Bridge: Backgrounded Islands → Island Classification

@cite{lu-degen-2025}

Connects the formal backgroundedness model (Theories/Semantics/Focus) to the shared island infrastructure in Phenomena/FillerGap/Islands/Data.lean.

The MoS island is classified as weak (ameliorable) and discourse-sourced, and we derive both properties from the formal model.

theorem Phenomena.FillerGap.BackgroundedIslandsBridge.discourse_source_from_model : Semantics.Focus.BackgroundedIslands.complementStatus Semantics.Focus.BackgroundedIslands.CommDimension.manner = Core.InformationStructure.DiscourseStatus.given ∧ constraintSource ConstraintType.mannerOfSpeaking = IslandSource.discourse

Discourse source derivation: The formal model predicts that MoS islands are discourse-sourced because the island effect arises from QUD-determined backgroundedness, not syntactic configuration or processing load.

The classification constraintSource.mannerOfSpeaking =.discourse is not stipulated arbitrarily — it follows from the formal model: the island effect is predicted by complementStatus (defaultDimension v) which depends on QUD selection (information structure), not syntax or processing.

theorem Phenomena.FillerGap.BackgroundedIslandsBridge.weak_strength_from_model (v : Semantics.Focus.BackgroundedIslands.VerbDecomp) (h : v.hasMannerWeight = true) : Semantics.Focus.BackgroundedIslands.complementStatus (Semantics.Focus.BackgroundedIslands.activeDimension v false) = Core.InformationStructure.DiscourseStatus.given ∧ Semantics.Focus.BackgroundedIslands.complementStatus (Semantics.Focus.BackgroundedIslands.activeDimension v true) = Core.InformationStructure.DiscourseStatus.new ∧ constraintStrength ConstraintType.mannerOfSpeaking = ConstraintStrength.weak

Weak classification derivation: The formal model predicts that MoS islands are weak (ameliorable) because prosodic focus overrides the default QUD, changing the complement from backgrounded to discourse-new.

The classification constraintStrength.mannerOfSpeaking =.weak follows from the existence of prosodic_focus_ameliorates.

theorem Phenomena.FillerGap.BackgroundedIslandsBridge.mos_unique_discourse_source : constraintSource ConstraintType.mannerOfSpeaking = IslandSource.discourse ∧ constraintSource ConstraintType.embeddedQuestion = IslandSource.syntactic ∧ constraintSource ConstraintType.complexNP = IslandSource.syntactic ∧ constraintSource ConstraintType.adjunct = IslandSource.syntactic ∧ constraintSource ConstraintType.coordinate = IslandSource.syntactic ∧ constraintSource ConstraintType.subject = IslandSource.syntactic ∧ constraintSource ConstraintType.sententialSubject = IslandSource.syntactic

MoS islands are unique among island types: they are the only islands whose effect is derived from information structure (discourse source) rather than syntactic configuration.

This makes a testable prediction: any manipulation that changes the QUD (not just prosodic focus) should ameliorate MoS islands, but manipulations that don't change the QUD (like D-linking) should not.