13.15.3 Tensor Contraction Type Pair Update

Tensor Contraction Type Pair Update modifies tensor pairs via contraction, simplifying complex algebraic expressions in tensor calculus.

Tensor Contraction Type Pair Update is the rule that specifies how the ordered pair describing a tensor's variance, its number of contravariant and covariant indices, changes as a direct consequence of performing a contraction. It formalizes the bookkeeping step that takes an input type pair and produces the exact output type pair, distinct from the general statement that order decreases, by tracking both components of the pair individually.

Definition

Let a tensor have type pair $(p, q)$ . The type pair update rule for a single contraction is the map:

(p, q) \mapsto (p - 1, q - 1)

This update is applied component-wise: the first entry of the pair decreases by one because one contravariant slot is consumed, and the second entry decreases by one because one covariant slot is consumed, in the same operation.

Update Rule Properties

Simultaneity

The two components of the pair are never updated independently in a single contraction; a contraction always requires one upper and one lower index, so both entries of the pair change together in the same step. There is no valid contraction that updates only $p$ or only $q$ alone.

Preconditions

The update is only defined when both entries of the current pair are strictly positive:

p > 0 and q > 0

If either entry is zero, no valid index of that variance exists to contract, and the update rule cannot be applied.

Idempotent Difference

The difference between the two entries of the pair, $p - q$ , is invariant under the update rule, since both entries decrease by the same amount:

(p - 1) - (q - 1) = p - q

Iterated Application

Sequence of Pairs

Applying the update rule $k$ times in succession to an initial pair $(p_{0}, q_{0})$ generates the sequence:

(p_{0}, q_{0}), (p_{0} - 1, q_{0} - 1), \dots, (p_{0} - k, q_{0} - k)

valid only while both entries remain non-negative at every step.

Boundary Pair

The sequence halts at the first pair where either entry reaches zero. If $p_{0} = q_{0}$ , the terminal pair is $(0, 0)$ , corresponding to a scalar. Otherwise the terminal pair has one zero entry and one strictly positive entry equal to $| p_{0} - q_{0} |$ .

Update Table

Relation to Residual Type and Order

The type pair update rule is the precise, componentwise statement underlying both the residual tensor type and the order reduction behavior: the residual type is exactly the output of this update rule, and the order reduction of two per contraction follows immediately by summing the two componentwise decreases of one each.