Marcolli & Larson (2025): Theta theory, operads, and coloring #
[ML25] gives the explicit generating set of the colored
operad that implements theta theory in mathematical Minimalism
([MCB25] §3.8): theta-role assignment is a
coloring algorithm on structures freely formed by Merge, and filtering by
the coloring rules is equivalent to structure formation by a colored
Merge. The generating system, its local conservation law, the derived
External/Internal Merge dichotomy, and the derivability of the bare theta
combs live in Syntax/Minimalist/Theta/Basic.lean; this file runs them on
the paper's ditransitive example — Brian gave the book to Mary, whose
verb assigns the grid (agent, theme, goal) with the internal hierarchy
theme before goal — on a movement generator, and on [Sil12]'s
parasitic assignment, recast as a language-specific generator whose
absence from the standard rule set is the sole-role requirement and whose
addition is the se marking.
The example's internal theta hierarchy: theme outranks goal, per the ditransitive grid the paper builds for give.
Equations
- MarcolliLarson2025.giveHier a b = (a = ThetaRole.theme ∧ b = ThetaRole.goal)
Instances For
The give comb: agent, then theme, then goal discharge one node at a time; the head leaf carries the full giver grid. Colors only — the lexical anchoring of terminal leaves is orthogonal to the theta structure.
Equations
Instances For
The give comb is derivable in the complete theta bud system: the three-role grid discharges against the theme-before-goal hierarchy.
Every vertex of the give comb is a generator instance: the recursive form of the theta criterion, checked structure-wide.
A movement landing site over the residual agent grid.
Equations
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Movement lands only in non-theta positions, at the example rule: the empty-tree input forces the non-theta output.
The give comb satisfies the theta criterion: agent, theme, and goal receivers exactly match the head's grid.
A language-specific generator: parasitic assignment #
[Sil12]'s syntactic reciprocalization assigns a retained internal
role together with the external role to one argument — parasitic
assignment, a last resort licensed by the se morphology. In the coloring
model this is a dual-receiver generator: one argument receives two roles
against a dual-giver residue, as in the ECM derivation where Jean ends
with the matrix experiencer role and the embedded verb's retained agent
role. The generator is absent from the standard rule set
(parasiticGen_notMem — the model's form of the sole-role requirement),
and adding it, as the se marking does, makes the dual-role structure
derivable (parasiticGen_derivable).
Parasitic assignment as a generator: a dual-receiver argument discharges a dual-giver residue in one step.
Equations
- One or more equations did not get rendered due to their size.
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The standard theta bud system contains no parasitic generator: every generator discharges at most one role per argument.
The system extended by the parasitic generator: the se-marked grammar.
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- One or more equations did not get rendered due to their size.
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With the marking, the dual-role discharge is derivable: Jean ends with both the matrix experiencer role and the retained embedded agent role.
Without the marking, the dual-role configuration is underivable: the structure is movement-free and its root receives no role, so the sole-role invariant forbids the dual-receiver argument. The sole-role requirement of [Sil12]'s "I"-reading account, as an underivability theorem.
Realization: from colored trees to syntactic objects #
An adjunct attaches to the give comb by colored Merge, staying derivable: the closure half of the paper's build-by-coloring equivalence, at the example.
The lexically anchored give comb.
Equations
- One or more equations did not get rendered due to their size.
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Pruning the colors realizes the anchored comb as the bare Merge structure of Brian gave the book to Mary.
A movement landing site above the anchored comb realizes to the same syntactic object: the empty-tree marker exists only for the coloring.