4. The Task Continuum Index Adapted to AI-IoT

Section 1 introduced the Task Continuum Index as the task-side counterpart to the judge-side Cognitive Continuum Index. Section 3 already produced the raw material needed to operationalize it: the "task card" from its Application Rule requires, for each of the eleven properties, a provisional classification — intuitive pole, analytical pole, or mixed — recorded separately for each of the three perspectives (physical ecology, AI representation, supervisor representation). What remains is to close the loop: turning that qualitative, property-by-property classification into a number that locates the task — or, more precisely, each of its three representations — on the continuum.

It is worth being direct about what this number can and cannot be. Hammond's (1988) original TCI was not derived from an algebraic formula applicable to any task; it was constructed experimentally, by ranking nine display conditions whose properties had been manipulated by the researcher himself. There is no validated, ready-to-import formula in the literature for AI-IoT. The index proposed in this section is, therefore, an original construction of this trilogy — an explicit, auditable heuristic aggregation device, not the recovery of an already-validated psychological instrument. This follows directly from the warning already given in Section 3, regarding Dhami and Thomson (2012): CCT does not specify how much each property shifts cognitive mode. The index that follows does not resolve that indeterminacy — it makes it explicit and manageable.

The formula. For each property $i \in \{1, \ldots, 11\}$ and each level $\ell \in \{\text{eco}, \text{AI}, \text{sup}\}$, the task card from Section 3 assigns a provisional classification. That classification is coded as:

$$s_i(\ell) = \begin{cases} -1 & \text{intuitive pole} \\ 0 & \text{mixed or insufficiently classifiable} \\ +1 & \text{analytical pole} \end{cases}$$

This scoring scale is deliberately simple — three values, not five or nine — not for lack of sophistication, but to avoid feigning a quantitative precision that the underlying qualitative classification cannot support. A more granular scale (for example, $-2$ to $+2$, distinguishing "strong" from "weak" intensity within each pole) would be a natural extension, subject to future empirical justification, but at this conceptual stage it would introduce exactly the false precision Section 3 already warned against.

The Task Continuum Index of a level is the mean of the eleven scores:

$$TCI_\ell = \frac{1}{11} \sum_{i=1}^{11} s_i(\ell)$$

bounded within $[-1, +1]$: $TCI_\ell \to -1$ indicates a task strongly inducing intuition at that level; $TCI_\ell \to +1$, strongly inducing analysis; $TCI_\ell$ near $0$, quasirationality — but with an important caveat, addressed next.

The confidence caveat. A $TCI_\ell$ near zero is ambiguous by construction: it can mean the task is genuinely quasirational, with properties pointing robustly and evenly in opposite directions; or it can mean, far less interestingly, that many properties were classified as "mixed" for lack of sufficient evidence to decide. These two scenarios should not be conflated, and reporting only $TCI_\ell$ hides the difference. The index should therefore always be accompanied by:

$$n_{\text{mixed}}(\ell) = \#\{i : s_i(\ell) = 0\}$$

A task with $TCI_\ell \approx 0$ and $n_{\text{mixed}}(\ell) = 0$ is quasirational with confidence; a task with $TCI_\ell \approx 0$ and $n_{\text{mixed}}(\ell) = 8$ is, in practice, still unclassified. The index without this second metric invites a precision the underlying classification does not support.

Divergence across levels. The real original contribution of this section is not $TCI_\ell$ on its own — it is the possibility of computing it separately for the three levels and comparing them. Define:

$$\Delta TCI(\ell_1, \ell_2) = TCI_{\ell_1} - TCI_{\ell_2}$$

The case of greatest interest for this trilogy is $\Delta TCI_{\text{AI-sup}} = TCI_{\text{AI}} - TCI_{\text{sup}}$: the difference between the task's position on the continuum as represented to the algorithmic policy, and its position as represented to the human supervisor. A $\Delta TCI_{\text{AI-sup}}$ near zero suggests that AI and supervisor face, structurally, similar tasks on the continuum, even through different cues. A high $\Delta TCI_{\text{AI-sup}}$ signals something more interesting and riskier: that the same physical ecology is inducing structurally distinct cognitive modes depending on who observes it — the AI operating on a task that its representation renders analytical, the supervisor operating, nominally on "the same problem," but on a representation that renders it intuitive, or vice versa.

Returning to the two Argus examples already described in Section 3: thermal anomaly detection, as characterized there — few calibrated sensors, a stable trend, known engineering thresholds, a comprehensible relationship, time to confirm — would tend toward a $TCI_{\text{eco}}$ and a $TCI_{\text{AI}}$ close to the analytical pole, with most of the eleven properties pointing in that direction. If the supervisor's interface preserves that context — trends, thresholds, room to confirm — $TCI_{\text{sup}}$ tracks them, and $\Delta TCI_{\text{AI-sup}}$ stays low. But if the same system reduces that task, at the human interface, to a single binary alert with no trend and no context, $TCI_{\text{sup}}$ may shift toward the intuitive pole — not because the ecological task changed, but because the representation available to the supervisor drastically changed its position on the continuum. The resulting high $\Delta TCI_{\text{AI-sup}}$ is exactly the kind of signature this index was designed to make visible.

Bearing-failure prediction from vibration — the second Argus example — illustrates the opposite, harder case. There, the ecology and the algorithmic representation are already structurally complex and poorly decomposable: $TCI_{\text{eco}}$ and $TCI_{\text{AI}}$ tend toward near zero, but with low $n_{\text{mixed}}$ — genuine quasirationality, not a lack of classification. Reducing that task to a single score does not, in this case, produce an artificial divergence across levels: $TCI_{\text{sup}}$ can remain equally close to zero, because the task was already, in its structure, resistant to decomposition — it merely becomes opaque, which is a different kind of failure (of organizing principle and of presentation), not necessarily one captured by $\Delta TCI$.

This asymmetry between the two examples is deliberately foregrounded in this section: $\Delta TCI_{\text{AI-sup}}$ detects a specific kind of risk — structural displacement of the induced mode across levels — and intentionally does not detect other, equally real risks, such as the opacity of an organizing principle that exists at depth without reaching the surface. This limitation is not a flaw to be fixed — it is a boundary of application to be respected. The two "mixed" cases already identified in Section 2's table — organizing principle and decision duration — are precisely where that boundary is most likely to show up.

What this index does not do. It bears repeating, because it is easy to forget in front of a number: $TCI_\ell$ and $\Delta TCI$ are structural measures of the task as classified — they do not demonstrate that the AI, the supervisor, or the hybrid system actually respond in the mode the task induces. A high $\Delta TCI_{\text{AI-sup}}$ is a warning sign worth investigating, not a diagnosis of failure; the algorithmic policy may be well matched to its own (analytical) task and the supervisor may be equally well matched to theirs (intuitive), with no problem implied — or it may not be, and it is precisely that correspondence between induced mode and actually enacted mode that Article 5 will audit through $G$, $R_s$, the cue weights $\hat{w}_{Z_{IA}}$, and the cue-criterion relation. This index locates the question; it does not answer it.

Two contributions, declared as such. This section proposes two original extensions to Hammond's (1988) apparatus: first, the decomposition of a single TCI into a $TCI_\ell$ per level, necessary because cyber-physical chains — unlike the isolated human judge classic CCT presupposes — place more than one decision-maker in front of potentially distinct representations of the same ecology; second, $\Delta TCI$ as an explicit measure of that divergence, which turns Section 2's surface/depth distinction from a qualitative observation into a number that Articles 4 and 5 can use as input.

(Bibliographic note: Hammond, K. R. (1988). "Judgement and decision making in dynamic tasks." Information and Decision Technologies, 14(1), 3–14 [also circulated as ART Research Note 88-81, U.S. Army Research Institute for the Behavioral and Social Sciences] — citation directly verified against two independent sources; identified here as the origin of the Task Continuum Index, the Cognitive Continuum Index, and the surface/depth distinction, earlier and more foundational than Dunwoody et al. (2000), which tests it empirically twelve years later. Section 2's bibliographic note has been corrected accordingly. Disambiguation note: the "Hamm (1989)" entry already present in the trilogy's core bibliography refers to Robert M. Hamm, co-author of Hammond, Hamm, Grassia & Pearson (1987) — a person distinct from Kenneth R. Hammond, the author of this 1988 article; the two citations should not be merged. The $TCI_\ell$ formula, the $n_{\text{mixed}}$ metric, and the definition of $\Delta TCI$ are original constructions of this trilogy, with no direct correspondence in Hammond (1988) or in any cited source — the original article does not propose a generalizable algebraic formula.)