Derivatives of tensor coordinate maps #
Slicing, gathering, and broadcasting all read input coordinates through a fixed map. The map need not be injective: repeated reads are precisely why the reverse pass must sum contributions.
These statements use Rep.pull, the coordinate kernel underlying the tensor representation, and
the existing Euclidean topology on real tensors. A derivative is therefore stated on the actual
tensor type. The coordinate formulas also apply to empty shapes; there is no chosen first element
or hidden positivity assumption.
The native coordinate pullback, bundled as a continuous linear map.
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The bundled map reads exactly the coordinates selected by the native kernel.
Reindexing is linear, so the same coordinate map acts on every input tangent.
A source coordinate receives the sum over all output coordinates which read it.
There is deliberately no distinctness hypothesis on index. For a gather with repeated indices,
every occurrence contributes; for a broadcast, every replicated batch contributes.
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Coordinate pullback and summation over its fibers are adjoint.
The analytic adjoint is the concrete accumulation formula, not an unspecified reduction.
The Euclidean inner product separates over the leading tensor axis.