The Topological Closure of the One-Spin phtn: Paired-Support Coequalization, Port-Labeled Seat Occupancy, and Persistent Ligamen Circulatus Formation

This paper derives an explicit receiver-induced topological map by which a freely transported one-spin phtn becomes a persistent, materially occupied half-spin Ligamen Circulatus. The free carrier is a normalized coherent state supported on two opposite-sphere, same-temporal-torque half-spin loxodromes. At one quantum moment, its lifted support is the pointed graph Dₕ S¹ S¹, with integral generators e₋ and e₊ and normalized carrier class μ_γ=(e₋+e₊)/2. The temporal-degree map satisfies wₕ(e₋)=wₕ(e₊)=h. Its kernel is generated by the support-difference class Δq=e₊-e₋, which is both temporal-null and transported-content-null. The freely transported phtn therefore possesses one paired-support closure defect, whereas a single occupied Ligamen cycle possesses none. The receiver owns its electrostatic-sector polarity qE,y, Ligamen handedness ηL,y, boundary parity by, triadic registration class, endpoint phase χy, six-port address, and occupancy mark ny. The phtn supplies its temporal-torque helicity h, species scale x, relative support phase δₕ, and complete phtn and angm content. Spectral admission requires byh=qE,yηL,y, but admission alone does not reduce the two free support cycles to one occupied cycle. A separate receiver support-routing map gives the occupied-mode weightPLC,y=12[1+sin(2θy)cos(δₕ-χy)].Complete no-remainder closure requires equal receiver coupling to both supports, θy=π/4, together with endpoint-phase matching, δₕ=χy2π. Under these conditions, the induced coequalizer acts asky,*(e₋)=ky,*(e₊)=oyy,where oy=byqE,y. Equivalently, the receiver adjoins the relation g₋g₊⁻¹=1. This removes only the temporal-null difference class, Δq↦0, while preserving the carrier temporal degree and transferring one complete phtnₓ=mₓλC³Fq² and one complete angmₓ=mₓλC²Fq into the occupied Ligamen. No independent half-phtn remainder is required. Seat availability is derived as vacancy within a port-labeled registry of admissible Ligamen closure classes. With structural projector Gy and occupancy marking Ny, the free-seat projector isAy= Gy(I- Ny) Gy.A successful direct closure changes the occupancy mark as ny:0→1. Receiver attachment replaces the terminal free-carrier boundary and yields the complete event ledger (cy,ry,fy)=(1,0,0). The terminal free support is canceled, one occupied Ligamen remains, and there is no independently active continuation of the source-rooted cardioid front. The source root and projected nose are two features of one normalized active carrier rather than two independently transported objects. Complete closure neither sends the projected nose back toward the source nor transmits a reverse notification to the origin. The intrinsic temporal state is represented by the half-angle lift ζₕ(Φ)=exp(ihΦ/2), which distinguishes the 2π branch boundary from complete 4π closure. Occupied half-spin matter registers the quadratic quotient QT(ζₕ)=ζₕ², for which QT(ζₕ)=QT(-ζₕ). The complete five-dimensional branch sign is therefore retained by the Aether state but is not simultaneously accessible to a four-dimensional material receiver. Laboratory detectors are themselves occupied half-spin structures and record the local receiver consequence of the free phtn rather than observing the complete five-dimensional carrier from an external vantage point. The occupied Ligamen is kinematically persistent under port-preserving, boundary-preserving, no-surgery receiver evolution. Release is the topological and conservation-ledger inverse of direct closure within the same deterministic, direct, one-seat, same-scale, no-remainder event class. Equality of capture and release amplitudes, transition rates, or environmental dynamics is not asserted. The accompanying reproducibility package contains the complete manuscript source, figures, derivation records, claims and prediction ledgers, symbolic and numerical validation programs, machine-readable outputs, package manifest, and checksums. The stage packages report 228,454 automated checks with zero failures and residuals below 7×10⁻¹⁶. These checks establish formal and computational consistency; they are not independent experimental confirmation. The topology determines which receiver closures are admissible but does not yet determine the receiver-specific coupling magnitude, transition threshold, lifetime, or competition among multiple admissible receivers. Those quantities require a subsequent dynamical coupling law.