13.6.1 Tensor Full Contraction All Slot Pairing

Tensor Full Contraction All Slot Pairing is complete contraction of all tensor indices, yielding a scalar through summed index pairings.

Tensor Full Contraction All Slot Pairing is the specific slot pair set underlying the full contraction operation, characterized by the property that every contravariant and every covariant slot of the input tensor belongs to exactly one member of the set, with no slot left outside of any pairing.

Defining Feature of the Pairing

Complete Coverage of Every Slot

All slot pairing requires that each contravariant index and each covariant index of the input tensor appear in precisely one member of the slot pair set, ensuring that the union of all pairs accounts for the tensor's entire index structure.

T_{i j}^{i j}

No Slot Appearing More Than Once

Consistent with the disjointness required of any valid slot pair set, all slot pairing additionally requires that no slot appear in more than one pair, so that complete coverage is achieved through a partition of the slots rather than through any overlapping assignment.

Existence Condition for a Valid All Slot Pairing

Requirement of Balanced Type

An all slot pairing can exist only when the input tensor's contravariant count equals its covariant count, since a pairing scheme assigning every contravariant slot to a distinct covariant slot requires the two counts to match exactly.

p = q

Nonexistence When Counts Differ

When a tensor's contravariant and covariant counts are unequal, no all slot pairing can be formed, since any attempted assignment would necessarily leave at least one slot of the majority category without a partner, preventing complete coverage from being achieved.

Multiplicity of Possible All Slot Pairings

More Than One Valid Pairing on the Same Tensor

When a tensor's balanced type permits more than one way of matching its contravariant slots to its covariant slots, several distinct all slot pairings may exist for that same tensor, each satisfying the coverage and disjointness requirements independently of the others.

T_{i j}^{i j} \neq T_{j i}^{i j}

Distinct Pairings Producing Distinct Scalars

Because different all slot pairings generally sum different combinations of the tensor's components together, distinct pairings applied to the same tensor typically yield different scalar results, so specifying which all slot pairing is intended remains necessary even once the input tensor's type is known to permit full contraction.

All Slot Pairing as the Basis of the Full Contraction Operation

Direct Correspondence with the Operation's Definition

Because the full contraction operation is defined by its slot pair set exhausting every available slot, an all slot pairing is precisely the object that must be specified before the full contraction operation can be carried out on a given tensor.

Determining the Summation Set for Full Contraction

Once an all slot pairing is fixed, the corresponding summation set is determined directly, with one summation index introduced for each member of the pairing, together defining the complete nested summation that produces the final scalar.

Relationship to Tensor Operation Notation

An all slot pairing is represented in tensor operation notation by an expression in which every index symbol, without exception, is repeated as a matched upper-lower pair, with the specific assignment of symbols to pairs indicating which particular all slot pairing, among possibly several available, has been chosen for that expression.