13.22.5 Tensor Contraction Error Pattern Boundary

Tensor Contraction Error Pattern Boundary is the dividing line separating the class of index-level mistakes that the standard verification procedure's structural checks are capable of detecting from the class of mistakes that lie outside their reach, marking the point beyond which additional or different verification techniques become necessary to catch a given kind of error.

Definition

For the set of checks comprising the tensor contraction verification procedure, the error pattern boundary separates errors into two categories:

Errors outside this set lie beyond the boundary: they are not caught by any check in the current procedure, even though they may still produce an incorrect result.

Errors Within the Boundary

Structural Errors

Errors such as a dimension mismatch, an imbalanced index, or an invalid same-variance pairing are reliably caught by the dimension check, the index balance check, and the pair validity check respectively, placing these error types clearly within the detectable region.

Type-Level Errors

Errors such as a miscounted number of contraction steps, producing a result of unexpected type, are caught by the result type check, since any discrepancy between the predicted and actual type pair is directly observable by this test.

Errors Beyond the Boundary

Numerically Compensating Errors

An error in one part of a computation that happens to be exactly canceled by a second, unrelated error elsewhere can produce a result that passes the component-level spot check purely by numerical coincidence, placing this class of error beyond the boundary of what spot checking alone can detect.

Errors Invisible to a Single Basis

An error that corrupts the transformation behavior of a result only in certain specific coordinate systems, while leaving it apparently well-behaved in the coordinate system originally used for computation, may escape detection unless the invariance check happens to test a basis change exposing the flaw.

Errors in the Verification Logic Itself

A flaw in the implementation of a check, rather than in the contraction being checked, lies entirely outside the scope of the boundary as defined, since the boundary characterizes only which properties of a correct contraction the existing checks are designed to test.

Diagram

Significance

Recognizing the error pattern boundary clarifies the actual guarantee provided by passing the standard verification procedure: a result surviving every listed check is confirmed free of the specific error classes those checks are designed to detect, but passing verification does not certify absolute correctness against every conceivable error, motivating supplementary techniques, such as testing multiple independent bases under the invariance check, whenever an error type suspected to lie beyond the current boundary is a genuine concern.