13.14.1 Tensor Contraction Covariant Count Change

Tensor Contraction Covariant Count Change is the specific decrease undergone by the covariant index count component of a tensor's type as a direct consequence of contraction, tracked in isolation from the parallel change occurring in the contravariant count. It isolates the other half of the overall type reduction produced by contraction, focusing exclusively on how the number of lower indices evolves as successive contractions are applied, independent of any accompanying discussion of the contravariant side.

Conceptual Basis

Isolating the Covariant Component of the Type Pair

A tensor's type is described by two counts considered jointly, but the covariant count change examines only the second of these two numbers, tracking how it decreases as contractions proceed, without folding in the simultaneous behavior of the contravariant count into the same discussion.

Why the Covariant Count Decreases

Each valid contraction, by requiring one covariant index to be consumed alongside one contravariant index, necessarily reduces the covariant count by exactly one for every contraction performed, making the change entirely predictable and directly tied to the number of contractions applied.

Relationship to Which Covariant Slots Are Consumed

The covariant count change records only the numerical reduction in count, not which specific covariant slots were selected for contraction; different choices of covariant slot pair selection can produce the same numerical count change while consuming different particular indices.

Formal Description

General Formula for the Count Change

For a tensor with original covariant count $$q$$ subjected to $$k$$ contractions, the resulting covariant count is:

with the change itself, the amount by which the count has decreased, equal to $$k$$.

Bound on the Possible Change

Since each contraction requires an available contravariant index as well, the covariant count change is bounded by the original contravariant count $$p$$, meaning:

so that the covariant count cannot be reduced below $$q-\min (p,q)$$ regardless of how many contractions are attempted.

Independence From Which Specific Indices Are Chosen

If a tensor has several covariant indices available, contracting any one of them produces the same magnitude of covariant count change, namely a decrease of one, regardless of which specific slot among the several candidates was selected for the contraction.

Properties

Direct Proportionality to Contraction Count

The covariant count change is always exactly equal to the number of contractions performed, establishing a direct and simple proportional relationship between these two quantities that holds regardless of the tensor's rank or the specific contractions chosen.

Zero Change Under Operations Not Involving Covariant Slots

If an operation performed on a tensor does not involve summing over any covariant index, the covariant count change associated with that operation is zero, distinguishing genuine contraction from other rank-preserving or structure-altering operations that leave covariant slots untouched.

Coupling With the Contravariant Count Change

Although considered in isolation here, the covariant count change is always numerically equal to the contravariant count change for any valid contraction, since both counts decrease by the same amount with each contraction performed, a coupling that follows directly from the requirement that contraction always removes one index of each variance.

Applications

Tracking Progress Toward a Scalar Result

Monitoring the covariant count change across a sequence of contractions provides a direct way of tracking progress toward the scalar result case, since this case is reached precisely when the covariant count has been reduced to zero, alongside the contravariant count.

Predicting the Type of Intermediate Tensors

In a multi-step computation, knowing the covariant count change expected from each planned contraction allows the type of every intermediate tensor in the sequence to be predicted in advance, aiding in the design and verification of complex tensor expressions.

Diagnostic Use in Verifying Contraction Counts

Comparing the observed covariant count change against the number of contractions actually performed provides a straightforward consistency check, since any discrepancy would indicate that an operation other than a standard contraction, or an incorrectly specified contraction, has been applied.