1+O(3^{-N}). Relative modular two-jets define the response fields (C,u,Q); their Gaussian covariance dynamics are Lie--Poisson with modular-spectrum Casimirs, and pseudodifferential commutators yield the mass-independent Q^{ij} deformation anchor. Spatial locality, branch smoothness, and two-derivative closure remain explicit conditional inputs.
Relational quantum causal processes formulate finite operational contexts as normal positive functionals on local completely positive maps. Response differences generate an influence algebra, and its central projections define jointly readable Boolean events. We develop this starting point through a sequence of exact and controlled models. Fresh-environment unitary collision circuits produce dephasing-exchange kinetics with an exact charge-center fixed algebra, a uniform finite-step limit at fixed response order, and graph-controlled metastable Markov dynamics. An absorbing-state model exhibits a sharp transition between non-Abelian quantum memory and Boolean records. A reversal-covariant defect dynamics generates a locally finite partial order on a restricted graph family without assuming a Lyapunov time. Conditional on a certified order, a positive additive record measure, compactness, and identifiability, we prove subsequential convergence to a Lorentzian metric-measure space, finite reconstruction bounds, and uniqueness of admissible smooth limits. Complementary finite regulators provide controlled tests of modular-to-boost response, null tomography, same-update variational identities, induced quadratic gravity, and compatible common-refinement limits. These results are exact or controlled within their stated models, but they do not yet constitute a single background-independent microscopic law that jointly generates adjacency, time, volume normalization, dimension, signature, nonlinear Einstein constraints, and quantum matter. We therefore present RQCP-QG as a theorem-indexed framework that separates established mechanisms, conditional compositions, and open assumptions.
A supplied rank-two Complementarity-First carrier has a positive duality h and a canonical complement #. The companion time paper used the local twist h(x,y^#) to obtain an index-one form. Here the same complement acts on one leg of the identity-pairing tensor, Γ_# = e ⊗ e - Σ_{i=1}^m e_i ⊗ e_i, with ω_# = (1/4)Γ_#. Relative to the supplied duality this construction is basis independent, product-positive, and normalized by the rank-two deterministic effect. It gives P(a,b | n,m) = (1/4)(1 - ab n·m). The m = 1 branch is separable; for m ≥ 2, physical admission with the sharp product interface gives CHSH magnitude 2√2. On the selected complex-qubit carrier, ω_# is the Bell singlet projector. The structural complement is positive but not completely positive. For a separately supplied exchange generator we derive G(X) = 2i[ω_#,X]. A partial-swap witness rejects one specific statically self-dual partial-transpose-image cone under the frozen process representation; the stronger carrier-relative PSD closure remains an inherited result. Entangled extensions of Euclidean-ball systems and canonical maximally entangled cone tensors are established prior art. The contribution claimed here is the typed local/tensor/process synthesis, not priority for singlet correlations, maximally entangled tensors, probability, or the physical composite.
Identifying causal order from restricted projective data is generally nontrivial. When two quantum players interact only through an unobserved environment, the available local measurement statistics are typically not tomographically complete, so the underlying process cannot in general be reconstructed exactly from the observed distribution. As a result, causal direction can be statistically identifiable in some cases but fundamentally indistinguishable in others.
In this work, we determine necessary and sufficient conditions for deciding when an observed distribution is compatible with a memoryless sequential quantum process in a fixed direction. We show that directional conditional-independence structure and the positivity criterion based on the pseudo-density matrix, as developed in recent work by We show that directional conditional-independence structure and PDM-based positivity criteria are not sufficient by themselves. We then specialize to the two-qubit Pauli setting, where the problem remains non-tomographic but becomes explicitly tractable. In this regime, we characterize when the two sequential directions are statistically indistinguishable, and we show by example that positivity alone does not exclude more general memoryful strategies, whereas the additional algebraic consistency requirement does.
We present a reconstruction of quantum correlation structure from six admissibility principles: compositional closure, locality factorization, norm-bounded growth, convex linearity, closure symmetry (self-duality), and local tomography. We show that these conditions force the state space to be a homogeneous self-dual cone, which by the Koecher–Vinberg theorem corresponds to a Euclidean Jordan algebra. Imposing physical composition constraints reduces this structure uniquely to complex matrix algebras. The Tsirelson bound then follows as a corollary of operator norm boundedness. Superquantum correlations (e.g., PR-box correlations) are excluded not by appeal to relativity but by violation of norm-bounded compositional closure. The result reframes quantum theory as the maximal globally admissible correlation theory under bounded composition