Computer Science and Mathematics

Abstract: Defect data attached to a morphism admit two orthogonal refinements. The first is factorization sensitive: a factorization \( A\xrightarrow{u}B\xrightarrow{v}A^\dagger \) with vu = R, gives a residual triple (Cone(u), Cone(v), Cone(R)) constrained by the octahedral axiom. The second is coefficient-sensitive: an excess object is the cone of a comparison morphism between integral or finite-coefficient residual data and its rational or mixed-Hodge realization. We organize these two directions into a residual–excess matrix. The main residual witness is a finite ordinary-double-point conifold degeneration. Saito divisor gluing produces a node-supported Hodge interface \( W^H_\Sigma=\bigoplus_{p_k\in\Sigma}i_{k*}\mathbb Q^H_{\{p_k\}}(-1) \) factoring finite-node monodromy through \( \psi^H_{\pi,1}\to W^H_\Sigma\to\psi^H_{\pi,1}(-1) \). The variation cone yields the corrected extension \( 0\to IC^H_{X_0}\to P^H_{\mathrm{var},\Sigma}\to W^H_\Sigma\to0 \), whose class decomposes into nodewise Ext residual classes. Ordinary double points calibrate the zero-excess regime because their Milnor fibers are integrally torsion-free. Diaz’s Enriques-product Bockstein mechanism calibrates the nonzero-excess regime through integral torsion killed by rationalization. The resulting framework compares finite-node Saito gluing with integral Hodge obstruction channels without conflating rational residual data with coefficient-change defects.

Abstract: Almost paracontact almost paracomplex manifolds equipped with a pair of structure-compatible metrics, one Riemannian and the other pseudo-Riemannian, are considered. The difference tensor between the two corresponding Levi-Civita connections is studied within the framework of the known Manev–Staikova classification of these manifolds. The obtained characterizations of the basic classes in terms of the difference tensor are illustrated by means of Lie groups of the lowest dimension.

Abstract: We initiate the study of solitons for the Gauss curvature flow on surfaces immersed in three-dimensional Lorentzian Walker manifolds. A soliton is a surface whose shape is preserved along the flow, evolving purely by the ambient isometries. Working with both the extrinsic and intrinsic Gauss curvature, we compute the fundamental forms of the relevant families of invariant surfaces and reduce the soliton condition to an ordinary differential equation on the generating curve. Our main contributions are threefold. First, we prove a rigidity theorem: a surface invariant under a one-parameter group of isometries is a soliton for that same group if and only if it is flat. Second, in the strictly Walker case we show that the only solitons with respect to the canonical parallel null field are the coordinate planes. Third, for the remaining Killing fields we classify the solitons via a phase-plane analysis of the associated ODE systems and describe the qualitative geometry of the solution surfaces. These results provide a Lorentzian analogue of the classification recently carried out by Belli and López in the Riemannian solvable Lie group Sol.

Abstract: In the historical development of various fields of mathematics, significant advances have occurred in areas such as algebra, abstract algebra, group theory, and numerous other mathematical and scientific domains. Contributions from mathematicians such as Dio- phantus, Goldbach, Euler, Girolamo Cardano, Johannes Kepler, Poncelet, Henri Poincaré, George Cantor, Felix Klein, David Hilbert, and Hermann Weyl have been fundamental, particularly in the pursuit of increasingly complex and deeper structures within geometry and topology. In this work, the division operation in the Alpha group is defined by analogy with the Kronecker tensor product. The representation of quaternion theory, based on De Moivre’s theorem, is employed for the construction of the matrices. The Alpha Group di- vision operation is then applied to analyze the various tensor metrics resulting from plane rotations over the interval from 0 to 2π radians. Since the general transformation kernel of the 4 × 4 matrix is defined within the Alpha group, it is possible to observe the variabil- ity associated with the tangent and cotangent functions that constitute the transformation matrix. The Alpha group, defined through a generalized division operation, thus provides a geometric and topological representation of infinity via the kernel transformation of the 4 × 4 matrix. Ultimately, this work seeks to connect the ideas developed by Poncelet and Cantor regarding the formation of imaginary elements in infinite projections with the con- cept of different types of infinity, as interpreted through the application of group theory.

Abstract: In this work, we present a new geometric framework that integrates quaternion-based rotation and translation operators with a generalized inner product and vector product framework defined on Gielis-type superquadrics. By incorporating the multiplicative shape factor ρ(ϕ), we construct a family of rotation matrices and quaternion mappings adapted to the elastic and non-Euclidean behavior of biological growth surfaces. This framework enables smooth, direction-dependent deformations and provides a unified representation for curvature-induced growth, differential thickening, and torsional motions observed in plants. The proposed model provides a mathematically tractable and biologically interpretable tool in many applications in plants, animals and biomolecules. Potential applications include computational botany, growth-based animation, and the design of biologically inspired structures.

Abstract: Many geometrical models have been created and applied in different subjects of sciences, such as physics, astronomy, biology,\ldots This paper presents a generalization of the recently defined $(\bar m,m)$-conformal mappings of Riemannian spaces. In this article, the affine connections of Riemannian spaces and symmetric affine connection spaces are combined. In this way, the characteristics of structures without external effects (Riemannian spaces) and with external effects (symmetric affine connection spaces) are geometrized. The main contributions of this research are: 1. Invariants of $(\bar m,m)$-conformal mappings of Riemannian spaces (review) and of geodesic mappings of symmetric affine connection spaces (review); 2. Actions and variational calculi with respect to some invariants obtained in 1; 3. Generalized Einstein's equations with respect to the analyzed transformations with clear reductions to standard ones.

Abstract: This study introduces new classes of fuzzy open sets, namely (p,q)-FΩ-open (resp. (p,q)-FΩP open, (p,q)-FΩS-open, (p,q)-FΩα-open, and (p,q)-FΩγ-open) sets in double fuzzy topological spaces (DFTSs) in view of Šostak. We conduct a detailed investigation of the relationships among these classes of open sets, supported by carefully constructed illustrative examples. Furthermore, we propose and characterize the associated DFΩ-interior and DFΩ-closure operators. Subsequently, we define and analyze new classes of fuzzy functions based on (p,q)-FΩ-open sets, referred to as DFΩ-continuous and DFΩ-irresolute functions within the framework of DFTSs (S,ϑ,ϑ^∗) and (Z,ζ,ζ^∗). We also introduce the notions of DFΩP-continuous, DFΩS-continuous, DFΩα continuous, and DFΩγ-continuous functions, which constitute weaker forms of DFΩ-continuity. As an application, we demonstrate that these newly defined continuity concepts generalize, extend, and unify several existing results in the theory of DFTSs. Finally, we propose and discuss the concepts of DFAΩ-continuity and DFWΩ-continuity as additional weaker variants of DFΩ-continuity. Moreover, we establish new separation axioms, termed (p,q)-FΩ-normal and (p,q)-FΩ-regular spaces, formulated via (p,q)-FΩ-closed sets.

Abstract: The study of the so-called Π-manifolds has continued. This is a short name for almost paracontact almost paracomplex manifolds with a pair of metrics — Riemannian and pseudo-Riemannian — that are mutually related and compatible with the structure of the manifold. The well-known classification of these manifolds with respect to the Riemannian metric has been used to derive an alternative classification of the same manifolds with respect to the pseudo-Riemannian metric. The interrelations between the two classifications and the corresponding classification tensors are investigated. Finally, a three-dimensional explicit example on a Lie group is given, which illustrates the most interesting case of the transition from one classification to the other.

Abstract: Standard cryptocurrency transaction cost models assume flat geometry and assign execution cost as a proportional fee. This paper tests whether replacing flat-fee models with a unified Riemannian execution cost framework improves out-of-sample prediction of realised execution costs. The framework identifies execution slippage as the geodesic arc length on the Fisher information manifold of a Markov-switching GARCH maximum-entropy model, augmented by a joint curvature-topological fragmentation alarm derived from the same parameter vector. Ablation confirms that each geometric component contributes uniquely: removing the geodesic increases mean squared prediction error by 2.9%, removing topological data analysis by 2.1%, and removing curvature by 1.5%. No subset matches the full framework. On five major cryptocurrency markets (BTC, ETH, XRP, LTC, BCH) over 2,253 daily observations, the integrated framework achieves the lowest prediction error on all five assets and is the sole model retained in the Model Confidence Set at the 10% significance level against six benchmarks, including Amihud, Kyle λ, and Almgren and Chriss. A joint curvature-topological alarm fires a median of two days before price-based circuit breaker thresholds across four crisis episodes, including the Terra collapse of May 2022 and the FTX bankruptcy of November 2022. The framework requires no additional data or free parameters beyond the upstream estimation pipeline.

Abstract: Quantumgateestimationandtomographypipelinesroutinelycombineintrinsicallydefined likelihoods with priors or regularization terms specified in local Euclidean coordinates. This practice implicitly replaces the Haar reference measure on SU(2) with Lebesgue measure, specifying a different statistical model rather than a reparametrization of the intended one. Weshowthat omitting the associated chart-volume factor alters the optimization objective itself, modifying its gradient field and stationary-point structure. The mismatch persists arbitrarily close to the identity, so that flat-coordinate surrogate objectives can converge to points that are non-stationary for the corresponding Haar-consistent objective even in regimes where local Gaussian approximations are assumed valid. We prove a formal non-equivalence proposition and validate a leading-order Fisher-information correction analytically and numerically. Large-scale multi-start optimization experiments (N = 11,900 runs) demonstrate that the discrepancy is regime-dependent and most pronounced under moderate-to-strong regularization or limited data. The fix requires a single-line modification to any gradient-based optimizer. These results identify reference-measure selection as an explicit modeling decision with direct consequences for optimization and inference in gate-set tomography, randomized benchmarking, and Bayesian gate estimation on curved parameter manifolds.

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.