Modern AI's deepest design principles — the manifold hypothesis, metrics on latent spaces, hyperbolic embeddings, equivariance — are one lesson learned repeatedly: meaning is what survives coordinate change. Hare learned that lesson in APL at Berkeley in the 1980s, was handed back-propagation by his advisor, and built the constructive version: a metric for coordination space, specified rather than learned, enforced rather than observed.
Each stop is a page in the graph. The order is an argument — but every one of them is also a door into the rest of it, so you can leave the path wherever it stops being useful.
Geometric deep learning constrains a network so that only symmetry-respecting transformations are expressible. The architecture constrains the substrate so that only governance-preserving derivations settle. Same mathematics, opposite direction — the field discovers geometry, the architecture specifies it. A reviewer who accepts the first has already accepted the formal basis of the second. And the widest-register statement of the same geometry now exists: One Mathematics, Many Heavens — the unification of all faiths and science — running the identical Hilbert-space argument from the IEOR classroom to the Fermi Paradox, with its claims graded in a closing ledger.