13.5.4 Tensor Multiple Contraction Type Effect

Tensor Multiple Contraction Type Effect is the cumulative change in a tensor's type, expressed as its paired contravariant and covariant counts, that results from applying a multiple index contraction operation, describing how the type shifts as a direct consequence of the number of pairs contained in the slot pair set employed.

Statement of the Cumulative Effect

Reduction of Both Counts by the Number of Pairs

A multiple index contraction operation built from a slot pair set of size $k$ transforms a tensor of type $(p,q)$ into a tensor of type $(p-k,q-k)$, since each of the $k$ pairs removes exactly one contravariant and one covariant index.

Derivation from the Single Contraction Type Effect

Because the multiple index contraction operation is composed of $k$ applications of the single index contraction type effect, each contributing a reduction of one to both components, the cumulative type effect follows by simple addition of the individual, elementary effects.

Consequences for the Order

Total Order Reduction

Summing the individual reductions in the two type components gives the familiar total order reduction of twice the number of pairs, consistent with the order effect associated with the multiple index contraction operation generally.

Distinguishing Type Effect from Order Effect Alone

Reporting the effect on order alone would obscure the fact that the reduction is distributed evenly between the two index categories, information the type effect preserves and that remains relevant whenever a subsequent operation depends specifically on the balance between contravariant and covariant indices.

Bound on the Achievable Effect

Limitation Imposed by the Smaller Original Count

The number of pairs $k$ comprising any valid slot pair set cannot exceed the smaller of the original contravariant and covariant counts, since each pair consumes one index from each category, and no more pairs can be formed than the more limited category permits.

Reaching the Maximal Possible Effect

When $k$ equals the smaller of the two original counts, and the two original counts happen to be equal, the multiple contraction type effect reduces the type entirely to $(0,0)$, coinciding with the type effect of the full contraction operation.

Type Effect for a Partial Slot Pair Set

Residual Type When Fewer Than the Maximum Pairs Are Chosen

When the slot pair set contains fewer pairs than the maximum permitted by the tensor's original type, the multiple contraction type effect leaves a nonzero residual type, with the surplus indices of each category persisting as free indices in the result structure.

Consistency with Partial Contraction Area

A multiple contraction type effect leaving a nonzero residual type corresponds to a result falling within partial contraction area, since the tensor produced retains free indices and has not been reduced to a scalar, distinguishing this outcome from the endpoint associated with a fully exhausted slot pair set.

Relationship to Tensor Operation Notation

The multiple contraction type effect is observed in tensor operation notation by counting the distinct upper index symbols and distinct lower index symbols removed through matched repeated pairs, with the number of such pairs present in an expression directly indicating the magnitude $k$ of the type effect produced by the corresponding operation.