Mereological Syntax: Angular Locality and Islands

@cite{adger-2025}

@cite{adger-2025} (Linguistic Inquiry Monograph 90, MIT Press) develops a mereological alternative to set-theoretic Bare Phrase Structure: syntactic objects are parts of one another (rather than members of sets), and the operation Subjoin makes one object a 1-part or 2-part of another. The book derives a range of locality phenomena from Angular Locality (AL): a structural condition on subjunction paths that fails when the path crosses dimensions (mixed 1-part/2-part transitivity).

Coverage of this file

This is a thin study layer that re-exports the AL substrate derivations from Theories/Syntax/SynGraph.lean (§10) and frames cross-framework engagement. The substrate covers:

• Subject islands (Ch 7 §7.7, book pp. 216–223): cross-dimensional path blocks sub-extraction; the subject DP itself, as T's direct 2-part, is reachable. Caveat: Adger's full §7.7 derivation depends on a [Fam]/[uFam] T feature triggering D-subjunction (which fills D's 2-part and produces the cross-dimensional path). The substrate models the blocked endpoint, not the [Fam] machinery that triggers it. Ch §7.8 emphasises that subject islands are "not consistent islands" — strength varies with definiteness/ topicality of the subject.

• Definite nominal islands (Ch 6 §6.3.2, book pp. 153–157): when Det/Dem/Poss subjoins to D, D's 2-part is "used up", blocking extraction through D. Indefinite DPs (D with a free 2-part) are transparent.

• Successive cyclicity (Ch 4 §4.3): wh requires intermediate stops at embedded C edges to traverse the right dimension chain at each step.

• Anti-locality, lowering, parallel merge, sideward subjunction (Ch 4 consequences (35a–e)): all blocked by AL.

Out of scope

• Wh-Islands (entire Ch 5, book pp. 117–142), including the WIRE Effect (Wh-Island Re-Emergence, book p. 125) — Adger's most novel cross-linguistic prediction. Formalising WIRE as a graph-decidable predicate over Wh-pair clausemate configurations is the most natural empirical extension.
• Adjunct islands beyond Ch 4 graph instantiation. Adger explicitly admits at the start of Ch 8 (book p. 225): "I owe at least a sketch of how these islands might be tackled." The substrate's adjunct_island_blocks exhibits the Ch 4 cross-dimensional mechanism on an AdvP 2-part of v; the Ch 8 Mod-headed Geis/Haegeman analysis is not formalised here.
• Concrete (§6.3.1) vs Relational (§6.3.2) nominal split.
• Articleless-language typology for nominal islands (§6.4): Mandarin/Persian/Japanese contrasts predicted by D-Interpretation.
• The [Fam]/[uFam] T-feature machinery (Ch 7 §7.7) driving subject- island gradience — substrate models the structural endpoint only.

Cross-framework engagement

§3 of this file articulates AL's relationship to one rival framework (@cite{marcolli-chomsky-berwick-2025} §1.6 algebraic Merge): both reach a false verdict on Sideward Merge from incompatible primitives.

The classification handles adgerSubjectIslandSource and adgerDefiniteNominalSources are exposed for use by later paper-anchored Studies files. Newer rivals make convergence/divergence claims against Adger's classification:

• @cite{cartner-et-al-2026} (Studies/CartnerEtAl2026.lean) converges with Adger on IslandSource.syntactic for subject islands, from cross- constructional invariance data.
• @cite{shen-huang-2026} (Studies/ShenHuang2026.lean) diverges from Adger on definite-nominal sources, arguing for a [.syntactic, .semantic] composite from English VOC effects + Mandarin wh-in-situ data.

Phase Theory (Theories/Syntax/Minimalist/Phase.lean, @cite{chomsky-2000}, @cite{chomsky-2008}) is the immediate theoretical rival — Adger's framing is to derive island effects "without stipulating phases, barriers, or subjacency." No formal cross-translation is provided here: AL operates on graph-theoretic parthood across dimensions; Phase Theory on PIC over derivational phases. The frameworks share no structural lemma; identifying a configuration where AL blocks but PIC permits (or vice versa) is left as a future critical experiment.

Core AL derivations live in Theories/Syntax/SynGraph.lean (§10):

Theorem	Phenomenon
al_blocks_superlocal	antilocality (35a)
al_blocks_lowering	no lowering (35b)
al_blocks_sideward	no sideward subjunction (35c)
al_blocks_parallel	no parallel merge (35d)
al_blocks_cross_dim / al_allows_within_dim	cross-dim transitivity (35e)
al_allows_rollup_2part / al_allows_rollup_1part	roll-up movement
succ_cyc_blocked_cross_clause	cross-clausal succ-cyc requires stops
succ_cyc_wh_reaches_C1_after_stop	with stops, succ-cyc allowed
subject_island_blocks / subject_itself_can_extract	subject islands
adjunct_island_blocks / adjunct_itself_can_extract	adjunct islands
nominal_island_definite_blocks / nominal_island_indefinite_allows	nominal islands
antilocality_sub1 / antilocality_sub12	general antilocality

The graphs g_subject_island, g_adjunct_island, g_definite_island, g_sideward are also public for downstream consumers.

theorem Adger2025.al_blocks_three_island_configurations : g_subject_island.satisfiesAL ⟨8, ⋯⟩ ⟨0, ⋯⟩ = false ∧ g_adjunct_island.satisfiesAL ⟨7, ⋯⟩ ⟨0, ⋯⟩ = false ∧ g_definite_island.satisfiesAL ⟨9, ⋯⟩ ⟨0, ⋯⟩ = false

The same satisfiesAL predicate fires false on three distinct configurations: subject (Ch 7 §7.7), adjunct (Ch 4 mechanism on AdvP), definite nominal (Ch 6 §6.3.2). The conjunction composes the substrate theorems rather than re-running native_decide on inlined copies of the same graphs.

The "same mechanism" claim is internal to Adger's account — all three blockings route through cross-dimensional path failure on the AL substrate. It is not a unification claim across all of CED:

• Adjunct islands receive only a Ch 8 sketch (book p. 225); the substrate graph instantiates the Ch 4 mechanism, not Ch 8's Mod-headed Geis/Haegeman analysis.
• Subject islands themselves are non-uniform per §7.8 — strength varies with definiteness/topicality of the subject.
• The definite-nominal case requires the Det-subjunction-fills-D machinery (book pp. 154–157), not just AL alone.

def Adger2025.adgerSubjectIslandSource : IslandSource

Adger's AL classifies subject islands as syntactically sourced — they arise from the structural cross-dimensional path failure on a graph (subject_island_blocks), not from binding (semantic), memory load (processing), or information-structural backgroundedness (discourse).

The classification is editorial in the sense that IslandSource.syntactic is the natural bin for any structural-configurational mechanism; subject_island_blocks is the structural fact this classification summarises. Exposed as a handle for cross-framework theorems in newer Studies files (e.g., CartnerEtAl2026.subjectIslandSource).

Equations

• Adger2025.adgerSubjectIslandSource = Phenomena.Islands.IslandSource.syntactic

def Adger2025.adgerDefiniteNominalSources : List IslandSource

Adger's AL classifies definite-nominal islands as single-source syntactic: the Det-subjunction-fills-D mechanism (Ch 6 §6.3.2) is itself structural — Det subjoins to D filling its 2-part, blocking extraction across the resulting cross-dimensional path (nominal_island_definite_blocks). No separate semantic mechanism is invoked.

@cite{shen-huang-2026} (Studies/ShenHuang2026.lean) argues from English VOC effects + Mandarin wh-in-situ data that this should be a [.syntactic, .semantic] composite — the divergence is recorded in that file's theorems.

Equations

• Adger2025.adgerDefiniteNominalSources = [Phenomena.Islands.IslandSource.syntactic]

Both @cite{adger-2025} (mereological Merge, this file) and @cite{marcolli-chomsky-berwick-2025} §1.6 (algebraic Merge) reach a false verdict on Sideward Merge from incompatible structural primitives:

• Adger: Sideward subjunction (sibling subjoin) violates Angular Locality — the would-be mover and target are both 1-parts of the same parent vertex, so the candidate-α set is empty (no within-dimension chain reaches the target). See al_blocks_sideward in SynGraph.lean, on the canonical g_sideward configuration. The book's own (35c) lists this as "Sidewards subjunction (to a 2-part / 'specifier')," book p. 91 (PDF p. 103); the substrate's g_sideward is a graph-level representation of (30) on book p. 91.

• MCB: Sideward operations 2(b), 3(a), 3(b) violate either MinimalYieldWeak (Δb₀ > 0 — workspace components increase) or InducedMapNCL (the canonical induced map decreases leafCount). See Theories/Syntax/Minimalist/Merge/MinimalYield.lean and NoComplexityLoss.lean.

The two frameworks share NO structural lemma. AL reasons about graph-theoretic parthood across dimensions; MCB reasons about Hopf- algebra coproduct counting and induced component maps. The shared verdict is convergent evidence from incompatible foundations — exactly the kind of theoretical cross-checking linglib is designed to make visible (CLAUDE.md: "high interconnection density … incompatibilities between theories … become visible across the codebase").

The bundled theorem below is honestly a verdict-comparison: a conjunction of two unrelated propositions about different structural objects, both true. A genuine reduction would require a translation SynGraph → TraceForest lifting satisfiesAL ↔ MinimalYieldWeak; no such bridge is in scope here.

theorem Adger2025.adger_and_mcb_both_reject_sideward (T_i Tnode T_iq : ConnesKreimer.TraceTree Minimalist.LIToken Unit) : ¬Minimalist.Merge.MinimalYieldWeak {T_i} {Tnode, T_iq} ∧ g_sideward.satisfiesAL ⟨2, ⋯⟩ ⟨1, ⋯⟩ = false

Verdict comparison: the canonical Sideward configuration is rejected by both frameworks. Two propositions about different structural objects, conjoined to make the cross-framework agreement visible at the type level. The Adger conjunct does not depend on the MCB parameters T_i Tnode T_iq; the MCB conjunct does not reference the AL graph. The bundling is documentation, not a reduction.