arXiv subjects

Chandrasekhar Gokavarapu

Publications and source records attributed to Chandrasekhar Gokavarapu.

At least 37 records · Page 2Linked to original sources

arXiv2026

Quantitative fixed-point theorems with verifiable hypotheses: rates and stability

Let $(X,\dist)$ be a complete metric space and let $C\subseteq X$ be a closed invariant set. We study fixed points of maps $T\colon C\to C$ governed by a \emph{verifiable} contractive modulus. The modulus is encoded by a contractive gauge $ω$ and a certified constant $κ=\sup_{0<r\le R}ω(r)/r<1$ on a computable working radius $R$. From this datum we derive explicit a priori bounds $\dist(x_n,x^\ast)\le Φ(n;κ,δ_0)$ for Picard iterates, a residual-to-error estimate, and a quantitative data dependence bound $\dist(x^\ast,y^\ast)\le (1-κ)^{-1}\sup_{x\in C}\dist(Tx,Sx)$. We further treat inexact evaluations $\dist(\tilde x_{n+1},T\tilde x_n)\le η_n$ and obtain certified resilience bounds with the same stability factor. The framework applies to Hammerstein--Volterra integral equations and to boundary value problems via Green operators, where kernel bounds yield certified convergence rates.

math.DS↗

arXiv2026

Certified Learning under Distribution Shift: Sound Verification and Identifiable Structure

Proposition. Let $f$ be a predictor trained on a distribution $P$ and evaluated on a shifted distribution $Q$. Under verifiable regularity and complexity constraints, the excess risk under shift admits an explicit upper bound determined by a computable shift metric and model parameters. We develop a unified framework in which (i) risk under distribution shift is certified by explicit inequalities, (ii) verification of learned models is sound for nontrivial sizes, and (iii) interpretability is enforced through identifiability conditions rather than post hoc explanations. All claims are stated with explicit assumptions. Failure modes are isolated. Non-certifiable regimes are characterized.

cs.LG↗

arXiv2026

Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings

Purpose: To develop the algebraic foundation of finite commutative ternary $Γ$-semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and $Γ$-ring frameworks. Methods: Finite models of commutative ternary $Γ$-semirings are constructed under the axioms of closure, distributivity, and symmetry. Structural and congruence lattices are analyzed, and subdirect decomposition theorems are established through ideal-theoretic arguments. Results: Each finite commutative ternary $Γ$-semiring admits a unique (up to isomorphism) decomposition into subdirectly irreducible components. Radical and ideal correspondences parallel classical results for binary semirings, while the classification of all non-isomorphic systems of order $\lvert T\rvert\!\le\!4$ confirms the structural consistency of the theory. Conclusion: The paper provides a compact algebraic framework linking ideal theory and decomposition in finite ternary $Γ$-semirings, establishing the basis for later computational and categorical developments.

math.RA↗

arXiv2026

Unified Geometric, Fuzzy, and Computational Framework for Ternary Gamma Semirings

Aim. This paper (Paper D) unifies the ideal-theoretic, computational, and homological layers developed in Papers A (Rao 2025), B (Rao 2025), and C (Rao 2025) into a geometric framework that includes fuzzy and computational geometries on the spectrum Spec_G(T) and derived invariants in TGMod. Scope. We construct structure sheaves and Grothendieck topologies adapted to ternary G-products, develop fuzzy and weighted sites, and prove dualities bridging primitive spectra, Schur-density embeddings, and derived functors Ext and Tor. Outcomes. We obtain comparison theorems between radical/primitive strata and cohomological supports, and supply computable criteria and algorithms for finite models.

math.GM↗

arXiv2026

Computational and Categorical Frameworks of Finite Ternary $Γ$-Semirings: Foundations, Algorithms, and Industrial Modeling Applications

Purpose: This study extends the structural theory of finite commutative ternary $Γ$-semirings into a computational and categorical framework for explicit classification and constructive reasoning. Methods: Constraint-driven enumeration algorithms are developed to generate all non-isomorphic finite ternary $Γ$-semirings satisfying closure, distributivity, and symmetry. Automorphism analysis, canonical labeling, and pruning strategies ensure uniqueness and tractability, while categorical constructs formalize algebraic relationships. \\ \textit{Results:} The implementation classifies all systems of order $|T|\!\le\!4$ and verifies symmetry-based subvarieties. Complexity analysis confirms polynomial-time performance, and categorical interpretation connects ternary $Γ$-semirings with functorial models in universal algebra. \\ Conclusion: The work establishes a verified computational theory and categorical synthesis for finite ternary $Γ$-semirings, integrating algebraic structure, algorithmic enumeration, and symbolic computation to support future industrial and decision-model applications.

math.RA↗

arXiv2026

Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum

The theory of ternary $Γ$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $Γ$. Building on the foundational axioms recently established by Rao, Rani, and Kiran (2025), this paper develops the first systematic ideal-theoretic study within this setting. We define and characterize prime and semiprime ideals for commutative ternary $Γ$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T/P$ is free of nonzero zero-divisors under the induced ternary $Γ$-operation. Semiprime ideals are shown to be stable under arbitrary intersections and coincide with their radicals, providing a natural bridge to radical and Jacobson-type structures. A correspondence between prime ideals and prime congruences is established, leading to a Zariski-like spectral topology on $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $Γ$-semirings of order $\leq 4$ confirms the theoretical predictions and reveals novel structural phenomena absent in binary semiring theory. The results lay a rigorous algebraic and computational foundation for subsequent categorical, geometric, and fuzzy extensions of ternary $Γ$-algebras.

math.RA↗

arXiv2026

A curvature-regularized variational problem with an area constraint

Interlocking interfaces are commonly employed to mitigate relative sliding under shear.Indeed, Their geometry is typically selected on grounds of fabrication convenience rather than analytical optimality. There is no reason to suppose that circular or polygonal profiles minimize localized stress concentration under fixed geometric constraints. We propose a variational model in which the interface is represented by a planar curve $y=f(x)$, and localized stress amplification is quantified by a curvature-sensitive functional \[ J[f] = \int_{-a}^{a} \bigl(1+γκ^2\bigr) \sqrt{1+f'(x)^2}\,dx, \] defined on the Sobolev space $W^{2,2}([-a,a])$. The functional is motivated by elasticity-theoretic considerations in which curvature enters the leading-order stress field near a singular interface.Indeed, any profile possessing discontinuous tangents yields a divergent integral, thereby rendering it energetically inadmissible within the Sobolev space $W^{2,2}$. An area constraint $\int_{-a}^{a} f(x)\,dx = A_0$ is imposed to model fixed material volume. Using the direct method of the calculus of variations, we establish the existence of a minimizer and derive the associated Euler--Lagrange equation, a nonlinear fourth-order boundary value problem. Note, however, that constant-curvature and piecewise-linear profiles fail to satisfy the necessary optimality conditions under the imposed constraint. Indeed, we are thus forced to conclude that analytical optimality necessitates a more complex variation in the local tangent angle The analysis indicates that commonly employed interlock geometries are not variationally optimal for minimizing localized shear stress within this class of admissible interfaces.

math.RA↗

arXiv2026

Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Random-Walk Laplacian Formulation)

The operator-theoretic dichotomy underlying diffusion on directed networks is \emph{symmetry versus non-self-adjointness} of the Markov transition operator. In the reversible (detailed-balance) regime, a directed random walk $P$ is self-adjoint in a stationary $π$-weighted inner product and admits orthogonal spectral coordinates; outside reversibility, $P$ is genuinely non-self-adjoint (often non-normal), and stability is governed by biorthogonal geometry and eigenvector conditioning. In this paper we develop a harmonic-analysis framework for directed graphs anchored on the random-walk transition matrix $P=D_{\mathrm{out}}^{-1}A$ and the random-walk Laplacian $L_{\mathrm{rw}}=I-P$. Using biorthogonal left/right eigenvectors we define a \emph{Biorthogonal Graph Fourier Transform} (BGFT) adapted to directed diffusion, propose a diffusion-consistent frequency ordering based on decay rates $\Re(1-λ)$, and derive operator-norm stability bounds for iterated diffusion and for BGFT spectral filters. We prove sampling and reconstruction theorems for $P$-bandlimited (equivalently $L_{\mathrm{rw}}$-bandlimited) signals and quantify noise amplification through the conditioning of the biorthogonal eigenbasis. A simulation protocol on directed cycles and perturbed non-normal digraphs demonstrates that asymmetry alone does not dictate instability; rather, non-normality and eigenvector ill-conditioning drive reconstruction sensitivity, making BGFT a natural analytical language for directed diffusion processes.

math.RA↗

arXiv2026

The Spectral Geometry of Ternary Gamma Schemes:Sheaf-Theoretic Foundations and Laplacian Clustering

This article develops a self-contained affine $Γ$-scheme theory for a class of commutative ternary $Γ$-semirings. By establishing all geometric and spectral results internally, the work provides a unified framework for triadic symmetry and spectral analysis. The central thesis is that a triadic $Γ$-algebra canonically induces two primary structures: (i) an intrinsic triadic symmetry in the sense of a Nambu--Filippov-type fundamental identity on the structure sheaf, and (ii) a canonical Laplacian on the finite $Γ$-spectrum whose spectral decomposition detects the clopen (connected-component) decomposition of the underlying space. We define $Γ$-ideals and prime $Γ$-ideals, endow $\SpecG(T)$ with a $Γ$-Zariski topology, construct localizations and the structure sheaf on the basis of principal opens, and prove the affine anti-equivalence between commutative ternary $Γ$-semirings and affine $Γ$-schemes. Furthermore, we demonstrate that the triadic bracket on sections is invariant under $Γ$-automorphisms and compatible with localization. The main spectral theorem establishes the block-diagonalization of the Laplacian under topological decompositions and provides an algebraic-connectivity criterion. The theory is verified through explicit computations of finite $Γ$-spectra and their corresponding Laplacian spectra

math.RA↗

arXiv2026

Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants

We establish a foundational homotopical framework for ternary $Γ$-modules by establishing that $\mathcal{T}\text{-Mod}$ is a Barr-exact, monoidal closed category. We resolve the long-standing "additivity obstruction" in non-binary algebra by constructing a cofibrantly generated Quillen model structure on the simplicial category $s(\mathcal{T}\text{-Mod})$. Our central discovery is that the derived category $D(\mathcal{T}\text{-Mod})$ constitutes a 3-angulated category, where the derived periodicity is governed by triadic quadrilaterals rather than binary triangles. We derive the 3-ary long exact sequence and characterize the connecting morphisms as invariants of the $Γ$-parameter space. This framework provides a rigorous homological bridge to Nambu mechanics and absolute geometry over $\mathbb{F}_1$.

math.RA↗

arXiv2026

Proof-Carrying Verification for ReLU Networks via Rational Certificates

Rectified Linear Unit (ReLU) networks are piecewise-linear (PWL), so universal linear safety properties can be reduced to reasoning about linear constraints. Modern verifiers rely on SMT(LRA) procedures or MILP encodings, but a safety claim is only as trustworthy as the evidence it produces. We develop a proof-carrying verification core for PWL neural constraints on an input domain $D \subseteq \mathbb{R}^n$. We formalize the exact PWL semantics as a union of polyhedra indexed by activation patterns, relate this model to standard exact SMT/MILP encodings and to the canonical convex-hull (ideal) relaxation of a bounded ReLU, and introduce a small certificate calculus whose proof objects live over $\mathbb{Q}$. Two certificate types suffice for the core reasoning steps: entailment certificates validate linear consequences (bound tightening and learned cuts), while Farkas certificates prove infeasibility of strengthened counterexample queries (branch-and-bound pruning). We give an exact proof kernel that checks these artifacts in rational arithmetic, prove soundness and completeness for linear entailment, and show that infeasibility certificates admit sparse representatives depending only on dimension. Worked examples illustrate end-to-end certified reasoning without trusting the solver beyond its exported witnesses.

cs.LO↗

arXiv2026

Massively Parallel Reductions in Multivariate Polynomial Systems: Bridging the Symbolic Preprocessing Gap on GPGPU Architectures

Gröbner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faugère-type F4 and signature-based F5) reduce many instances to large sparse linear algebra over finite fields, their dominant cost is not merely elimination but the symbolic preprocessing that constructs Macaulay-style matrices whose rows encode shifted reducers. This phase is characterized by dynamic combinatorics (monomial discovery, sparse row assembly, and deduplication) and is typically memory-latency bound, resisting naive parallelization. This article develops a rigorous synthesis that reframes S-polynomial reduction as syzygy discovery: row construction is a structured map from module relations to the kernel of a massive, sparse, highly non-random Macaulay matrix A over Fp. Building on this viewpoint, we propose a GPU-targeted architecture that (i) converts dynamic symbolic data structures into static, two-pass allocations via prefix-sum planning; (ii) enforces coalesced memory access through structure-of-arrays polynomial layouts and sorted monomial dictionaries; and (iii) integrates finite-field arithmetic kernels (Montgomery/Barrett-style reduction) at register granularity. On the linear-algebra side, we explore the transition from classical Gaussian elimination to parallel structured Gaussian elimination (PSGE) and to Krylov-type kernel solvers (Block Wiedemann/Lanczos) that better match GPU throughput while controlling fill-in. The result is a principled bridge between algebraic syzygy theory and SIMT hardware constraints, isolating the true bottleneck and providing a pathway to massively parallel reductions for multivariate polynomial systems.

math.RA↗

arXiv2025

Incremental Certificate Learning for Hybrid Neural Network Verification . A Solver Architecture for Piecewise-Linear Safety Queries

Formal verification of deep neural networks is increasingly required in safety-critical domains, yet exact reasoning over piecewise-linear (PWL) activations such as ReLU suffers from a combinatorial explosion of activation patterns. This paper develops a solver-grade methodology centered on \emph{incremental certificate learning}: we maximize the work performed in a sound linear relaxation (LP propagation, convex-hull constraints, stabilization), and invoke exact PWL reasoning only through a selective \emph{exactness gate} when relaxations become inconclusive. Our architecture maintains a node-based search state together with a reusable global lemma store and a proof log. Learning occurs in two layers: (i) \emph{linear lemmas} (cuts) whose validity is justified by checkable certificates, and (ii) \emph{Boolean conflict clauses} extracted from infeasible guarded cores, enabling DPLL(T)-style pruning across nodes. We present an end-to-end algorithm (ICL-Verifier) and a companion hybrid pipeline (HSRV) combining relaxation pruning, exact checks, and branch-and-bound splitting. We prove soundness, and we state a conditional completeness result under exhaustive splitting for compact domains and PWL operators. Finally, we outline an experimental protocol against standardized benchmarks (VNN-LIB / VNN-COMP) to evaluate pruning effectiveness, learned-lemma reuse, and exact-gate efficiency.

math.RA↗

arXiv2025

Derived Gamma Geometry II: Stable $\infty$-Categories of Gamma-Modules, Derived Monoidal Structures, and Obstructions to Binary Shadows

Let $\T$ be a commutative ternary $\Gm$-semiring in the sense of the triadic, $\Gm$-parametrized multiplication $\{a,b,c\}_γ$. Building on the affine $\Gm$-spectrum $\SpecG(\T)$, the structure sheaf, and the equivalence between $\Gm$-modules and quasi-coherent $\Gm$-sheaves on affine $\Gm$-schemes, we construct and organize the derived formalism at the level of stable $\infty$-categories. Our first contribution is a technically explicit construction of a stable $\infty$-category $\Dinfty(\T,\Gm)$ enhancing the unbounded derived category of $\Gm$-modules, obtained by dg-nerve and $\infty$-localization of chain complexes. We further explain the derived monoidal structure induced by the ternary $\Gm$-tensor product and the corresponding internal $\RHom$, under standard exactness/projectivity hypotheses. Our second contribution is an obstruction theory to \emph{binary reduction}: we formalize the nonexistence of any conservative ``binary module shadow'' compatible with the cubic localization calculus intrinsic to ternary $\Gm$-semirings. In particular, any attempt to represent the triadic $\Gm$-action by binary scalars forces $\Gm$-mode data to be absorbed into the scalars, hence ceases to be a genuine reduction. Finally, we give a detailed affine derived equivalence between derived quasi-coherent $\Gm$-sheaves on $X=\SpecG(\T)$ and $\Dinfty(\T,\Gm)$, and we include worked examples illustrating the cubic localization relation and its derived consequences.

math.RA↗

arXiv2025

Fundamental Theorems in the K-Theory of Gamma Semirings: Additivity, Localization, and Dévissage

Building on the Waldhausen and Quillen models of higher algebraic $K$-theory for exact categories and Waldhausen categories attached to a non-commutative $n$-ary $\Ga$-semiring $(T,\Ga)$, we establish the fundamental formal properties of $K$-theory in this $\Ga$-parametrised, slot-sensitive setting. For the exact/Waldhausen categories of finitely generated bi-positional $n$-ary $\Ga$-modules, perfect complexes in the derived category, and perfect quasi-coherent complexes on the non-commutative $\Ga$-spectrum $\SpecGnC{T}$, we prove Waldhausen Fibration and Additivity theorems and Quillen-type Localization for Serre and Waldhausen pairs. Under natural hypotheses on $\Ga$-stable filtrations we obtain dévissage and Approximation theorems, together with cofinality and Karoubi invariance, showing that idempotent completion does not change $K$-theory and that cofinal subcategories control $K_n$ in positive degrees. We further derive a Bass--Quillen fundamental triangle for polynomial extensions in the $n$-ary $\Ga$-context and prove nilpotent invariance for two-sided $\Ga$-ideals. In geometric terms, these results yield localization and Mayer--Vietoris sequences for the $K$-theory of $\Perf(X)$ on $X=\SpecGnC{T}$ and its admissible open covers. Altogether, the paper shows that the higher $K$-theory of non-commutative $n$-ary $\Ga$-semirings enjoys the same formal properties as in the classical ring and scheme cases, providing a robust foundation for subsequent computational and homotopy-theoretic applications.

math.KT↗

arXiv2025

Harmonic Analysis on Directed Networks: A Biorthogonal Laplacian Framework for Non-Normal Graphs

Classical spectral graph theory relies on the symmetry of the adjacency and Laplacian operators, which guarantees orthogonal eigenbases and energy-preserving Fourier transforms. However, real-world networks are intrinsically directed and asymmetric, resulting in non-normal operators where standard orthogonality assumptions fail. In this paper, we develop a rigorous harmonic analysis framework for directed graphs centered on the \emph{Combinatorial Directed Laplacian} ($L = D_{out} - A$). We construct a \emph{Biorthogonal Graph Fourier Transform} (BGFT) using dual left and right eigenbases, and introduce a directed variational semi-norm based on the operator norm $\|Lx\|_2$ rather than the quadratic form. We derive exact Parseval-type bounds that quantify the energy distortion induced by the non-normality of the graph, explicitly linking signal reconstruction stability to the condition number of the eigenvector matrix, $κ(V)$. Finally, we present experimental validation comparing normal directed cycles against non-normal perturbed topologies, demonstrating that while the BGFT provides exact reconstruction in ideal regimes, the geometric departure from normality acts as the fundamental limit on filter stability in directed networks.

math.RA↗

arXiv2025

Projective Modules and Classical Algebraic K-Theory of Non-Commutative Gamma Semirings

In this paper, we initiate the study of algebraic K-theory for non-commutative $Γ$-semirings, extending the classical constructions of Grothendieck and Bass to this setting. We first establish the categorical foundations by constructing the category of finitely generated projective bi-$Γ$-modules over a non-commutative $Γ$-semiring $T$. We prove that this category admits an exact structure, allowing for the definition of the Grothendieck group $K_0^Γ(T)$. Furthermore, we develop the theory of the Whitehead group $K_1^Γ(T)$ using elementary matrices and the Steinberg relations in the non-commutative $Γ$-semiring context. We establish the fundamental exact sequences linking $K_0$ and $K_1$ and provide explicit calculations for specific classes of non-commutative $Γ$-semirings. This work lays the algebraic groundwork for future studies on higher K-theory spectra.

math.RA↗

arXiv2025

Sheaf Theory and Derived Gamma Geometry over the Non-Commutative Gamma Spectrum

We develop the geometric and homological framework for non-commutative $n$-ary $Γ$-semirings by constructing a sheaf and derived theory over their non-commutative $Γ$-spectrum. Starting with a non-commutative $n$-ary $Γ$-semiring $(T,+,Γ,μ)$ and its bi-$Γ$-modules, we define the space $\Spec_Γ^{\mathrm{nc}}(T)$, equip it with a Zariski-type topology, and build the structure sheaf $\mathcal{O}{\SpecΓ^{\mathrm{nc}}(T)}$ via localization at prime $Γ$-ideals. We introduce quasi-coherent $Γ$-sheaves, show that their category is exact with enough injectives, and interpret the derived functors $\Ext^Γ$ and $\Tor^Γ$ as global cohomological invariants on this non-commutative $Γ$-space. On the derived side, we construct the category $\mathbf{D}(\QCoh(\Spec_Γ^{\mathrm{nc}}(T)))$, establish a local--global principle for $\Ext^Γ$ and $\Tor^Γ$, and prove a non-commutative local duality theorem assuming a dualizing complex. We further introduce derived non-commutative $Γ$-stacks and a dg-enhancement of the spectrum, giving a spectral and motivic interpretation of homological invariants. Structural consequences include a Wedderburn--Artin type decomposition in the $n$-ary $Γ$-setting, a derived Morita theory for semisimple $n$-ary $Γ$-semirings, and a duality between the primitive $Γ$-spectrum and simple objects of the derived category. These results extend our earlier commutative derived $Γ$-geometry to a fully non-commutative $n$-ary context.

math.RA↗