7.14.5 Tensor Component Assignment Consistency

Tensor Component Assignment Consistency is the requirement that every value assignment made while constructing or modifying a tensor's component table agrees with all other assignments already made, with any declared structural properties of the tensor, and with the fixed basis context under which the assignments are performed.

Dimensions of Consistency

Consistency With Declared Symmetry

When a tensor is declared symmetric across certain indices, consistency demands that any two index targets related by permuting those indices always receive equal assigned values, so an assignment sequence that ever places different values at such related targets has violated consistency.

Consistency With Declared Antisymmetry

When a tensor is declared antisymmetric across certain indices, consistency instead demands that related targets receive values that are exact negatives of one another, and that any target with a repeated index within those positions receives a value of exactly zero.

Consistency Across a Sequence of Assignments

Non-Contradiction Between Successive Assignments

If the same index target is assigned a value more than once during a construction process, consistency requires that every such assignment agree on the same final value, since conflicting assignments to a single target leave the tensor's true intended value at that entry ambiguous.

Consistency With Basis Context

Every assignment made under a given basis context must remain compatible with that same context throughout the construction process; introducing a value computed relative to a different, unreconciled basis without applying the appropriate transformation constitutes an inconsistency, even if each individual assigned number appears reasonable in isolation.

Detecting Inconsistency

Cross-Checking Related Entries

A practical method for detecting inconsistency involves cross-checking entries that structural properties predict should match, or should be related by a known factor, and flagging any discrepancy as evidence that an assignment error has occurred somewhere in the construction process.

Verifying Against Independent Computation

When possible, recomputing a subset of the assigned values independently, using a different but equivalent method of evaluation, provides an additional check that the assignments as a whole are internally consistent and free from arithmetic or bookkeeping mistakes.

Consequences of Inconsistency

Resulting Object May Not Be a Valid Tensor

An array of numbers that fails to satisfy the consistency required by its declared structural properties does not, strictly speaking, represent the type of tensor it was intended to represent, since violating a declared symmetry or antisymmetry breaks the defining characteristic of that tensor type.

Propagation of Errors Into Later Computation

Inconsistent assignments that go undetected can silently propagate into any subsequent operation performed on the tensor, such as contraction or comparison with another tensor, producing results that appear numerically valid but rest on a flawed foundation.

Diagrammatic Illustration

Two related index targets checked against each other to confirm that their assigned values remain consistent with a declared symmetry.

Broader Significance of Assignment Consistency

Preserving the Integrity of the Tensor Concept

Maintaining assignment consistency throughout construction is what preserves the tensor's claim to satisfy its declared structural properties and its stated basis context, without which the constructed object risks being little more than an arbitrary, uninterpretable grid of numbers.

A Discipline Rather Than an Automatic Guarantee

Because assignment is an active process carried out by a person or a program, consistency does not arise automatically; it must be actively maintained through careful bookkeeping, cross-checking, and adherence to the rules implied by the tensor's declared type, symmetry, and basis context.