13.22.2 Tensor Component Operation Contraction Boundary

Tensor Component Operation Contraction Boundary defines limits on tensor contractions, constraining operations within algebraic tensor spaces.

Tensor Component Operation Contraction Boundary is the limiting condition observed at the level of individual stored numerical entries, marking the point at which every computed component of a partially contracted tensor has been produced by summing over its full assigned index range, so that no additional per-component summation work remains to be performed for the contraction currently in progress.

Definition

For a contraction producing a result tensor with components indexed by the surviving free indices, the component operation boundary is reached once every entry in the component array has been computed by summing over the entire range of the contracted index:

R_{j}^{i} = \sum_{a = 1}^{n} A_{a}^{i} B_{j}^{a}

for every admissible pair of values of the free indices $i$ and $j$ ; once all such pairs have been evaluated, the operation boundary for that contraction has been reached.

Distinguishing From the Structural Boundary

Computation-Level Versus Type-Level

Where the general contraction boundary and its variants describe when no further contraction is possible in terms of remaining index types, the component operation boundary describes a distinct, lower-level milestone: the completion of the arithmetic work required to produce the numerical values of a single already-specified contraction, regardless of whether further contractions remain possible afterward.

Reachable Mid-Sequence

Because it concerns only the completion of one contraction's arithmetic, the component operation boundary can be reached and passed at every single step of a longer contraction sequence procedure, occurring once for each intermediate tensor produced along the way, rather than only once at the very end of the entire sequence.

Partial Completion Before the Boundary

In-Progress Summation

Before the component operation boundary is reached for a given output entry, that entry's summation may be only partially accumulated, having processed some but not all values of the contracted index's range; the entry's final, correct value is only guaranteed once the full range has been summed.

Entry-by-Entry Progress

Different entries of the same output tensor may reach their own individual completion at different times during an ongoing computation, so the component operation boundary for the tensor as a whole is reached only once every one of its entries has individually reached its own summation completion.

Diagram

Significance

The component operation contraction boundary provides the practical, arithmetic-level completion criterion that a direct implementation of a single contraction step must satisfy, complementing the more abstract, type-based general contraction boundary by specifying exactly when the numerical work underlying one particular contraction, whether standalone or embedded within a longer sequence, has been fully carried out.