13.22.4 Tensor Contraction Product Operation Boundary

Tensor Contraction Product Operation Boundary sets limits on index contraction, ensuring valid tensor algebra computations and operational clarity.

Tensor Contraction Product Operation Boundary is the limiting condition specific to a product of several tensors, marking the point at which no remaining pair of indices, drawn from any two factors in the product or from a single factor's own indices, satisfies the pair validity check, so that the entire product admits no further contraction regardless of which factors are considered together.

Definition

For a product of tensors $T_{1} T_{2} \dots T_{m}$ , the product operation boundary is reached when, across all factors jointly, no upper index of any factor remains available to pair with a lower index of any factor, including itself:

for every pair (i, j), no valid upper-lower pairing remains among T_{1}, \dots, T_{m}

Contrast With the Single-Tensor Boundary

Broader Scope of Pairing Opportunities

Where the general contraction boundary considers only a single tensor's own upper and lower index counts, the product operation boundary must additionally account for cross-factor pairings, since an upper index belonging to one factor in the product may validly contract against a lower index belonging to a different factor, extending the pool of available pairings well beyond what any single factor offers alone.

Later Arrival Than Single-Factor Boundaries

Because of this broader pool, the product operation boundary is generally reached later, in terms of total contractions performed, than the individual boundary of any single factor considered in isolation, since cross-factor contractions can continue even after a given factor's own indices have all been paired with indices from other factors.

Reaching the Boundary

Total Index Budget

If the factors in the product collectively have $P$ total upper indices and $Q$ total lower indices across all of them, the maximum number of contractions achievable before the boundary is reached is at most $min (P, Q)$ , mirroring the single-tensor formula but applied to the combined index budget of the entire product.

Residual Variance at the Boundary

Once the boundary is reached, the surviving free indices across the product consist entirely of one variance type if the original totals were unequal, or none at all if $P = Q$ and full contraction to a scalar has been achieved.

Diagram

Significance

The product operation boundary provides the correct termination criterion for a contraction sequence procedure applied to an entire multi-tensor product, ensuring that the sequence is not halted prematurely by mistakenly treating each factor's index budget in isolation, and instead correctly accounts for the full range of contraction opportunities that arise only when the factors are considered together as a single combined expression.