Proposition 86 ([Terminal path bound]). There is a length n₀ such that every segment of T of length at least n₀ has trivial, cyclic, or parabolic pointwise stabilizer. Moreover there is a uniform bound on the length of a segment fixed by …

We first rule out arbitrarily long monotone paths with nonparabolic stabilizers. Suppose they exist. Normalize a centered short window to one fixed W_0 , put H_0=(W_0) , and let R_0 be its relative free-product tree. Each longer window group has a loxodromic axis in R_0 . Orient such an axis and mark one edge at index zero. Translate by H_0 to make this edge fixed. For each finite extension length there are now only finitely many actual possibilities: there are finitely many extension orbits, each relevant minimal subtree has a finite edge quotient, and a regular edge has trivial stabilizer. Passing diagonally therefore fixes every finite extension of W_0 . At the same time, diagonalize the oriented turn labels of the axes, indexed relative to the marked edge. At each index the label is eventually constant or leaves every finite set of labels. We call the latter an escaping turn. A turn at a finite-valence trivial vertex cannot escape. There cannot be two escaping turns. Choose consecutive escaping indices, so all intermediate turn labels are constant, and normalize the intervening edge path. Its edge markings determine actual coordinates because its edge stabilizers are trivial. Its two end vertices are then fixed peripheral vertices labeled by two distinct actual objects d,e . Distinctness here is the label assertion following Proposition 85 ([Simultaneous relative factors]), not merely distinctness of conjugacy classes. Diagonalize the window extensions in these new coordinates, retaining a regular edge of the intervening path as marking. For every fixed depth the corresponding minimal subtree contains infinitely many directions at each of these vertices. Since it has finitely many edge orbits, the stabilizer of each vertex is nontrivial: otherwise each global edge orbit could contribute at most one incident edge at that vertex. Thus every finite-depth window stabilizer intersects both S_d and S_e nontrivially. Lemma 78 ([Bounded peripheral incidences]) forces the interiors of all these windows to lie on both object lines. Their limiting monotone ends coincide, contradicting the terminal system. Consequently at least one half of the axis coordinates has no escaping turn. The constant turns and the marked edge define an actual infinite ray ρ⊂R_0 . Every finite arc of ρ is carried on lines in the minimal subtree of every finite-window stabilizer of the limiting bilateral trajectory τ . The first diagonal argument has produced an infinite carried ray. It has not yet produced a gate event that recurs on that ray. We obtain repeated finite arcs by a second normalization, now of disjoint blocks. Choose infinitely many disjoint blocks in ρ , each of length greater than 8R+4 , where 2R is the gate-event length. Normalize a regular edge in each block by H_0 , and diagonalize its finitely many turn positions. The preceding two-escape argument applies again: together with each normalized block translate the whole trajectory, and diagonalize its finite extensions before using infinite valence. Every block is carried through every finite depth, so two escaping positions would again force distinct object lines to share an end. Removing the possible single escaping turn leaves a subblock of length at least 2R whose turn type is constant along a subsequence. There are therefore infinitely many disjoint H_0 -equivalent arcs of length 2R on ρ . Fix a depth large enough for phase uniqueness in both signs. The corresponding window stabilizer has finite edge quotient on its minimal subtree. Among the repeated arcs, infinitely many marked initial edges belong to one orbit under this smaller stabilizer. Its transporter carries the entire arc, since the arcs already have the same H_0 -type and the marked edge has trivial stabilizer. Choose one of these arcs. Flaring supplies an expanding sign. Phase uniqueness, applied to τ and each of its translates, says that every such transporter preserves the expanding end of τ ; all required initial comparisons lie in the chosen fixed window. Bounded equivariance errors are included in the gate-event thresholds. Each transporter has height zero and fixes that initial window. Preserving its expanding end therefore makes it fix the ray to that end pointwise. Lemma 77 ([Capture of a fixed monotone ray]) puts every nonidentity transporter in the same S_d , because only one object line has that end. But H_0∩S_d fixes a vertex of R_0 . Its orbit of any fixed arc has bounded distance from that vertex and cannot contain arcs going arbitrarily far along the single ray ρ . This is the required contradiction. It follows that sufficiently long paths in T have at most cyclic, trivial or parabolic stabilizer (nonmonotone paths have a reversal). In fact a path fixed by a height-zero element outside every S_d has bounded length: its sufficiently long blocks would all have nonparabolic cyclic stabilizers, with finite orbit dictionary for fixed block size, and repeating an oriented block would commensurate the element by a tree translation.