The Pacioli Combinator Library: A Universal Domain-Specific Language for Financial and Economic Computation on the Pacioli Manifold

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Abstract

Peyton Jones, Eber, and Seward [1] identified a structural problem in financial engineering: practitioners maintained a large catalogue of hand-coded contract types with no compositional structure and no formal semantics, making systematic analysis impossible. Their solution—a combinator library with a denotational valuation semantics embedded in Haskell—is the direct intellectual ancestor of this paper.

The same problem recurs across every application domain built on the Pacioli manifold: derivative pricing, accounting consolidation, credit risk, XVA, regulatory capital, agent-based simulation, and climate risk modelling each maintain their own isolated catalogue of hand-coded objects, with no shared primitives and no common language.

We introduce the Pacioli Combinator Library (PCL): a Python embedded domain-specific language (eDSL) that is the universal expression language for Financial and Economic Gauge Theory (FGT/EGT, Papers 291–305). PCL is to the Pacioli manifold what SQL is to the relational model: a single compositional language whose terms compile, under domain-specific models, to computations appropriate for pricing, accounting, risk, simulation, or optimisation—from one codebase, all differentiable. PCL provides:

1. A compositional denotational semantics $\Phi(\cdot )$ mapping Behaviour terms to flow processes on the Pacioli manifold, with the $\partial^2 = 0$ bookkeeping invariant guaranteed by construction at every combinator;

2. The choose$(\beta, \dots)$ combinator: a Maslov–Gibbs softmax over alternatives, differentiable at all finite $\beta$ and recovering classical winner-take-all choice in the limit $\beta \to \infty$;

3. An observable layer carrying the full EGT connection coefficient vocabulary: yield, credit spread, FX holonomy, off-diagonal climate curvature, and carbon liability (Papers 295–300);

4. A JAX compiler translating any well-formed PCL term into a differentiable JAX computation graph under a caller-supplied Model, enabling gradient-based optimisation via jax.grad across all application domains.

The same PCL term zcb(t, 100, GBP, edge) compiles to a discounted cash flow under a PricingModel, to a balance sheet entry under an AccountingModel, and to a credit-adjusted exposure under a RiskModel. The design follows Peyton Jones et al. exactly in spirit: identify the minimal set of orthogonal primitives, give each a precise denotational semantics, and let the algebra emerge. The choose combinator, the EGT observable layer, and the multi-domain model architecture are the three primary contributions relative to their contract DSL.