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On the Structure of Optimal Tours in the Euclidean Traveling Salesman Problem
Axel Abdel Rahamane Meghezzi
— 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
Axel Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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
Axel Abdel Rahamane Meghezzi
— 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.
