arXiv subjects

Jean-Pierre Magnot

Publications and source records attributed to Jean-Pierre Magnot.

At least 19 recordsLinked to original sources

arXiv2026

Chern--Simons Fluctuations and Information Geometry in Discrete Electromagnetism

We construct helicity-conditioned statistical states for a Whitney-discretized electromagnetic field on a closed oriented three-manifold. The simplicial de Rham complex provides exact discrete gauge symmetry, while the Whitney inner product separates exact, harmonic, and coexact sectors. The spatial Abelian Chern--Simons functional is gauge invariant and depends only on the coexact potential. After fixing harmonic modes, we introduce a helicity-biased Gaussian ensemble on the reduced electromagnetic phase space and derive explicit formulas for its admissible parameters, partition function, mean helicity, relative entropy, and Fisher information. The distribution uniquely minimizes relative entropy under a prescribed mean-helicity constraint. Its helicity susceptibility equals the variance of the discrete Chern--Simons functional and controls the local distinguishability of neighboring statistical states. Because magnetic helicity is generally not conserved under unconstrained Maxwell dynamics, these states represent conditioned inference rather than dynamical equilibrium.

math-ph↗

arXiv2026

Information Loss under Gauge Reduction of Discrete Electromagnetic Fields

We study relative entropy under gauge reduction in Whitney-discretized electromagnetism. The discrete Hodge decomposition separates exact gauge directions from harmonic and coexact physical degrees of freedom. For Gaussian distributions on the auxiliary potential space, relative entropy splits exactly into the relative entropy of the physical marginals and a nonnegative conditional gauge contribution. We express this contribution through Schur complements, conditional means, and physical--gauge correlations, and characterize equality between auxiliary and physical relative entropies. The physical divergence is also shown to be the minimum over all Gaussian auxiliary extensions. At the infinitesimal level, Fisher information admits an analogous decomposition. A two-variable model illustrates how statistically distinct auxiliary potentials may represent identical physical statistics. The construction provides an information-geometric interpretation of discrete gauge reduction without assigning physical significance to the eliminated gauge variables.

math-ph↗

arXiv2026

Diffeological Tangent Spaces and Distributional Linearization for Lifted Euler--Reynolds Limits

We develop a diffeological framework for the geometry of Euler--Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler--Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler--Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.

math.AP↗

arXiv2026

Entropy Geometry and Normalized Means on Infinite-Dimensional Hamiltonian Manifolds

We propose a geometric--analytic framework for equilibrium statistical mechanics on infinite-dimensional Hamiltonian systems. In situations where no suitable $\sigma$-additive invariant measure is available, we use \emph{normalized means}, which generalize probability measures and normalized integrals. This construction yields entropy and free-energy functionals on weak symplectic Fr\'echet manifolds and gives existence and uniqueness of exponential-family equilibrium states under explicit admissibility and separation assumptions. These states are stationary under Hamiltonian flows preserving both the reference mean and the equilibrium weight, and satisfy a classical Poisson--KMS identity when the reference mean is Poisson invariant. Under a local exponential regularity assumption, the logarithmic partition functional is smooth and convex, with Hessian given by the covariance form. It is strictly convex modulo thermodynamically null directions and, through Legendre--Fenchel duality, induces a concave entropy on the domain of extensive variables. We illustrate the framework with $H^s$-geodesic equations on current groups $\operatorname{Map}(M,G)$ and diffeomorphism groups $\operatorname{Diff}(M)$, including hydrodynamic and field-theoretic examples.

math-ph↗

arXiv2026

Graph-Series Semantics and Abel Regularization for Recursive Hybrid Quantum Programs

We introduce a graded graph-series semantics for recursive hybrid quantum programs interpreted in the quantum orchestra monad. Finite terminating executions are represented by directed paths whose edges carry normal completely positive subunital maps and whose terminal vertices carry classical results. Path concatenation defines a graded execution category, while continuation grafting models outcome-dependent sequential composition. We construct a semantic evaluation from admissible execution-graph series to quantum orchestras and prove that it is compatible with both channel composition and Kleisli composition. For finitary recursive programs, the truncation of the execution series at degree $n$ is shown to coincide with the $n$-th Kleene approximant of the associated Scott-continuous recursion functional. Consequently, evaluation of the complete graph series recovers the ordinary least-fixed-point denotation. Weighting a graph of degree $n$ by $q^n$, with $0<q<1$, yields an Abel-regularised semantics whose Scott limit as $q\to 1^{-}$ is the unregularised recursive denotation. Equivalently, the parametrisation $q=e^{-t}$ exponentially suppresses long executions and reconstructs the denotation as $t\to 0^{+}$. In a supplementary linear feedback sector, repeated recursion is represented by the execution resolvent $(I-qST)^{-1}$. We identify $I-qST$ with an algebraic cross-ratio of graph subspaces. Under Hilbert--Schmidt assumptions, the associated return operator is trace class and defines the Fredholm feedback determinant $ \operatorname{det}_{F}(I-qST), $ whose zeros detect singular feedback configurations and whose logarithmic expansion records closed loop traversals.

cs.LO↗

arXiv2026

Diffeologies on Locally Convex Spaces and Smooth Multiplication of Distributions

We investigate the canonical and $c^\infty$-diffeologies on Hausdorff locally convex spaces and their applications to Schwartz distributions. We prove that a Hausdorff locally convex space, endowed with its canonical diffeology, is convenient if and only if the canonical map to its internal tangent space at each point is a linear isomorphism. This yields a geometric characterization of Mackey completeness. We also compare several natural diffeologies on locally convex spaces and identify conditions under which they are preserved under completion, dualization, and the formation of inductive limits. As an application, we realize the space of microlocally multipliable distributions as a diffeological colimit and show that H\"ormander-admissible multiplication is smooth. This establishes a framework for nonlinear distribution theory beyond the classical manifold setting.

math.DG↗

arXiv2026

Spectral Cut-off Oscillatory Integrals for Non-Autonomous Hamiltonian Evolution Equations

We develop a spectral cut-off and time-slicing construction for non-autonomous Hamiltonian evolution equations. Let $H_0$ be a positive self-adjoint reference operator with compact resolvent on a Hilbert space $\Hilb$, and let $$ P_N=\mathbf{1}_{[0,N]}(H_0). $$ For a time-dependent family of generally unbounded symmetric Hamiltonians (H(t)), we consider the finite-dimensional cut-off Hamiltonians $$ H_N(t)=P_NH(t)P_N. $$ Their time-sliced propagators admit finite-dimensional state-sum representations and, when suitable configuration-space or phase-space kernels are available, oscillatory integral realizations. We establish commutator conditions implying uniform stability of the cut-off dynamics and construct the full unitary propagator as the strong limit of the finite-dimensional propagators. Additional $H_0$-regularity yields the quantitative estimate $$ \sup{s,t\in I} |U_N(t,s)P_Nu-U(t,s)u| \leq C(1+N)^{-\sigma}|u|{\mu+\sigma}. $$ For Hamiltonians that are H"older continuous of exponent $\alpha$ in time, we also prove the joint spectral and time-slicing bound $$ |U{N,M}(t,s)P_Nu-U(t,s)u| \leq C\left((1+N)^{-\sigma}+(1+N)^\mu M^{-\alpha}\right) |u|_{\mu+\sigma}. $$ The assumptions are verified for time-dependent Schr"odinger operators and symmetric first-order pseudodifferential Hamiltonians. In the periodic case, the construction is compatible with finite-order Floquet--Magnus coefficients for unbounded operators.

math.SP↗

arXiv2026

On spherical Milnor Classifying Spaces I: differential geometry

We develop a geometric framework for generalized Milnor classifying spaces in the setting of diffeological spaces and infinite-dimensional geometry. Starting from Milnor's construction, we introduce spherical and projective models endowed with natural diffeological structures compatible with gluing operations. We then investigate their tangent structures, with particular attention to the behavior at the boundary of simplices and to the distinction between tangential and higher-order normal directions. A natural Riemannian metric is constructed from a barycentric energy functional, leading to a consistent differential calculus on these spaces. This allows us to define differential forms, a Hodge-type theory based on formal adjoints, and a Laplacian without relying on the Hodge star operator. We further introduce Clifford structures and Dirac operators adapted to this framework, and study twisted versions associated with $\mathbb{Z}_2$-local systems. These structures naturally relate to non-abelian extensions and gerbe-type geometries in the sense of infinite-dimensional Lie theory. The results provide a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.

math.DG↗

arXiv2026

Spherical Milnor Spaces II: Projective Quotients and Higher Topological Structures

We introduce a spherical variant of Milnor's classifying construction for diffeological groups, based on quadratic normalization of barycentric coordinates. This construction gives rise to a contractible diffeological space endowed with commuting actions of a group $G$ and of $\mathbb{Z}_2$, leading naturally to a hierarchy of quotient spaces. We investigate the topological and geometric properties of these quotients, including a projective model and a double quotient space which encodes twisted and higher structures. In particular, we show that this framework provides a natural setting for the study of principal bundles with $\mathbb{Z}_2$-twists, and leads to obstruction classes in low-degree cohomology. The construction is further related to non-abelian gerbes and higher bundles, providing a bridge between diffeological geometry, classifying space theory, and higher topological structures.

math.DG↗

arXiv2026

Remarks on pairwise comparisons, transition amplitudes, and qubit states

We discuss a pairwise-comparison viewpoint on finite families of qubit states. Starting from transition amplitudes between pure states, we distinguish three associated levels of comparison data: complex amplitudes, transition probabilities, and phase-valued pairwise comparisons. In the non-orthogonal case, the phase data define a $U(1)$-valued reciprocal pairwise comparison structure. We show that the corresponding triangular defects are naturally related to normalized Bargmann invariants and therefore to geometric phases. This gives a simple interpretation of inconsistency-type quantities in terms of quantum kinematics. We also comment on realizability constraints coming from Gram matrices of rank at most two, and on the passage from unitary phase data to more general transition data. The aim of the paper is mainly conceptual: to isolate a common language between pairwise comparisons and elementary quantum geometry.

math-ph↗

arXiv2026

Pairwise-comparison-valued cosurfaces: a projective framework for multi-scale relational structures

We introduce cosurfaces with values in the group $\PC_n(H)$ of $H$-valued reciprocal pairwise comparison matrices. The composition law is covariant on upper triangular coefficients and contravariant on lower triangular coefficients, which makes $\PC_n(H)$ a natural target for oriented gluing constructions. Starting from a directed family of finite oriented discretizations, we define finite configuration spaces, coarse-graining maps induced by ordered refinements, and the associated universal projective limit. This yields a multi-scale organization of local comparative data in which global objects are reconstructed only through compatibility across scales. In the stochastic setting, projectively compatible probability laws define a cylindrical semantics on the limit space. We also introduce inconsistency observables, interpreted as discrete curvature-type defects measuring obstructions to global coherence. The resulting framework is simultaneously geometric, algebraic, and probabilistic, and suggests a foundational perspective on relational structures built from local comparisons rather than absolute observables.

physics.gen-ph↗

arXiv2026

Noisy Nonreciprocal Pairwise Comparisons: Scale Variation, Noise Calibration, and Admissible Ranking Regions

Pairwise comparisons are widely used in decision analysis, preference modeling, and evaluation problems. In many practical situations, the observed comparison matrix is not reciprocal. This lack of reciprocity is often treated as a defect to be corrected immediately. In this article, we adopt a different point of view: part of the nonreciprocity may reflect a genuine variation in the evaluation scale, while another part is due to random perturbations. We introduce an additive model in which the unknown underlying comparison matrix is consistent but not necessarily reciprocal. The reciprocal component carries the global ranking information, whereas the symmetric component describes possible scale variation. Around this structured matrix, we add a random perturbation and show how to estimate the noise level, assess whether the scale variation remains moderate, and assign probabilities to admissible ranking regions in the sense of strict ranking by pairwise comparisons. We also compare this approach with the brutal projection onto reciprocal matrices, which suppresses all symmetric information at once. The Gaussian perturbation model is used here not because human decisions are exactly Gaussian, but because observed judgment errors often result from the accumulation of many small effects. In such a context, the central limit principle provides a natural heuristic justification for Gaussian noise. This makes it possible to derive explicit estimators and probability assessments while keeping the model interpretable for decision problems.

stat.ML↗

arXiv2026

Effective dynamics and defect expansions for polynomial PDEs on thin annuli

We develop a geometric and analytic framework for polynomial partial differential equations posed on thin annuli in the plane. Using renormalized Sobolev inner products, we construct Sobolev orthogonal polynomial bases adapted to the thin geometry and use them to define stable Galerkin approximations. We prove a general dimension-reduction theorem for polynomial Hamiltonian and dissipative PDEs, showing that solutions converge to effective one-dimensional dynamics on the limiting circle. Beyond the leading-order limit, we identify transverse defect correctors and derive cell problems describing anisotropic dispersive and homogenized effects. Our framework applies uniformly to integrable models (KdV, modified KdV, nonlinear Schr\"odinger, sine--Gordon), anisotropic dispersive systems such as Zakharov--Kuznetsov, and non-integrable perturbations including dissipation, forcing, and rapidly oscillating coefficients. We establish stability of the effective dynamics under changes of Sobolev order and of polynomial Hilbert geometry, and show robustness of the associated Galerkin schemes. The results provide a unified geometric perspective on dimension reduction, homogenization, and integrability in thin geometries, and introduce Sobolev orthogonal polynomial methods as a constructive tool for multiscale PDE analysis.

nlin.SI↗

arXiv2026

A Lorentzian SU(3)-covariant noncommutative KP hierarchy and hypercomplex gauge fields

We propose a formal framework for a noncommutative Kadomtsev--Petviashvili (KP) hierarchy which is covariant under the action of $SU(3)$ and compatible with a Lorentzian structure encoded in a twisted quaternionic (or Clifford) algebra. The starting point is a formal pseudodifferential operator $L$ built from an abstract derivation $D$ of Dirac type and coefficients in an associative algebra $\A$ that combines spin degrees of freedom (twisted quaternions, Clifford algebras) and color degrees of freedom (an internal $SU(3)$ factor, possibly realized via the octonions). In this way we obtain a hierarchy of formal partial differential equations which are Lorentz invariant and $SU(3)$ covariant and can be interpreted as integrable sectors of nonabelian gauge theories in $(3+1)$ dimensions and of their dimensional reductions.

math-ph↗

arXiv2025

Preliminaries on Pre-Hilbert Structures on Polynomial Spaces and Associated Laplacians

We study orthogonal polynomial systems arising from general pre-Hilbert inner products on polynomial spaces, beyond the classical framework of measures. To each such inner product we associate a canonical Laplacian defined from an abstract derivation, and we investigate the operator-theoretic structures induced by this construction. Our main contribution is the introduction of a resolvent-based distance between polynomial Hilbert geometries, and the proof of quantitative stability results for finite-degree orthogonalization procedures. In particular, we show that norm-resolvent closeness of the associated Laplacians implies stability of Gram--Schmidt orthogonal bases, orthogonal projectors and reproducing kernels on all finite-dimensional polynomial subspaces. The general theory is illustrated by several explicit examples. We analyze in detail the case of orthogonal polynomials on the unit circle, comparing classical $L^2$ geometries associated with finite Radon measures and Sobolev-type regularizations via Fourier methods. We also revisit the thin annulus problem, showing that its asymptotic regime admits a natural interpretation as a resolvent limit of polynomial geometries. These results provide a unified operator-theoretic framework for the study of stability, degenerations and geometric limits of orthogonal polynomial systems.

math.GM↗

arXiv2025

On the geometry of profinite diffeological spaces

We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes inductive limits of finite-dimensional manifolds, as well as solution spaces of differential relations and spaces that agree with cylindrical approximations. We analyze tangent and cotangent spaces, differential forms, Riemannian metrics, connections, differential operators, Laplacians, symplectic forms, momentum maps and the relation between de Rham and singular cohomology on them, unifying constructions partially developed in various contexts.

math.DG↗

arXiv2025

Poisson structures on weak Sobolev loop spaces and applications to integrable systems

We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces $W^{s,p}(\mathbb{S^1},\mathbb{R}^m)$ with $o < s \frac{1}{2}$ and $1 < p < \infty.$ Within this setting we construct presymplectic and Poisson structures of hydrodynamic type, as well as their weakly non local deformations involving inverse derivatives. The analytic backbone relies on the boundedness of fractional multipliers, Hilbert transforms, and Lipschitz Nemytski operators on $W^{s,p}$, which ensures that all operations used in Mokhov's formalisn remain well defined at this level of regularity. We further show that teh horizontal-vertical bicomplex underlying variational Poisson geometry can be extended to $W^{s,p},$ so that the cohomological arguments proving skew-symmetry and the Jacobi identity carry over verbatim. As an application, we embed the Hamiltonian formalisms of several integrable PDEs (KdV, nonlinear Schr\"odinger, Camassa--Holm, and hydrodynamic systems of Dubrovin--Novikov type) into this weak Sobolev setting. Local order-one Poisson operators and their weakly nonlocal extensions are shown to be well posed for $W^{s,p}$ loops, while higher-order operators (e.g. the second KdV bracket) require stronger regularity. Our results provide a rigorous analytic foundation for Poisson geometry on weak loop spaces and open the way for extending the Hamiltonian theory of integrable systems beyond the smooth category.

math-ph↗

arXiv2025

Contextuality, Holonomy and Discrete Fiber Bundles in Group-Valued Boltzmann Machines

We propose a geometric extension of restricted Boltzmann machines (RBMs) by allowing weights to take values in abstract groups such as $ \mathrm{GL}_n(\mathbb{R}) $, $ \mathrm{SU}(2) $, or even infinite-dimensional operator groups. This generalization enables the modeling of complex relational structures, including projective transformations, spinor dynamics, and functional symmetries, with direct applications to vision, language, and quantum learning. A central contribution of this work is the introduction of a \emph{contextuality index} based on group-valued holonomies computed along cycles in the RBM graph. This index quantifies the global inconsistency or "curvature" induced by local weights, generalizing classical notions of coherence, consistency, and geometric flatness. We establish links with sheaf-theoretic contextuality, gauge theory, and noncommutative geometry, and provide numerical and diagrammatic examples in both finite and infinite dimensions. This framework opens novel directions in AI, from curvature-aware learning architectures to topological regularization in uncertain or adversarial environments.

cs.LG↗