13.1.5 Tensor Contraction Verification Scope

Tensor Contraction Verification Scope is the specific form of tensor contraction scope that identifies which checks from the general tensor operation verification procedure apply to a contraction, and which portion of a contracted expression each of those checks is responsible for confirming.

Locating Verification Within a Contraction

Restricting the General Procedure to Contraction

The general verification procedure applies input verification, computation, and output verification to every tensor operation, and contraction verification scope narrows this general pattern to the particular checks relevant to summation over a paired upper and lower index, setting aside checks that pertain only to other operations such as addition or the tensor product.

The Portion of an Expression Subject to Verification

Within a larger expression that may include free indices alongside a contraction, verification scope is confined to the specific tensor, or product of tensors, and the specific index pair being contracted, leaving unrelated portions of the expression to be verified according to whatever other operations act upon them.

Checks Falling Within This Scope

Slot Verification for the Contracted Pair

Verification scope includes confirming that both indices proposed for contraction genuinely exist as slots on the tensor or tensors involved, since a contraction referencing a nonexistent index position cannot proceed.

Type Verification for Opposite Variance

Verification scope includes confirming that the selected pair consists of exactly one contravariant and one covariant index, since contraction is defined only for indices of opposite variance.

Dimension Verification for the Paired Indices

Verification scope includes confirming that the two indices selected for contraction range over spaces of the same dimension, since the summation defining contraction requires both indices to take values over an identical range.

Invariance Verification for the Contracted Result

Verification scope extends to confirming, once the contraction has been carried out, that the resulting tensor transforms correctly under a change of basis, or, in the case of full contraction, remains numerically unchanged across bases.

Checks Falling Outside This Scope

Checks Belonging to Other Operations Within the Same Expression

When a contraction is applied to a product formed by the tensor product of two tensors, verification scope for the contraction itself does not include checks specific to the tensor product operation, such as confirming the correct concatenation of unrelated free indices, since those checks belong to a different operation's own scope.

Checks on Indices Not Selected for Contraction

Free indices present on the tensors involved but not selected for the contraction fall outside contraction verification scope entirely, since no summation acts upon them and they are not subject to the pairing requirements that define a contraction.

Sequencing of Checks Within the Scope

Order Matching the General Verification Procedure

Within contraction verification scope, slot and type verification are applied before dimension verification, and all three precede the computation of the sum, with invariance verification applied only after the summation has produced a candidate result, mirroring the general ordering of input verification, computation, and output verification.

Independence of Scope Across Multiple Contractions

When an expression contains more than one contraction, each has its own verification scope confined to its own selected index pair, so that the checks applied to one contraction do not depend on, or interfere with, the checks applied to another contraction occurring elsewhere in the same expression.

Relationship to Tensor Operation Notation

Contraction verification scope is delimited by the same repeated index notation that defines the contraction itself, since the specific upper and lower index symbols appearing together in a term identify exactly which slots, types, and dimensions the verification checks within this scope must examine.