Lemma 128 ([A surviving horizontal edge]). The chain α² contains a physical horizontal edge with odd multiplicity. A final central system whose cut and switch thresholds exceed the new regular comparison bounds has a slot tree with odd …

First choose the final system with its central cut separations, satellite switch bends, and, in the line setting, outer switch bends all above the bounds supplied by the connectors after the horizontal replacements. Two distinct horizontal midpoint anchors of α^2 cannot belong to one completed tree. A comparison between them forbids switches at either level. They would therefore lie in one W_0 ; its path between distinct slots has an internal panel bend, also forbidden by the same comparison. Likewise a ray asymptotic to a line endpoint can neither switch nor continue indefinitely inside W_0 , because it would have such an interior bend. This proves the limit-set assertion for every tested slot tree. We prove nonzero physical parity without assuming that the modified chain is connected. In the circle setting, assign each participating physical vertex its position on the original parameter circle, using its occurrence or the connector to an original site. All ambiguities have bounded cyclic diameter: the connectors and replacement patches avoid the parent outer apex p , so the angle dictionary applies. Choose a single position for each physical vertex and send each edge along its short parameter arc. This is the short-arc construction used in the proof of Proposition 115 ([Odd inner seeds]). Every replacement polygon lies in a bounded parameter interval and has degree zero for sufficiently large cycle order. The original boundary cycle has degree one. The final chain therefore has degree one modulo two and cannot be the zero physical chain. In the line setting, assign each participating physical vertex a single real parameter, chosen from its occurrence or connector positions on α . Quasigeodesicity and bounded tails bound the ambiguity. Cut the parameter line at a generic finite point and give each physical vertex a single binary sign according to which side its chosen parameter lies on. For an edge, count one if the endpoint signs differ. Every replacement loop has even crossing count, because its sign changes telescope modulo two. Only bounded-parameter loops can meet both signs, so only finitely many replacements affect this count. The original bi-infinite path has opposite signs at its two ends and hence odd crossing count. The final physical chain is consequently nonzero. At a horizontal midpoint the cycle condition says that its two half-edge counts agree, in both settings; local finiteness justifies this condition in the line setting. Thus some full physical horizontal edge has odd multiplicity. Choose a slot at its midpoint in the final system. On a purely horizontal chain only marked slots contribute. No other midpoint of the chain is marked by this tree, as proved above, and node uniqueness allows just this one slot over the chosen midpoint across all constituent pieces. The evaluation is therefore odd. All evaluations pass back through the substitutions by triangle consistency. In the line setting the previously proved limit avoidance ensures that the support misses sufficiently distant tails and their bounded-parameter patches. Thus only finitely many substitutions affect this particular evaluation, as required.