8.8.5 Tensor Repeated Index Validity Constraint

Tensor Repeated Index Validity Constraint is the set of formal requirements that a repeated index must satisfy in order for the expression containing it to be considered well formed under the standard tensor notation conventions, covering the exact occurrence count permitted, the required opposition of variance, and the restrictions on how repeated letters may coexist with other indices in the same term.

The Core Constraints

Exactly Two Occurrences

The first constraint restricts a repeated index to appearing exactly twice within any single term; an occurrence count of three or more is never valid under the standard convention and always requires the expression to be rewritten with distinct letters before it can be interpreted.

This expression violates the occurrence constraint, since (i) appears three times, and one occurrence must be renamed before the expression is valid.

Opposite Variance

The second constraint requires that, of the two occurrences, one must be upper and the other lower; two occurrences sharing the same variance do not satisfy the standard summation convention and either indicate an error or require an explicit metric factor to convert one occurrence before contraction is justified.

Constraints Relating Repeated Indices to Free Indices

No Letter Serves Both Roles Simultaneously

A given letter within a single term cannot simultaneously function as a free index and as a member of a repeated pair; if a letter appears once, it is free, and if it appears twice in opposite variance, it is a repeated pair, but it cannot be both at once within the same term.

Distinctness from Other Indices in the Term

A repeated index pair must not reuse a letter that is already serving as a free index elsewhere in the same term, since doing so would create an unintended third occurrence and collapse the clear distinction between the free index and the paired index.

Here (j) is validly repeated as a pair while (i) remains a validly distinct free index; had (i) instead been chosen as the letter for the repeated pair, it would collide with the intended free index of the same name.

Constraints Across Multiple Terms in a Sum

Free Index Balance Is Independent of Repeated Index Validity

While the repeated index validity constraint governs individual terms, a separate but related requirement demands that whichever free indices survive after all repeated pairs have been accounted for must match, in name and variance, across every term of a sum, ensuring the overall equation remains structurally consistent.

Repeated Letters May Differ Freely Between Terms

The validity constraint on repeated indices applies independently within each term, so different terms of the same sum may use entirely different letters for their internal repeated pairs without any conflict, since each pair's validity is judged only within its own local term.

Consequences of Constraint Violations

Violating any of these constraints, whether through excessive occurrence count, matching variance, or unintended collision with a free index, renders the affected term ambiguous or outright ill-formed, and no summation, contraction, or further manipulation can be reliably carried out until the offending indices are renamed or otherwise corrected to restore a valid pairing structure.

Practical Checklist

Applying this checklist to every term of a tensor expression before attempting any further manipulation is the standard practical safeguard against the most common notational errors in index-based tensor algebra, since each of these constraints, taken together, defines precisely what it means for a repeated index to be valid under the conventional rules of the notation.