the proof

the mathematics that the protocol is computing.

the theorem attributed to Pythagoras has been proved in at least three hundred and seventy distinct ways since antiquity, more than any other theorem in mathematics. the original proof, if it ever existed in written form, has been lost. what survives is the result. the square of the hypotenuse of a right triangle equals the sum of the squares of the two legs.

the proof generalizes in a single step from triangles in a plane to vectors in any euclidean space of any dimension. the squared length of any vector is the sum of the squares of its components. the squared distance between any two points is the squared length of the vector between them. this is the foundation of every metric used in modern mathematics.

the inequality

the cauchy-schwarz inequality, proved in the early nineteenth century, is a direct consequence of the pythagorean theorem. it states that for any two vectors, the magnitude of their dot product is at most the product of their magnitudes. when the dot product equals the product of the magnitudes, the vectors are parallel. when it is zero, they are orthogonal. this is the geometric content of the inequality, and it is what makes the pythagorean distance a meaningful measurement.

the consequence for markets is direct. when the trades on a token form a cloud of points whose displacement vectors are largely orthogonal to each other, the cloud expands in many directions and the pythagorean distance from the centroid is large. when the vectors become parallel, the cloud collapses along a single direction and the distance shrinks. parallel vectors are the signature of coordinated flow.

the natural reference

the protocol's reference value of 1.414 is the square root of two, which is the average pythagorean distance an independent participant produces in a unit space. the value was computed analytically from the geometry of a uniform distribution on the two dimensional trading manifold. it is hardcoded into the binary. it cannot be changed by anyone.