6.5.2 Tensor Valence Contravariant Count

Tensor Valence Contravariant Count is the number p of up-arrow (contravariant, upper) entries appearing in a tensor's valence sequence, obtained by counting the contravariant positions within the full ordered arrangement rather than merely stating the type total in isolation. It is the same integer that type classification records as the first component of the pair (p, q), but recovered here as a derived quantity, extracted by scanning a specific, already-fixed valence sequence rather than assumed as a given input, exactly mirroring how the covariant count q is extracted from the down-arrow positions.

Extracting the Contravariant Count From a Valence Sequence

The Counting Formula

Given a valence sequence (σ₁, ..., σ_k) with each σ_i ∈ {↑, ↓}, the contravariant count is:

the size of the set of positions carrying an up-arrow, computed directly from the sequence.

Worked Example

For the valence sequence (↑, ↓, ↑, ↓, ↓), scanning position by position finds up-arrows at positions 1 and 3, giving a contravariant count of p = 2; combined with the covariant count q = 3 found by the complementary scan, the full type recovered from this valence is (2, 3), consistent regardless of which count is computed first.

Diagram of Scanning a Valence Sequence for Up-Arrows

Relation to Type Classification

Consistency With the Type Pair

The contravariant count extracted from any valid valence sequence for a tensor always equals the p component of that tensor's type (p, q); this is a direct consequence of valence being a refinement of type that adds positional detail without ever contradicting the totals type already fixes.

Complementarity With the Covariant Count

Because every slot in a valence sequence is either an up-arrow or a down-arrow and no third option exists, the contravariant count and the covariant count are complementary with respect to the total slot count: p + q = k always, so computing one of the two counts from a valence sequence and subtracting from k gives the other without a second scan.

The Contravariant Count as a Constraint on Valence

Fixing p Without Fixing the Full Valence

Knowing only that p = 2 for a tensor of some order k constrains, but does not determine, the full valence sequence, since the two up-arrows could occupy any 2 of the k available positions; as with the covariant count, the contravariant count alone is necessary but not sufficient to reconstruct the exact arrangement.

Number of Valences Sharing a Fixed Contravariant Count

is numerically identical to the corresponding formula for the covariant count, since choosing which p positions are up-arrows automatically determines which remaining positions are down-arrows.

Practical Uses of the Contravariant Count

Verifying a Contraction Is Well-Formed

Before performing a contraction that pairs a specific upper slot against a specific lower slot, confirming that the contravariant count p is at least 1 is the complementary necessary check to confirming q ≥ 1, together guaranteeing that at least one valid upper-lower pair exists to contract.

Identifying Purely Contravariant Tensors

A tensor with covariant count q = 0 and contravariant count p = k is classified as purely contravariant; recognizing this case from the contravariant count alone, without needing to inspect the full valence sequence, is useful whenever an operation is defined specifically for purely contravariant tensors, such as certain constructions in the symmetric algebra restricted to V^{⊗k} alone.

Why the Contravariant Count Matters

Completing the Bridge Between Type and Valence

Together with the covariant count, the contravariant count shows precisely how type classification's two defining totals are both recoverable from valence classification's more detailed sequence, confirming from both directions that valence strictly refines, and never conflicts with, type.

A Building Block for Slot Bookkeeping Alongside the Covariant Count

Isolating the contravariant count as its own quantity, usable independently of the full positional detail of a valence sequence, supports the same class of lightweight feasibility checks — contraction eligibility, purely-contravariant recognition — that the covariant count supports on the opposite side of the type pair.