6.6.4 Tensor Arity Evaluation Context

Tensor Arity Evaluation Context is the state of how many, and which, of a k-ary tensor's argument slots have already been assigned specific vectors at a given point in a calculation, ranging from the fully unevaluated tensor with all k slots open, through partially evaluated intermediate results with some slots filled and some still open, to the fully evaluated scalar obtained once every slot has received an argument. The evaluation context is what distinguishes an abstract k-ary tensor, defined independently of any particular input, from a specific number obtained by supplying it with concrete vectors, and tracking this context correctly is what keeps a multi-step calculation involving several partial evaluations well-organized.

Degrees of Evaluation

Fully Unevaluated: All k Slots Open

A k-ary tensor with no arguments yet supplied is fully unevaluated, existing as an abstract multilinear map T : V₁ × ... × V_k → F or an element of a tensor product space, with every slot ready to receive an argument and none yet committed.

Partially Evaluated: Some Slots Filled

Supplying specific vectors to m of the k slots (via currying) produces a partially evaluated result, itself a (k - m)-ary tensor with the remaining k - m slots still open; the evaluation context at this stage records exactly which m slots have been filled and with what values.

illustrates a 4-ary tensor with slots 1 and 3 filled, leaving a binary tensor over slots 2 and 4.

Fully Evaluated: Zero Slots Open

Once all k slots have received specific vectors, the result is a single scalar, the fully evaluated output; the evaluation context at this final stage is simply the full list of k vectors supplied, one per slot, with no open positions remaining.

Diagram of Progressive Evaluation

Order-Independence of Which Slot Is Filled First

Multilinearity Guarantees the Final Result Is Context-Order Independent

Because a k-ary tensor is multilinear, filling its k slots in any order eventually produces the same final scalar, provided the same vector ends up in the same slot regardless of the sequence in which the slots were filled; the evaluation context tracks which vector went into which slot, and it is this slot assignment, not the order of filling, that determines the outcome.

Why the Distinction Still Matters for Intermediate Results

Although the final fully evaluated scalar does not depend on fill order, the intermediate partially evaluated tensors obtained along the way do depend on which slots have already been filled at each step; two different fill orders pass through different sequences of intermediate arities and intermediate tensors, even though they converge on the same final answer.

Symbolic Versus Concrete Evaluation Context

Evaluating Against Basis Vectors

Supplying basis vectors e_{i₁}, ..., e_{i_k} to all k slots of a tensor yields exactly the component T_{i₁...i_k}; this is the most concrete form of evaluation context, tying the abstract tensor directly to one specific number in its array of components.

Evaluating Against General Symbolic Vectors

Supplying general vectors v = Σ a_i e_i expressed in terms of basis coefficients, rather than basis vectors themselves, produces a symbolic evaluation context in which the final scalar is expressed as a multilinear expansion over all combinations of basis components, weighted by the corresponding coefficients — the standard expanded formula relating a tensor's action on general vectors to its components.

Evaluation Context in Multi-Step Calculations

Tracking Context Across Several Operations

In a calculation involving several tensor products and contractions performed in sequence, the evaluation context at each intermediate step records both the current arity (how many slots remain open) and, where relevant, which specific values or symbols occupy the slots already filled, information that must be tracked accurately to correctly interpret each subsequent operation.

Avoiding Slot Confusion Across Partial Evaluations

When several different vectors are supplied to a multi-slot tensor across separate steps, maintaining a clear evaluation context prevents a common error: mistakenly supplying a new vector to a slot that has already been filled, or supplying it to the wrong one of the several slots still remaining open.

Why the Evaluation Context Matters

Distinguishing the Abstract Tensor From Any One of Its Evaluations

A single abstract k-ary tensor supports infinitely many different evaluation contexts, one for every possible choice of arguments for its slots; keeping the abstract tensor and its evaluation context conceptually distinct prevents conflating a general multilinear object with any one particular number it can be made to produce.

Supporting Correct Bookkeeping in Currying-Based Calculations

Because currying reduces arity one slot at a time, explicitly tracking the evaluation context at each stage — how many slots remain, and what has already been supplied to the rest — is the direct bookkeeping mechanism that keeps a multi-step, partially evaluated calculation both correct and interpretable at every intermediate stage.