13.13.5 Tensor Contraction Pair Selection Criterion

Tensor Contraction Pair Selection Criterion is the formal standard against which any candidate contravariant and covariant slot combination is measured to determine whether it qualifies as an admissible contraction pair, stated as a single unified principle rather than as a checklist of separately named conditions. It provides the definitional statement from which the more specific checks of variance opposition and dimensional agreement are derived, functioning as the master rule invoked whenever a selection among candidate slots must be justified as valid.

Conceptual Basis

A Single Principle Underlying Multiple Checks

Rather than treating variance matching and dimension matching as two independent, unrelated rules, the selection criterion frames them as two necessary clauses of one overarching principle: a candidate pair is admissible for contraction if and only if it satisfies both clauses simultaneously, with the criterion itself being the statement of this joint necessity.

Criterion as a Filter Applied Before Any Computation

The selection criterion operates strictly at the stage of choosing which indices to contract, prior to any arithmetic being carried out, functioning as a gate that determines whether a proposed contraction is even eligible to be attempted, independent of what values the tensors involved might take.

Universality Across All Contraction Instances

Because the selection criterion is stated in fully general terms, referencing only variance and dimension rather than any particular tensor or rank, it applies without modification to every instance of contraction, whether within a single tensor or across several, and regardless of which named contraction case is ultimately being constructed.

Formal Description

Statement of the Criterion

A candidate contravariant slot with variance $$\sigma$$ and dimension $$m$$, paired with a candidate covariant slot with variance $$\tau$$ and dimension $$n$$, satisfies the selection criterion if and only if:

with both statements required to hold for the candidate pair to be accepted.

Applying the Criterion to Distinguish Valid From Invalid Pairs

Given a tensor with contravariant indices of dimension $$3$$ and covariant indices of dimension $$3$$, any candidate pair drawing one index of each variance from this tensor automatically satisfies the criterion, while a hypothetical candidate pairing two contravariant indices, or pairing indices of differing dimension, fails the criterion regardless of any other property of the tensor.

Derivation of Specific Checks From the General Criterion

The separately discussed notions of index compatibility and dimension match are both direct specializations of this single selection criterion, obtained by isolating one of its two clauses for individual verification during the practical process of examining a candidate pair.

Properties

Necessity and Sufficiency

The selection criterion is both necessary and sufficient for a candidate pair to be eligible for contraction: satisfying it guarantees that a well-defined summation can be constructed from the pair, while failing it guarantees that no valid contraction can be formed from that specific pair as proposed.

Invariance of the Criterion Itself

Unlike the outcome of applying the criterion to any particular candidate pair, the criterion itself does not change from one contraction instance to another, remaining a fixed standard applied uniformly regardless of the specific tensors or ranks under consideration.

Independence From the Number of Available Candidates

The selection criterion evaluates a single candidate pair at a time and does not depend on how many other candidate pairs might also be available on the same tensor, meaning its application to one candidate has no bearing on whether a different candidate pair also satisfies the criterion.

Practical Considerations

Guiding Systematic Enumeration of Valid Pairs

When multiple candidate pairs exist on a tensor with several indices of each variance, applying the selection criterion systematically to each candidate in turn allows the complete set of valid, admissible pairs to be enumerated without omission or error.

Foundation for Automated Validation

Symbolic and computational tensor systems implement the selection criterion as an automated check performed whenever a contraction is requested, rejecting or flagging any proposed operation whose designated pair fails to satisfy either clause of the criterion.

Basis for Teaching and Formalizing Contraction

Presenting the selection criterion as a single unified statement, rather than as two disconnected rules, offers a clearer and more economical way of communicating the essential requirement underlying every instance of tensor contraction, from the simplest vector covector pairing to the most elaborate multi-tensor expressions.