12.22.5 Tensor Error Pattern Boundary

Tensor Error Pattern Boundary is the classification boundary that separates recognizable, recurring categories of failure in tensor operations from failures that do not fit an established pattern, marking the line beyond which a violation of an operation's boundary is understood as a distinct, named type of error rather than an unclassified anomaly.

Purpose of Classifying Errors by Pattern

Recurrence Across Different Operations

Certain failures in tensor operations recur across many different specific operations in structurally similar forms, and the error pattern boundary exists to group these recurring failures into named categories so that a failure can be diagnosed by which category it falls into rather than treated as an isolated incident.

Connecting Failures to the Boundary They Violate

Each recognized error pattern corresponds to the violation of a specific condition drawn from the general tensor operation boundary, so that classifying an error by pattern amounts to identifying precisely which condition of the operation's boundary was not satisfied.

Recognized Error Patterns

Order Mismatch Pattern

This pattern covers failures in which operands presented to an operation such as addition possess differing numbers of indices, violating the order agreement condition of the general operation boundary.

Variance Mismatch Pattern

This pattern covers failures in which corresponding indices of two operands, or the two indices selected for a contraction, carry the wrong combination of contravariant and covariant placement, violating the type agreement condition of the boundary.

Dimension Mismatch Pattern

This pattern covers failures in which corresponding indices agree in variance and count but range over spaces of differing dimension, violating the dimension agreement condition of the boundary.

Slot Misidentification Pattern

This pattern covers failures in which an operation references an index position that does not exist on the tensor in question, or refers to the correct slot under an incorrect assumption about its variance, violating the slot existence condition of the boundary.

Vector Space Mismatch Pattern

This pattern covers failures in which operands are built from unrelated underlying vector spaces despite matching order, variance, and dimension, violating the vector space agreement condition of the boundary.

Basis Inconsistency Pattern

This pattern covers failures in which operands are expressed relative to different, unreconciled bases, or in which a proposed basis change relies on a noninvertible transformation, violating the basis consistency and invertibility conditions of the boundary.

Using the Pattern Boundary in Diagnosis

Matching an Observed Failure to a Pattern

When a tensor operation fails during verification, the specific check that failed indicates which recognized error pattern the failure belongs to, allowing the failure to be reported using the established terminology for that pattern rather than a generic description of noncompliance.

Failures Outside Any Recognized Pattern

A failure that does not correspond to any of the established patterns lies outside the error pattern boundary, indicating either an unusual combination of otherwise-valid conditions or a case not yet incorporated into the recognized classification scheme.

Relationship to the Verification Procedure

Correspondence with Verification Stages

Because each error pattern corresponds to a specific condition checked during input or output verification, the stage at which a failure occurs during the verification procedure narrows which error patterns are possible causes of that failure.

Value for Consistent Reporting

Classifying failures according to established error patterns allows the verification procedure to report failures in a consistent form across different operations, since the same named pattern, such as a dimension mismatch, can arise in addition, contraction, or the tensor product alike.

Relationship to Tensor Operation Notation

Error patterns are described using the same indicial notation used to express tensor operations generally, since a mismatch pattern is most precisely stated by comparing the specific index labels, placements, and dimension annotations appearing on the operands, which is exactly the information that tensor operation notation makes explicit.