11.20.4 Tensor Variance Verification Type Preservation

Tensor Variance Verification Type Preservation is the final stage of the verification procedure that confirms whether an operation performed on one or more confirmed tensors, such as a sum, product, or contraction, yields a result whose variance type matches exactly what the arithmetic rules for that operation predict, ensuring that tensor status itself is not silently lost or altered during a composite calculation.

Foundational Setting

Why This Stage Comes Last

Earlier stages of the verification procedure confirm that individual objects genuinely transform as tensors of a claimed type. Type preservation addresses a different question: once several verified tensors are combined through an operation, does the resulting object still qualify as a tensor, and does its variance type match the value predicted by the known arithmetic of that operation.

The Predicted Type as a Benchmark

Each standard operation carries a predictable effect on variance type: tensor product adds type pairs, contraction subtracts one from each count, and raising or lowering converts one index kind into the other while preserving total rank. Type preservation checks the actual result of a computation against this predicted benchmark.

Verifying Preservation Under the Tensor Product

The Predicted Type

For tensors of type $$(p_1,q_1)$$ and $$(p_2,q_2)$$, the predicted type of their product is $$(p_1+p_2,q_1+q_2)$$.

Checking the Actual Result

Verification confirms this by directly transforming the product under an explicit basis change and checking that exactly this many factors of the direct and inverse matrices appear, with no missing or extra factors:

Verifying Preservation Under Contraction

The Predicted Type

Contracting one upper index against one lower index of a tensor of type $$(p,q)$$ is predicted to yield a tensor of type $$(p-1,q-1)$$.

Confirming Invariance of the Contracted Result

Verification checks that the contracted quantity, computed in two different bases, yields consistent components related exactly by the predicted, reduced transformation law, with the matrix factors associated with the contracted pair of indices having correctly canceled:

Visual Overview

Diagram of the Preservation Check

Preservation Under Raising and Lowering

Checking Rank Conservation

When an index is raised or lowered using a metric tensor, type preservation confirms that the total rank $$p+q$$ remains unchanged, with one contravariant index converted to covariant, or vice versa, and that the newly repositioned index now satisfies the transformation law appropriate to its new position.

Catching Errors in Multi-Step Derivations

In a derivation involving several successive operations, checking type preservation after each individual step allows an error, such as an accidentally uncontracted index or a misapplied metric factor, to be localized to the specific step at which the predicted and actual types first diverge.

Summary of Key Traits

Defining Characteristics

• Type preservation compares the variance type predicted by an operation's known arithmetic rule against the type confirmed by direct transformation analysis of the actual result.
• Tensor product, contraction, raising, and lowering each have a specific predicted effect on the type pair that this stage verifies.
• A mismatch between predicted and actual type indicates the operation was misapplied or that an intermediate object was not a genuine tensor.
• Applying this check after each step of a multi-step derivation localizes errors to the specific step where they first arise.