Public Cash Money
A Formal Proof That the Current Monetary System Is Mathematically Configured to Generate Unpayable Debt
This chapter does not describe P.C.M. It describes what is wrong with the current system. P.C.M. is proposed in subsequent chapters.
Davide Serra · Systems Analyst · publiccashmoney.com · May 2026 Published as Open Source. Peer review actively welcomed. Mathematical level required: secondary school algebra and compound interest. Nothing more.
Abstract
This chapter demonstrates, through three verifiable axioms and elementary algebra, that any monetary system in which money is created as interest-bearing debt contains a structural impossibility: the total debt generated always exceeds the total money supply, making complete systemic debt repayment mathematically impossible by design. This is not a political observation. It is a mathematical consequence of the architecture. We then trace the three historical phases through which this structural bug became universal.
1. Three Axioms
We begin with three statements about how money is created in the current global monetary system. Each is verifiable in the official documentation of central banks, including the Bank of England, the Federal Reserve, and the European Central Bank.
Axiom 1 — Money is created as debt Every unit of currency currently in circulation was created through a lending operation. A bank issues a loan, and in doing so creates new money by crediting the borrower’s account. Money does not pre-exist the loan. It is created at the moment of lending.
Formal: M_created = L (where L = loan principal issued)
Source: Bank of England Quarterly Bulletin 2014 Q1, “Money Creation in the Modern Economy.” Federal Reserve Bank of Chicago, “Modern Money Mechanics.” ECB Working Paper Series.
Axiom 2 — Every loan generates an interest obligation Every loan carries an obligation to repay the principal plus interest. The total repayment obligation D for a loan of principal M at interest rate r over time t is given by the compound interest formula.
Formal: D = M × (1 + r)^t, where r > 0, t > 0
Source: Standard compound interest formula. Verified in any financial mathematics textbook.
Axiom 3 — Interest is never created alongside the principal When a bank creates M units of money through a loan, it creates only M. The interest obligation is not created at the moment of the loan. The borrower must find the interest in the existing money supply, which was itself created through other loans, each carrying their own interest obligations.
Formal: M_created = L | M_owed = L × (1+r)^t | Gap = M_owed − M_created > 0
Source: This follows directly from Axioms 1 and 2. No central bank documentation contradicts it.
2. The Formal Proof
Theorem: In a debt-based monetary system, total debt D is always greater than total money supply M.
- Let M be the total money supply at time t=0. By Axiom 1, M was created through loans of equal aggregate principal.
- By Axiom 2, the total repayment obligation is D = M × (1+r)^t for any r > 0 and t > 0.
- By Axiom 3, the interest component (D − M) was never created as money. It does not exist in the money supply.
- Therefore: D − M = M(1+r)^t − M = M[(1+r)^t − 1] > 0 for all r > 0 and t > 0.
- This gap grows continuously as t increases. It cannot be eliminated within the current architecture without creating new debt to cover the existing gap, which generates new interest, which widens the gap further. Q.E.D.
D = M(1+r)^t > M ∀ r>0, t>0 The Evil Formula. The structural impossibility at the heart of the current global monetary system.
3. Two Corollaries
Corollary 1 — Structural Insolvency In a debt-based monetary system, there always exists a quantity x = D − M > 0 of debt that cannot be repaid from the existing money supply. Some actor in the system must default. The identity of the defaulting actor is determined by economic and political conditions, but the existence of at least one defaulting actor is determined by the mathematics of the architecture. This is not a risk. It is a certainty.
Corollary 2 — The Growth Trap Any attempt to repay the gap x by creating new money generates new interest obligations that exceed the new money created. The gap does not close. It widens. This is why economic growth does not solve the debt problem in a debt-based system: growth requires more money, more money requires more debt, more debt generates more interest, the gap widens further.
4. The Three Historical Phases of the Bug
Venice · 1374 — The bug is written. Local. Relatively innocent.
Medieval trade in Venice had outpaced the physical supply of gold. The Venetian banking system formalized fractional reserve lending, issuing paper claims on gold that exceeded physical holdings. The Evil Formula was written into financial practice for the first time at scale.
Critically: the bug was local and partial. Most of the global economy still operated on metallic currencies or systems like the English Tally Sticks, which did not contain the structural interest obligation. The bug existed but had not yet escaped its local context. It was, in software terms, a known issue in a limited deployment.
Bretton Woods · 1944 — The bug is universalized by decree. This is the critical moment.
In July 1944, representatives of 44 nations agreed that the US dollar, issued by the Federal Reserve, a debt-based system, would become the world’s reserve currency. Every other currency was pegged to the dollar. Every central bank was required to hold dollars as reserves.
The consequence: the Evil Formula was now the operating system of the entire global economy. By decree. Simultaneously. Without exception.
This is the moment the bug became universal. Not 1374. Not 1913. 1944.
Washington · 1971 — The last physical constraint fails. Not an act of power. An act of necessity.
The Bretton Woods system included one partial constraint on the bug: the dollar was convertible to gold at $35 per ounce. By 1971, US gold reserves had fallen to approximately $10 billion against $40 billion in outstanding dollar liabilities. European central banks were literally sending ships to collect gold.
The gold window was not closed as an act of monetary aggression. Fort Knox could not honor the existing commitments. The constraint had already been overwhelmed by the logic of the bug itself. After 1971, the Evil Formula operated without any physical constraint. The compounding began in earnest.
1374: The bug is written. Local. The world is still mostly hybrid. 1944: The bug is universalized by decree. Inescapable for every nation on earth. 1971: The last physical brake fails, not by choice, but by mathematical necessity.
4a — The Gold Standard: The Wrong Brake on the Right Problem
Before 1944 universalized the bug, there was a partial constraint. For most of the period between 1374 and 1944, the Evil Formula operated under a physical limitation: the gold standard. Currencies were convertible to gold at fixed rates. A government that wanted to create more money had to have — or credibly claim to have — gold to back it. This created friction. It slowed the compounding. It did not eliminate the bug, but it put a ceiling on how fast it could grow.
The gold standard was, in its intention, a reasonable response to a real problem: how do you prevent governments and banks from issuing more money than the economy can absorb? The answer — anchor the currency to a scarce physical commodity — had an intuitive logic. You cannot print gold. But the brake was wrong for three reasons. First, gold supply is geologically determined, not economically determined. The amount of gold available to back a currency has no relationship to the productive capacity of the economy it serves. When economies grew faster than gold could be mined — which was almost always — the system produced deflation, credit contraction, and depression. The 1930s are the empirical proof. The Great Depression was not caused solely by the gold standard, but the gold standard prevented the monetary response that could have shortened it. Second, gold is itself subject to speculation and hoarding. A commodity whose scarcity is its value attracts exactly the behavior it is supposed to prevent: accumulation by those with access to it, withdrawal from circulation at moments of uncertainty, and speculative flows that have nothing to do with productive economic activity. The gold standard did not eliminate speculation. It concentrated it in one metal. Third, and most importantly: gold does not solve the Evil Formula. A gold-backed loan still carries interest. The interest is still never created alongside the principal. D = M(1+r)^t > M is true whether M is denominated in paper or in gold. The bug runs in any medium. The gold standard slowed the visible symptoms. It did not touch the structural cause.
When Keynes proposed the Bancor in 1944 — a supranational unit anchored not to gold but to a basket of commodities — he was trying to escape this trap. He was overruled. The dollar, a debt-based currency backed by the full faith and credit of a government running the Evil Formula at national scale, became the global reserve. The wrong brake was not replaced with a better one. It was removed entirely. The Austrian School’s proposal to return to gold, or to a gold-like hard money standard, is therefore a proposal to restore the wrong brake. It addresses the symptom — unlimited money creation — without addressing the cause — the interest obligation that is never created alongside the principal. A gold-backed PCM would still be subject to the Evil Formula. The anchor must be real productive capacity, not geological scarcity. That is what the Base Basket in Chapter 6 is designed to provide.
5. Empirical Confirmation
The theorem makes a testable prediction: in a debt-based monetary system, total debt should grow faster than total economic output, continuously, without a structural mechanism that reverses the trend.
US economy 1944 to 2026:
- GDP growth: from approximately $2 trillion to $27 trillion. A 13-fold increase.
- National debt growth: from approximately $260 billion to $39 trillion. A 150-fold increase.
- Ratio: debt grew approximately 12 times faster than the economy over 82 years.
- Dollar purchasing power loss since 1950: approximately 87-88%.
Sources: US Treasury Fiscal Data; Federal Reserve FRED database; Bureau of Labor Statistics CPI-U series; US Congress Joint Economic Committee (April 2026).
The data confirms the theorem.
6. What This Chapter Does Not Claim
It does not argue that all debt is bad. It does not argue that all economic crises are caused solely by the structural bug. It does not propose a solution. That is the subject of subsequent chapters.
To confute this chapter, a critic must demonstrate that at least one of the three axioms is false, or that the algebraic derivation contains an error. The axioms are sourced from central bank documentation. The algebra is elementary. The peer review invitation is genuine.
The three axioms are verifiable. The algebra is elementary. The empirical data is official and public. The conclusion is unavoidable:
D = M(1+r)^t > M for all r>0 and t>0.
The system is not failing. It is operating exactly as its architecture requires. That is the problem.
Coming in Chapter 2: The Corrected Formula: Why MV = PQ omits the variables that make the structural bug visible, and what the corrected formula MV = PQ + T + I predicts about the trajectory of US national debt from 1944 to $39 trillion today.
$2+2=4. Period.