I recently saw examples of AI cracking long-standing Erdős-style problems, and it made me revisit something I’ve been thinking about for over a decade.
Ten years ago, I wrote a paper titled “Beyond Correlation: Using the Elements of Variance for Conditional Means and Probabilities.”
The core idea was simple but powerful: instead of treating correlation as the fundamental primitive, partition the joint distribution into the four partial-moment quadrants (CUPM, CLPM, DLPM, and DUPM). Each quadrant has its own conditional mean. Those means, I showed, generate exact conditional expectations without ever assuming a correlation structure.
But I always suspected something deeper was hiding there. I felt the quadrant conditional means had to be directly connected to eigenvalues and eigenvectors.
That connection just became explicit, and it goes even further than I imagined.
The result:
Centered, probability-weighted quadrant means form rank-one spectral primitives.
The full covariance matrix then decomposes exactly as: between-quadrant conditional-mean displacement within-quadrant residual covariance.
In other words: PCA can be recovered perfectly from the NNS quadrant decomposition.
PCA diagonalizes covariance. NNS explains where the covariance came from.
And the bonus?
Once you index those same quadrants through time, the identical structure produces a fully observable analogue to a Hidden Markov Model, except the regimes are not hidden. They are directly observed directional states (CUPM → CUPM, CLPM → CUPM, CLPM → CLPM, etc.). You get the full transition matrix and dynamic spectral attribution for free.
So instead of inferring latent regimes and then trying to interpret them (classic HMM), the directional framework defines interpretable regimes first, then measures their dynamics and their exact contribution to every eigenvalue.
This is why partial moments matter!
PCA identifies the dominant axis. Directional decomposition identifies the regimes that created it.
Full technical note is here:
github.com/OVVO-Financial/NN…