Harmonic externalization

A head function on a binary tree determines a planar embedding of it, and conversely (Marcolli et al. 2025 Lemmas 1.13.4 and 1.13.5), and the book reads the head function's value on a tree as the ordered leaves of that embedding (eq. (1.13.3)). Lemma 1.13.7 identifies head functions with bare-phrase-structure projection (Chomsky 1995); composed with Adger 2003's identification of the projecting item as the selecting item (a synthesis step of this formalization, not a claim of the book), c-selection computes the embedding. The harmonic linearization SyntacticObject.linearize? places the projecting daughter on a language's head side at every node. It is the linearized component of the head function's value, a morphism of commutative magmas out of the carrier (SyntacticObject.headHom), and a section of forgetting the order wherever every node has a head (SyntacticObject.toSyntacticObject_of_linearize?): the harmonic candidate for the book's externalization section σ_L, which must extend it noncanonically off that domain.

Main declarations

• Minimalist.LinearizationState: the value of the head function in its two descriptions, the selection state and the linearized object the head determines, 0 off Dom(h); a CommMagma and MulZeroClass under the Merge-local product.
• Minimalist.LinearizationState.sel: the projection morphism to SelectionState.
• Minimalist.SyntacticObject.headHom: the head function as a morphism of magmas SyntacticObject →ₙ* LinearizationState side, refining selCheckHom.
• Minimalist.SyntacticObject.linearize?: the harmonic linearization.

Main results

• Minimalist.SyntacticObject.sel_comp_headHom: fusion, the selection component of the head function's value being the selection check, by SyntacticObject.hom_ext.
• Minimalist.SyntacticObject.toSyntacticObject_of_linearize?: the harmonic linearization is a section of forgetting the order on its domain.
• Minimalist.SyntacticObject.linearize?_merge: a daughter merged on an edge is linearized there when it projects exactly if the edge is the head side.

Implementation notes

The head side is an Edge, left for head-initial and right for head-final. Head functions are partial (Marcolli et al. 2025 §1.13.2): at exocentric nodes no daughter projects and no order is determined, the book rejecting an order induced from a total order on labels as "not a realistic linguistic assumption", so the state is 0 there, the absorbing element of the book's renormalization reading of partial head functions. The selection state and the linearized object are the book's two descriptions of one head function: eq. (1.13.3) reads h(T) as "a single leaf (the head)" or as the ordered leaves of the planar embedding it determines, "switch[ing] between these two descriptions without changing the notation". LinearizationState fuses them into one partial value, so the linearization is defined exactly on Dom(h). Naming: the head-function maps (headHom and the leaf data it lifts) carry the book's name for h; the value type does not, the book restricting "the term head to terminal elements" (quoting Chomsky 1995 §4), and the strict head is recovered via sel.

Only the two harmonic sections are realized: uniform head-side placement is right for head–complement structure but does not model specifier placement, which needs a head side per category.

References

• Marcolli et al. 2025, Mathematical Structure of Syntactic Merge: An Algebraic Model for Generative Linguistics formalized
• Chomsky 1995, Bare Phrase Structure
• Adger 2003, Core Syntax: A Minimalist Approach