This paper presupposes three papers — the foundations paper, "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811); the unified-description paper, "A Unified Description via the Triadic Structure" (DOI: 10.5281/zenodo.21131636); and the Tomita–Takesaki paper, "Selection and Heat" (DOI: 10.5281/zenodo.21161282) — and decomposes the quantum state itself in the vocabulary of the triad: direction and calibration. The scope is the minimal case (the two-level system); d > 2 is explicitly reserved. The claims of this paper are the theorems of the decomposition and a dictionary; no unification of dynamics and no new predictions are claimed. The central results are the following. First, the three degrees of freedom of the two-level state ρ = ½(id + r n·σ) are exhausted, with nothing left over, by the direction n (2) and the strength r (1) (Proposition 1.1). This is the same counting 3 = 2 + 1 as Proposition 2.5 of the unified-description paper — the three degrees of freedom of the pointwise metric are exhausted by direction (the two null directions) and calibration (a positive factor) (Corollary 1.2; the counting is a theorem on both sides, the identification is a reading). Second, for 0 < r < 1 the modular flow of the state is a rotation of the Bloch sphere about the axis n with angular speed 2 artanh(r) (Proposition 2.1): the direction selects the axis of time, and the strength sets its speed (Remark 2.4). At r = 0 the flow vanishes; for the thermal state, r = tanh(βε) and the speed is 2βε — the calibration encodes the inverse temperature (Proposition 2.3). Third, the update between two selections is implemented by the Connes cocycle u_t = ρ_ω^{it} ρ_φ^{−it} (Proposition 3.1 and Remark 3.2; the properties are theorems, the identification carries the tag). Fourth, non-selective measurement projects the direction and shrinks the strength, while selective measurement jumps to r = 1 — outside faithfulness. The mechanism lives on 0 < r < 1 and degenerates at both ends (no selection; pure selection) (Remark 4.2, the naming of a condition). What is a theorem, and what is an identification, a reading, a citation, or a reservation, is stated explicitly in the demarcation table. Theorem-level claims are machine-verified (verify_bloch_2plus1.py 8/8, verify_flow_axis_speed.py 8/8, verify_update_cocycle.py 7/7; numpy/scipy).
This paper presupposes three papers — the foundations paper, "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811); the unified-description paper, "A Unified Description via the Triadic Structure" (DOI: 10.5281/zenodo.21131636); and the Tomita–Takesaki paper, "Selection and Heat" (DOI: 10.5281/zenodo.21161282) — and decomposes the quantum state itself in the vocabulary of the triad: direction and calibration. The scope is the minimal case (the two-level system); d > 2 is explicitly reserved. The claims of this paper are the theorems of the decomposition and a dictionary; no unification of dynamics and no new predictions are claimed. The central results are the following. First, the three degrees of freedom of the two-level state ρ = ½(id + r n·σ) are exhausted, with nothing left over, by the direction n (2) and the strength r (1) (Proposition 1.1). This is the same counting 3 = 2 + 1 as Proposition 2.5 of the unified-description paper — the three degrees of freedom of the pointwise metric are exhausted by direction (the two null directions) and calibration (a positive factor) (Corollary 1.2; the counting is a theorem on both sides, the identification is a reading). Second, for 0 < r < 1 the modular flow of the state is a rotation of the Bloch sphere about the axis n with angular speed 2 artanh(r) (Proposition 2.1): the direction selects the axis of time, and the strength sets its speed (Remark 2.4). At r = 0 the flow vanishes; for the thermal state, r = tanh(βε) and the speed is 2βε — the calibration encodes the inverse temperature (Proposition 2.3). Third, the update between two selections is implemented by the Connes cocycle u_t = ρ_ω^{it} ρ_φ^{−it} (Proposition 3.1 and Remark 3.2; the properties are theorems, the identification carries the tag). Fourth, non-selective measurement projects the direction and shrinks the strength, while selective measurement jumps to r = 1 — outside faithfulness. The mechanism lives on 0 < r < 1 and degenerates at both ends (no selection; pure selection) (Remark 4.2, the naming of a condition). What is a theorem, and what is an identification, a reading, a citation, or a reservation, is stated explicitly in the demarcation table. Theorem-level claims are machine-verified (verify_bloch_2plus1.py 8/8, verify_flow_axis_speed.py 8/8, verify_update_cocycle.py 7/7; numpy/scipy).
This paper gives a unified description of quantum theory and relativity as readings of different axes of a single triadic structure (Δ, Λ, Ψ), presupposing the preceding paper "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811). The unification claimed is a unification of description, not of dynamics: no new dynamics are introduced, and "unification" here does not refer to gauge grand unification (GUT). The central new result: the light-cone coordinates of the 1+1-dimensional Minkowski plane form the standard two-dimensional triad as they stand (Proposition 1.1). The two poles V₊, V₋ are the two null directions; parity is P = Ψ; time reversal is T = −Ψ; the total spacetime inversion is PT = −id = Δ²; the boost of rapidity s is exp(sσ) = diag(eˢ, e⁻ˢ); and a boost composed with parity is the scaled exchange Ψ_λ (λ = e⁻ˢ). Under boosts the interval x₊x₋ is invariant while only the ratio x₊/x₋ moves (Proposition 2.1): the invariance of the interval is "a fixed product with a free ratio" — the same structure as the uncertainty relation (the fixed product ℏ with a free ratio). On the quantum side the identifications of the preceding paper are inherited: the complex unit i as the generator Δ (in the sense of right multiplication), the spinor's 2π as Δ², and the KMS antiperiodicity as a conditional reading. A dictionary maps the basic objects of both theories onto the elements and axes of one triad, and the junction identity −id = Δ² = PT ties them to one and the same half-period residue (Theorem 4.1). Gravity is placed as follows: by axiom T2 of the preceding paper, the scale cannot be fixed from inside the structure; the metric field is read as the field that supplies this externally required calibration at each point. The Einstein equations, the gauge content, and the values of the constants are external inputs and are not derived. A demarcation table states the status of every claim (theorem / citation / identification / reading / conditional / unconstructed), and criteria of attainment (U1–U3) for a unification of dynamics are defined, with a self-assessment of this paper against them. Files: Triadic_Unified_Description.pdf (the paper, Draft v2); Triadic_Unified_Description_verification_scripts.zip (three verification scripts — verify_lightcone_triad.py 12/12, verify_interval_product.py 8/8, verify_reversal_T.py 7/7 — with README; dependencies: numpy, scipy). Related: the foundations paper (DOI: 10.5281/zenodo.20986811); the axiomatic proposal for a non-orientable spacetime (DOI: 10.5281/zenodo.20678390).
On commence dans le chapitre d'introduction par rappeler les résultats majeurs sur l'intrication et les systèmes quantiques ouverts. Puis en particulier on prouve la désintrication en temps fini pour deux qubits (systèmes quantiques à deux niveaux d'énergie) en interaction avec des bains thermiques distincts à température positive. On propose dans le premier chapitre de cette thèse une méthode pour empêcher la désintrication en temps fini basée sur des mesures continues sur les bains et utilisant la théorie des sauts quantiques et celle des équations différentielles stochastiques. Dans le deuxième chapitre on étudie un sous-ensemble des états de deux qubits : celui des états qu'on peut représenter dans la base canonique pour une matrice ayant une forme de X. Cela nous permet d'obtenir des formules explicites pour la décomposition d'un état X séparable en au plus cinq états purs produits. On généralise ensuite cette étude à l'ensemble des états obtenus à partir d'états X par conjugaison avec des unitaires locaux. Puis on donne un algorithme pour décomposer tout état séparable de cet ensemble en une combinaison convexe de cinq états purs produits. Le troisième chapitre de cette thèse propose l'étude de l'évolution de l'intrication de deux qubits dans un modèle d'interactions répétées avec la même chaîne de spins dans les limites de van Hove et de couplage singulier. En particulier on observe une intrication asymptotique non nulle quand la chaîne est à température infinie et des phénomènes de création d'intrication quand la chaîne est à température nulle.
We address the problem of time in quantum mechanics: time is an external classical parameter, not a quantum observable, yet quantum gravity requires it to be emergent. We show that a canonical preferred temporal flow arises from within the algebraic structure of quantum mechanics, without presupposing a Hamiltonian, a Schrödinger equation, or a background time parameter. The starting point is a pair (M, ω) — a von Neumann algebra of observables and a faithful normal state — with no other input. From these, Tomita–Takesaki modular theory derives a canonical strongly-continuous one-parameter group of automorphisms σ_t : M → M (the modular flow). The group parameter t labels the spectral flow of the modular operator Δ = S^*S constructed from ω, and is not assumed as background structure. Three results are established. First (Structural Inevitability): any non-tracial quantum system unavoidably carries a non-trivial modular flow — the Tomita–Takesaki modular automorphism group, derived from algebraic data alone. Under the Connes–Rovelli thermal time hypothesis (adopted as an explicit physical postulate, supported by exact agreement with Heisenberg evolution for Gibbs states and with Rindler time via the Bisognano–Wichmann theorem), this flow constitutes physical time. Second (Cocycle Colimit): the directed colimit of all subsystem modular flows exists in the category W*-Dyn, interacting or not; for interacting systems the colimit is twisted by Araki's Connes cocycle, and the assembled flow equals the modular flow of the limiting algebra. Third (Classical Recovery): a structural outline of how, in the decoherence limit under density, separability, and the second law, the framework recovers a separable dense linear order from which n-dimensional spatial structure is derived; full topological reconstruction is left as an open program. This paper is self-contained and requires no companion papers. It extends the Connes–Rovelli thermal time hypothesis and the Page–Wootters relational framework to a universal directed cocycle colimit.
Mature theories of physical science associate states with systems, for instance with particles or waves in classical physics. The quantities whose values comprise states in classical mechanics, namely position and momentum, are chosen, first of all, for predictive purposes. Everything that one would want to say about classical systems can, it seems, be said in terms of properties: specification of the properties of a system gives a complete specification of its state, and conversely. Quantum mechanics associates states with physical systems; whatever else these might be, systems are the particulars which bear physical properties. The former is non-reduced state, the quantum state proper, while the latter is the reduced state, the class of quantitative properties possessed by the system. The state therefore determines which quantities and which properties are instantiated. The non-reductive account of states was initially appealing because it seemed to promise some additional explanatory scope.
This is the demonstration layer of the Phase III central proposition of the research programme on the rigidity of geometric incidence. The predecessor framework supplies the quantum-measurement collapse arrow as a purely geometric datum — the causal structure and temporal orientability of Minkowski space — while keeping the global evolution unitary, and records the irreversibility of a measurement event as a causal monotone tied to the four-volume of the causal diamond between event and observer. The same data admit two readings: a geometric reading, in which the functional form of the irreversibility is rigidly determined by the modular geometry of the diamond (fixed up to an overall normalization) and is invariant under a change of geometry without recalibration; and an effective reading, in which the law is merely carried by a thermodynamic registration channel whose coefficient is free and must be recalibrated for each geometry. The paper poses the decision between them as a single theorem of empirical non-equivalence, built symmetrically so that either verdict — rigidity (new physics) or equivalence (reduction to einselection with an imported thermodynamic arrow) — would be a valid, publishable outcome. Working in algebraic quantum field theory, where the local algebras are type III₁, it establishes (on a reference state with geometric modular action — the vacuum, via Bisognano–Wichmann) that the irreversibility functional is determined by the diamond's modular data up to one normalization constant (Tomita–Takesaki, Casini–Huerta–Myers, Araki relative entropy), with the entire residual freedom localized in the modular status of the record-carrying state. Whether that normalization is itself geometric (the value π/24) is graded as a conjecture and is not load-bearing; the decisive content is invariance under a change of geometry, and the value is adjustable. In the genuine type III₁ continuum, both halves of the discriminating apparatus return a third verdict: the registered antisymmetric orientation is geometric and irreducible to any CPTP channel (in type III₁ there is no reduced density matrix to dilate, so no Stinespring dilation can readmit it "as only an ancilla"), yet it coincides with the Borchers–Wiesbrock spectral condition (energy positivity) — already-known QFT serving as the arrow, rather than a new irreducible mechanism. Any genuinely new positive is localized to two named sites: the interacting / non-geometric-flow sector (β ≠ 2π/a, where the Connes–Rovelli thermal-time reading must be rebutted, not assumed) and the orientation of the modular inclusion. A preregistered finite-N Gaussian discriminator returns equivalence in the simulable (type I) class — reported honestly as inconclusive for the fundamental reading, since the decisive type III₁ regime is not reached at any finite N. Scope: special-relativistic (Minkowski); gravity and quantum gravity are deferred. All results are explicitly graded (theorem / proposition / conjecture / computation / inherited), and the type III₁ instruments and verdict data are included with the deposit. Changes in v2 (2026-07-08). This version adds Phase IV, step 3 — the two-axis closure of the region left open by the free-sector verdict. Route (b), non-geometric Gaussian flow (multi-interval): the multi-interval vacuum modular flow is genuinely non-geometric (bilocal strength ρbi ∈ [0.004, 0.064]) yet carries a single modular temperature on every channel (D ≡ 0 at βmod = 1 to 10-12, continuum-stable), with the independent-scales einselection foil detected at D = 0.148 — NULL. Route (a), perturbative non-Gaussianity (tree-level λφ3): the T-odd connected cumulant onsets at first connected order (|A3| ∝ λ) but its separating content is zero (D = 0, dD/dλ = 0), while the multi-scale and FDT-breaking foils fire at 0.138 and 0.177 — NULL / recalibrable; the third verdict extends to the perturbative interacting sector. Both axes were pre-registered before running (frozen 2026-06-26 and 2026-06-25 respectively); no v1 result is altered. New verified references: Peschel (2003), Casini–Huerta (2009), Peschel–Eisler (2009), Arias–Casini–Huerta–Pontello (2018).