Why does Econiac use sheaves?

Sheaves sound like advanced algebraic topology. In practice they solve one ubiquitous problem: local information that may not fit together globally. That problem appears in at least four distinct places in Econiac.

The one problem sheaves solve

Imagine you have a network of agents — banks, markets, proteins, neurons — and each agent holds a piece of local information: a capital ratio, a price, an energy level. The question a sheaf answers is:

"Do all the local pieces of information fit together into a consistent global picture?"

If they do, the system is in a coherent state. If they do not, there is a measurable obstruction — a "gap" between what different parts of the network believe. That gap is what the sheaf cohomology group H¹ measures.

This is not an abstract concern. Every use case below is a situation where local consistency fails in an economically or scientifically important way, and where detecting the failure early matters.

The four uses of sheaves in Econiac

1. Contagion early-warning (current)

File: econiac/finance/contagion/sheaf.py

The question: Do different parts of the interbank or repo network agree on which dealers are solvent / which repos are safe?

The section: Capital ratios (Paper 332) or funding ratios (Paper 333).

What H¹ measures: The extent to which bilateral assessments of solvency are inconsistent across the network. When lenders disagree on the creditworthiness of the same borrower — MMFs rolling while LDI funds withdraw — the H¹ signal rises before any individual threshold is breached.

Lead time: 2–3 periods before distress in the CHZ and repo models.

2. Sheaf neural networks (planned)

The question: Can a neural network's weights be made globally consistent with a geometric constraint — a symmetry, a conservation law, a gauge invariance?

A standard neural network assigns weights to edges of a computation graph. A sheaf neural network assigns a vector space (stalk) to each node and a linear map (restriction map) to each edge. Gradient descent then optimises not just the weights but the consistency of the stalks across the graph.

What H¹ measures: Whether the network's representations are globally consistent with the imposed geometric structure. A network that has learned a good representation of a symmetric physical system should have H¹ = 0 (the learned stalks form a globally consistent section).

Why this matters for Econiac: Financial networks have natural geometric structures — gauge invariance of the unit of account, SFC conservation, interest rate term structure constraints. A sheaf NN trained on such a network should respect these structures, and H¹ tells us whether it does.

Reference: Hansen & Ghrist (2020), Sheaf Neural Networks. arXiv:2012.06333.

3. Topological signal processing on market data (planned)

The question: Is a market microstructure signal globally consistent across trading venues, time zones, and asset classes?

A price signal observed on exchange A and exchange B should, under the no-arbitrage condition, be consistent: they should agree (up to bid-ask spread and latency) on the same underlying value. When they do not, the inconsistency is a trading signal — or a sign of market stress.

The sheaf encodes the expected consistency relationship between observations at different nodes (venues, time-stamps, asset classes). H¹ detects when the market is generating inconsistent signals — which precedes liquidity crises in the same way it precedes solvency crises in the contagion model.

Reference: Robinson, M. (2014). Topological Signal Processing. Springer.

4. Biological network coherence (Papers 324, 325, 328)

The question: Is energy, information, or mechanical force being transferred coherently across a biological network?

In the FMO complex (Paper 325), the chromophores form a graph. Energy transferred along the Fano line (the "broken" edge connecting chromophores 1 and 7) creates a measurable H¹ obstruction. When the Fano symmetry is intact, H¹ = 0 and efficiency is zero. When it is broken (Tyr16 coupling), H¹ ≠ 0 and the heat engine runs.

The same construction appears in:

Ribosomal decoding (Paper 324): tRNA accommodation as a sheaf consistency problem on the A-site network
Microtubule CST (Paper 328): tubulin conformational states as sections of a sheaf on the protofilament lattice

What H¹ measures in biology: A topological obstruction to coherent global function. H¹ ≠ 0 means the network cannot assign consistent local states everywhere simultaneously — which is exactly the condition for directed, asymmetric energy or information transfer.

The shared mathematics

All four uses share the same computational core, now in contagion/sheaf.py:

graph       = FinancialGraph (or BiologicalGraph, MarketGraph, ...)
section     = local information vector (capital ratios, energy levels, prices)
L_F         = sheaf_laplacian(graph, section)
signal      = h1_signal(L_F, section)       # scalar: 0 = consistent
decomp      = harmonic_decomposition(...)   # which nodes contribute most

The function names are application-agnostic. FinancialGraph is the current concrete implementation; a BiologicalGraph and MarketGraph will follow the same interface.

Why one library, not four

The temptation is to write separate sheaf code for each application. The reason not to:

The mathematics is identical. The coboundary operator, the Laplacian, the H¹ dimension count, and the harmonic decomposition are the same computation regardless of whether the nodes are banks, chromophores, or trading venues.
Cross-model comparison becomes trivial. compare_h1_series(ts_finance, ts_biology) in sheaf.py measures whether two H¹ time series are structurally isomorphic. Paper 334 §6 uses this to prove the three-way isomorphism between the CHZ cascade, the repo run, and the FMO heat engine. That result requires a single shared library.
Bugs are fixed once. The harmonic_decomposition() function had a subtle sign convention issue in an earlier draft. Fixing it in one place fixes it for all four applications.

The deeper reason: sheaves are the right language for consistency

The recurring theme across all four uses is:

Local information + a notion of "how local pieces should relate" + a measurement of how far reality departs from that relation.

That is precisely what a sheaf is: a functor from the category of open sets of a space (or faces of a simplicial complex, or nodes and edges of a graph) to the category of vector spaces, together with restriction maps saying how the local pieces relate.

In all four Econiac applications, the "right relation" is determined by the conservation law:

Finance: Pacioli ∂²=0 (double-entry accounting)
Biology: energy conservation on the chromophore graph
Signal processing: no-arbitrage / no-energy-creation

The sheaf is what makes these conservation laws geometric rather than merely algebraic — it gives them a spatial structure that H¹ can measure.