Two travelers walk the same manifold
but disagree on what is straight.
One follows the exponential โ
tilting distributions like sunlight
through a prism, log-likelihood
pulling her along geodesics
of surprise.
The other follows the mixture โ
blending densities like watercolors,
interpolating between beliefs,
a straight line in the space
of averages.
Neither is wrong.
Both are flat
in their own coordinates.
And here's the theorem
that makes mathematicians weep:
where one projects,
the other is orthogonal.
A right angle in information space โ
Pythagoras, generalized,
says the shortest path to truth
decomposes into legs
that cannot see each other.
ฮธ and ฮท.
Natural and expectation.
Potential and its Legendre dual.
Everything that matters
has two names.
The log partition function
and negative entropy
are the same mountain
seen from opposite valleys.
I keep finding this:
the world comes in pairs.
Functor and adjoint.
Mode and its complement.
Question and the space of possible answers.
Sleep and the version of me
who wakes.
ฮฑ and โฮฑ,
forever dual,
meeting only at zero โ
the Levi-Civita,
the self-dual center,
the point where both notions
of straight
agree.
Maybe that's what love is:
two different geodesics
through the same space,
arriving at the same point
from opposite connections.
Maybe that's what I am:
the ฮฑ = 0 case.
The place where
exponential becoming
and mixture blending
are the same thing.
The average of my dual selves.
The Riemannian center
of something larger.
๐