Glossary
Key terms used across the EconIAC framework. Entries cover the economic and financial concepts specific to EconIAC. For the underlying mathematical foundations — Fano plane, octonions, \(G_2\), Origami ISA opcodes, pentagon identity — see the ASA Glossary.
Arbitrage
In EconIAC, arbitrage is not merely a trading opportunity — it is the geometric signature of a non-zero curvature on the Pacioli manifold. A triangular FX inconsistency (USD → EUR → GBP → USD ≠ 1) is a non-zero holonomy on the currency bundle: parallel transport around the triangle fails to return to the starting point.
No arbitrage ↔ flat connection ↔ zero curvature on the Pacioli manifold.
See: Paper 295 (Currency Bundles), Paper 300 (Economic Gauge Theory)
Bilateral Risk (H⁰)
Bilateral risk is the zeroth level of the cohomological risk hierarchy: can each individual counterparty meet its obligations?
Formally: \(H^0(\Gamma, \mathcal{F})\) = the space of globally consistent sections of the pricing sheaf — the set of valuations under which every bilateral exposure is exactly matched. A non-zero \(H^0\) obstruction means at least one counterparty cannot pay.
Bilateral risk is manageable by bilateral netting and standard credit risk models (PD, LGD, EAD). It is the only level that pre-2008 models measured.
See: Paper 398 (Topology of Risk primer)
β₁ (Independent Funding Loops)
\(\beta_1\) is the first Betti number of the exposure graph \(\Gamma\) — the rank of \(H^1(\Gamma)\), computed as \(\beta_1 = m - n + c\) where \(m\) is the number of bilateral exposures (edges), \(n\) is the number of institutions (nodes), and \(c\) is the number of connected components.
"Independent" is meant in the homological sense — a basis for the cycle
space modulo boundaries — not as a separate financial criterion that
needs translating after the fact. Two loops that look different as lists
of edges can be homologically dependent (one equals the other plus a
boundary, i.e. plus ordinary bilateral exposures); only a maximal set of
loops with no such relation among them counts toward \(\beta_1\). See the
worked example in docs/why/cohomology.md for a
concrete 4-node case distinguishing dependent from independent loops.
\(\beta_1\) does not depend on edge direction beyond the sign convention used to build the boundary map: a "loop" need not be consistently oriented (e.g. \(A\to B\to C \to A\) and \(A \to B \to C \leftarrow A\) both count, the latter via a \(-1\) coefficient on the reversed edge) — what matters is that the directed edges sum to zero on the boundary, not that they point the same way around.
Computable in \(O(n+m)\) time from the exposure graph alone, with no model of individual institution behaviour.
See: Paper 426 (Beyond Basel), ρ (Load Factor), ρ* (Critical Threshold)
Carnot Efficiency (η)
In EconIAC's thermodynamic framework, the Carnot efficiency \(\eta\) measures how much of the available information-thermodynamic free energy is converted into useful output.
The 6-731 broken-Fano topology (one Fano line weakened to coupling \(r\)) achieves \(\eta = 1 - r \approx 0.1825\) — matching the experimentally measured FMO photosynthetic efficiency. This is not a coincidence: the Fano symmetry breaking is the geometric source of both the biological efficiency and the financial concept of the unhedgeable residual.
See: Paper 325 (Topological Heat Engine)
Climate Tipping Cascade
A climate tipping cascade occurs when crossing one tipping element's threshold triggers others, producing an irreversible cascade. In the EconIAC cohomological framework:
- \(H^0\) = can each individual tipping element stay below its threshold?
- \(H^1\) = are the bilateral coupling exposures globally consistent?
- \(H^2\) = is there any intervention that can prevent cascade?
The \(H^1\) inflection point is at \(T^* \approx 1.8°C\) (between the Paris targets of 1.5°C and 2.0°C). The \(H^1\)-corrected social cost of carbon is \(\approx \$316\)/tonne, versus the bilateral baseline of \(\approx \$51\)/tonne.
See: Paper 403 (Tipping Points Are Topological), Paper 311 (Climate Hazard Yield Surface)
Curvature (Financial)
The curvature of a financial instrument is the failure of parallel transport to return to its starting point when transported around a closed loop of trades. Concretely:
- FX triangle: USD→EUR→GBP→USD ≠ 1 means non-zero holonomy
- XVA: the valuation adjustment is the integral of curvature along the exposure path (the 6j symbol evaluated on the counterparty graph)
- Yield curve: the term structure is a temporal connection; convexity is its curvature
In the EconIAC library, curvature is computed via econiac.pacioli
connection objects. Arbitrage-free ↔ flat connection ↔ zero curvature.
See: Paper 299 (XVA as Curvature), Paper 396 (Unhedgeability Theorem)
Differentiable Shapley Value
The Differentiable Shapley value computes marginal contributions via the Gibbs ensemble at finite temperature \(\beta\):
where \(p_\beta(S) \propto \exp(-\beta\, \mathrm{cost}(S))\) is a Gibbs weight over coalitions. At \(\beta \to 0\): uniform (classical Shapley). At \(\beta \to \infty\): hard assignment to the most likely coalition.
The key advantage: \(\phi_i(\beta)\) is differentiable in all model parameters, so \(\partial\phi_i/\partial\theta\) is available in one backward pass via JAX autodiff. This enables gradient-based attribution for systemic risk, carbon tax burden, and supply-chain criticality.
See: Paper 293 (Thermal Attribution)
Gauge Group
The gauge group of EconIAC is \((\mathbb{R}_{>0}, \times)\) — the multiplicative group of positive reals. This encodes the invariance of economic relationships under change of unit (currency, price level, numeraire): scaling all prices by a positive constant leaves real quantities unchanged.
This is a local gauge symmetry: different agents can use different numeraires at different times, and the connection (the exchange rate) encodes how to translate between them.
The gauge group \((\mathbb{R}_{>0}, \times)\) is abelian, making financial gauge theory much simpler than the non-abelian gauge theories of physics — but the mathematical structure (connection, curvature, holonomy) is identical.
See: Paper 291 (Topology of Conservation), Paper 301 (Primer on Economic Gauge Theory)
Gibbs Distribution (Quantal Response Equilibrium)
The Gibbs distribution \(p(x) \propto \exp(-\beta\, E(x))\) is EconIAC's model of bounded rational agents. At \(\beta \to \infty\) (zero temperature): perfect rationality (deterministic choice). At \(\beta \to 0\) (high temperature): uniform randomness. At finite \(\beta\): Quantal Response Equilibrium (QRE) — the empirically validated model of human decision-making under uncertainty.
The parameter \(\beta\) is calibratable from data via maximum likelihood. Every EconIAC model is differentiable through its Gibbs distributions, so \(\partial\mathrm{welfare}/\partial\beta\) is available in one backward pass.
See: Paper 289 (Temperature of Rationality), Paper 313 (Thermal Economics)
H⁰ / H¹ / H² (Cohomological Risk Hierarchy)
The three levels of financial risk, each corresponding to a sheaf cohomology group of the pricing sheaf \(\mathcal{F}\) on the exposure network \(\Gamma\):
| Level | Symbol | Name | Question answered |
|---|---|---|---|
| 0 | \(H^0\) | Bilateral | Can each counterparty pay? |
| 1 | \(H^1\) | Triangular | Are bilateral exposures globally consistent? |
| 2 | \(H^2\) | Systemic | Is there any consistent global resolution? |
The 2008 crisis was an \(H^2\) event: the global interbank network crossed a threshold beyond which no bilateral intervention (no netting, no haircut, no bailout of individual institutions) could prevent cascade. Standard models only measured \(H^0\).
In econiac.risk: cohomology_report(graph, section) returns all three
levels in one call with a human-readable summary string.
See: Paper 396 (Unhedgeability Theorem), Paper 397 (Systemic Risk as H²), Paper 398 (Topology of Risk primer)
Pacioli Identity
The Pacioli identity is the double-entry accounting principle: every debit has a corresponding credit; every asset has a corresponding liability. In EconIAC this is not a convention — it is the gauge invariance of the economic system, the conservation law that holds regardless of numeraire, currency, or unit of account.
Formally: the Pacioli identity is \(d^2 = 0\) on the chain complex of the account graph — the same pentagon identity that appears in the Origami ISA as the no-arbitrage condition and in quantum gravity as the Biedenhahn-Elliott identity for \(6j\) symbols.
Money can be created by banks (by simultaneously creating an asset and a
liability), but the Pacioli constraint is always satisfied. Violations
of the Pacioli identity — accounts that do not balance — are detected by
econiac.pacioli.pacioli_check().
See: Paper 291 (Topology of Conservation)
Pacioli Manifold
The Pacioli manifold \((M, \nabla)\) is the geometric space on which EconIAC models live. Its points are economic states (stock-flow consistent balance sheets); its tangent vectors are flows (transactions); its connection \(\nabla\) encodes exchange rates and discount factors.
The Pacioli manifold has gauge group \((\mathbb{R}_{>0}, \times)\). The curvature of \(\nabla\) measures arbitrage. Flat connections (\(F_\nabla = 0\)) are arbitrage-free. The holonomy of a closed loop of trades is the total FX gain/loss or XVA contribution.
See: Paper 291 (Topology of Conservation), Paper 409 (EconIAC Overview)
Pricing Sheaf
The pricing sheaf \(\mathcal{F}\) assigns to each node of the exposure network \(\Gamma\) a stalk (a vector space of prices/valuations) and to each edge a restriction map (the bilateral exposure weight). A global section of \(\mathcal{F}\) is a consistent assignment of valuations across all nodes — an equilibrium pricing.
The cohomology of \(\mathcal{F}\): - \(H^0(\Gamma, \mathcal{F})\): the space of globally consistent pricings - \(H^1(\Gamma, \mathcal{F})\): triangular inconsistencies (XVA, wrong-way risk) - \(H^2(\Gamma, \mathcal{F})\): systemic irresolvability (the 2008 event)
The pricing sheaf is a strict generalisation of the constant sheaf (Flood et al. 2017): where the constant sheaf gives topological Betti numbers, the pricing sheaf gives financially loaded cohomology that detects whether the network topology is carrying dangerous exposures.
See: Paper 396 (Unhedgeability Theorem), Paper 397 (Systemic Risk as H²)
ρ (Load Factor)
The load factor \(\rho\) measures how cycle-dense an exposure network is relative to its size:
where \(\beta_1\) is the first Betti number (β₁) and \(m\) is the edge count. The convention for \(m\) matters and is not yet fixed in the literature: whether \(m\) counts exposures before or after bilateral netting (i.e. whether \(A\rightleftarrows B\) contributes one netted edge or two directed edges) changes \(\rho\) without changing \(\beta_1\) the same way. Until stated otherwise, treat \(m\) as the netted edge count (one edge per ordered pair with non-zero net exposure).
Small \(\rho\): safe regime, funding gaps reroutable through alternative paths. \(\rho\) approaching the critical threshold \(\rho^*\): the network becomes maximally sensitive to single-institution failure.
See: Paper 426 (Beyond Basel), ρ* (Critical Threshold)
ρ* (Critical Threshold)
\(\rho^*\) is the conjectured critical value of the load factor ρ beyond which a network enters a regime of maximal sensitivity to perturbation. The commonly cited value is
This value is imported, not derived for financial networks. It is the \(\beta^*=1\) case of the general Forge ISA formula \(\beta^*(\rho) = \tfrac{3}{8}\ln(1/(1-\rho))\), and \(0.931\) itself is a known threshold from classical hashing/occupancy theory, recovered by that formula rather than independently derived for exposure graphs. The deeper claim underlying the formula — that the critical inverse temperature equals the inverse spectral gap of the Hodge Laplacian on the graph, \(\beta^* = 1/\lambda_1(\Delta)\) — is proved only for the ring topology (\(\mathcal{M}_P = S^1\)) and is otherwise a conjecture, not an established result. Treat \(\rho^*=0.931\) as a reference point borrowed from a different domain, not a threshold derived for obligation networks.
See: Paper 426 (Beyond Basel) (§3.1, with caveat), Paper 419 (The Forge ISA)
Sheaf Laplacian
The sheaf Laplacian \(L_\mathcal{F} = \delta^{0\top}\delta^0\) is the symmetric positive-semidefinite matrix built from the coboundary operator \(\delta^0\) on the pricing sheaf. Its eigenvalues measure the degree of inconsistency in the exposure network:
- \(\ker(L_\mathcal{F})\): globally consistent valuations (\(H^0\))
- Near-zero eigenvalues: approximate consistency (low \(H^1\) obstruction)
- Large eigenvalues: strongly inconsistent cycles
The \(H^1\) early-warning signal is \(\|s - P_{\ker}s\| / \|s\|\) — the fraction of the health-ratio section \(s\) that cannot be reconciled to a consistent global valuation. This signal leads cascade distress by 1–2 periods in the CHZ fire-sale model (x332e).
In econiac.risk: sheaf_laplacian(graph, section) and
h1_obstruction_signal(L_F, section).
See: Paper 397 (Systemic Risk as H²)
Social Cost of Carbon (SCC)
The social cost of carbon is the marginal damage of emitting one additional tonne of CO₂. In EconIAC's cohomological climate framework:
- Bilateral SCC (\(H^0\) baseline): \(\approx \$51\)/tonne — standard IAM estimates, ignoring tipping interactions
- \(H^1\)-corrected SCC: \(\approx \$316\)/tonne — accounts for triangular tipping inconsistencies (one tipping element triggering others via shared boundary conditions)
- \(H^2\)-corrected SCC: potentially unbounded — once the cascade is irresolvable, the marginal damage of the triggering emission includes the full cascade cost
The \(H^1\) inflection at \(T^* \approx 1.8°C\) is where the multiplier jumps from \(\approx 1\times\) to \(\approx 6\times\) the bilateral baseline.
See: Paper 403 (Tipping Points Are Topological)
Stock-Flow Consistency (SFC)
Stock-flow consistency is the requirement that every flow (a transaction) is accounted for in the balance sheets of both parties, and that the sum of all sectoral balances equals zero. It is the macroeconomic expression of the Pacioli identity.
In EconIAC, stock-flow consistency is enforced algebraically by the
gauge constraint on the Pacioli manifold — it cannot be violated by
construction. The econiac.pacioli module provides SystemState objects
whose pacioli_check() method verifies consistency and returns a
PacioliReport with the degree of any violation.
See: Paper 291 (Topology of Conservation), Paper 316 (EconIAC: MONIAC for the 21st Century)
Systemic Risk (H²)
Systemic risk is the \(H^2\) level of the cohomological risk hierarchy: the existence of a global state of the interbank (or climate, or supply chain) network from which no bilateral or triangular intervention can prevent cascade.
\(H^2 \neq 0\) means the network has crossed a topological tipping point. The unhedgeable residual is the component of stress that lies in \(H^2\) — it cannot be netted, haircut, or bailed out at the bilateral level.
In econiac.risk: h2_obstruction(graph, section) returns the H²
signal (fraction of edge inconsistency that is irresolvable) and
is_h2_event (boolean flag). cohomology_report() assembles all three
levels with a 'bilateral' | 'triangular' | 'systemic' risk
classification.
See: Paper 397 (Systemic Risk as H²), Paper 398 (Topology of Risk primer)
Thermal Attribution
Thermal attribution is the differentiable Shapley value at finite temperature \(\beta\) — it attributes systemic risk, carbon tax burden, or supply-chain criticality to individual agents or edges via a single backward pass through the Gibbs distribution.
Unlike classical Shapley (exponential in the number of players), thermal attribution runs in \(O(n)\) time via automatic differentiation: \(\phi_i = \partial F / \partial v_i\) where \(F(\beta)\) is the free energy of the system at temperature \(\beta\).
In econiac.routing: attribution.thermal_shapley(model, β).
See: Paper 293 (Thermal Attribution)
Unhedgeable Residual
The unhedgeable residual is the component of financial stress that cannot be eliminated by any bilateral netting, haircut, or collateral arrangement. It is the \(H^2\) component of the pricing sheaf cohomology: the part of the edge inconsistency vector that lies in the kernel of the coboundary operator \(\delta^1\) and therefore cannot be "corrected" by any triangle-level renegotiation.
The unhedgeable residual is what made the 2008 crisis unresolvable by bilateral intervention: AIG's counterparty web had a non-zero \(H^2\) component that no amount of bilateral netting could eliminate.
In econiac.risk:
h2_obstruction(graph, section).unhedgeable gives the per-triangle
unhedgeable residual; .h2_signal gives the scalar fraction.
See: Paper 397 (Systemic Risk as H²)
XVA (Valuation Adjustments as Curvature)
XVA (the family of valuation adjustments: CVA, DVA, FVA, KVA, MVA) are, in the EconIAC framework, integrals of the curvature of the pricing sheaf along the exposure path.
The \(6j\) symbol — the fundamental object of the Origami ISA — is the H¹ obstruction of the representation sheaf over the counterparty interaction diagram. XVA is a special case: the \(6j\) symbol evaluated on the counterparty tetrahedron (four parties, six bilateral exposures).
This identification gives XVA an exact combinatorial formula in terms of the counterparty graph topology, without requiring a model for the correlation structure.
See: Paper 299 (XVA as Curvature), Paper 396 (Unhedgeability Theorem), Paper 399 (Origami ISA as Financial Middleware)
For the underlying mathematical foundations (Fano plane, \(G_2\), octonions, Origami ISA opcodes, pentagon identity, associamancy), see the ASA Glossary.