We compute the complete relative entropy of a faithful quantum family controlled by spatial translations and Lorentz boosts. The prescribed contrast has a minimal three-dimensional positive Hessian realisation. Finite comparisons determine every minimal single-chart embedding up to an affine transformation and fix the potential's first jet, leaving its normal Hessian free. No positive single-chart realisation on a globally convex domain exists in any finite dimension for the entire control plane, whereas every bounded rapidity strip admits one. An exact dephasing channel preserves the energy-momentum law while reducing the contrast to classical rapidity distinguishability. Calibrating a preparation Hamiltonian gives sharp work-susceptibility and finite-work control-area bounds. Within the specified four-parameter preparation family, intrinsic Bogoliubov-Kubo-Mori (BKM) length bounds entropy change; paths outside that family need not obey this bound, even at the same observable law. A genuine Born lift preserves the commutator form with an exact area remainder independent of normal-potential freedom. These results separate finite-contrast rigidity, calibrated energy response, and geometric compatibility without identifying statistical transport with physical dynamics.
We compute the complete relative entropy of a faithful quantum family controlled by spatial translations and Lorentz boosts. The prescribed contrast has a minimal three-dimensional positive Hessian realisation. Finite comparisons determine every minimal single-chart embedding up to an affine transformation and fix the potential's first jet, leaving its normal Hessian free. No positive single-chart realisation on a globally convex domain exists in any finite dimension for the entire control plane, whereas every bounded rapidity strip admits one. An exact dephasing channel preserves the energy-momentum law while reducing the contrast to classical rapidity distinguishability. Calibrating a preparation Hamiltonian gives sharp work-susceptibility and finite-work control-area bounds. Within the specified four-parameter preparation family, intrinsic Bogoliubov-Kubo-Mori (BKM) length bounds entropy change; paths outside that family need not obey this bound, even at the same observable law. A genuine Born lift preserves the commutator form with an exact area remainder independent of normal-potential freedom. These results separate finite-contrast rigidity, calibrated energy response, and geometric compatibility without identifying statistical transport with physical dynamics.
We prove a controlled completion of dual-affine entropy geometry connecting a supplied amplitude-carrier complex structure, constructed tangent-bundle Born complex structures, non-Abelian compact transport, and endpoint action through typed relative-entropy reconstruction and defects. On the supplied Hermitian carrier, Umegaki relative entropy reconstructs the entropy potential up to affine gauge and the relative logarithmic cotangent class; diagonal jets yield the Bogoliubov-Kubo-Mori (BKM) metric, Hessian cubic, and dual connections, which construct Born lifts on real doubled statistical carriers. Determinant normalization selects a principal-log traceless Gibbs representative. A quantum channel induces support-reduced output geometry and a Bregman loss whose Hessian and cubic are metric and response losses with sequential cocycles; one CPTP map recovering every state of an embedded faithful model annihilates its restricted loss and intertwines its intrinsic geometry and holonomy. Proximal families yield Rényi and Hopf-Lax identities; BKM relaxation needs mobility and scale. From a separately chosen affine-invariant metric, the compact branch constructs canonical principal SU(n) and associated tangent PSU(n) transport, compared with BKM transport by connection and curvature defects rather than conjugacy. The indexed architecture is dimension-stratified: its general branches are available for each fixed finite n≥2 under their stated inputs, full BKM tangent holonomy is proved only for n≥3, and the qutrit and independently graded AIII packages occur only at n=3 and n=5, respectively. For qutrits, the tracial BKM metric-cubic pair has oriented stabilizer PSU(3); alignment with normalized affine-invariant holonomy lets the central cubic seed a distinct, unique parallel comparison cubic defining a global tangent-frame reduction. A fixed action conversion gives an exact endpoint canonical relation; the parameterized Rényi-Moreau family gives a branch-local scale-Hamiltonian isotopy. Mechanical interpretation needs inertia and a clock. Relative entropy does not construct the amplitude-carrier complex structure or supply physical calibrations.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.
This preprint investigates structural relations between relative entropy, modular theory, and information geometry in algebraic quantum field theory (AQFT). In finite‑dimensional quantum systems, the Hessian of Umegaki relative entropy coincides with the Bogoliubov–Kubo–Mori (BKM) metric, and modular automorphisms act by isometries of this geometry; these standard results are reviewed as a mathematically controlled reference model. Motivated by this setting, the work formulates minimal structural conditions under which a perturbative modular information‑geometric programme for Type III1_11 von Neumann algebras could become mathematically meaningful. The analysis focuses on admissible perturbation sectors, Araki relative entropy, modular covariance, and the functional‑analytic obstructions caused by unbounded modular operators. A general consequence of positive semidefiniteness of the perturbative entropy Hessian is recorded: whenever the Hessian defines a well‑defined positive semidefinite bilinear form, its null directions annihilate the full form. In the Rindler wedge model, the corresponding regularity problem for candidate perturbations is formulated in terms of spectral properties and domain issues of the Lorentz boost generator. Throughout, explicit separation is maintained between rigorous results, conditional statements, heuristic motivations, and unresolved analytic problems. A conjecture concerning degenerate modular directions is stated together with a necessary falsifiable condition. This document is exploratory in nature and has not been peer reviewed.
In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.
A holographic code assigns a state-dependent entropy remainder to every boundary region, but such remainders need not be the extremal lengths of any one bulk metric. We study this common-source question for the proto-area of a calibrated recovery, the entropy of a boundary region minus that of the matter recovered from it. At zeroth order the reference values must match the lengths of one reference geometry in a fixed renormalization scheme, and this matching also determines the entropy-to-length factor. On a graph with a fixed minimum-cut chamber the data derivatives must lie in the image of the cut-incidence matrix, whose left null vectors are witnesses. On the hyperbolic disk the first variation of renormalized length is the rank-two geodesic X-ray transform, with closed range in an adapted data space, and the geodesic displacement, fixed by a forced Jacobi equation, enters at second order. The resulting normal Hessian gives a gauge-invariant criterion, necessary and sufficient, for a regular pair of response coefficients to come from one metric two-jet. For the constant-curvature family generated by a boundary reparametrization the Hessian has a nonzero rank-four mode, so the image of the length map is curved. For a fixed recovery the proto-area Hessian is minus the Bogoliubov-Kubo-Mori information lost in recovery, which the Petz map decomposes orthogonally at a product reference, and which, on a finite-dimensional space of logical directions with positive minimum retained fraction, is at most a sharp multiple of the retained information. Geometric realization requires this loss to reproduce the normal length Hessian on every interval. Two codes show that exact regional area operators can fail to sew into one graph geometry, and that a matched reference with vanishing first response can fail at second order. All criteria refer to one recovery branch and one extremal-surface branch.