Meaning Becomes Geometry

Mon, 07 Sep 2026 09:10:00 +0000

Part I — A Vector Is Not Meaning

A space you can draw

Give six words two coordinates each, by hand:

              (x = royalty/power,  y = gender: -1 female … +1 male)
king      ( 0.9,  0.8)
queen     ( 0.9, -0.8)
man       ( 0.1,  0.9)
woman     ( 0.1, -0.9)
apple     (-0.8,  0.0)
orange    (-0.8,  0.1)

Now every semantic question has a geometric form:

Are king and queen related? → they are close (distance 1.6, mostly along y).
What distinguishes king from queen? → a direction: king − queen ≈ (0, 1.6), the “gender axis.”
Is apple like king? → the angle between them is wide; they point different ways.
Which words form a group? → {apple, orange} cluster tightly, far from the royalty cluster.

This is the whole promise of embeddings in one picture:

Retrieval Is Not Geometry

Wed, 09 Sep 2026 01:50:00 +0000

Research Coda — What Alignment Really Preserves

The result that looks like success

Suppose a bridge translates vectors from space A into space B. On a held-out set, the true B-space counterpart appears in the translated vector’s top ten 99.7% of the time.

That sounds finished.

Now ask a different question: does the translated vector have the same ten neighbors as the native B-space vector?

The answer can be barely 59%.

Geometry on Programmer.ie: Modern AI programming

Meaning Becomes Geometry

A space you can draw

Retrieval Is Not Geometry

The result that looks like success