13.22.1 Tensor Contraction Indexed Notation Boundary

Tensor Contraction Indexed Notation Boundary is the specific manifestation of the general contraction boundary as it appears directly in written indexed notation, marking the point at which an expression contains no remaining index symbol eligible to be repeated as an upper-lower dummy pair, so that no additional application of repeated index notation can be performed on the expression as written.

Definition

For an indexed expression $$E$$, the indexed notation boundary is reached when every index symbol appearing in $$E$$ occurs exactly once, meaning no symbol satisfies the repeated-pair condition required by repeated index notation:

At this point, the expression written in indexed form admits no further contraction without introducing a new tensor factor from outside the current expression.

Distinguishing From the General Contraction Boundary

Property of the Written Expression

While the general contraction boundary is a property of a tensor's type pair, becoming unreachable once either the upper or lower count hits zero, the indexed notation boundary is a property of the specific written expression: it is reached whenever no eligible repeated pair remains in the notation currently on the page, even if, considered abstractly, the tensor's type would in principle admit further contraction against some other tensor.

Local Versus Global View

The indexed notation boundary reflects a purely local, syntactic view of a single expression, whereas the general contraction boundary reflects the global, type-based view of what contractions remain mathematically possible for the underlying tensor, independent of how it happens to be currently written down.

Recognizing the Boundary in Practice

Scanning Procedure

To determine whether an expression has reached its indexed notation boundary, scan every index symbol in the expression and tally its occurrences; if no symbol has exactly two occurrences of opposite variance, the boundary has been reached for that expression as written.

Boundary After Simplification

Applying the tensor contraction simplification procedure to an expression can either preserve or eliminate eligible repeated pairs; once the simplification procedure has fully terminated, the resulting expression necessarily sits at its own indexed notation boundary, since no further repeated pair remains by the very definition of the procedure's terminal condition.

Diagram

Relation to the Broader Boundary Concept

The indexed notation boundary is the notational instance of the general tensor contraction boundary, situating the abstract stopping condition defined by exhausted upper or lower index counts within the concrete, symbol-by-symbol scanning process a reader or verification procedure actually performs when examining a specific written expression for any remaining contraction opportunity.