Chapter 7: The Hollow Tetrahedron
Four firms. Six bilateral contracts. Four consistent triangles. One irresolvable conflict.
The example
Four institutions: a bank (A), an insurer (B), a money market fund (C), and a hedge fund (D). Each pair has a bilateral contract — six edges forming a complete graph \(K_4\).
Each triangle is internally consistent: - {A, B, C}: A's swap with B, B's repo with C, C's commercial paper held by A — all consistent - {A, B, D}: consistent - {A, C, D}: consistent - {B, C, D}: consistent
Every bilateral stress test passes. Every triangle closes. And yet: the four triangles together form a hollow tetrahedron — a 2-sphere with no 3-simplex filling it. There is no single settlement mechanism that can resolve all six bilateral claims simultaneously, because the four triangular faces are consistent individually but mutually contradictory as a whole.
This is \(H^2 \neq 0\).
What makes it hollow
A filled tetrahedron has a 3-simplex inside it: a four-way settlement mechanism that can resolve all claims at once. Think of a CCP that clears all four firms across all products simultaneously — the CCP is the interior point.
A hollow tetrahedron has no such interior. The four firms each have bilateral close-out rights, but exercising A's rights against B interferes with C's rights against B, which interferes with D's rights against C. The claims are mutually inconsistent. No sequence of bilateral settlements resolves them all; some creditor always ends up with less than their bilateral contract entitles them to.
The technical condition: the tetrahedron is hollow iff the 2-cycle formed by the four triangular faces is not a boundary — it does not bound any 3-chain in \(\Gamma\). This is precisely the statement that it represents a non-trivial element of \(H^2(\Gamma)\).
Why bilateral close-out fails
When Lehman Brothers failed in September 2008, its prime brokerage clients (hedge funds) held assets at LBIE (the UK entity). Those assets had been rehypothecated: lent out to other counterparties by Lehman as collateral. The four parties — Lehman, the hedge funds, Barclays (which acquired parts of the US business), and the LBIE administrators — held mutually inconsistent claims.
Each bilateral claim was valid. Each triangular sub-claim was consistent. But the four-party structure was a hollow tetrahedron: the LBIE administration lasted over a decade because there was no interior point — no settlement mechanism that could satisfy all claims simultaneously.
This is \(\beta_2 > 0\). The Impossibility Theorem says it could not have been resolved by any netting or clearing arrangement operating on the individual bilateral contracts. The only resolution was exogenous: a court-supervised process that imposed a settlement from outside the network.
Measuring \(\beta_2\)
\(\beta_2(\Gamma)\) counts the number of independent hollow tetrahedra in the obligation complex. Each one represents an irresolvable four-party conflict — a potential Lehman LBIE situation.
A system with \(\beta_2 = 0\) can always be resolved by some combination of bilateral close-out and multilateral netting. A system with \(\beta_2 > 0\) cannot: some residual will always remain. The size of \(\beta_2\) measures the irreducible systemic complexity — the part of systemic risk that no private arrangement can eliminate.
The SIFI theorem
An institution \(i\) is a systemically important financial institution (SIFI) in the topological sense if \(\Delta\beta_2(i) = \beta_2(\Gamma) - \beta_2(\Gamma \setminus \{i\}) > 0\): removing \(i\) from the network reduces \(\beta_2\).
This is a precise, model-free definition. It does not depend on size, leverage, or any parametric model of contagion. An institution is systemically important if and only if it is a vertex of a hollow tetrahedron — if its obligations complete an irresolvable four-party conflict.
Under this definition, size is neither necessary nor sufficient for SIFI status: a small institution that bridges two otherwise-disconnected subnetworks may have \(\Delta\beta_2 > 0\), while a very large institution operating entirely within a well-connected subnetwork may have \(\Delta\beta_2 = 0\).