The Unhedgeability Theorem: Sheaf Cohomology of Interaction Diagrams and the Topology of Financial Risk

We give a topological classification of financial risk. A risk factor propagating on an interaction diagram Γ — a simplicial complex whose vertices are instruments and whose edges are pairwise exposures — is assigned a sheaf $\mathcal{F}$ recording the local pricing data at each vertex and the consistency conditions between them. The Čech cohomology of $\mathcal{F}$ stratifies risk into three levels: $H^0$ (bilateral risk, always locally consistent), $H^1$ (triangular risk, the first obstruction to a globally consistent pricing scheme), and $H^2$ (systemic risk, a secondary obstruction that vanishes whenever a no-arbitrage condition holds).

We work this out in full for interest rate risk. The discount factor sheaf on the time axis is exact — $H^1 = 0$ — in a deterministic world. Once rates are stochastic, Itô's lemma produces a non-trivial $H^1$ class: the Heath–Jarrow–Morton (HJM) convexity adjustment $$A(t,T) = \sigma(t,T)\int_t^T \sigma(t,u),du$$ is exactly the value of this cocycle on the elementary triangle $(t,s,T)$, and the HJM no-arbitrage condition is exactly the vanishing of the corresponding $H^2$ obstruction.

This gives the paper's central result, the Unhedgeability Theorem: a risk factor is hedgeable with a finite portfolio of bilateral instruments (forwards, futures, vanilla swaps) if and only if its Čech class is trivial in $H^1(\Gamma, \mathcal{F})$. Convexity risk, basis risk, smile/skew risk, and CDO correlation risk are all non-trivial $H^1$ classes, and are therefore topologically — not merely practically — unhedgeable with bilateral instruments. This is proved directly from the Čech complex and holds for any sheaf on any interaction diagram; it requires no group representation theory and no 6j-symbol or recoupling combinatorics of any kind.

Finally, the Origami ISA opcodes (SPLIT, SPLAT, FLIP, FLOP, TWIST) are identified with the Čech cohomology operations on $\mathcal{F}$ — SPLIT is the coboundary map $\delta^0$, SPLAT is fibre integration back to $H^0$, and the Pentagon identity is $\delta^1\circ\delta^0=0$ — giving the theorem a direct computational realisation. The paper is explicit about the scope of this identification (§5.4): in the two-instrument case worked out here, SPLIT/SPLAT classify a quantity already derived from Itô calculus, rather than computing it in a new way — a contrast with the non-abelian, spin-foam instance of the same opcodes (companion paper, Spin Foams as Origami, doi:10.5281/zenodo.20680634), where they evaluate 6j/15j amplitudes that have no closed form otherwise.