Axabra

Transcender les Algorithmes, Libérer l'Intelligence

Conseil stratégique en sciences et mathématiques avancées. Nous accompagnons les organisations visionnaires dans leur transformation technologique.

Nos Domaines d'Excellence

CE QUE NOUS FAISONS VRAIMENT

Analyse & Optimisation

Vous nous confiez vos données. Nous explorons leur structure.

Axabra analyse vos données comme un système : invariants, variations, zones stables, zones exploitables. L'objectif n'est pas de prédire à l'aveugle, mais de comprendre ce qui est réellement structurant.

Audit Algorithmique

Est-ce que votre système fait ce qu'il prétend faire ?

Nous auditons vos modèles, algorithmes ou pipelines existants. Pas pour juger, mais pour comprendre : stabilité, sensibilité, zones de fragilité, marges d'amélioration.

Exploration & Recherche

Quand les solutions classiques ne suffisent plus.

Axabra explore les structures cachées de vos problèmes complexes. Là où d'autres cherchent une solution unique, nous analysons l'espace des solutions.

Restitution & Rapport

Pas de boîte noire.

Chaque mission se conclut par un rapport compréhensible, structuré et actionnable. Ce que nous avons observé. Ce qui est stable. Ce qui peut être amélioré.

Discuter de vos données

Axabra opère sous stricte confidentialité.
Notre travail parle par sa structure, pas par son exposition.

Technological Depth
Long-Term Vision

Our expertise is not defined by tools, disciplines, or labels.

It lies in the ability to read complex systems as a whole:
to distinguish structure from noise,
invariants from fluctuations,
and signals from coincidence.

We work where problems resist simplification.
Where optimization alone is not enough.
Where understanding precedes action.

Axabra engages with complexity patiently,
without forcing outcomes,
without compressing reality into premature answers.

We are interested in what remains when models converge,
when solutions repeat,
when variations stop being accidental.

Intelligence, for us, is not about speed.
It is about orientation.

Scientific Excellence
In Service of Ambition

Axabra is not driven by trends, noise, or short-term performance.

It was born from a simple conviction:
when intelligence is treated with rigor, patience, and clarity,
it becomes a force of direction rather than reaction.

We approach artificial intelligence as a field of understanding before a field of application.
Structure matters. Constraints matter.
What remains stable under pressure matters most.

Axabra operates quietly.
We do not showcase clients.
We do not chase visibility.

We build systems meant to endure.

Excellence

Scientific rigor is not a claim. It is a requirement.

Confidentiality

What we explore stays contained.

Long Term

We design for continuity, not momentum.

Contact

Parlons de
Votre Vision

Une question, un projet, une ambition ? Notre équipe est à votre écoute pour explorer ensemble les possibilités.

Notre source d'inspiration

Research — Combinatorial Optimization, Differential Geometry, Arithmetic Geometry & Applied Mathematics

On the Structure of Optimal Tours in the Euclidean Traveling Salesman Problem

— Axabra, Florence Project —

Empirical study of the structure of optimal solutions for the Euclidean Traveling Salesman Problem (TSP) on the TSPLIB benchmark instance xqf131 (131 cities, EUC_2D metric, optimal cost 564). Through 50,000 independent runs using Florence, an experimental structural analysis tool, we identified exactly 14 distinct optimal tours sharing a massive invariant backbone of 116 edges (88.5% of the tour). Variations are confined to 7 independent local permutation modules. We introduce the concept of Meghezzien — the geometric and combinatorial superposition of all optimal tours — as the maximal representation of optimality for a given instance.

  • Traveling Salesman Problem (TSP)
  • xqf131 — TSPLIB benchmark
  • Structural degeneracy
  • Optimal tour backbone
  • Combinatorial optimization
  • Meghezzien decomposition
  • Permutation modules
  • Solution space analysis
  • Florence — structural exploration tool
  • Euclidean TSP
  • NP-hard problems
  • Basin theory
  • Spectral analysis of tours
  • Concorde TSP solver verification

Geometric Hydrodynamics on Non-Metric Weyl Manifolds: Variational Derivation and Well-Posedness

— Axabra —

A geometrically rigorous framework for fluid dynamics on manifolds equipped with non-metric affine connections of Weyl type. Starting from a variational principle on a 3-dimensional Riemannian manifold (M, g) with a pure-trace Weyl connection, we derive modified Navier-Stokes equations — the Weyl affine hydrodynamic equations — that include geometric correction terms arising from non-metricity. We establish the divergence theorem for Weyl connections, compute the Ricci tensor with correct coefficients, prove local existence and uniqueness of smooth solutions in Sobolev spaces, and derive explicitly all geometric correction terms. The framework reduces to classical incompressible Navier-Stokes in the limit where the Weyl 1-form vanishes.

  • Navier-Stokes equations
  • Modified Navier-Stokes
  • Weyl geometry
  • Weyl connections
  • Non-metric affine geometry
  • Riemannian manifolds
  • Differential geometry
  • Variational calculus
  • Action principle
  • Euler-Lagrange equations
  • Ricci tensor
  • Christoffel symbols
  • Curvature tensor
  • Riemann curvature
  • Affine Laplacian
  • Laplace-Beltrami operator
  • Divergence theorem
  • Integration by parts on manifolds
  • Sobolev spaces
  • Well-posedness
  • Picard iteration
  • Energy estimates
  • Quasilinear parabolic PDE
  • Conformal geometry
  • Torsion-free connections
  • Non-metricity tensor
  • Levi-Civita connection
  • Strain tensor
  • Fluid mechanics on curved spaces
  • Geometric hydrodynamics
  • Turbulence modeling
  • Geometric Reynolds stresses
  • Volume form
  • Lagrangian-Eulerian correspondence
  • Weyl 1-form
  • Incompressible flow
  • Viscous dissipation

A Physics-Based Framework for Lightning Damage in Painted Composite Structures: From Arc Confinement to Mechanical Retention

— Axabra —

A unified physics-based framework explaining enhanced lightning damage on painted composite aircraft structures (CFRP). The analysis links arc attachment geometry, in-plane current redistribution, transient thermal response, gas generation, pressure build-up, and mechanical retention through a chain of causality. Geometric confinement of the arc root by insulating paint layers is identified as a key amplification mechanism. Reduced-order electrical, thermal, and fluid-mechanical models yield design-oriented scaling laws, a pyrolysis-based damage depth criterion, and a pressure-controlled delamination criterion, translated into a residual strength formulation and practical design envelope.

  • Lightning strike protection (LSP)
  • CFRP composite structures
  • Carbon-fiber reinforced polymer
  • Aerospace engineering
  • Arc confinement
  • Joule heating
  • Elliptic PDE
  • Sheet conduction model
  • Heat equation
  • Transient thermal conduction
  • Thermal diffusion
  • Pyrolysis criterion
  • Matrix degradation
  • Delamination
  • Darcy flow
  • Pressure evolution
  • Scaling laws
  • Dimensional analysis
  • Boundary value problems
  • Dirichlet conditions
  • Residual strength
  • Design envelope
  • Reduced-order models
  • Current conservation
  • Expanded copper foil (ECF)
  • Interlaminar fracture

On the parity bridge between algebraic and root numbers for elliptic curves of rank one

— Axabra —

A precise and self-contained exposition of the arithmetic-analytic bridge underlying the Birch-Swinnerton-Dyer conjecture for elliptic curves of algebraic rank one. The parity of the algebraic rank is transferred via the p-adic Selmer group to the global root number. Under standard hypotheses — including finiteness of the p-primary part of the Tate-Shafarevich group — this yields a conditional proof that elliptic curves of rank one have root number -1, forcing the vanishing of L(E,1) by the functional equation. Numerical illustration on the elliptic curve 37a1.

  • Elliptic curves
  • Birch-Swinnerton-Dyer conjecture
  • L-functions
  • Selmer groups
  • p-adic Selmer group
  • Root numbers
  • Parity conjecture
  • Tate-Shafarevich group
  • Mordell-Weil rank
  • Functional equation
  • Algebraic number theory
  • Arithmetic geometry
  • Iwasawa theory
  • Bloch-Kato conditions
  • Weierstrass equations
  • Galois representations
  • Exact sequences
  • Gross-Zagier theorem
  • Kolyvagin Euler systems

Structural limitations of non-singular fractional operators with bounded memory

— Axabra —

A minimal axiomatic framework for bounded-memory fractional operators based on causality, Volterra structure, passivity, and non-oscillatory relaxation. The non-evasion principle establishes that any admissible bounded-memory operator necessarily reduces to a superposition of purely relaxing modes with a completely monotone kernel admitting a Bernstein-Stieltjes representation. No oscillatory, resonant, or energy-storing memory effects are compatible with the admissibility axioms. Implications for Navier-Stokes-type models, dissipative PDEs, viscoelasticity, and anomalous diffusion.

  • Fractional calculus
  • Fractional derivatives
  • Volterra operators
  • Memory kernels
  • Non-singular kernels
  • Bounded memory
  • Non-evasion principle
  • Complete monotonicity
  • Bernstein-Stieltjes representation
  • Bernstein functions
  • Laplace transform
  • Spectral measures
  • Functional analysis
  • Banach spaces
  • Operator theory
  • Caputo derivative
  • Riemann-Liouville operators
  • Dissipative systems
  • Dissipative PDE
  • Viscoelasticity
  • Anomalous diffusion
  • Relaxation modes
  • Passivity
  • Causality

The Ricci Tensor of Torsion-Free Weyl Connections in Dimension Two

— Axabra —

Complete derivation of the Ricci tensor for torsion-free Weyl connections on Riemannian surfaces (dimension two). The symmetric part of the Ricci tensor collapses identically to pure trace due to the (n−2) prefactor in the quadratic Weyl contribution, eliminating all local anisotropic information at the differential level. Only a scalar curvature channel and a global non-metricity energy survive. Five structural obstructions are identified: symmetric Ricci collapse, conformal exactness of the scalar channel, modified Gauss-Bonnet rigidity, and a gauge-invariant energy obstruction.

  • Ricci tensor
  • Torsion-free Weyl connections
  • Dimension two
  • Riemannian surfaces
  • Pure-trace collapse
  • Conformal geometry
  • Non-metricity tensor
  • Weyl covector
  • Curvature decomposition
  • Symmetric-antisymmetric splitting
  • Levi-Civita connection
  • Christoffel symbols
  • Difference tensor
  • Riemann curvature tensor
  • Scalar curvature
  • Gauss-Bonnet theorem
  • Euler characteristic
  • Exterior derivative
  • Differential 2-forms
  • Non-metricity energy
  • Dimensional rigidity
  • Trace decomposition
  • Index contraction
  • Einstein summation convention

Structural Obstructions in Two-Dimensional Torsion-Free Weyl Geometry

— Axabra —

Identifies five structural obstructions in two-dimensional torsion-free Weyl geometry: (1) symmetric Ricci collapse to pure trace, (2) conformal exactness of the scalar curvature channel, (3) Gauss-Bonnet rigidity forcing a topological constraint on the Weyl energy, (4) a de Rham cohomological obstruction distinguishing exact from non-exact Weyl structures, and (5) a gauge-invariant energy obstruction detecting non-trivial non-metricity globally. Dimension two is established as a regime of maximal geometric rigidity where all local affine freedom collapses.

  • Weyl geometry
  • Structural obstructions
  • Torsion-free connections
  • Dimension two rigidity
  • Symmetric Ricci collapse
  • Conformal exactness
  • Gauss-Bonnet rigidity
  • de Rham cohomology
  • Gauge invariance
  • Gauge transformations
  • Euler characteristic
  • Obstruction theory
  • Weyl energy functional
  • Non-metricity
  • Differential forms
  • Closed 2-forms
  • Exact vs non-exact 1-forms
  • Global invariants
  • Conformal classes
  • Trace-free tensors
  • Topological constraints
  • Riemannian surfaces
  • Affine geometry

Torsion-Free Weyl Connections in Dimension n ≥ 3: Explicit Anisotropy, Canonical Ricci Decomposition, and Global Energy Functionals

— Axabra —

Complete and explicit derivation of the Ricci tensor for torsion-free Weyl connections in dimension n ≥ 3. The linear contribution is purely antisymmetric and given by n(dQ), while the quadratic contribution is purely symmetric and proportional to (n−2)(QμQν − |Q|2gμν). This structural decomposition identifies dimension two as a rigid threshold where all symmetric anisotropic contributions vanish, whereas for n ≥ 3 genuine local anisotropy survives. Canonical global energy functionals measuring non-metricity and antisymmetric curvature are introduced. The scalar curvature shift is negative definite and proportional to (n−1)(n−2)|Q|2.

  • Weyl geometry
  • Torsion-free connections
  • Ricci tensor decomposition
  • Non-metricity
  • Curvature decomposition
  • Dimensional rigidity
  • Riemannian manifolds
  • Levi-Civita connection
  • Christoffel symbols
  • Difference tensor
  • Riemann curvature tensor
  • Exterior derivative
  • Weyl covector
  • Scalar curvature shift
  • Global energy functionals
  • Non-metricity energy
  • Antisymmetric Ricci curvature
  • Integral identities
  • Divergence theorem on manifolds
  • Tensor algebra
  • Index contraction
  • Symmetric-antisymmetric sectors
  • Anisotropy persistence
  • Conformal geometry
  • Trace-free Ricci tensor
  • Quadratic curvature invariants

Weyl Connections on Surfaces: Complete Curvature Expansion

— Axabra —

Complete line-by-line derivation of all curvature tensors (Christoffel symbols, Riemann tensor, Ricci tensor, scalar curvature) associated with torsion-free Weyl connections on Riemannian surfaces. Establishes the structural collapse of the symmetric Ricci tensor to pure-trace form in dimension two, derives the modified Weyl–Gauss–Bonnet formula linking the integrated Weyl scalar curvature to the Euler characteristic minus the Weyl energy, and proves a global dichotomy: exact Weyl structures are conformally trivial, while non-exact structures are obstructed by a strictly positive energy functional.

  • Weyl connections on surfaces
  • Curvature expansion
  • Christoffel symbols
  • Riemann curvature tensor
  • Ricci tensor
  • Scalar curvature
  • Torsion-free connections
  • Non-metricity
  • Weyl covector
  • Weyl–Gauss–Bonnet formula
  • Euler characteristic
  • Weyl energy functional
  • Conformal reduction
  • Conformal rescaling
  • Exact vs non-exact Weyl structures
  • Global obstruction
  • Pure-trace collapse
  • Riemannian surfaces
  • Connection shift identity
  • Quadratic curvature terms
  • Derivative terms
  • Covariant derivatives
  • Diffeomorphism invariance
  • Orientation-preserving maps
  • Symplectic forms (outlook)
  • Complex structures (outlook)

Axabra operates under strict confidentiality. Research findings are shared selectively. For inquiries regarding combinatorial optimization, differential geometry, Weyl geometry, arithmetic geometry, fractional operator theory, aerospace engineering, or the Florence project, contact us.