Formal Proof
I wish convincing my wife was this easy....
r <- matrix(c(1,2,4,8,16), nrow = 1)
E <- t(r) %*% (1/r)
Print(E)
1 0.5 0.25 0.125 0.0625
2 1 0.5 0.25 0.125
4 2 1 0.5 0.25
8 4 2 1 0.5
16 8 4 2 1
Print(E %*% E)
5 2.5 1.25 0.625 0.3125
10 5 2.5 1.25 0.625
20 10 5 2.5 1.25
40 20 10 5 2.5
80 40 20 10 5
Let E be an n x n matrix of exchange rates, such as those between 30 or so currencies actively traded on the FX (Forex) market. We use the notation E_T to refer to the Hadamard inverse (element-wise inverse) of the transpose of E, i.e.,
E_T = (E^T) ^{\circ (-1)}
The constraint E = E_T results in a condition where E becomes a matrix equal to the outer product of its first column and its first row. Let r represent the first row of E. Then E can be written as:
E = r^T \times \left( {1 \over r} \right)
E = r^T \times \left( {1 \over r} \right) \times r^T \times \left( {1 \over r} \right)
My First Attempt At a Formal Proof
by Joseph Mark Haykov
October 23, 2024
No-Arbitrage Constraint on Exchange Rates
In this analysis, we explore the foreign exchange (Forex) market, where approximately 30 of the most actively traded currencies are exchanged. Exchange rates in this market can be structured as a matrix, denoted by E, where each element eij in row i and column j represents the exchange rate from currency i to currency j. This matrix provides a framework for understanding how exchange rates are organized to prevent arbitrage opportunities—market inefficiencies that allow risk-free profit.
Arbitrage is prevented when a uniform pricing structure is maintained across different markets. Specifically, in the Forex market, the exchange rate from currency A to currency B must be the reciprocal of the exchange rate from currency B to currency A. For instance, if 1 USD buys 0.50 GBP (or 50 pence), then 1 GBP must buy 2 USD. This reciprocal relationship is essential for eliminating arbitrage opportunities, which could otherwise allow an arbitrageur to make risk-free profits by exploiting discrepancies between exchange rates.
Exchange Rate Matrix and No-Arbitrage Condition
Let E be a matrix representing the exchange rates between major currencies in the Forex market. The no-arbitrage condition — eij · eji = 1 — imposes a constraint on the elements eij of E, which is dually defined for all i, j as follows:
Because eij · eji = 1:
Not only: eij = 1 ÷ eji
but also: eji = 1 ÷ eij
This condition ensures that the product of exchange rates in both directions between any two currencies equals 1. In other words, it enforces the symmetry needed to prevent arbitrage.
The Concept of the Evolutory Matrix
A Hadamard inverse of a matrix
is already defined in mathematics as the element-wise reciprocal of itself:
Mathematically, the element-wise no-arbitrage condition can be dually defined in linear algebra matrix form using the existing transpose and Hadamard inverse operations as follows:
not only:
\(E = \left( E^{\circ (-1)} \right)^T\)but also:
\(E = \left( E^T \right)^{\circ (-1)}\)
This holds true because transpose and Hadamard inverse are commutative operations.
Let us use the notation ET to refer to the Hadamard inverse of the transpose of a matrix, or equivalently, the transpose of its own Hadamard inverse, as explained above. Formally, ET is dually defined as:
not only:
\(E_T = \left( E^{\circ (-1)} \right)^T\)but also:
\(E_T = \left( E^T \right)^{\circ (-1)} \)
This states that:
but now formally, dually-defined, and in matrix form.
The no-arbitrage constraint, E = ET, ensures the absence of arbitrage by enforcing symmetry and reciprocity in exchange rates. This constraint is analogous to a matrix being involutory—that is, equal to its own inverse. However, we refer to matrices that satisfy the condition of being the Hadamard inverse of their own transpose, ET, as evolutory, rather than involutory.
This distinction is important because, while for an involutory matrix A, we have A⋅A-1 = I (the identity matrix), for an evolutory matrix E, the relationship is different.
In this case, because E = ET, we have:
E ⋅ E = E2 = n ⋅ E
which means E2 is a scalar multiple of E. This occurs because of the symmetry and reciprocity of the exchange rates. This implies that the first row vector, let's call it r, defines the entire matrix as follows: rT⋅ (1÷r)
To be thorough, this condition is also dually defined as follows:
not only: E2 = (E ⋅ E) = n ⋅ E
but also: E2 = (ET ⋅ ET)T = (n ⋅ ET)T
We note in passing that E⋅ET and ET⋅E do not multiply to form n⋅E, but instead result in two other distinct matrices whose properties are outside the scope of this proof.
As we can see, when multiplied by its reciprocal transpose, the evolutory matrix does not produce the identity matrix but rather a scalar multiple of E, scaled by the row count n, effectively becoming E2. This occurs because, under the constraint E=ET, the matrix E exhibits certain structural properties. Specifically, E has a single eigenvalue equal to its trace, which is n.
This is due to the fact that the exchange rate of a currency with itself is always 1, meaning that the diagonal entries of E are all equal to 1. Thus, the trace of E—which is the sum of the diagonal elements—is n, the number of currencies.
This structure implies that E is not an identity matrix but is instead scalar-like, in the sense that its eigenvalues are tied to its trace.
Simplification of E Through Evolutory Constraints
Imposing the constraint E = ET simplifies the matrix E, leaving it with a single eigenvalue, n, and reducing it to a vector-like structure. This occurs because any row or column of E can define the entire matrix, significantly reducing the dimensionality of the information required to quote exchange rates.
For example, the matrix E can be expressed as the outer product of its first column and first row, with each row being the reciprocal of the corresponding column. Consequently, all rows and columns of E are proportional to one another, making them scalar multiples.
This property renders E a rank-1 matrix, meaning all its information can be captured by a single vector.
Higher Powers and Roots of E
An intriguing property of the constrained matrix E=ET is its behavior when raised to higher powers. In theory, an unconstrained matrix raised to the fourth power would have four distinct roots. However, due to the constraint E=ET, E has only two fourth roots: E and ET. This can be expressed as:
not only: E4 = (E ⋅ E ⋅ E ⋅ E) = n2 ⋅ E
but also: E4 = (ET ⋅ ET ⋅ ET ⋅ ET)T = (n2 ⋅ ET)T
This suggests a deep connection between the structure of ET and the physics of symmetry. In this framework, the relationship E4 = n2 ⋅ E = m ⋅ c2 suggests a potential analogy to Einstein’s famous equation E = m ⋅ c2, where mass could be viewed as the fourth root of energy—compressed energy that can be released, for example, in a nuclear explosion.
What’s interesting here is that Einstein’s famous equation becomes dually defined:
not only: E4 = (E ⋅ E ⋅ E ⋅ E) = n2 ⋅ E = m ⋅ n2 = m ⋅ c2
but also: E4 = (ET ⋅ ET ⋅ ET ⋅ ET)T = (n2 ⋅ ET)T = m ⋅ n2 = m ⋅ c2
While E theoretically has four roots, in reality, only two roots, E and ET, exist due to the E = ET evolutory constraint imposed on E by quantum entanglement. Although we are not experts in physics, this concept could be explored further by those familiar with the mathematical properties of complex numbers and quantum systems. But it seems to us that under this evolutory constraint on energy, mass is equivalent to energy but exists as a strictly constrained subset of all possible energy states, limited by the E = ET evolutory condition.
Conclusion: Dual Consistency
Einstein famously remarked that he did not believe God "plays dice" with the universe, expressing his discomfort with the inherent randomness in quantum mechanics. Upon closer reflection, however, this view may not fully capture the nature of the universe. If God did not "play dice"—if there were no randomness at all—even God would be constrained by monotony, a hypothesis not worth testing.
Our analysis offers a different perspective: God does indeed "play dice," but these dice are loaded in such a way that they ensure fairness. This mechanism guarantees that all interactions remain Pareto-efficient and balanced over time, ensuring that, in the long run, everyone receives what they are due, effectively restoring equilibrium in all exchanges.
This leads us to speculate about the deeper implications of Einstein’s famous equation, E = mc². When restated correctly, in a sound formal system in a dually defined way, it represents two dice as:
not only: E4 = E ⋅ n2 = m ⋅ n2
but also: E4 = (ET ⋅ n2)T = m ⋅ n2
Where ET represents the transpose of the Hadamard inverse of matrix E, and ET denotes the transpose of matrix E, we uncover a potential new relationship between energy, mass, and the structural properties of the universe.
Under the constraint E = ET, we know that there are, in reality, two recursively entangled energy states: E=ET and ET. This suggests a deeper connection between energy, mass, and time, hinting at an intrinsic link between temporal dynamics and the fundamental equations that govern the cosmos.
Moreover, these two energy states, when superimposed under E=ET conditions, are interesting in and of themselves—from a formal systems perspective. We posit, as a self-evident axiom—the first, one-truth postulate of all applied mathematics—that the reason why logical inference rules work correctly is that they represent the physical inference rules that operate in our shared objective reality, that of universal causality, which is never violated. If cause, then effect is how logical inference rules work in all formal systems, across all mathematics, under dual consistency.
Errors in representing reality can occur in two fundamental ways: a Type I error (a false positive, rejecting a true claim—akin to disbelieving an honest person) or a Type II error (a false negative, failing to reject a false claim—akin to believing a liar). These two categories are commonly understood in statistical hypothesis testing and illustrate potential pitfalls in scientific and mathematical reasoning. In a sound formal system, such errors do not arise if the rules of deduction are properly followed, leading to correct conclusions derived from the axioms and inference rules.
When evaluating any logical claim, whether within a formal system or in real-world scenarios, there are only four possible outcomes:
Correct Decision I: Accepting a true claim.
Correct Decision II: Rejecting a false claim.
Type I Error: Rejecting a true claim.
Type II Error: Accepting a false claim.
What we are claiming here, under the maximum likelihood principle, is that these four outcomes correspond to the four possible roots of E4—the four possible logical outcomes. However, since there are only two roots, E=ET and ET, the following conclusions hold:
Correct Decision I: ET=E — Accepting a true claim. This is because ET=E is how we initially defined the no-arbitrage condition, which we posit represents a balance in the universe, akin to "goodness."
Correct Decision II: ET≠ET — Rejecting a false claim. This arises because, in fact, the matrix E=ET is not equal to its own transpose but rather equal to the transpose of its Hadamard inverse. Rejecting this false equivalence would correspond to rejecting a false claim.
Type I Error: Rejecting a true claim would be equivalent to rejecting the equality E=ET, which contradicts reality. It’s like believing a false claim instead of trusting what you see with your own eyes. Your own eyes represent an "honest" person—the objective reality that does not lie. In this context, rejecting the truth of E=ET – denying the real, observable balance in the system—would be akin to disbelieving an honest person. This is a classic Type I error – rejecting a true claim: E=ET.
Type II Error: Accepting a false claim would be equivalent to believing that ET=ET, when in fact, this is mathematically false by definition. In this context, accepting the false claim that ET=ET – just like believing a liar—constitutes a Type II error. In statistics and decision-making, a Type II error occurs when one fails to reject a false claim, which in this case would be incorrectly accepting the equality ET=ET.
The other two roots correspond to inherently false claims—the kind that are rejected under the law of excluded middle and the law of non-contradiction within the inference rules of all formal systems and logic. This completes the set of four possible outcomes. As a result, our formal system remains sound—though, admittedly, somewhat incomplete, as we have yet to formally define "God." Nonetheless, the system exhibits dual consistency, ensuring it is both internally consistent and aligned with reality, particularly as it pertains to the physical interpretation of E=mc2.
This connection resonates with principles from supersymmetry in theoretical physics and echoes the ancient Hermetic axiom, "as above, so below." Additionally, this idea aligns with the geometry of the Egyptian pyramids and touches on the humorous notion from The Hitchhiker’s Guide to the Galaxy that "42" is the "answer to the ultimate question of life, the universe, and everything." While this reference is not directly tied to quantum physics, it humorously reflects the probabilistic nature of existence.
QED