arXiv subjects

Abdul Rahman

Publications and source records attributed to Abdul Rahman.

At least 19 recordsLinked to original sources

arXiv2026

Cyber Dynamics I: Finite Macrostates for Behavioral Anomaly Detection in Network Telemetry

Entropy-based methods have long been used for network anomaly detection, but most existing approaches treat entropy as a scalar statistic on narrow observables rather than as part of a broader behavioral state-space for cyber systems. We propose a finite-dimensional macrostate framework for network telemetry, instantiated over the Canonical Security Telemetry Substrate (CSTS), so that coarse-graining is performed over persistent entities, typed relations, and temporal state rather than isolated event records. The resulting macrostate captures activity, distributional disorder, structural organization, temporal volatility, persistence, and deviation from benign baselines. Rather than scoring only unusual states, we model window-to-window macrostate transitions and define regime structure, stability, and anomalous change. This supports discrimination between benign workload drift and adversarial reorganization. We evaluate the framework on benchmark network telemetry datasets and compare it against Shannon-, R\'enyi-, and Tsallis-style entropy baselines, as well as standard anomaly detectors. The proposed representation improves anomaly discrimination and supports more interpretable behavioral analysis of cyber telemetry.

cs.CR↗

arXiv2026

Towards an Agent-First Web: Redesigning the Web for AI Agents

The World Wide Web was built on an assumption held for three decades: the primary consumer of web content is a human being. This permeates every layer; its access model presumes human visitors, its economics rest on human attention, and its content targets human perception. The rapid emergence of AI agents as intermediaries between humans and web content invalidates this assumption. Yet the web resists agents through blanket blocking, CAPTCHA-based exclusion, and economic models that treat agent access as extraction rather than legitimate interaction. This paper proposes a principled redesign across three layers. At the access layer, agents acting for humans should inherit equivalent access rights, governed by rate limiting and agent identification metadata in HTTP requests, analogous to browser headers, alongside a dual-layer architecture serving human-readable and agent-optimized content from the same domain. At the economic layer, we propose an intent-based tier framework grounded in the agent-as-human-proxy principle: an agent's economic obligation mirrors that of the human it represents. A token-based subscription model meters content in tokens rather than pageviews, alongside a commissioned content economy anchoring AI content production in human intentionality. At the content layer, we identify epistemic recursion, the self-referential loop in which AI-generated content is consumed by agents to produce further content, progressively detaching web knowledge from human ground truth. We propose the Agent Text Markup Language (ATML), a four-level human supervision tier model, and a cryptographic provenance chain to counter this threat. Together these constitute ten design principles for an agent-first internet, one in which agents are first-class citizens whose integration requires renegotiating the web's foundational social contract across access, economics, and content.

cs.AI↗

arXiv2026

From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples

We reinterpret Diaz's construction of Chow-trivial smooth projective varieties violating the integral Hodge conjecture as the level-two case of an $n$-fold cup-product Bockstein mechanism. Diaz's dimension-four example is $V=S_1\times S_2$, where $S_1,S_2$ are Enriques surfaces, and its obstruction is the Bockstein of $\pi_1^*\alpha_1\cup\pi_2^*\beta_2\in H^3(V,\mathbb Z/2(2))$. Here $\alpha_1$ is the K3 double-cover class and $\beta_2$ is an Enriques Brauer-detecting class. We extend the finite-coefficient source construction to $X_n=S_1\times\cdots\times S_n$ by forming $\Theta_n=\pi_1^*\alpha_1\cup\pi_2^*\beta_2\cup\cdots\cup\pi_n^*\beta_n \in H^{2n-1}(X_n,\mathbb Z/2(n))$,with Bockstein $\Delta_n=\delta(\Theta_n)\in H^{2n}(X_n,\mathbb Z(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson--Vilonen boundary, we prove unconditionally that $\Delta_n$ has nonzero image in a distinguished Enriques--Brauer component of the MV obstruction channel. Under the Brauer-separation hypothesis, which asserts that algebraic codimension-$n$ cycle classes have zero image in this same component, the class $\Delta_n$ is a non-algebraic $2$-torsion integral Hodge class. We verify this separation for decomposable algebraic cycles and reduce the remaining non-decomposable case, via integral even Chow--K\"unneth projectors on the Enriques factors, to a single coefficient-level algebraic-control problem involving the $H^1(S_1,\mathbb Z/2(1))$ Enriques double-cover direction. We also record a motivic finite-coefficient lift of the tower via the finite-coefficient cone $\mathbf 1_X(n)/2:=\operatorname{Cone}(\mathbf 1_X(n)\xrightarrow{\times 2} \mathbf 1_X(n))$, and explain which formal part of the MacPherson--Vilonen zig-zag construction lifts motivically under Betti realization.

math.AG↗

arXiv2026

From Finite-Node Conifold Geometry to BPS Structures III: Mediated Triangle Transport and Graded Interaction Data

In previous work, we extracted from a finite-node conifold degeneration the state-data package $A_\Sigma=(V_\Sigma,E_\Sigma,c_\Sigma)$ and then constructed the support-level interaction package encoded by a binary incidence structure and finite quiver-theoretic skeleton \cite{RahmanQuiverDataI,RahmanQuiverDataII}. The present paper introduces the next layer: a graded pairwise interaction package refining binary support. Since the support matrix records where a mediated channel is present, but not its derived size, cohomological degree, or exact-triangle behavior, we introduce \emph{mediated triangle transport} (MTT). An MTT datum combines bulk-mediated schober transport, localized probes, corrected-extension shadow compatibility, and derived interaction profunctors. For each ordered pair $(i,j)$, it produces $\mathbb T_{ij}(X,Y):=\RHom_{\mathcal C_{p_j}}(\Psi_j\Phi_i(X),Y)$ and the probe interaction complex $\mathsf H_{ij}:=\mathbb T_{ij}(L_i,L_j)=\RHom_{\mathcal C_{p_j}}(\Psi_j\Phi_i(L_i),L_j)$. We prove exactness and long exact interaction sequences, isolate a triangle-visible nonvanishing criterion, and formulate a conditional bridge theorem showing that supported channels yield nontrivial pairwise interaction complexes under the stated probe, content, and detector hypotheses. Under a bounded Hom-finite convention, the cohomology of $\mathsf H_{ij}$ defines $P_{ij}(q)=\sum_m \dim H^m(\mathsf H_{ij})q^m$, and these polynomials assemble into $I_\Sigma^{\mathrm{gr}}$. Thus $(A_\Sigma,I_\Sigma^{(0/1)},I_\Sigma^{\mathrm{gr}})$ provides the first graded interaction input for later stability, BPS, and wall-crossing theory.

math.AG↗

arXiv2026

Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds

We prove a projection-triangle statement for projective Calabi--Yau threefold conifold degenerations and use it to organize an intersection-space Hodge atom shadow package. For a nodal central fiber $X_0$, assume the relevant Banagl--Budur--Maxim, multi-node gluing, mixed-Hodge-module, and specialization-splitting hypotheses, so that $\psi_\pi(F)\simeq \mathcal{IS}^{H}_{X_0}\oplus\mathcal C^H_\Sigma$. Projection of the variation morphism to the intersection-space summand defines $\operatorname{var}_I:\phi_\pi(F)\to\mathcal{IS}^{H}_{X_0}$, and the octahedral axiom gives $P^H\to P^H_I\to\mathcal C^H_\Sigma\xrightarrow{+1}$, where $P^H=\operatorname{Cone}(\operatorname{var})[-1]$ and $P^H_I=\operatorname{Cone}(\operatorname{var}_I)[-1]$. This realizes the intersection-space atom shadow package $\mathsf{HA}^{I}(X_0)$ and compares it with the intersection-homology package $\mathsf{HA}^{IH}(X_0)$. Under the self-dual specialization-splitting hypothesis, the projected variation object satisfies $\mathbb D P^H_I\simeq Q^H_I(3)$, where $Q^H_I=\operatorname{Cone}(\operatorname{can}_I)[-1]$. Under the mixed-Hodge realization of Banagl's middle exact sequence, we isolate a rigid--vanishing filtration and identify the IIB vanishing atom with the realized kernel. For the classical $125$-node quintic, the middle-degree IC--intersection-space defect has rank $202$. The construction remains at Hodge-realization level and identifies $\mathcal C^H_\Sigma$ and $\Delta_{I/IC}(X_0)$ as geometry-side handoff objects for future DT/BPS comparisons.

math.AG↗

arXiv2026

Torsion Trajectories from Local Discriminants to Global Obstructions

For a normal surface singularity, the discrepancy between the ordinary and dual middle-perversity intersection complexes over $\mathbb Z$ is measured by a finite group $E$. In previous work, $E$ was identified with link torsion, the exceptional-lattice discriminant group $\Lambda^\vee/\Lambda$, a resolution-neighborhood boundary quotient, and, in the hypersurface case, $\operatorname{coker}(T-\mathrm{id})_{\mathrm{tors}}$. This paper tracks the trajectory of this torsion from local singularity data to global obstruction theory. We follow the discriminant package $(E,q)$ through support cohomology, excision, global torsion, Brauer comparison, Bloch--Ogus residues, and rationalization. The method is example-driven: trajectory tables are computed for $A_1$, $A_k$, $D_4$, $E_8$, a non-ADE Brieskorn singularity, the threefold ordinary double point, nodal threefolds, nodal quintics, and the Benoist--Ottem benchmark. The computations reveal a sharp distinction: a surface $A_1$ singularity has local $\mathbb Z/2$-torsion, whereas a threefold ordinary double point has torsion-free link $S^2\times S^3$ and contributes free vanishing-cycle data instead. Thus finite discriminant torsion is naturally a codimension-two phenomenon, not a generic feature of nodes. The resulting pattern motivates a specialization problem: whether the Enriques $2$-torsion in Benoist--Ottem integral Hodge counterexamples is genuinely global, or can arise after degeneration from transverse $A_1$-type discriminant data along codimension-two strata.

math.AG↗

arXiv2026

Think Before You Act -- A Neurocognitive Governance Model for Autonomous AI Agents

The rapid deployment of autonomous AI agents across enterprise, healthcare, and safety-critical environments has created a fundamental governance gap. Existing approaches, runtime guardrails, training-time alignment, and post-hoc auditing treat governance as an external constraint rather than an internalized behavioral principle, leaving agents vulnerable to unsafe and irreversible actions. We address this gap by drawing on how humans self-govern naturally: before acting, humans engage deliberate cognitive processes grounded in executive function, inhibitory control, and internalized organizational rules to evaluate whether an intended action is permissible, requires modification, or demands escalation. This paper proposes a neurocognitive governance framework that formally maps this human self-governance process to LLM-driven agent reasoning, establishing a structural parallel between the human brain and the large language model as the cognitive core of an agent. We formalize a Pre-Action Governance Reasoning Loop (PAGRL) in which agents consult a four-layer governance rule set: global, workflow-specific, agent-specific, and situational before every consequential action, mirroring how human organizations structure compliance hierarchies across enterprise, department, and role levels. Implemented on a production-grade retail supply chain workflow, the framework achieves 95% compliance accuracy and zero false escalations to human oversight, demonstrating that embedding governance into agent reasoning produces more consistent, explainable, and auditable compliance than external enforcement. This work offers a principled foundation for autonomous AI agents that govern themselves the way humans do: not because rules are imposed upon them, but because deliberation is embedded in how they think.

cs.AI↗

arXiv2026

Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy

For a germ $(X,0)$ of a normal complex analytic surface, let $E:=H^0({}^p_+IC_X\mathbb Z)_0$, where ${}^pIC_X\mathbb Z$ and ${}^p_+IC_X\mathbb Z$ denote the ordinary and dual middle-perversity intersection complexes with integral coefficients. This finite abelian group measures the integral discrepancy between the two middle extensions. Motivated by work of Jung--Saito, we study $E$ as a local invariant of the singularity. We prove that $E$ admits a topological realization as $H^2(L,\mathbb Z)_{\tors}$, where $L$ is the link of the singularity, and a geometric realization as the discriminant group of the exceptional lattice of the minimal resolution. In particular, if $M$ is the intersection matrix of the irreducible exceptional curves, then $|E|=|\det(M)|$. If $(X,0)$ is an isolated hypersurface surface singularity, we further prove that $E\cong \coker(T-\id)_{\tors}$, where $T$ is the Milnor monodromy on integral vanishing cohomology. Under the additional hypothesis that $(T-\id)\otimes_{\mathbb Z}\mathbb Q$ is an isomorphism, this yields $|E|=|\det(T-\id)|$. Thus the same local integral obstruction admits compatible perverse, topological, resolution-theoretic, and monodromy-theoretic realizations.

math.AG↗

arXiv2026

From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly

In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma)$, where $V_\Sigma$ is the finite node-indexed vertex set, $E_\Sigma$ is the nodewise coupling space, and $c_\Sigma$ is the coefficient vector of the corrected global extension class. The purpose of the present paper is to construct the corresponding interaction and incidence layer. Starting from the finite-node schober package $S_\Sigma := (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{\Phi_k,\Psi_k\}_{k=1}^r,Sh(S_\Sigma))$, we define the extended vertex set $V_\Sigma^{\mathrm{ext}} := V_\Sigma \sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_\Sigma := (V_\Sigma^{\mathrm{ext}},\rightsquigarrow_\Sigma)$, and its canonical binary decategorification $\mathcal I_\Sigma := (V_\Sigma^{\mathrm{ext}},I_\Sigma)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma,\mathcal F_\Sigma,I_\Sigma)$, where $\mathcal F_\Sigma := \{(\Phi_k,\Psi_k)\}_{k=1}^r$ is the functorial coupling datum. We prove that this package is canonically determined by the finite-node schober datum, compatible with the corrected perverse extension and its mixed-Hodge-module refinement, and invariant under equivalence of finite-node schober realizations. This yields the interaction and incidence layer required for later graded interaction, stability, BPS, and wall-crossing structures.

math.AG↗

arXiv2026

Interacting Multi-Node Conifold Light Sectors

We study finite-node conifold degenerations of Calabi--Yau threefolds from the point of view of interacting light sectors. Although each ordinary double point contributes a rank-one local vanishing sector, the corrected global object need not assemble as a freely independent sum of nodewise pieces. Using the corrected perverse and mixed-Hodge-module degeneration package, the global gluing law for corrected extension classes, and the rigid-flexible atom decomposition on the $F$-bundle side, we define an interacting multi-node light-sector package and prove a block-reduced structure theorem. In the block-separated cycle family, the finite-node package separates into two logically distinct layers: relation collapse, controlled by a common relation lattice on the corrected-extension, smoothing, and resolution sides, and residual interaction among the surviving global sectors, controlled by a reduced block interaction matrix on the transport and atom sides. The result isolates the geometric and Hodge-theoretic precursor of coupled conifold light states and provides the mathematical input for a later multi-node reformulation of Strominger's conifold mechanism.

math.AG↗

arXiv2026

From Finite-Node Conifold Geometry to BPS Structures I: Algebraic State Data

Let $\pi:X\to \Delta$ be a one-parameter degeneration whose central fiber $X_0$ is a complex threefold with finitely many ordinary double points $\Sigma=\{p_1,\dots,p_r\}\subset X_0$. Associated with this degeneration is the corrected finite-node perverse extension, together with its mixed-Hodge-module refinement and a finite-node schober datum whose perverse-sheaf shadow is identified with the corrected perverse sheaf $\mathcal P$. The purpose of the present paper is to extract from these finite-node geometric, extension-theoretic, mixed-Hodge, and categorical inputs the intrinsic algebraic state data carried by the degeneration. More precisely, we isolate the finite localized quotient $Q_\Sigma:=\bigoplus_{k=1}^r i_{k*}\Q_{\{p_k\}}$, the nodewise coupling space $E_\Sigma:=\Ext^1_{\Perv(X_0;\Q)}(Q_\Sigma,IC_{X_0})$, its canonical nodewise decomposition $E_\Sigma\cong\bigoplus_{k=1}^r \Q e_k$, and the coefficient vector $c_\Sigma=(c_1,\dots,c_r)\in\Q^r$ defined by $[\mathcal P]_{\mathrm{perv}}=\sum_{k=1}^r c_k e_k$. We then prove that these state variables are compatible with both the mixed-Hodge-module lift and the schober realization of $\mathcal P$, so that the same finite-node architecture appears simultaneously in perverse, mixed-Hodge, and categorical form. The resulting package $(V_\Sigma,E_\Sigma,c_\Sigma)$ is the intrinsic algebraic state data attached to the finite-node conifold degeneration. It provides the first algebraic layer in the passage from finite-node geometry to later incidence, quiver, stability, BPS-spectral, and wall-crossing structures.

math.AG↗

arXiv2026

Hodge Atoms at Conifold Degenerations: F-Bundles, Limiting Mixed Hodge Modules, and the Rigid-Flexible Decomposition

We extend the Hodge atoms framework of Katzarkov--Kontsevich--Pantev--Yu to one-parameter conifold degenerations of Calabi--Yau threefolds. For a degeneration $\pi\colon X \to \Delta$ whose central fiber $X_0$ has $r$ ordinary double points, we construct a canonical rigid-flexible decomposition of the Hodge atoms of the nearby smooth fiber attached to the corrected degeneration object. The rigid atom $A(\IC^H_{X_0})$ is preserved across the degeneration, while the flexible atoms $A(i_{k*}\QQ^H_{\{p_k\}}(-1))$ are rank-one contributions, one for each vanishing cycle. The total degeneration atom $A(P^H)$ is the atom of the corrected mixed Hodge module $P^H\in\MHM(X_0)$ and fits into an exact sequence of atoms whose non-split structure is controlled by the intersection matrix $(\langle\delta_i,\delta_j\rangle)$. The technical core is the Stokes--Extension Identification, which identifies the Stokes matrix of the Dubrovin connection at the conifold locus with the matrix of the variation morphism $\varF\colon\varphi_\pi(F) \to \psi_\pi(F)$ under mixed Hodge module realization.

math.AG↗

arXiv2026

Cycle Relations and Global Gluing in Multi-Node Conifold Degenerations

We study projective one-parameter conifold degenerations whose central fiber has finitely many ordinary double points. Existing finite-node theory isolates one rank-one local sector per node on the perverse-sheaf, mixed-Hodge-module, and categorical sides, but does not determine which global extension classes are actually realized by geometry. We show that when the nodes are linked by common cycle geometry or homological relations, the corrected extension is not free nodewise data, but is forced into a smaller relation-controlled subspace. To formalize this, we introduce a cycle-node incidence datum and the associated geometrically realized subspace of the ambient nodewise extension space. Under geometrically admissible and block-adapted incidence hypotheses, we prove that the corrected perverse extension factors through this subspace, with incidence compatibility derived from propagation of local variation data along admissible cycle components, and we show that the same relation law lifts compatibly to the mixed-Hodge-module setting. We then compare this relation law with the resolution and smoothing sides and, in the block-separated cycle family, obtain $R_{\mathrm{res}}=R_{\mathrm{sm}}=R_{\mathrm{ext}}=R_{\mathrm{blk}}$.

math.AG↗

arXiv2026

GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural K\"ahler Potentials

We present \emph{GlobalCY}, a JAX-based framework for globally defined and symmetry-aware neural K\"ahler-potential models on projective hypersurface Calabi--Yau geometries. The central problem is that local-input neural K\"ahler-potential models can train successfully while still failing the geometry-sensitive diagnostics that matter in hard quartic regimes, especially near singular and near-singular members of the Cefal\'u family. To study this, we compare three model families -- a local-input baseline, a globally defined invariant model, and a symmetry-aware global model -- on the hard Cefal\'u cases $\lambda=0.75$ and $\lambda=1.0$ using a fixed multi-seed protocol and a geometry-aware diagnostic suite. In this benchmark, the globally defined invariant model is the strongest overall family, outperforming the local baseline on the two clearest geometric comparison metrics, negative-eigenvalue frequency and projective-invariance drift, in both cases. The gains are strongest at $\lambda=0.75$, while $\lambda=1.0$ remains more difficult. The current symmetry-aware model improves projective-invariance drift relative to the local baseline, but does not yet surpass the plain global invariant model. These results show that global invariant structure is a meaningful architectural constraint for learned K\"ahler-potential modeling in hard quartic Calabi--Yau settings.

hep-th↗

arXiv2026

Finite-Node Perverse Schobers and Corrected Extensions for Conifold Degenerations

We study one-parameter conifold degenerations whose central fiber has finitely many ordinary double points. Working within a deliberately minimal finite-node bulk/localized-sector formalism, we identify the first categorical layer suggested by the corrected finite-node perverse extension and its mixed-Hodge-module package. Assuming that the local ordinary-double-point coupling pattern admits categorical realization in this finite-node setting, we formalize the corresponding local and finite-node data over a chosen bulk category, prove compatibility of their specified shadows with the corrected finite-node perverse extension established in earlier work, and isolate one localized categorical sector per node. We also extract a first finite combinatorial skeleton encoding the nodewise coupling pattern. The paper does not claim a universal perverse-schober theory for arbitrary singular Calabi--Yau degenerations, nor a categorical wall-crossing theory. Rather, it provides the foundational finite-node categorical formalization layer above the corrected perverse and mixed-Hodge-module packages in the conifold degeneration.

math.AG↗

arXiv2026

Mixed Hodge Modules and Canonical Perverse Extensions for Multi-Node Conifold Degenerations

We study one-parameter conifold degenerations whose central fiber has finitely many ordinary double points and construct a mixed-Hodge-module refinement of the canonical corrected perverse object associated with the degeneration. We build a rank-one point-supported mixed-Hodge-module block at each node, identify the global singular quotient as $\bigoplus_{k=1}^r i_{k*}\Q^H_{\{p_k\}}(-1)$, and assemble these local blocks via Saito's divisor-case gluing formalism into a global object $\mathcal P^H \in MHM(X_0)$. We prove that $\mathcal P^H$ realizes the corrected perverse object, fits into an exact sequence $0 \to IC^H_{X_0} \to \mathcal P^H \to \bigoplus_{k=1}^r i_{k*}\Q^H_{\{p_k\}}(-1) \to 0$, and that the same quotient realizes the finite local vanishing sector in the nearby-cycle formalism. We further relate the mixed-Hodge-module extension, its realized perverse extension, and the induced extension on hypercohomology carrying the limiting mixed Hodge structure. This gives a theorem-level Hodge-theoretic refinement of the corrected perverse extension in the finite multi-node ordinary double point setting.

math.AG↗

arXiv2026

Flowr -- Scaling Up Retail Supply Chain Operations Through Agentic AI in Large Scale Supermarket Chains

Retail supply chain operations in supermarket chains involve continuous, high-volume manual workflows spanning demand forecasting, procurement, supplier coordination, and inventory replenishment, processes that are repetitive, decision-intensive, and difficult to scale without significant human effort. Despite growing investment in data analytics, the decision-making and coordination layers of these workflows remain predominantly manual, reactive, and fragmented across outlets, distribution centers, and supplier networks. This paper introduces Flowr, a novel agentic AI framework for automating end-to-end retail supply chain workflows in large-scale supermarket operations. Flowr systematically decomposes manual supply chain operations into specialized AI agents, each responsible for a clearly defined cognitive role, enabling automation of processes previously dependent on continuous human coordination. To ensure task accuracy and adherence to responsible AI principles, the framework employs a consortium of fine-tuned, domain-specialized large language models coordinated by a central reasoning LLM. Central to the framework is a human-in-the-loop orchestration model in which supply chain managers supervise and intervene across workflow stages via a Model Context Protocol (MCP)-enabled interface, preserving accountability and organizational control. Evaluation demonstrates that Flowr significantly reduces manual coordination overhead, improves demand-supply alignment, and enables proactive exception handling at a scale unachievable through manual processes. The framework was validated in collaboration with a large-scale supermarket chain and is domain-independent, offering a generalizable blueprint for agentic AI-driven supply chain automation across large-scale enterprise settings.

cs.AI↗

arXiv2026

Perverse Extensions and Limiting Mixed Hodge Structures for Conifold Degenerations

Let $pi:X\to\Delta$ be a one-parameter degeneration whose central fiber $X_0$ has a single ordinary double point. The nearby- and vanishing-cycle formalism determines a canonical perverse sheaf on $X_0$, obtained from the variation morphism and fitting into an extension of the intersection complex by a point-supported rank-one contribution. We study this object from the perspective of limiting mixed Hodge theory and Saito's theory of mixed Hodge modules. In the ordinary double point case, we show that the corrected perverse object is the unique minimal Verdier self-dual perverse extension of the shifted constant sheaf across the node, and that its rank-one singular contribution and the corresponding rank-one vanishing contribution in the limiting mixed Hodge structure arise from the same nearby-cycle formalism. We also formulate the analogous structural statements for multi-node degenerations and for more general stratified singular loci. Finally, we explain how Saito's divisor-gluing formalism provides the natural framework for a fuller mixed-Hodge-module refinement of these constructions.

math.AG↗