BlueberryML treats knowledge as a curved space. Students are objects moving through it. The best path is not recommended — it is derived from the shape of the space itself.
Each learning outcome — fractions, phonics, read aloud — is a point. Related outcomes sit close together, so the curriculum stops being a list and becomes a map.
But it is not a semantic location. The point is not where a language model happens to embed the words. Its coordinates are the named things a teacher already measures.
Where prerequisites are dense and the abstraction leap is large, the space bends. A hard concept doesn’t sit in flat space and somehow make it curve. The difficulty is the curvature.
Cognitive cost equals complexity density times the square of how fast a given student can propagate understanding. Small differences in λ become large differences in cost exactly where the space is densest — which is why ability gaps widen at higher abstraction.
The straight line between two outcomes crosses the densest territory. The geodesic — the path of least action — bends around it, picking up nearby mastery that flattens the curve ahead. Not minimum time. Not minimum difficulty. Minimum action.
A depth-first tradition digs narrow, deep wells before moving on. A breadth-first tradition spirals, letting partial mastery next door flatten the hill here. The geometry does not pick a pedagogy — it derives the right one per student.
The manifold is universal. The metric is cultural. The geodesic is personal.
Every assessment updates a position. Practice moves it forward; forgetting pulls it back. The system sees the drift before a test would — and surfaces the review that stops it.
Outcomes pin the coordinates. Textbooks supply the distances — twelve pages and six exercise types on one outcome is a measurement of ρ. Student data hardens the estimate into an empirical map.
Built so that every learner who passes through makes the map sharper for the next. The tenth curriculum is far more valuable than the first.
These are not embedding dimensions that compress themselves and mean nothing to a teacher. The axes are fixed, named and human — and they change with the subject. Same structural framework, different axes.
Accuracy is on every subject. Conceptual Understanding and Representation recur across Maths and Science. Pronunciation belongs only to English, Procedural Fluency only to Maths, Inquiry only to Science. The shape changes; the geometry does not.