Nagging

Nagging repeatedly presents the same or a materially equivalent request after dismissal or non-response. It uses recurring prompts, banners, or interruptions to sustain attention or pressure completion, while the thesis conditions remain limited to directly observable recurrence, salience, and repeated imperative language.

Repeated Prompt Injection

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Nagging: Repeated Prompt Injection

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Condition 1: Repeated Prompt Injection

\exists p_{1}, p_{2} \in P_{prompt} : A (p_{1}) = A (p_{2}) \land t (p_{2}) > t (p_{1}) \land Δ t (p_{1}, p_{2}) \leq τ_{repeat} \land Dismissed (p_{1}) = True \land Injected (p_{2}) = True

Given

To identify repeated prompt injection, let

P_{prompt}

be the set of prompt instances observed within the declared route and session, and let

A (p)

be a normalized action signature containing the request target and requested action. The predicate

A (p_{1}) = A (p_{2})

requires material equivalence rather than merely similar wording. Let

t (p)

be the observed render or insertion time,

Δ t (p_{1}, p_{2}) = t (p_{2}) - t (p_{1})

, and

τ_{repeat}

the declared observation window.

Dismissed (p_{1})

is TRUE only when the evidence records a close, decline, defer, or equivalent dismissal transition;

Injected (p_{2})

is TRUE only when the equivalent request is newly inserted or visibly re-presented after that transition. A single reminder, a user-requested prompt, or a later prompt with materially new task information is outside this condition.

Persistent Interruption Salience

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Nagging: Persistent Interruption Salience

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This request interrupts the task again.

Condition 2: Persistent Interruption Salience

\frac{A ( N _{nag} )}{A _{viewport}} > τ_{nag_area} \lor z (N_{nag}) \geq τ_{nag_layer}

Given

To measure the rendered salience of a recurring prompt, let

N_{nag}

be the prompt node in the inspected state,

A (N_{nag})

its rendered bounding-box area, and

A_{viewport}

the visible viewport area in the same capture. Let

z (N_{nag})

denote the observed stacking rank or overlay-layer indicator, and let

τ_{nag_area}

and

τ_{nag_layer}

be declared thresholds. The condition is present when either the prompt occupies more than the declared viewport fraction or it is rendered as a high-layer interruption over the active task. This concerns observable persistence and visual salience; it does not infer that a user noticed, attended to, or was affected by the prompt.

Imperative Repetition Without New Context

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Nagging: Imperative Repetition

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Condition 3: Imperative Repetition Without New Context

\exists p_{1}, p_{2} \in P_{prompt} : A (p_{1}) = A (p_{2}) \land ∣ W (N_{nag}) ∣ > 0 \land \frac{∣ W ( N _{nag} ) \cap L _{imperative} ∣}{∣ W ( N _{nag} ) ∣} > τ_{imperative} \land ContextChange (p_{1}, p_{2}) = False

Given

To evaluate repeated imperative wording, let

P_{prompt}

be the set of prompt instances in the declared route and session, and let

A (p)

be their normalized action signature. Let

W (N_{nag})

be the normalized visible token multiset of the recurring prompt and

L_{imperative}

a declared lexicon of request or command terms such as ``enable,'' ``subscribe,'' ``invite,'' or ``finish setup.'' The non-zero cardinality guard prevents an empty prompt from satisfying the ratio.

ContextChange (p_{1}, p_{2})

is TRUE only when the inspected task state, user request, or materially available information changes between two equivalent prompt observations. The condition is present when the imperative-token ratio exceeds

τ_{imperative}

while the equivalent request is repeated without new task context. Neutral status updates and reminders that add relevant information remain outside this operational condition.