Abstract: Sobolev spaces and their implications for nonlinear nonlocal Dirichlet problems governed by the fractional g−Laplacian. The analysis begins with a detailed investigation of the underlying structure of the functions N− and Orlicz functions, which constitute the functional setting for these spaces. Fundamental features such as completeness, separability, reflexivity, and their limiting behavior as s ↑ 1 are rigorously addressed. Within this setting, a fractional counterpart of the compact Rellich-Kondrachov embedding theorem is established. As a principal application, the existence and uniqueness of weak solutions to a non-linear Dirichlet problem are obtained through a variational approach, relying on monotonicity methods and the Minty–Browder framework. The results highlight the role of fractional Orlicz–Sobolev spaces in extending the functional analytic tools required for the treatment of non-local differential models.

Abstract: Any inference system satisfying the TEAG axioms must obey a tropical Hamilton–Jacobi equation in the max-plus semiring, and the Epistemic Support-Point Filter (ESPF) is that solution. This paper proves the necessity and derives the complete dynamical and causal geometry that follows. The necessity has three steps, each forced. Popperian contraction requires that evidence can only increase impossibility, never decrease it: under the log-admissibility transformation this forces the max-plus operation. The evidence-referencing axiom requires that survivor selection depends only on innovation geometry, not on prior structure: this forces the Hamiltonian to be momentum-independent. A momentum-independent tropical Hamiltonian forces the Lax–Oleinik reduction to a pointwise max. The result is unique: no alternative update structure consistent with these axioms exists. The ESPF is not one admissible filter among many — it is the unique admissible inference dynamics under TEAG. Two scalar fields live on hypothesis space. The impossibility field \( Phi_\varnothing = -\log\pi \) encodes accumulated epistemic history: zero where a hypothesis enjoys full prior support, growing without bound as evidence withdraws that support. The surprisal field \( \Phi_S(h) = \tfrac{1}{2}\|L_e^{-1}(y - g(h))\|^2 \) encodes the tension between each hypothesis and the current observation in MVEE-whitened measurement space. The conjunctive (Popperian) update produces the posterior impossibility field as their pointwise max-plus upper envelope: \( \widetilde{\Phi}_\varnothing = \Phi_\varnothing \oplus \Phi_S = \max(\Phi_\varnothing, \Phi_S). \) This equality follows from \( -\log\min(a,b) = \max(-\log a, -\log b): \) it is an algebraic identity, not a modeling choice or an analogy. The active deformation front — the tropical variety of this two-term polynomial, where both fields achieve the maximum simultaneously — is the exact locus where evidence begins to deform the posterior impossibility field. It is a necessary condition for falsification. Sufficient falsification requires exit from the PRCB-admissible basin, whose threshold is determined by the PRCB at each step. A scalar example derives every quantity in closed form, making the front geometry and basin structure visible without probabilistic machinery. The ESPF predict–update recursion is the Lax–Oleinik operator of max-plus optimal control: the one-step solution operator of the tropical Hamilton–Jacobi equation, with the surprisal field as Hamiltonian and the Possibilistic Cramer–Rao Bound (PCRB) as the minimum action per update cycle. This structure is not chosen: Proposition 5.2 proves it is forced by the TEAG axioms — specifically, Popperian contraction forces the max-plus operation, and the evidence-referencing condition forces momentum independence of the Hamiltonian. No alternative update structure consistent with these axioms exists. Falsification is wavefront propagation: the surprisal field radiates outward from each observation, and a hypothesis enters the active deformation front at the moment the surprisal wavefront overtakes the prior impossibility field. The active deformation front is the epistemic Lagrange point — the locus of exact balance between prior epistemic history and current evidence tension — and is a necessary condition for falsification. Sufficient falsification occurs when the wavefront has pushed a hypothesis outside the PRCB-admissible basin: the isotropic equipotential region whose threshold is governed by the PRCB at each step. The term \emph{wavefront} denotes level-set evolution under max-plus dynamics; no physical medium is assumed. The gravitational language is structural: it reflects equivalence of governing equations, not shared physical ontology. The PCRB emerges as a minimum action principle: no measurement can compress the surviving well below the PCRB floor per update. The zero-temperature limit of the classical Hamilton–Jacobi equation — passing from log-sum-exp (probabilistic) to max (possibilistic) aggregation — recovers this framework exactly, making precise the passage from Bayesian to possibilistic inference as a thermodynamic degeneration. The whitened minimax medioid is proved to be the geodesic attractor of the surviving well: the unique support point nearest the center of the PCRB-defined epistemic geoid in the MVEE-whitened metric. The correct geometric primitive of epistemic phase space is identified as a contact manifold rather than a symplectic one: the irreversibility of Popperian falsification forces a contact structure, the PCRB is a contact energy floor, and the ESPF implements a contact Hamiltonian system with discrete projection onto the admissible basin. These results constitute the dynamical foundation of the Theory of Epistemic Abductive Geometry (TEAG) (Jah, 2026b).

Abstract: The Gielis superformula is a powerful parametric tool that generates an infinite variety of natural and organic curves and surfaces through a compact set of parameters. However, classical differential geometry has lacked a unified framework for analyzing their curvature, torsion, and intrinsic geometric properties. This study addresses this gap by developing a novel superelliptic geometric framework that integrates the superformula 6with the differential geometry of curves and surfaces. We define the superelliptic inner and cross products, the star derivative, and the superelliptic Frenet frame to extend Euclidean and Riemannian interpretations of curvature and torsion to a more flexible parametric structure. The framework provides a uniform geometric characterization of all Gielis curves and surfaces, independent of their classical parametric expressions; even singular cases are regularized so that their curvature and torsion reduce exactly to those of a circle. This unifies the entire family under a common, robust foundation while preserving orthonormality and differentiability. This superelliptic approach offers a consistent and computationally 14tractable model that bridges mathematical abstraction with real-world morphology, with 15the superformula serving as a representative example of the framework’s broad generality for diverse geometric structures.

Abstract: We investigate the geometrical structure underlying the notion of Inferential Scattering, which was formulated by E. T. Jaynes in the 1980s using the language of equilibrium statistical mechanics. We show that inferential scattering can be naturally defined on a dually flat Riemannian manifold equipped with dual coordinate systems, a differential- geometric structure that occupies a central place in information geometry. We find that the evolution of the system on the dually flat manifold can be expressed as the horizontal lift of an integrable connection. We stress that the notion of inferential scattering has a wide range of applications, being a form of inference and therefore applicable to any statistical system with insufficient information.

Abstract: This paper introduces the Morphological Participation Index (MPI), a substrate-agnostic framework for estimating whether a system’s morphology can plausibly support strongly integrated, coherence-sensitive, trace-rich, and temporally scaffolded dynamics. “Participation” refers to the degree to which morphology actively contributes to, constrains, and scaffolds the integrated, trace-bearing, and temporally organized dynamics available to a system. The immediate motivation comes from two adjoining lines of work: spectral approaches to resistance to decomposition, and recent proposals by Schneider and Bailey concerning prototime, quantum Darwinist stabilization, and the selective emergence of conscious basins [2,16–18]. MPI evaluates the structural conditions under which a system might sustain unified dynamics, stable internal traces, and organized temporal regimes, without presupposing a human, cortical, or even purely biological baseline. Formally, MPI represents morphology as a weighted constraint hypergraph [4,24], or as an explicit multilayer family of such hypergraphs [11], and returns a score bundle rather than a single undifferentiated scalar. The core bundle consists of six components: integration geometry, multiscale nesting, resonant-mode support, trace geometry, temporal scaffolding, and robustness. An optional contextual patchiness module is provided for domains in which a defensible predicate family is available. The integration component is anchored in a balanced-cut spectral formalism: it uses sweep cuts over the Fiedler vector of the normalized Laplacian rather than raw minimum-cut objectives or simple sign cuts, thereby avoiding familiar degeneracies and linking MPI directly to contemporary spectral proxies for resistance to informational decomposition [6,19,23]. The principal contribution of MPI is a structural profile: seam maps, multiscale partitions, trace-capacity maps, temporal breadth measures, and perturbation-stability diagnostics, in a form that remains useful across biological, artificial, collective, and other nonstandard architectures. More generally, the same diagnostics may be useful in AI alignment. Seam topology, trace geometry, and temporal scaffolding provide a way to screen for architectures that may be difficult to audit, prone to distributed lock-in, or vulnerable to hidden coordination through narrow bottlenecks or persistent externalized traces. MPI can also serve as a screening tool for artificial systems whose structural profile merits closer safety and oversight attention.

Abstract: The Quantum Darwinist Theory of Consciousness (QDT) and the Prototime Interpretation (PT) characterize localized conscious basins in terms of spectral integration, PT-participation, recursive coherence, witness redundancy, and temporally ordered record formation [7–9]. An upstream question has remained largely implicit in that program: what sort of morphology makes such dynamics structurally plausible? This paper argues that morphology constitutes the architectural precondition of basin formation, and that the Morphological Participation Index (MPI; Montes 5) can make that precondition operational. Architecture, realized integration, carrier structure, and witnessed temporality each answer a different question about the same candidate system. MPI contributes the architectural prior: it localizes where balanced seams lie, where redundant trace or witness surfaces are available, where carrier-sensitive assays are worth running, and where temporally thick, record-supported basins—integrated regimes whose stability depends on redundant internal records or traces that persist across behaviorally relevant time windows—are plausible. A structural factorization of candidate basins connects MPI’s score bundle to downstream Φ_s, PT-participation, and clock indices. Expected dissociations—cases where high MPI coexists with low realized integration, or where trace-rich architectures lack the carrier geometry for PT-participation—sharpen experimental design and help distinguish genuine basin formation from structural mimics. The result is a bridge from morphology to the empirical core of QDT/PT, grounded in the same balanced-cut spectral formalism that underlies Φ_s itself.

Abstract: Starting from the representation of De Moivre’s matrix formula combined with the Kronecker product, we derive a 4 × 4 matrix M_θ that encodes algebraic properties with topological implications in four dimensions. This matrix describes trans- formations associated with symmetry and antisymmetry, encompassing distortion, dilation/contraction, and combined shear in a four-dimensional framework. We de- compose M_θ into its symmetric (S) and antisymmetric (A_anti ) components, a result that characterizes key features of the Alpha Group, including the formation of a tan- gent plane at infinity and its corresponding topological and geometric consequences. To investigate the system’s dynamics, we perform Monte Carlo simulations for varying values of the parameter θ, revealing attractor behavior and geometric transitions between Euclidean and hyperbolic regimes in projected three-dimensional trajectories.

Abstract: We provide an intrinsic construction of the central conics in the real hyperbolic plane H2 whereby each conic C is the composition of a unique pair of Steiner conics (those generated by collineations). The composition is achieved by el- liptic curve addition on intersection points of the two components with their orthogonal trajectories, which have a natural representation as genus 1 curves in any inversive model of H2. The central Steiner conics that have a focal axis L are identified with the subgroup G(L) of collineations generated by reections in the lines perpendicular to L. We define the fiber over g 2 G(L) to be the set of compositions C such that Pi (C) = g. Here, Pi (C) is the unique Steiner conic tangent to C at the points on L, and we show that Pi (C) is the product of the two elements in G(L) that represent the components of C. The central conics are partitioned into these fibers, which are acted upon transitively by G(L). The geometry and algebra of the fiber bundle are emphasized, without topological considerations.

Abstract: In this paper, we introduce a new set-theoretic operator $(\cdot)^{\sharp}_{\omega}$ in the framework of ideal topological spaces and investigate its fundamental properties, including its connections with the classical $\sharp$-operator and the $\omega$-local function. Using this operator, we define a closure-type operator $\mathrm{Cl}^{\sharp}_{\omega}$ and show that it satisfies the Kuratowski closure axioms. Consequently, a topology $\mathcal{T}^{\sharp}_{\omega}$ is obtained, which is strictly finer than the topology induced by the $\sharp$-operator. Furthermore, the structural relationships among these topologies are examined, and some applications of the $\omega^\sharp$-operator are presented. Finally, we introduce the notions of $\omega^\ast$-continuity and $\omega^\sharp$-continuity, investigate their relationship, and establish a new decomposition of continuity. We also compare these notions with related concepts such as $\ast$-continuity and $\sharp$-continuity.

Abstract: We study real-valued transition profiles on the real axis that admit holomorphic extension to a horizontal strip in the complex plane. The functions considered have a continuously differentiable and nondecreasing real trace and are normalized to take values strictly between zero and one. We assume that the associated conformal transform obtained by rescaling and shifting the profile extends holomorphically to the strip and maps it into the unit disk. Under these conditions strip analyticity imposes a sharp pointwise bound on the rate of change along the real axis. The bound depends only on the width of the analytic strip and is optimal. We further prove a rigidity result: if the bound is attained at any real point then the profile is uniquely determined up to translation and coincides with the logistic transition. The argument is purely analytic and follows from the Schwarz–Pick contraction principle applied to the strip geometry. No classification of non-saturating profiles is attempted.

Abstract: We propose a structural framework for organizing the submanifold content of compact Calabi--Yau manifolds through the notion of a {Topological Slice Structure} (TSS), a coherent collection of calibrated submanifolds compatible with the Ricci-flat metric data. The central result is a decomposition principle asserting that, under mild conditions on the K\"ahler polarization, such a structure exists, its cohomology classes span the full integer homology, and it is covariant with respect to mirror symmetry. Special cases recover special Lagrangian torus fibrations, divisors, and holomorphic curves as natural constituents of a unified geometric datum. We illustrate the framework through worked examples, introduce a numerical slice complexity invariant, and discuss implications for D-brane wrapping and moduli stabilization in string compactifications.

Abstract: Unlike geometry, spheres in topology have been seen as topological invariants, where their structures are defined as topological spaces. Forgetting the exact notion of geometry, and the impossibility of embedding one into another, homotopy theory relates how a sphere of one dimension can wrap around, or map into, a sphere of another dimension. This paper revisits the classical theory of homotopy groups of spheres, providing a detailed exploration of their computation and structure. We place special emphasis on the pivotal role of Hopf fibrations in revealing the higher homotopy groups of spheres, particularly the exotic and fascinating case of π₃(S²). Furthermore, we explore the elegant geometric connection to Villarceau circles, demonstrating how these circles on a torus are intimately linked to the Hopf fibration of S³. This work serves as a comprehensive guide, bridging abstract algebraic topology with tangible geometric phenomena. This version expands significantly on the foundational ideas, providing deeper insights and connections to contemporary research, including stable homotopy theory, the Adams conjecture, and generalizations to Calabi-Yau manifolds. (An earlier version of this work was published as: D. Bhattacharjee, S. Singha Roy, R. Sadhu, “Homotopy group of spheres, Hopf fibrations and Villarceau c ircles”, EPRA International Journal of Research & Development (IJRD), Vol. 7, Issue 9, September 2022. DOI: https://doi.org/10.36713/epra11212)

Abstract: This study investigates the dripping and leakage problem in kitchenware known as the "teapot effect" through a multidisciplinary experimental approach encompassing fluid mechanics, material science, and ergonomic design. Unlike previous studies confined to idealized geometries and single-fluid analyses, this work systematically examines 32 distinct spout geometries from commercially available teapots, coffee pots, and milk jugs under realistic operating conditions. Experiments were performed using four fluids with contrasting rheological properties water, boiling black tea, cow's milk, and Turkish coffee on a precision rotating platform operating at 1°/s to isolate surface tension, gravitational, and geometric effects from inertial forces. Three quantitative parameters were measured for each specimen: capillary dome angle, teapot effect angle range, and optimum pouring angle. Results demonstrate that spout tip geometry is the dominant controlling parameter. Thin-lipped elliptical cross-sections effectively suppressed dripping, whereas triangular and wide curved geometries produced the teapot effect across broad pouring angle ranges reaching up to 70°. A spout outlet extension length of 4–5 mm combined with a spout tip radius below 4 mm was found necessary and sufficient for clean flow separation. Furthermore, suspended particles and proteins in milk and Turkish coffee were shown to intensify the teapot effect by disrupting contact line dynamics at the spout tip. These findings provide quantitative design thresholds directly applicable to industrial kitchenware development.

Abstract: The aim of the article is to introduce a few variants of generalized quasi-continuity of multifunctions defined on a bitopological space and to study their mutual relationship. The results known for functions are extended to multifunctions which provide a wider range of relationships, mainly in terms of upper and lower semi continuities and corresponding continuities with respect to a dual bitopology. The proof procedures are based on a notion of pseudo refinement of two topologies and the Baire property in a bitopological space. A characterization of some continuities depending on two topologies by continuities depending only on one topology and the structure of the sets of semi discontinuity points are given. The end of the article is dedicated to several interpretations that facilitate and clarify orientation in the achieved results.

Abstract: This paper investigates cylindrical sphere packings, that is, patterns of uniform spheres with mutually disjoint interiors which are all tangent to a common cylinder. The key unifying themes are existence and uniqueness of hexagonal packings, in which each sphere is tangent to six others. Constructions are both intuitive and subtle, but result in the complete characterization in term of integer parameter pairs $(m,n)$. Interesting questions in rigidity and density are encountered. Density questions arise because the packings, being of equal diameter, lie within the space between inner and outer cylinders. This density problem hoovers between the 2D and 3D sphere packing cases, and though it is not solved here, it is conjectured that the hexagonal packings are densest for the countable number of cylinders which support them. Other geometric objects are along for the ride, including equilateral triangles and the packings' dual graphs, which are associated with patterns of carbon atoms forming buckytubes. Interesting structural rigidity questions also arise.

Abstract: We study involutions on complex K3 surfaces and their quotients, focusing on the emergence of Enriques and Kummer surfaces. Emphasis is placed on lattice-theoretic structures, geometric invariants, and projective realizations via the Kodaira embedding theorem. Examples are provided to illustrate the relationships among these surface classes and their induced geometric properties.

Abstract: This work investigates a mechanism for topology-altering geometric behavior induced by angular group actions within the framework of the Alpha Group, demonstrating that a systematic angular sweep can drive spontaneous transitions between Riemannian-like and sub-Riemannian regimes. Motivated by the group-theoretic foundations of geometry established by the Göttingen school and by the central role of invariants in topology, we introduce a setting in which geometric organization emerges from the interplay between an invariant, idempotent operator µ and an angular matrix M(θ ). While µ provides a stable structural background, the angular action associated with M(θ ) induces anisotropy and directional dependence. A systematic angular sweep is performed to probe the geometric response of the in- duced structure, revealing the spontaneous emergence of distinct isotropic, moderate, and strong regimes. These regimes are not imposed a priori, but arise dynamically from the interaction between the invariant background and the angular action. As the angular pa- rameter departs from stable configurations, anisotropic effects activate a sub-Riemannian Carnot–Carathéodory geometric framework, leading to a qualitative reorganization of the underlying topology. The strong regime is characterized by localized and persistent geometric features, ac- tivated only within bounded angular intervals and producing symmetric bifurcation pat- terns around θ = 90◦ . Despite these anisotropic reorganizations, global coherence and connectivity are preserved by the invariant action of µ. These results demonstrate that sub-Riemannian geometry can act as a natural driver of dynamic topological restructur- ing within group-based geometric frameworks, providing a coherent alternative to classical Riemannian descriptions.

Abstract: We investigate a potential route to the Riemann Hypothesis based on de Branges positivity and wild isomonodromic geometry, focusing on Painlevé III of type D6. Rather than proposing a proof, we reduce any such route to four explicit conditions (C1)–(C4), isolating a single analytic bottleneck: the existence of a global positivity normalization for the associated wild Riemann–Hilbert problem. Using the decorated character variety framework of Chekhov–Mazzocco–Rubtsov and the embedding t = s(1 − s), we show that symmetry, gauge freedom, and growth constraints of the completed zeta function are all compatible with this setting. We further perform a quantitative density test based on the Weyl–Levinson law for canonical systems, showing that the zeta-induced spectral growth is highly selective yet not excluded by the Painlevé III_D6 Hamiltonian. The result is a falsifiable and discriminating framework that identifies where a de Branges-based realization of the Riemann Hypothesis must succeed or fail. We further analyze the analytic regularity condition (C4), show that the symmetry-compatibility condition (C3) is automatically satisfied for the natural embedding t = s(1− s), and isolate the global positivity condition (C2) as the decisive remaining analytic obstacle. In particular, we reduce (C2) to the absence of a single explicit Weyl–Herglotz obstruction for the associated canonical system, and develop falsifiable diagnostics, including a quantitative density test based on Weyl–Levinson asymptotics.

Abstract:

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are \( \textit{gauge structures on these vector bundles} \). We are interested in dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e. properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection affine connection? (P2-Riemannian Geometry): When is a Koszul connection metric connection? (P3-Fedosov Geometry): When is a Koszul connection symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of \( \textit{how to produce labeled foliations the most studied of which are Riemannian foliations} \). On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemeted to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce \( \textit{Koszul Homological Series}\textit{Koszul Homological Series} \). This notion is a machine for converting Obstructions whose nature is vector space into Obstructions whose nature is Homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2) and (P3). \( \textit{In the abundant literature on Riemannian foliatins we have only cited references directely related to the open problems which are studied using the tools which are introduced in this work. Thus the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How to produce Riemanian foliations?} \). See our Theorem 7.4 and Theorem 7.5 which are fruits of a happy conjunction between the gauge geometry and the differential topology.

Abstract:

We study a purely local inverse problem for non-reversible Randers metrics \( F = \|\cdot\|_g + \beta \) defined on smooth oriented surfaces. Using only the lengths of sufficiently small closed curves around a point \( p \), we prove that the exterior derivative \( d\beta(p) \) can be uniquely and stably recovered. Moreover, we establish that \( d\beta(p) \) is the only second-order local invariant retrievable from such local length measurements. Our approach is entirely metric-based, independent of geodesic flows or boundary data, and naturally extends to general curved surfaces.