What is money a claim on?

Double-entry accounting says every liability has a matching asset. But when a central bank creates reserves under fiat money, the liability is redeemable in... more of the same liability. This is not a bug in the accounting. It is a deep fact about what money is — and it has consequences for how EconIAC models monetary systems.

The three monetary regimes

Money has existed in three structurally distinct regimes, each corresponding to a different answer to the question "what is this liability ultimately a claim on?":

1. Commodity money (gold standard)

The central bank holds gold. Banknotes are claims on that gold. The liability is redeemable in a physical asset that exists independently of the monetary system.

∂²=0 holds: the balance sheet balances, and the liability has external substance. The constraint on money creation is real: you cannot issue more notes than you have gold.

Gauge-theory framing: the unit of account is fixed to the commodity. The connection has a preferred gauge — gold — and all other prices are measured relative to it. Abandoning the gold standard is a gauge transformation: the unit of account floats free, and the "zero of potential" is no longer pinned.

2. Credit money (commercial banking)

A commercial bank makes a loan. It simultaneously creates: - Asset: the loan receivable (a claim on the borrower) - Liability: a deposit in the borrower's account (a claim on the bank)

The deposit is redeemable in central bank reserves, which are redeemable in banknotes, which are claims on the central bank. The chain terminates at the central bank — but as long as the bank is solvent, the deposit is backed by real assets (the loan portfolio).

∂²=0 holds throughout: every new claim has a matching counter-claim. Money is created endogenously by the banking system, not exogenously by the central bank. This is the core insight of post-Keynesian monetary theory (Minsky \cite{minsky1986}, Keen \cite{keen2011}, Wray \cite{wray2012}).

Gauge-theory framing: credit money floats relative to the central bank liability. The "gauge group" is the set of credit relationships. Defaults (loan impairment) are gauge-field singularities — points where the connection breaks down and the holonomy becomes non-trivial.

3. Fiat money (modern central banking)

The central bank creates reserves by purchasing an asset (a government bond, a mortgage-backed security, a foreign currency). The accounting is:

Central bank asset	Central bank liability
Government bond	Reserves credited to commercial banks

The reserves are claims on the central bank. But what is the central bank's liability redeemable in? In more reserves — or in banknotes, which are themselves central bank liabilities. The chain is circular.

∂²=0 still holds: the balance sheet always balances. But the substance of the liability is different from any private-sector liability. It is not backed by a commodity (no gold) and not backed by a specific loan portfolio. It is backed by:

The taxing power of the state: the government accepts its own currency in payment of taxes, which creates demand for the currency (Chartalism / MMT — see Wray \cite{wray2012}).
Legal tender laws: the state mandates acceptance of the currency in settlement of debts.
The institutional topology of the monetary system as a whole: the currency functions as long as the network of agents that accepts it continues to accept it.

The third point is the deepest. Under fiat money, the "backing" of the currency is a topological property of the monetary network — not a balance-sheet entry.

The gauge-theory formalisation

EconIAC's Financial Gauge Theory (Paper 295, \cite{buckley2026_295}) treats the unit of account as a connection on the Pacioli manifold.

In gauge theory, a connection describes how to compare quantities at different points of a manifold. The curvature of the connection measures path-dependence — whether the result of a sequence of transactions depends on the order in which they are executed.

The three monetary regimes correspond to three types of connection:

Regime	Connection type	Curvature	What "zero" means
Commodity money	Flat (zero curvature)	\(F = 0\)	Pinned to gold
Credit money	Mildly curved	\(F \neq 0\) locally	Pinned to central bank
Fiat money	Gauge-free	\(F \neq 0\) globally	No external anchor

Switching between regimes is a gauge transformation: it changes the reference point for the unit of account without changing the real economic relationships. This is why models calibrated under a gold standard give different answers from models calibrated under fiat — not because the underlying economics changed, but because the gauge changed.

EconIAC makes this explicit: Paper 295 proves that the apparent "anomalies" of covered interest parity failure, FX basis persistence, and cross-currency swap spreads are all curvature effects — consequences of operating in a curved monetary manifold rather than the flat commodity-money ideal.

Why ∂²=0 is stronger than "money is conserved in quantity"

A common misstatement of the Pacioli identity is "money cannot be created or destroyed." This is wrong, and importantly wrong.

Under credit money, commercial banks create money every time they make a loan. Under fiat money, central banks create reserves whenever they purchase assets. Money creation is not prohibited — it is happening continuously.

What ∂²=0 says is subtler and more powerful:

Every new financial claim must be accompanied by a matching counter-claim.

Money can be created, but only symmetrically. A bank that creates a deposit without creating a matching loan receivable has violated ∂²=0 — not by creating money, but by creating an unmatched claim. This is what fraud looks like in EconIAC's type system: a flow that has no counter-flow, a liability with no asset, a transaction that does not balance.

The conservation law is about form, not quantity. It constrains the topology of creation, not the amount. This is why it is a genuinely different kind of law from physical conservation laws (energy, charge, momentum) — those do constrain quantity. ∂²=0 constrains structure.

Implications for modelling

Endogenous money is the default

EconIAC's SFC models (econiac.economics.sfc) treat money creation as endogenous by default: loans create deposits, deposits fund spending, spending generates income. The money supply is a consequence of credit decisions, not a policy variable set by the central bank. This follows Godley and Lavoie \cite{godley2007} directly.

The central bank controls the price of money (the interest rate) and provides an elastic supply of reserves on demand — it does not control the quantity of money in circulation. Quantity-theoretic models (MV=PQ) are a special case that requires additional assumptions EconIAC does not make by default.

Different regimes need different gauge fixings

A model of the gold-standard era needs a flat connection (zero curvature). A model of the post-2008 QE era needs a curved connection reflecting the excess reserves created by asset purchases. Fitting the same structural model to both periods without changing the gauge will produce spurious parameter instability — the model will attribute to behavioural change what is actually a change in monetary regime.

EconIAC's CurvedBalanceSheet and the curvature module of Paper 295 provide the language to make this explicit.

The central bank's liability is topological

Under fiat money, the central bank's reserve liability is not backed by any specific asset in the way a commercial bank deposit is. Its "backing" is the continued functioning of the monetary network — a topological property.

This has a concrete modelling consequence: the failure of a central bank (as distinct from a commercial bank) is not a balance-sheet event. It is a topological event — the collapse of the network of agents that accepts the currency. EconIAC's sheaf H¹ signal (Paper 335, \cite{buckley2026_335}) is the natural instrument for detecting this: \(H^1 \neq 0\) on the monetary network indicates that local acceptances of the currency can no longer be globally reconciled — the precursor to a currency crisis, not merely a solvency crisis.