Lemma 60 ([Identity on the nearby tied endpoint graph]). Every transfer constructed above computes the actual scalar mismatch for all nearby proposed endpoints on its argument graph. The terminal state coordinate has nonzero transverse …

All time and profile primitives above are identities of analytic functions on the prepared physical piece. Invariants in a time change are constant for every nearby time, by the weight-zero identity in Lemma 59 ([Rational diagonal primitives]). The pure-state invariant is constant for every nearby state by. The linearizer, power maps, and profile inversions have nonzero Jacobian on their specified domains. Clock swaps use a nonzero scalar velocity. The only apparent dependence of the computed flow on a proposed final state occurs in right amplitudes. For the additive annulus, the tie is equivalent, on its positive real domain, to z_+=v_+-v_- -κS . If d_iκ=|d_i| , it gives D_i=χ_iQ^ p_i (d_iv_-+|d_i|S). This expression is independent of the proposed v_+ . Substitution in, at every intermediate state z , gives H_i h^ -p_i e^ d_i z =χ_iw_0^ p_i (d_i[v_-+κ(A-q)+z]), exactly the original profile. The left identity is immediate. Using Q , rather than p , in the amplitude is exact because the layer contains (w/Q)^ p_i . For a mixed right layer with positive ρ_i=a_i+β_iκc , use D_i=χ_i(a_iθ_++β_iv_+-β_i z_+). The tie z_+=v_+-v_- -κW gives D_i=χ_i(a_iθ_++β_iv_-+β_iκW). Multiplication by e^ -ρ_i(L-t) e^ β_i z recovers χ_i e^ a_i(θ_-+t)+β_i(v_-+κct+z) . The calculation uses no sign restriction on β_i or κ . For the small-slope case replace v by v , which is regular by; invariance makes its p=A_0/b independent of the proposed other endpoint. Thus the flow side is independent of that proposed endpoint before it is equated with z_+ . For clarity, an endpoint normalization such as U_* obeys a slightly different but equally exact rule. Let C_a be its regular coordinate map, with a held temporarily independent. The transfer is a conjugacy for every nearby fixed a , so its mismatch is C_a(U_ flow )-C_a(U_+) . At equality, C_a(U_ flow )-C_a(U_+) =C_a'(U_+)\,(U_ flow -U_+). The two a -variations cancel even when a is then chosen from a proposed endpoint. For a clock swap, the nonzero clock velocity likewise determines the hitting-time variation; the remaining state derivative is transverse and nonzero. This proves the assertion.