8.17.5 Tensor Matching Consistency Check

Tensor Matching Consistency Check is the ordered, practical procedure by which an author or reader verifies that a proposed tensor equation satisfies the full index matching rule, proceeding through a fixed sequence of increasingly detailed tests — type, then position, then letter identity — so that each stage can catch its own class of error before more detailed and time-consuming verification is attempted. It is the operational checklist form of the matching rule, turning an abstract requirement into a concrete sequence of comparisons that can be applied mechanically to any candidate equation.

The Ordered Sequence of Checks

Stage One: Type Count

The check begins by counting, for every term on both sides of the proposed equation, the number of upper and lower free indices, and confirming that this count — the type $(p,q)$ — agrees across every term. Given

every term reduces, once dummy indices are excluded, to exactly one free upper index, so type $(1,0)$ is uniform throughout and the equation passes this first stage.

Stage Two: Position Agreement for Shared Letters

Having confirmed matching type, the check proceeds to verify that wherever the same letter recurs as a free index across different terms, it occupies the same position in each occurrence. In the same example, $i$ is upper in $D^{i}$, upper in $A^{i}$, and upper in the free-index result of $B^{i}{}_{j}C^{j}$, so position agreement holds and the equation passes this second stage.

Stage Three: Full Letter Correspondence

The final and most detailed stage confirms that the specific letters used for free indices correspond one-to-one across every term, with no term substituting a different, uncoordinated letter for what should be the shared free index. Since $i$ — not some other letter — appears consistently across all three terms of the example, this final stage is satisfied, and the equation is certified as fully consistent with the matching rule.

Why the Checks Are Ordered This Way

Coarse Checks Eliminate Obvious Failures Quickly

Type counting requires only tallying index occurrences and classifying them as free or dummy, a comparatively fast operation that can immediately disqualify equations with a mismatched number of free indices, without the need to trace individual letters at all. Performing this check first avoids wasted effort scrutinizing letter-by-letter correspondence in an equation that fails on the much coarser question of how many free indices it even has.

Position Checks Precede Letter Checks Because They Are More Restrictive Per Letter

Once type has been confirmed to match, checking position agreement for whichever letters happen to repeat across terms is generally quicker than fully tracing letter correspondence, since it can often be done by visual inspection of superscript versus subscript placement, catching positional errors before the more exhaustive final comparison of full letter identity is undertaken.

Detailed Letter Correspondence Is Reserved for Last

Because full letter-by-letter correspondence is the most exacting of the three stages, and because a failure at either of the first two stages already disqualifies the equation, this final and most labor-intensive check is only necessary for equations that have already survived the coarser filters, making the overall procedure more efficient than checking full correspondence from the outset on every candidate equation.

Handling a Failure at Any Stage

Type Failure Signals a Rank Mismatch

A failure at the type-counting stage indicates that the two sides, or some subset of terms, differ in how many free indices they carry altogether, typically pointing to a missing or extraneous contraction somewhere in how one of the failing terms was constructed.

Position Failure Signals an Unaccounted Transformation Change

A failure at the position stage, given that type already matches, typically indicates that an index has been raised or lowered in one term without a corresponding, explicit application of the metric tensor recorded elsewhere in the equation, and resolving it usually involves inserting the missing metric factor.

Letter Failure Signals a Derivation Discrepancy

A failure at the full letter-correspondence stage, given that both type and position already agree, indicates that two terms use different, seemingly compatible-looking letters for what should be the same underlying free index, a discrepancy that usually traces back to independent derivation of the two terms without cross-referencing their intended free-index labeling.

Applying the Check to Multi-Term and Multi-Index Equations

Repeating the Sequence for Each Distinct Free Letter

When an equation carries more than one free index, the full three-stage sequence is applied independently for each distinct letter expected to be free, since agreement for one letter does not imply agreement for another; a thorough consistency check audits every free-index letter present in the equation through all three stages before certifying the equation as a whole.

Applying the Sequence Within Each Side Before Comparing Across Sides

For equations whose sides are themselves sums of several terms, the three-stage check is first applied internally within each side — confirming that every term on the left agrees with every other term on the left, and likewise for the right — before the two sides are compared against each other, isolating internal errors from cross-side errors.

Role Within the Index Matching Rule

The matching consistency check is the concrete, sequential procedure that operationalizes the abstract index matching rule, breaking a single overarching requirement into three ordered, individually tractable stages. Applying the check in its prescribed order — type, then position, then letter — allows errors to be caught at the earliest and least labor-intensive stage possible, making it the standard practical method by which the correctness of a proposed tensor equation's index structure is verified.