Wesley Phoa | Searcharxiv

arXiv2026

A Double Categorical Framework for Multi-Stage Portfolio Construction and Alignment

We construct a thin double category HS (Hub-and-Spoke) whose objects are closed subsets of standard simplices, horizontal morphisms are continuous maps representing portfolio re-implementation processes, and vertical morphisms are closed relations representing alignment constraints. This framework models industrial portfolio construction pipelines -- hierarchical structures in which a single investment strategy is translated through multiple stages into thousands of client portfolios. We establish four structural theorems: compositionality of alignment (functoriality), a pre-trade safety guarantee (adjunction), an order-independence result for compliance checking (lax Beck--Chevalley), and a filter-commutation law (Frobenius reciprocity). The topological requirement that permissible portfolio spaces be closed and compact -- ruling out ``phantom portfolios'' that arise from open constraint specifications -- is shown to be essential for coherence. Extensions to set-valued re-implementations via the Double Operadic Theory of Systems, stochastic re-implementations via Markov kernels on Polish spaces, and transport-based safety metrics via Wasserstein distances are developed. An abstract axiomatic treatment identifies the equipment axioms sufficient for the main results. The mathematical content is elementary -- no novel category theory is required. The contribution is the modelling claim: that these particular objects and morphisms formalise portfolio re-implementation correctly.

math.CT↗

arXiv2026

Decorated Cospans at Work: Coupling Heterogeneous Dynamical Systems via Pushouts and Particle Filters

Decorated cospans provide a categorical framework for composing open systems along shared interfaces. This paper is a computational proof of concept: we show that the framework produces a working coupled dynamical system when the decorations are quantitative models from different mathematical traditions. Specifically, we couple a linearised New Keynesian DSGE, a stochastic compartmental epidemic (multi-strain SEIR), and a nonlinear vaccine adoption model with hysteresis into a single sequential Monte Carlo sampler. Each model is a decorated cospan -- interior dynamics as decoration, exposed variables as interfaces. The composite system is the pushout along variable identifications, with coupling functions encoded as factor graph constraints. The coupled system produces a rejection bifurcation: some trajectories escape via vaccination, others enter a self-reinforcing cycle of mandate backlash, vaccine refusal, sustained infection, and recession. This is a structural property of the coupling, not an input assumption. Coupling shifts the output gap by 0.78 pp and rejection by 22 pp relative to the uncoupled system. A fourth narrative -- fiscal/political dynamics, calibrated to the US COVID fiscal response -- attaches via a second pushout and introduces the first positive coupling channel. With pandemic-scale spending parameters, 14% of trajectories overshoot into positive output gap territory; the bearish bias shrinks, but persists. A computable bias decomposition separates three sources of this asymmetry -- sampling, structural, and observational -- and localises the structural component to specific coupling functions whose directional asymmetry can be tested against historical analogues.

math.CT↗