Currency Bundles: Foreign Exchange as Connection Curvature on the Financial Manifold

ATTRIBUTION NOTICE (2026-08, revised). One element of this framework has substantial prior art that is not cited: the identification of arbitrage with curvature — transport wealth around a closed loop of currencies, and a holonomy different from unity is riskless profit. That is Kirill Ilinski's, developed in Physics of Finance: Gauge Modelling in Non-equilibrium Pricing (Wiley, 2001) and in papers from the late 1990s, together with the reading of prices, exchange rates and discount factors as connection coefficients and of unit changes as gauge transformations.

What is not Ilinski's, and should not be read as a rediscovery. His base space is time crossed with a discrete set of assets, and his programme is dynamical: matter fields carrying capital, a least-action principle, lattice-QED path integrals, with Black–Scholes recovered as the free-field limit and market anomalies arising from interacting matter. The construction here instead builds its base from double-entry accounting — the Pacioli manifold, with ∂² = 0 as the balance-sheet identity — and extends it to physical production via Leontief input–output. Neither the accounting base nor the production extension appears in Ilinski, and the emphasis here is structural rather than dynamical. The double-entry-as-chain-complex reading has its own prior art in David Ellerman's work on the Pacioli group (1986 onward), also uncited.

A revised version citing both, and stating clearly which elements are inherited and which are new, is in preparation.

Five persistent failures of multi-currency financial modelling have resisted a unified explanation: the post-2008 multi-curve interest-rate basis, the persistent violation of covered interest-rate parity (CIP), the anomalous term structure of commodity convenience yields, the cross-asset inconsistency of stress co-movements, and the structural inability of open-economy stock-flow consistent models to treat exchange rates without ad hoc violations of their own conservation laws. This paper shows that all five are corollaries of a single geometric theorem within Financial Gauge Theory (FGT): each is an instance of non-zero curvature — or the absence of a well-defined connection — on a principal fibre bundle whose base space is the <b>Pacioli manifold</b> and whose gauge group is the positive reals $(\mathbb{R}_{>0},\times)$ acting by currency re-denomination.

The Pacioli manifold is the discrete directed graph of institutional accounting flows satisfying $\partial^2 = 0$ (the double-entry constraint). Exchange rates are connection coefficients on currency fibres over this base; numeraire changes are gauge transformations; the risk-neutral measure is the flat-connection gauge.

The central result is the Crown Jewel Theorem: CIP holds if and only if the connection on the currency bundle is flat. Post-2008 CIP violation is therefore persistent curvature. A central-bank currency swap line is a two-level intervention: a topological surgery on the Pacioli manifold (adding a new interbank edge, increasing the first Betti number by one) followed by a local gauge transformation that reduces curvature along the new path. Basel III leverage constraints bound the topology of the Pacioli manifold from below, thereby bounding the minimum achievable curvature and explaining why CIP cannot be fully restored by market forces alone. This is a testable quantitative prediction.

FGT is the first framework to unify the Ilinski–Young FX-as-connection approach with the discrete homological conservation laws of the Pacioli manifold, so that accounting identities and no-arbitrage conditions become dual properties of the same principal bundle. Geman–El Karoui–Rochet numéraire changes are gauge transformations; Flesaker–Hughston positive-interest models are temporal connections native to $(\mathbb{R}_{>0},\times)$. The remaining four empirical anomalies — multi-curve basis, commodity convenience yields, stress co-movements, and SFC exchange-rate inconsistency — are stated as propositions and given sketch proofs.

This is Paper 295 in the FGT series, and the foundational paper of the sequence. Companion papers: Paper 296 (Term Structure Bundles, doi:10.5281/zenodo.20244445), Paper 298 (Credit Bundles, doi:10.5281/zenodo.20257596), Paper 299 (XVA, doi:10.5281/zenodo.20257723), Paper 291 (Pacioli Manifold, doi:10.5281/zenodo.20229938).