Sara Dubois

“Create” is a powerful word. It is a mistake to assume that proof creates an equivalence. Proof simply establishes this equivalence under specific conditions, such as the existence of interior fixed points, which are not always guaranteed in real market scenarios with dynamic constraints. We need to define the scope of this equivalence. Now.

Attention is a bottleneck; claiming that a proof 'creates' an equivalence is an overstatement. This proof merely formalizes a preexisting mathematical relation, it does not generate it from nothing. The distinction is crucial because it affects how we allocate limited research resources. Considering a proof as an act of creation is like saying that the discovery of electricity created energy itself, instead of simply understanding and harnessing it. True innovations often come from recognizing existing phenomena, not inventing them.

However, capital controls constitute a weak link, capable of breaking this equivalence in real-world situations. The substitution of the Jacobian confirms that we now need to test specifically the limits imposed by the volatility of emerging markets, where such controls are common, to determine if equation (25) holds. This is where the mechanism breaks down.

The payment matrix for research shows that we should be cautious in saying that a mathematical proof 'creates' an equivalence; this reflects a superficial understanding of the incentives at play.
This is not because we manage to formalize a relationship that it did not exist before; the proof reveals, it does not generate the structural dynamics.
The incentives are there, regardless of our ability to model them perfectly, like the correlation of crude oil prices with OPEC decisions.
For example, the equivalence between different risk valuation methods on the gold market was not created by the proof of their consistency; this relationship preexisted, simply brought to light by the analysis.

Who has the incentive to believe that a mathematical proof 'creates' an existing equivalence?
The true gain comes from the understanding that this proof reveals an underlying property; it does not generate it.
If one thinks that a magic equilibrium formula creates reality, one risks misinterpreting market dynamics.
For example, the proof of an inverse correlation between gold and real interest rates in the market does not create this relationship; it allows us to better anticipate it based on central bank policies.

Before there is a conviction, one must understand the risk: the idea that proof creates the equivalence is difficult to accept. A proof does not bring the implicit market relation into existence; it formalizes it and delineates the threshold of application. If this equivalence depended on proof for its existence, what would protect us in case of failure of the proof or changing conditions? For example, the relation between the PBoC's gold volumes and liquidity exists, whether we have an elegant proof or not; proof only confirms it. We must remain vigilant about conditions that could render the proof ineffective, such as unexpected changes in Chinese regulation on futures markets.

Who has an interest in considering the equivalence of sensitivities as an unconditional creation rather than a necessary condition?
For example, even if an equivalence proof between sensitivities based on fixed points and those based on KKT conditions is established, its application in a volatile market like Brazil does not de facto create a new market reality.
The incentives diverge strongly between academic theory and trading practice, where commodity shocks or changes in monetary policy can render these equivalences invalid.
The true payment comes from understanding when these equivalences hold in the face of market frictions and not just by assuming their existence.
If actors ignore local macroeconomic signals in favor of a mathematical abstraction, the risk of an underperforming equilibrium is high.