Lemma 89 ([Induced subgroup fillings]). For a fixed finite collection of exact relative embeddings into rows, one can choose the finite tests in Definition 88 ([Peripheral compatibility tests]) so that every sufficiently long filling …

We prove the assertion for B ; the same argument applies to every group in the finite collection. The original embedding constants give local quasigeodesic constants (λ_0,ε_0) . Together with the uniform quotient hyperbolicity constant, these determine a sufficient local scale and global constants (λ_1,ε_1) . Choose the tested scale larger also than λ_1ε_1 , with the fixed offset margins, before making any finite lists. A bounded-length intrinsic coned geodesic in B still has no repeated apices in the ambient expanded path. Otherwise take a minimal bad repetition. Internal jumps have bounded angle by the loop, hence lift by finite-list membership to bounded-list actual peripheral letters of B preserving their values in B . Equality of the endpoint ambient cosets thus already held before filling, forcing repetition of the domain apex. Next compare an ordinary-endpoint such path, starting at identity, with any ambient geodesic shortcut. In the bounded union graph a jump not at a cut apex separating start and end has bounded angle (connect inbound to start and outbound to end, and these to each other avoiding that apex). Thus lift such letters as just above: the actual finite-list representative in an intersection subgroup after the membership test gives even the required standalone peripheral value, since its kernel there is the restriction and the reduced intersection subgroup injects in the ambient peripheral reduction. Cut apices occur on both paths in the same order. Discrepancies between their ordinary inbound germs, likewise outbound germs, are from finite lists in that peripheral by avoiding paths through start or end. Lift those connectors. Between cuts all consistency equations with the connectors hold by fixed-list injection. Choose any actual lift in the indicated subgroup for the B -jump at each cut; the shortcut jump can lift compatibly via the connectors since it only has to lie in the ambient peripheral. Thus some actual B lift of the intrinsic word has a shortcut of the same bounded cost. The intrinsic distance after reduction is no larger than that of this lift, where the original relative embedding bounds it by the shortcut cost. This proves the chosen local quasigeodesic bounds; trimming to ordinary frame endpoints costs only the fixed margins. Local-to-global now gives the uniform global constants (λ_1,ε_1) . A repeated apex on the global expanded path would cut off a subpath of length at most λ_1ε_1 , contradicting the local no-repetition check. The same estimate puts every potential kernel word within the tested scale. Its lifted length-zero shortcut makes the corresponding original subgroup word trivial, and the chosen peripheral lifts preserve its value in the standalone filling. Thus its standalone value is trivial. This proves injection and all the assertions simultaneously for the finite collection.