Triangular normal arcs admit braid position stable under partial regular switches

Label the vertices $A,B,C$ and choose a smaller homothetic central triangle with corresponding vertices $a,b,c$. The spokes $Aa,Bb,Cc$ divide the region outside it into three convex quadrilaterals, one along each original edge. Regard each quadrilateral as a rectangular strip from its outer original edge to the corresponding central edge, using [F2]. All original arc endpoints lie in the relative interiors of the outer strip edges.

In the central triangle put disjoint normal chords with the combined numbers of all three types. They may be chosen in small disjoint corner neighborhoods of $a,b,c$. For each type, designate one of its two original endpoint edges, using a fixed ordering of the three edges. Order that type's combined $\mathcal{X}$ and $\mathcal{Y}$ chords by the order of their corresponding original endpoints away from the common vertex on the designated edge. Assign their colors and individual labels in that order. By [F1], this order respects the order within each original family on both endpoint edges. The central chords therefore impose compatible same-color orders on their two central strip gates. On a central gate the two incident types occupy their separate canonical corner blocks.

In every strip join each original endpoint to the gate endpoint of its labeled central chord by the straight wire in the rectangular parametrization. For either color separately, its wire order at the outer edge equals its wire order at the gate: order within a type agrees by step 2.1, and the two types have the same separate corner blocks by [F1]. Thus wires of the same color do not cross by [F2]. A wire from the other color crosses it at most once. Distinct crossings cannot coincide, since that would make two same-color wires meet. All intersections are consequently transverse double points. Join the wires to their central chords. Each resulting arc has exactly one central chord and one wire in each of two distinct strips, so is embedded and has its original type.

Two original arcs of different types share only one strip, so intersect at most once. Two original arcs of the same type share two strips, but on that type's designated strip their combined endpoint and gate orders coincide, by step 2.1. They cannot intersect there, so can intersect only once in their other strip. This proves the first assertion, including the original pairwise bound.

At a wire crossing, choose the uncrossing that reconnects each incoming wire from the outer side to the other outgoing wire toward the gate. In a small rectangle this is the reconnection by two horizontal-monotone strands. The other uncrossing joins the two outer-side branches to one another and the two gate-side branches to one another. After any subset of the chosen reconnections, every parametrizing strand within a strip is still strictly monotone in its horizontal coordinate: following it from an outer endpoint always goes toward a gate, and conversely. It cannot form a circle or self-intersect, and no component can have both ends on the same side of that strip. All strips still have the same gate endpoints, and the central chords have not changed. Hence each full parametrizing component has exactly one central chord and wires in its two distinct strips. It is an embedded normal arc. No remaining crossing can be a self-crossing of such a component: it lies inside one strip, where that component has exactly one monotone wire. The central chords count its type, so all type counts are unchanged.

To check the local characterization, the two crossing arcs share the strip's original edge $e$. The chosen uncrossing pairs each outer branch, whose endpoint is on $e$, with one gate branch. The other endpoint of that gate branch lies on a different original edge, determined by its unchanged central chord. This recovers the original unordered pair of types. The alternative joins the two outer branches and therefore has both endpoints on $e$, an edge-return arc. Thus exactly one uncrossing preserves the two normal types. Since the preceding argument worked for any subset of crossings, all assertions follow without altering the original face or its edge endpoints.