arXiv subjects

Chandrasekhar Gokavarapu

Publications and source records attributed to Chandrasekhar Gokavarapu.

At least 19 recordsLinked to original sources

arXiv2026

Universal Operator Envelopes and Independent Arithmetic Defects of Additive Ternary Gamma-Rings

For an additive ternary $Γ$-ring $T$, we construct the associative rng generated by the two-element, two-index operators on ternary modules and compare it with its realization on the regular module. For a finite-free two-step system $T_μ=E\oplus W$, two integral matrices control complementary parts of this comparison: the middle-relation matrix $B_μ$ presents the hidden quotient occurring in the integral regular kernel, while the visible flattening $Θ_μ$ controls the visible image, its saturation and its nonflat base-change correction. The torsion cokernels of these matrices define an ordered arithmetic defect pair; this pair records selected arithmetic data and is not asserted to be the whole regular kernel. The principal result is a constructive realization theorem: for every ordered pair $(G,H)$ of finite abelian groups, there is a torsion-free two-step ternary $\mathbb Z$-ring whose visible-saturation torsion is $G$ and whose middle-relation torsion is $H$. Thus the two torsion mechanisms, including their prime supports and invariant factors, can be prescribed independently. The proof builds two rank-$(2,1)$ atomic tensor families with complementary Smith profiles and establishes an orthogonal-sum theorem for their presentation matrices. We also identify the standard base-change obstruction attached to these two matrices, describe their determinantal loci, realize classical rank-one matrix-factorization periodicity in an example not induced by an associative $\mathbb Z$-bilinear binary product on its underlying group, and classify all $256$ integral $2\times2\times2$ sign tensors by the resulting defect data. All finite calculations use exact integer arithmetic and are independently reproducible.

math.RA↗

arXiv2026

Morita Transport, Marked Projective Loci, and Dedekind Classification for C4-Type Conditionss

Objectwise Morita invariance of $\Cfour$ is known. We prove the corresponding statements for $\Cfourstar$, semi-weak-CS, and strongly $\Cfourstar$, including the chain joins in the definition of semi-weak-CS. For a module property $\mathcal Q$ preserved and reflected by equivalences, we consider the subset $\Vmon_{\mathcal Q}(R)$ of the finitely generated projective monoid $\Vmon(R)$. This marked pair is transported by Morita equivalence. The associated regular-module ring property is Morita invariant if and only if membership in $\Vmon_{\mathcal Q}(R)$ is constant on the order units. The matrix profile satisfies $μ_{\mathcal Q}(M_m(R))=\{n\geq1:mn\inμ_{\mathcal Q}(R)\}$. For a nondivision right Ore domain, the profiles of $\Cfour$, $\Cfourstar$, and strongly $\Cfourstar$ are $\{1\}$, whereas the semi-weak-CS profile is $\mathbb N_{\geq1}$. Over a nonfield Dedekind domain $D$, the first three marked loci consist of the projectives of rank at most one, while every finitely generated projective is semi-weak-CS. For a finitely generated $D$-module $M$, we prove that every $\Cfour$ defect at rank at most one can be transferred between $M$ and $τ(M)$, and we construct a defect when the rank is at least two. Hence $M$ is $\Cfour$ if and only if $\operatorname{rank}_D M\leq1$ and $τ(M)$ is $\Cfour$; the analogous criterion holds for $\Cfourstar$. Combined with the finite-length torsion criterion, these reductions give invariant-factor tests for the four conditions and criteria for their behaviour under direct sums.

math.RA↗

arXiv2026

Typed Congruences and Quotient-Critical Irreducibility in Complete Ternary $Γ$-Semirings

We study congruences and binary reducibility in completely additive ternary $Γ$-semirings through their two-sorted atomic cores. For an odd $m\geq3$, an abelian group $G$ of exponent dividing $m-1$, and $u,v\in G$, we construct a two-branch symmetric $m$-ary band $F_m(G;u,v)$. We prove \[ \Con(F_m(G;u,v))\cong\operatorname{Sub}(G)\times B_2 \] and classify every carrier--index congruence pair: $(θ,ϕ)$ is typed exactly when $θ$ is an ordinary congruence and $ϕ$ refines $θ$. We also give a quotient-by-quotient reducibility criterion. In the finite case, $F_m(G;u,v)$ is quotient-critically irreducible exactly when $G$ is a cyclic $p$-group and $u-v$ has order $p$. The specialization $G=\mathbb Z_2$ yields a four-point family $H_m$ extending the irreducible ternary example of Devillet and Mathonet. Its congruence lattice is $B_3$, its alternating core has exactly $36$ typed congruence pairs, and its automorphism group is $C_2$. Its seven proper quotients are reducible and form five isomorphism types; in arity five their exact reduction counts are determined. The powerset lift of $H_5$ is a $16$-element atom-total complete atomic Boolean ternary $Γ$-semiring with no atomic binary collapse, whereas each proper diagonal strong atom-saturated quotient has one. Its parity congruence is recovered on atoms by one explicitly specified depth-one polynomial inequation.

math.RA↗

arXiv2026

Operational Regimes of Ternary $Γ$-Semiring Chemical Systems

Ternary $Γ$-semiring chemical systems provide an algebraic language in which mediators such as catalysts, solvents, inhibitors, thermodynamic conditions, and external fields enter the transformation law as internal arguments. Building on the published MATCH foundation article, this paper studies operational regimes of the compressed TGS notation rather than proposing a second axiomatization. Distributivity is interpreted as observable non-interference of mediated channels, while associativity is interpreted as observable path-independence of bracketed mediated protocols. Since the state space is only semigroup- or semiring-level, subtraction is not used at the state level; instead, observable defect and ratio indices are defined after applying a non-negative observable. We prove that distributivity implies observable non-interference under observable additivity, and that associativity implies observable path-independence. Nonzero defects detect observable distinguishability. To strengthen the diagnostic layer, we prove recovery theorems showing that separating families of observables can recover algebraic distributivity and associativity from vanishing observable defects. We also introduce approximate operational regimes and prove ratio bounds controlled by uniform defect estimates. Under strict convexity, uniqueness, and common global-constraint assumptions, a potential-generated equilibrium operation is shown to be associative. A symbolic intermediate-dependent Michaelis--Menten type construction shows how non-associativity may arise in kinetic regimes, and regulatory/allosteric-type non-distributivity is discussed cautiously through observable coupling.

physics.chem-ph↗

arXiv2026

Barr-Exactness and Congruence-Based Homological Algebra in Ternary $Γ$-Semimodules

Let $T$ be a ternary $Γ$--semiring and let $\mathsf{SMod}_{T,Γ}$ be the category of ternary $Γ$--semimodules with the intrinsic five--ary action $T\times Γ\times M \times Γ\times T\to M$.This paper develops a congruence--first calculus in $\mathsf{SMod}_{T,Γ}$.Kernels and quotients are taken in the sense of congruences, not in the sense of cosets of subobjects, because coset quotients generally fail without subtraction. We prove that every morphism admits a canonical factorization through its kernel congruence, establish congruence versions of the three isomorphism theorems, and describe coequalizers as quotients by generated congruences. Exactness is formulated in the regular/Barr-exact sense: kernels are kernel pairs and exactness is expressed by equality of kernels with regular images. Finally, we record a projective generation criterion and a concrete failure of a classical diagram lemma in this Barr-exact but non-abelian setting.

math.RA↗

arXiv2026

Smith-Orbit Classification of Extensions and Exact-Sequence Asymmetry for C4*-Modules

Let $R$ be a ring and let a $C4^{\ast}$-module mean a module all of whose submodules are $C4$-modules. We first determine its asymmetric exact-sequence behavior: kernel closure is unconditional, while explicit commutative examples disprove split-extension and cokernel closure. We then classify the finite-length torsion objects over a Dedekind domain. A primary component is $C4$ exactly when it is homocyclic, and it is $C4^{\ast}$ exactly when it is cyclic or semisimple; in finite length $C4^{\ast}$ and strongly $C4^{\ast}$ coincide. The main result is a complete orbit calculation over a discrete valuation ring $V$. Put $A=(V/(π^a))^r$, $C=(V/(π^c))^s$ and $m=\min\{a,c\}$. Then \[ \Ext^1_V(C,A)\cong \operatorname{Mat}_{r\times s}(V/(π^m)), \] and the $\operatorname{Aut}(A)\times\operatorname{Aut}(C)$-orbits are classified by a Smith profile $0\leqν_1\leq\cdots\leqν_q\leq m$, $q=\min\{r,s\}$. The corresponding middle term is \[ \bigoplus_{i=1}^{q} \bigl(V/(π^{ν_i})\oplus V/(π^{a+c-ν_i})\bigr) \oplus (V/(π^a))^{r-q}\oplus(V/(π^c))^{s-q}, \] where $V/(π^0)=0$. This gives exact orbit-level criteria: the middle term is $C4$ precisely for the zero orbit with $a=c$, or for an invertible square orbit with $r=s$; it is $C4^{\ast}$ precisely for the zero orbit with $a=c=1$, or for a unit scalar orbit with $r=s=1$. We reformulate the Smith data as a valuation--rank profile, determine the complete geometry and additivity of the preservation loci, and globalize the result prime by prime. Consequently every extension between arbitrary finite-length torsion $C4^{\ast}$-modules over a Dedekind domain is decided by explicit local orbit invariants.

math.RA↗

arXiv2026

Finite diagonalizable torsors and Kummer cohomology over semiring schemes

Classical descent treats a finite diagonalizable torsor as a higher-rank locally free object. Over semirings that route is incomplete: fpqc-local freeness is not known to imply Zariski-local freeness in arbitrary rank. We show that diagonalizable symmetry bypasses this obstruction. The coordinate semialgebra splits into character pieces, each piece descends in rank one, and the torsor identity makes their multiplication maps invertible. This gives a natural equivalence between fpqc $D_X(Q)$-torsors and Picard-strong $Q$-gradings, and represents every torsor by a finite locally free morphism of rank $|Q|$. The same viewpoint proves that finite-index monomial maps of split tori are finite locally free torsors and yields matrix Kummer classification sequences. A concrete example is the nontrivial Boolean torsor $\Spec\mathbb B[v^{\pm1}]\to\Spec\mathbb B[u^{\pm1}]$, $u\mapsto v^m$, whose unit-power classes contribute $\mathbb Z/m\mathbb Z$ to degree-one cohomology. A two-term resolution of the character group then gives a presentation-independent universal-coefficient filtration for diagonalizable fpqc cohomology. This formalism organizes, but does not compute, $\mathbf G_{\mathrm m}$-cohomology. In degree two it identifies the Kummer boundary of a line bundle with its root gerbe. On projective space over the Boolean or real tropical semifield all $μ_m$-torsors vanish, whereas the root gerbe of $\mathcal O(1)$ has exact order $m$. Thus torsors and gerbes retain information that can disappear under ordinary ring completion.

math.RA↗

arXiv2026

Central Splitting, Division-Stable Residuals, and Opposite-Ring Transfer for Strongly C4*-Rings

The known decomposition theorem for strongly $\Cfourstar$-modules gives a semisimple summand and a summand-square-free residual summand, with two-way Hom-orthogonality when the ambient module is projective. Applied to the regular module, the two cross-corners vanish, so the idempotent defining the decomposition is central. Thus every strongly right $\Cfourstar$-ring splits as $R\cong Σ\times T$, where $Σ$ is semisimple artinian and $T_T$ is summand-square-free. The splitting need not be unique. We organize all admissible central splittings into a join-semilattice and prove a comparison theorem: any two residual factors have a common direct factor, and their complementary factors are finite products of division rings. Consequently the right-to-left defect is independent of the chosen splitting. If the central idempotents satisfy the ascending chain condition, there is a unique greatest admissible idempotent; it captures every central semisimple artinian direct factor and hence yields a canonical residual with no further such factor. An infinite product of fields shows that this finiteness hypothesis cannot simply be omitted. We also prove anti-isomorphism transport for $\Cfour$, $\Cfourstar$, semi-weak-CS, and strongly $\Cfourstar$ modules. It yields transfer whenever a residual factor is a product of a semisimple summand-square-free ring and a self-opposite core. This criterion does not assume regularity, exchange, or a left-sided hypothesis. Skew Laurent rings provide noncommutative nonregular examples, while the known injective-nonsurjective skew-polynomial construction gives a sharp one-sided residual obstruction.

math.RA↗

arXiv2026

Finite-sample certification and operating envelopes for spectral clustering and graph centrality

Spectral clustering and node rankings are commonly reported from one observed network without a finite-sample statement of what the observation supports. We develop a certification protocol that either returns a coverage-guaranteed set or explicitly returns ``no nontrivial certificate.'' For an inhomogeneous Bernoulli graph, a matrix-Bernstein quantile with all numerical constants and its ambient-dimension factor retained is combined with a one-sided spectral-gap certificate. The resulting Grassmann ball is valid at finite $n$, but is reported as informative only when its radius is below the diameter of the Grassmannian. We propagate the ball through a certificate-bearing approximate $k$-means map under declared population separation and minimum-cluster envelopes, derive simultaneous bands and an observed-gap certificate for degree centrality, and give a corrected normalized-Katz extension. A $12$-cell simulation study with $1{,}000$ graphs per cell maps the difference between coverage and usefulness. The submitted $n=200$ block-model example is shown to be necessarily vacuous after the dimension factor is restored; in the benchmark $p=0.30,q=0.10$, the subspace radius first falls below one at $n=\ExactRadiusThreshold$, whereas the mean-square clustering certificate remains unavailable until $n=\HammingThreshold$. An unequal-block example produces a genuine centrality certificate, while an analysis of the Zachary karate-club network correctly declines to certify despite $97.1\%$ agreement with the observed factions. These results separate algorithmic success, coverage validity and inferential informativeness.

math.ST↗

arXiv2026

The Spectral Topology of Global Imbalances:A Graph-Theoretic Framework for Systemic Risk in the Balance of Payments

We model global balance-of-payments linkages as a directed weighted exposure network and study systemic fragility through the spectrum of the associated operators. From a signed bilateral-flow matrix, we construct a nonnegative exposure lift, derive a spectral stability criterion for linear propagation, and define node- and edge-level risk measures using Perron-Frobenius geometry, PageRank, and marginal spectral impact. We further give two complementary extensions: a directed Laplacian formulation for shock diffusion and a non-backtracking threshold for cascade formation on sparse exposure networks. The framework identifies systemic risk as a topological property of the global network rather than a collection of isolated country indicators, and it yields mathematically explicit diagnostics for resilience, concentration of stress, and targeted stabilization.

physics.soc-ph↗

arXiv2026

Cyclic crossed-product envelopes and arity-detecting relative $K$-theory of automorphism-derived $n$-ary algebras

Let $m=n-1$. For a unital associative algebra $A$, an automorphism $σ$, and $c\in A^\times$ satisfying $σ(c)=c$ and $σ^m=\operatorname{Ad}_c$, we study the Hosszú--Gluskin-type product $[a_0,\ldots,a_m]=a_0σ(a_1)\cdotsσ^{m-1}(a_{m-1})ca_m$. We prove that its endpoint-anchored all-slot envelope is the cyclic crossed product generated by $A\otimes A^{\mathrm{op}}$ and $u$, with $ux=σ^e(x)u$ and $u^m=c\otimes(c^{-1})^{\mathrm{op}}$. For the finite-order case $c=1$ and permutation algebras $A=k^X$, this gives an orbit--stabilizer block decomposition. A general dimension-defect theorem then computes the homotopy fibre of the forgetful $K$-theory map. In characteristic zero, transitive $C_m$-sets of every divisor length $d\mid m$ have the same ordinary spectrum $\K(k)^m$ but boundary $\mathbb Z/d$. The regular action gives $π_{-1}\cong\mathbb Z/m$, recovering the arity; over finite fields, the remaining torsion is governed by multiplicative orders.

math.RA↗

arXiv2026

Fundamental comparison, base-change, and descent theorems in the $K$-theory of non-commutative n-ary Gamma-semirings

We develop a comparison, base-change, and descent framework for the algebraic $K$-theory of non-commutative $n$-ary $Γ$-semirings. Working in the Quillen-exact (and Waldhausen) setting of bi-finite, slot-sensitive $Γ$-modules and perfect complexes, we construct functorial maps on $K$-theory induced by extension and restriction of scalars under explicit $Γ$-flatness hypotheses in the relevant positional slots. We prove derived Morita invariance (via tilting bimodule complexes), establish Beck-Chevalley type base-change for cartesian squares, and deduce a projection formula compatible with the multiplicative structure coming from positional tensor products. Passing to the non-commutative $Γ$-spectrum Spec$^{\mathrm{nc}}_Γ(T)$, we show locality for perfect objects and derive Zariski hyperdescent for $\mathbb{K}(\mathrm{Perf})$, together with excision and localization sequences for closed immersions and fpqc descent for $Γ$-flat covers. Finally, we interpret $K_Γ(X)$ geometrically as the $K$-theory of the stable $\infty$-category of $Γ$-perfect complexes, describe its universal property in $Γ$-linear non-commutative motives, and record compatibility with cyclotomic and Chern-type trace maps.

math.KT↗

arXiv2026

Axiomatic Foundations of Chemical Systems as Ternary $Γ$-Semirings

Chemical transformations depend not only on the identities of the reacting species but also on the catalytic, environmental, and intermediate conditions under which they occur. Classical binary reaction formalisms usually treat such conditions as external annotations, which obscures the genuinely multi-state and multi-parameter character of real chemical processes. In this paper we introduce an axiomatic framework in which a chemical system is modelled by a ternary $Γ$-semiring. The elements of the state set represent chemical states, while the parameter set encodes catalytic and environmental conditions. A $Γ$-dependent ternary operation is used to describe mediated transformations, treating reactants, intermediates, and mediators as intrinsic arguments of the transformation law. We develop the algebraic axioms governing these mediated interactions and interpret their associativity, distributivity, and $Γ$-linearity in terms of multi-step pathways, parallel processes, and controlled environmental dependence. We introduce chemical ideals and $Γ$-ideals as algebraic structures modelling reaction-closed sub-systems and pathway-stable domains, and study their prime and semiprime forms. Homomorphisms between TGS-chemical systems are shown to preserve reaction pathways and describe consistent changes of chemical environment. Abstract examples from catalysis, thermodynamic phase control, and field-induced quantum transitions illustrate how familiar chemical phenomena fit within this framework. The resulting theory provides a unified algebraic foundation for multi-parameter chemical behaviour and establishes the structural basis for subsequent developments involving kinetics, geometric methods, and computational or AI-assisted models.

math.RA↗

arXiv2026

Localization, Local--Global Transfer, and Hull Theory for $C4^{\ast}$-Modules over Commutative Rings

Let $R$ be a commutative ring and $M$ an $R$-module. We develop a localization and local-global theory for $C4$-modules, $C4^{\ast}$-modules, strongly $C4^{\ast}$-modules, $C4$-hulls, and pseudo-continuous hulls over commutative rings. The problem is structural: these notions are defined through decompositions, summand conditions, and minimal extensions, while localization changes decomposition data, support, and hull minimality. We prove forward localization theorems for the $C4$, $C4^{\ast}$, and strongly $C4^{\ast}$ conditions under exact lifting hypotheses formulated through decomposition lifting, morphism lifting, and submodule lifting. We also prove converse local-global theorems under descent and patching hypotheses, showing when primewise or maximal-local $C4^{\ast}$ behavior implies global $C4^{\ast}$ behavior. In addition, we establish obstruction results showing that no unrestricted local-global principle can hold. We compare the localization of a global $C4$-hull or pseudo-continuous hull with the hull formed after localization. We show that hull commutation requires both localization stability of the hull class and envelope-type axioms for hull minimality and uniqueness, and we prove conditional patching theorems for reconstructing global hulls from compatible local hulls. Our method is purely algebraic and support-theoretic, based on summand descent, patching of local witnesses, support control, and dimension-stratified transfer on $\operatorname{Spec} R$. As applications, we show that for commutative artinian rings these properties are detected exactly on the local factors, and that for finitely generated torsion modules over a Dedekind domain they are detected exactly on the primary components, equivalently on the localizations at maximal ideals in the support.

math.RA↗

arXiv2026

Harmonic Analysis on Directed Networks via a Biorthogonal Laplacian Calculus for Non-Normal Digraphs

Spectral graph signal processing is traditionally built on self-adjoint Laplacians, where orthogonal eigenbases yield an energy-preserving Fourier transform and a variational frequency ordering via a real Dirichlet form. Directed networks break self-adjointness: the combinatorial directed Laplacian $L=D_{\mathrm{out}}-A$ is generally non-normal, so eigenvectors are non-orthogonal and classical Parseval identities and Rayleigh-quotient orderings do not apply. This paper develops a Laplacian-centric harmonic analysis for directed graphs that remains exact at the algebraic level while explicitly quantifying the geometric distortion induced by non-normality. We (i) define a Biorthogonal Graph Fourier Transform (BGFT) for $L$ using dual left/right eigenbases and show that vertex energy equals a Gram-metric quadratic form in BGFT coordinates, (ii) introduce a directed variational semi-norm $TV_{\mathcal{G}}(x)=\|Lx\|_2^2$ and prove sharp two-sided BGFT-domain bounds controlled by singular values of the eigenvector matrix, and (iii) derive sampling and reconstruction guarantees with explicit stability constants that separate sampling-set informativeness from eigenvector geometry. Finally, we provide reproducible simulations comparing a normal directed cycle to perturbed non-normal digraphs and show that filtering and reconstruction robustness track $κ(V)$ and the Henrici departure-from-normality $Δ(L)$, validating the theoretical predictions.

cs.CE↗

arXiv2026

Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras

Binary idempotent semirings govern classical path algebras. Their multiplicative structure is dyadic. We examine whether this restriction is structural or accidental. We define ternary idempotent $Γ$-semirings as higher-arity ordered algebraic systems admitting associative ternary composition compatible with idempotent addition. We prove that such structures strictly extend classical semiring path algebras. In particular, we construct a ternary associative operation which cannot be represented as an iterated associative binary operation. This establishes non-reducibility. We formulate a higher-order path problem in directed graphs with weights in a ternary idempotent $Γ$-semiring. The associated relaxation operator is shown to be monotone on a complete lattice and to admit a least fixed point. Convergence follows under a finite acyclicity condition. The combinatorial growth of interaction windows yields a distinct complexity class relative to binary path schemes. These results indicate that dyadic semiring frameworks do not exhaust algebraic path formalisms. Higher-arity composition introduces structural phenomena absent in binary systems.

math.RA↗

arXiv2026

Certified Reduced-Order Surrogates and Stability Margins in Viscous Incompressible Flow and Fluid--Structure Interaction

Let $(u,p)$ solve the incompressible Navier--Stokes equations in a regime in which an energy inequality is available and each constant in that inequality is computable from declared data. We construct a reduced-order model $u_n$ constrained so that its discrete evolution satisfies a certified energy inequality. This certificate yields global-in-time boundedness of the ROM energy and a regime-of-validity test that fails when a stated hypothesis fails. It follows that one can attach a computable residual functional $\mathcal{R}_n$ to the ROM trajectory. We prove an a posteriori bound of the form \[ \norm{u-u_n}_{\mathsf{X}(0,T)} \le C(\text{declared data})\,\mathcal{R}_n, \] with $C$ explicit and with $\mathcal{R}_n$ computed from the ROM and the discretization operators. Conversely, if the certificate constraint is relaxed, the bound can fail even for stable full-order dynamics, by an explicit instability mechanism recorded in the text. We then derive transition indicators from rigorous energy and enstrophy budgets in simplified geometries. Each indicator is an inequality involving declared quantities such as forcing norms, viscosity, Poincaré-type constants, and a computable resolvent surrogate. These inequalities provide thresholds that preclude transition, or else certify the presence of transient growth beyond a stated level. Finally, for a class of fluid--structure interaction models, we identify a parameter regime that implies existence and uniqueness of weak solutions. We derive discrete coupled energy estimates that produce computable stability margins. These margins yield explicit constraints on time step and mesh parameters. They are stated as inequalities with constants determined by fluid viscosity, structure stiffness, density ratios, and interface trace bounds.

math.NA↗

arXiv2026

Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation)

Classical spectral graph theory and graph signal processing rely on a symmetry principle: undirected graphs induce symmetric (self-adjoint) adjacency/Laplacian operators, yielding orthogonal eigenbases and energy-preserving Fourier expansions. Real-world networks are typically directed and hence asymmetric, producing non-self-adjoint and frequently non-normal operators for which orthogonality fails and spectral coordinates can be ill-conditioned. In this paper we develop an original harmonic-analysis framework for directed networks centered on the \emph{adjacency} operator. We propose a \emph{Biorthogonal Graph Fourier Transform} (BGFT) built from left/right eigenvectors, formulate directed ``frequency'' and filtering in the non-Hermitian setting, and quantify how asymmetry and non-normality affect stability via condition numbers and a departure-from-normality functional. We prove exact synthesis/analysis identities under diagonalizability, establish sampling-and-reconstruction guarantees for BGFT-bandlimited signals, and derive perturbation/stability bounds that explain why naive orthogonal-GFT assumptions break down on non-normal directed graphs. A simulation protocol compares undirected versus directed cycles (asymmetry without non-normality) and a perturbed directed cycle (genuine non-normality), demonstrating that BGFT yields coherent reconstruction and filtering across asymmetric regimes.

math.RA↗