Monetary Macroeconomic Accounting Theory (MoMaT)

• MoMaT is a framework that defines money as a means for settling obligations with clear legal principles like the Separation and Abstraction principles.
• It organizes accounting across micro, meso, and macro levels, linking double-entry practices with category theory to ensure systemic consistency.
• The theory emphasizes debt vortices and contractual instruments, such as Bills of Exchange, to model dynamic financial obligations and risk management.

Monetary Macroeconomic Accounting Theory (MoMaT) is a framework for consistent national accounting in which money functions primarily as a medium of payment for obligations and debts rather than as a medium of exchange. It is grounded in the claim that money originates from the temporal misalignment between producers’ payments to suppliers and workers and the later receipt of sales revenue, and it models monetary systems across interconnected micro, meso, and macro levels (Menéndez et al., 26 Jun 2025). In subsequent categorical formulations, MoMaT is presented as a compositional macroeconomic accounting system built from microeconomic double-entry systems with real and monetary units of account, using category theory to lift micro consistency to macro consistency (Menéndez et al., 19 Aug 2025).

1. Conceptual definition and legal foundations

MoMaT begins from two legal principles. The Separation Principle states that “an obligation contract (which creates rights and duties) is legally distinct from the disposition contracts (which transfer ownership to fulfill those obligations).” The Abstraction Principle states that “the validity of the obligation contract is independent of the validity of the disposition. Even if one is void, the other still stands until explicitly unwound by restitution” (Menéndez et al., 26 Jun 2025). Within this scheme, obligations and transfers are analytically distinct, and money is defined by its role in extinguishing obligations rather than by barter-like exchange.

The formal definition used in MoMaT expresses this directly. Let $O_{ij}^t$ denote the obligation of agent $i$ to agent $j$ at time $t$, and let $D_{ij}^t$ denote a disposition transferring ownership of a money-unit from $i$ to $j$. Then money is the token $m$ such that $D_{ij}^t(m)$ extinguishes exactly one unit of $O_{ij}^t$ (Menéndez et al., 26 Jun 2025). This places temporal settlement at the center of monetary theory.

A recurrent misconception addressed by MoMaT is that the primary analytical object should be “money-circulation loops.” MoMaT instead focuses on debt vortices, defined as the ongoing creation and resolution of financial obligations. This shift is substantive rather than terminological: it relocates explanatory priority from exchange to settlement and from circulation to debt relations. The categorical extension sharpens this point by stating that money’s main function is for the repayment of loans and not for the exchange of goods, thereby bridging the desynchronisation of input and output payments of producers (Menéndez et al., 19 Aug 2025).

2. Multi-level accounting architecture

MoMaT organizes monetary systems at three interconnected levels: micro, meso, and macro. The abstract motivation is division of labor at the micro level, banking for risk-sharing at the meso level, and GDP sharing with money issuance at the macro level (Menéndez et al., 26 Jun 2025).

Level	Core entities	Main relation
Micro	producers, suppliers, bilateral links	$i$0
Meso	banks, deposits, loans, equity	$i$1
Macro	central bank, base money, bank balance sheets	$i$2

At the micro level, each bilateral producer-supplier link $i$3 is represented by a receivable $i$4 at the seller and a liability $i$5 at the buyer. Macro-invariance at the micro network requires

$i$6

Over a period $i$7, expenditures and revenues are written as

$i$8

with the stated relation $i$9 (Menéndez et al., 26 Jun 2025). The micro layer is therefore a network of mirrored claims and obligations.

At the meso level, each bank $j$0 maintains deposit liabilities $j$1 and loan assets $j$2, with the state-variable invariance

$j$3

where $j$4 is bank equity. Default losses $j$5 reduce $j$6, and interest rates $j$7 are treated as insurance premiums satisfying the break-even condition

$j$8

The loan relation is represented diagrammatically as a cospan

$j$9

This is the level at which MoMaT places risk-sharing.

At the macro level, the central bank issues base money $t$0 as its own liability. The aggregate relation is

$t$1

The fiat issuance sequence is stated as: the central bank grants a loan $t$2 and creates a deposit $t$3; the bank withdraws cash $t$4; firms borrow $t$5, creating deposits $t$6; and firms then pay wages and suppliers (Menéndez et al., 26 Jun 2025). In MoMaT, these levels are not independent modules but linked accounting strata.

3. Debt vortices and the Bill of Exchange framework

The central dynamic object of MoMaT is the set of outstanding obligations,

$t$7

Its evolution is written as

$t$8

Each individual instrument traces a path $t$9 through agents, and in category-theoretic terms obligations are objects while issuance, endorsement, and settlement are morphisms. A full chain ends in the “zero” object, meaning no outstanding debt (Menéndez et al., 26 Jun 2025). The emphasis on lifetime dynamics distinguishes debt vortices from a circulation-centered description.

The Bill of Exchange (BoE) is the unifying contractual instrument in MoMaT, linking debt processes and monetary issuance across fiat and gold-based systems (Menéndez et al., 26 Jun 2025). It is written as

$D_{ij}^t$0

plus an endorsement list $D_{ij}^t$1. The BoE lifecycle is specified at all three monetary levels. At the micro level, product delivery creates receivable and liability entries, and BoE issuance or acceptance creates $D_{ij}^t$2 and $D_{ij}^t$3. At the meso level, a seller sells the BoE to a bank, increasing the bank’s asset $D_{ij}^t$4 and the seller’s deposit $D_{ij}^t$5, and banks may transfer BoE claims among themselves. At the macro level, banks clear BoE exposures via central-bank deposits $D_{ij}^t$6.

The endorsement and settlement chain is summarized by

$D_{ij}^t$7

Liquidity extraction, described as monetization, is represented by the paired entries

$D_{ij}^t$8

This makes the BoE both a contractual and accounting bridge across levels. A plausible implication is that MoMaT treats negotiable instruments not as peripheral finance objects but as central operators in monetary synchronization.

4. Categorical and homological formalization

MoMaT’s mathematical program uses category theory, sheaf theory, homology theory, and open games to guarantee consistency across levels and to structure macroeconomic analysis (Menéndez et al., 26 Jun 2025). In the categorical formulation, a micro double-entry system for an agent $D_{ij}^t$9 is a pair of T-accounts,

$i$0

and each booking is a state transition $i$1 subject to the binary accounting rule $i$2 (Menéndez et al., 19 Aug 2025). Real-unit and nominal-unit accounts are kept in parallel.

At the macro level, the categorical construction introduces

$i$3

and defines the macro system as

$i$4

A macro-booking is then a morphism in $i$5 obtained by pullbacks for validation and pushouts for aggregation (Menéndez et al., 19 Aug 2025). This is the formal expression of the claim that macroeconomic accounting systems are composed from microeconomic double-entry systems.

The time evolution of the economy is encoded by an endofunctor

$i$6

which on objects applies $i$7, $i$8, and $i$9, and on morphisms carries each booking morphism in period $j$0 into the corresponding booking in period $j$1. The defining square is specified to commute. Universal constructions are assigned distinct tasks: the limit verifies sectoral invariances and macro-invariance, while the colimit aggregates the eight local bookings into a global account update (Menéndez et al., 19 Aug 2025). The dual language in the abstract states that the universal constructions of a limit verify all constraints, while the dual colimit computes aggregated informations at the macro level.

The sheaf-theoretic version attaches a local accounting double system $j$2 to each node or edge of a graph $j$3. A presheaf $j$4 assigns these local data, and the sheaf condition states that local bookings can be glued if and only if they agree on overlaps; global sections are therefore consistent macroaccounting states (Menéndez et al., 26 Jun 2025). In the homological formulation, one builds a chain complex

$j$5

where $j$6 is the space of agents, $j$7 the obligations, and $j$8 the triangles of mutual debt. Exactness at $j$9,

$m$0

encodes that every net cycle of obligations comes from internal net-able triplets; failures of exactness identify local crises requiring policy intervention (Menéndez et al., 26 Jun 2025). Open games then provide a multi-agent semantics: an open game has type

$m$1

and bank strategies $m$2 solve $m$3 in, for example, two-bank coordination problems.

5. Sectoral dynamics, stability, and implementation

The categorical MoMaT model exemplifies the framework with five sectoral agents: labor owners, resource owners, a production company, a capitalist as dividend recipient, and a bank as financial intermediary (Menéndez et al., 19 Aug 2025). The dynamics is described by eight sectoral macroeconomic bookings in each period. These bookings include wages, goods purchases, resource purchases, loan creation, dividends, repayment, and goods flows involving the capitalist sector. The stated economic role of money in this sectoral model is synchronization: paying inputs in Bookings $m$4 and $m$5, receiving outputs in $m$6 and $m$7, using bank loans in $m$8 to provide immediate liquidity, and closing the loop via repayments in $m$9.

The simulation setup uses parameters $D_{ij}^t(m)$0, including investment length $D_{ij}^t(m)$1, markup $D_{ij}^t(m)$2, and sectoral consumption rates $D_{ij}^t(m)$3. The state is

$D_{ij}^t(m)$4

updated via $D_{ij}^t(m)$5 (Menéndez et al., 19 Aug 2025). The recursive sketch specifies: $D_{ij}^t(m)$6

$D_{ij}^t(m)$7

$D_{ij}^t(m)$8

$D_{ij}^t(m)$9

$O_{ij}^t$0

$O_{ij}^t$1

The convergence proposition is stated as follows: under parameter ranges such as $O_{ij}^t$2, $O_{ij}^t$3, and $O_{ij}^t$4, the endofunctor $O_{ij}^t$5 is a contraction on account-space norms, implying a unique fixed point $O_{ij}^t$6 and convergence of accounts, flows, and memory variables. The proof sketch is by Banach fixed point (Menéndez et al., 19 Aug 2025). The same source states that the categorical viewpoint yields a terminal coalgebra capturing infinite-horizon stability. This presents stability not merely as a numerical property but as a property of the compositional architecture.

The implementation program in MoMaT specifies a software stack with a data layer, a smart-contract layer, and a policy-and-analytics layer (Menéndez et al., 26 Jun 2025). ERP connectors ingest firm $O_{ij}^t$7, bank systems feed deposit and loan books, and a central-bank database stores issued money and interbank positions. In the smart-contract layer, blockchain or permissioned DLT implements BoE issuance, endorsement, sale, and settlement while enforcing double-entry at each step. In the analytics layer, on-chain AI agents compute liquidity-demand forecasts, and homology alerts fire when local invariants break. The sample Solidity-style pseudocode for a BillOfExchange contract is therefore not incidental; it functions as a concrete software specification for a machine-readable debt-instrument layer.

6. Monetary growth order as a MoMaT extension

A distinct line of work proposes the monetary growth order model and explicitly presents it in a form “that can be plugged directly into a Monetary Macroeconomic Accounting Theory (MoMaT)” (Kiedrowski et al., 2012). The standard continuous-compounding law

$O_{ij}^t$8

is generalized to

$O_{ij}^t$9

where $i$00 is the monetary principal, $i$01 is the continuous interest rate, and $i$02 is the monetary growth order. For $i$03, the closed-form solution is

$i$04

while the limit $i$05 yields the exponential case $i$06.

The interpretation of $i$07 is explicitly economic. If $i$08, growth is standard exponential compounding and is size-neutral on a relative basis. If $i$09, growth is subexponential, and smaller principals grow relatively faster than larger ones; this is stated to dampen wealth-polarization and to help monetary aggregates track real-economy growth patterns such as linear or saturating paths. If $i$10, growth becomes superexponential or hyperbolic, large principals grow both absolutely and relatively faster than small ones, and finite-time blow-ups may occur in a “financial Malthusian catastrophe” (Kiedrowski et al., 2012). In this formulation, fixing $i$11 when real GDP slows is said to sow the seeds of asset-price bubbles and debt crises.

The MoMaT embedding specifies that the law $i$12 must coexist with stock-flow and balance-sheet identities. These include the monetary aggregate

$i$13

the generalized Fisher identity

$i$14

the flow-of-funds relation

$i$15

and a nonlinear public-debt equation

$i$16

if nonlinear compounding is allowed for government debt. It also includes a reserve relation $i$17, with the growth law for deposits propagating through the money multiplier $i$18 (Kiedrowski et al., 2012). The implementation guidelines are to choose a targeted real-economy reference path $i$19, estimate $i$20, calibrate $i$21, amend bank-accounting software so that posted interest flows obey $i$22, and monitor distributional statistics and systemic-risk metrics.

This extension does not redefine MoMaT’s legal and accounting core. Rather, it adds a nonlinear compounding parameter to debt and asset dynamics within a balance-sheet-consistent framework. A plausible implication is that MoMaT can serve as the accounting and contractual substrate, while the monetary growth order supplies an adjustable law for the evolution of loan and deposit balances. The papers describe this as a possible macroprudential instrument for central banks, especially in boom or crisis conditions, but they also note that $i$23 combined with sufficiently large $i$24 risks hyperbolic instability and finite-time blow-up (Kiedrowski et al., 2012).