Theorem Map

Theorem map

The current public core forms a dependency chain rather than a bag of unrelated facts:

norm derivative
     |
normalization derivative
     |
radial / tangent geometry
     |
kernel / range / rank / spectrum
     |
composition and gradient boundaries
     |
scale-invariant / radial-memory consequences

Normalization derivative

The library formalizes the derivative of x / ||x|| away from the origin and its projection-like radial/tangent structure.

Radial and tangent behavior

Radial perturbations are annihilated by the local normalization derivative; tangent perturbations have an exact inverse-radius gain.

Spectral structure

Finite-dimensional results characterize kernel, range, rank, and singular values.

Loss and gradient consequences

The public core records exact consequences for differentiable losses composed with normalization, while keeping assumptions explicit.

Information boundary

Positive rescalings that map to the same normalized state establish one precise kind of information loss at a normalization boundary. This does not imply all information ever associated with radius disappears from a surrounding recurrence.

Composition boundary

The library records conditions under which local radial/tangent conclusions do or do not survive surrounding maps.

For stable theorem identifiers, names, statements, and source files, INDEX.md is authoritative.