Lemma 16 ([Comparison of terminal coefficients]). Suppose a complete exponent vector has been isolated on the two sides and coupled, with all earlier subtrees passed as in Lemma 15 ([One finite transition]). Its two terminal ordinary …

By terminal growth propagation the normalized function is bounded on the whole inner domain after a fixed backstep. For a target family with real anchors u_a→∞ , choose safe left columns u_b in fixed backstep intervals. The finite raw, background, and source data used by this terminal coupling have common bounds on their full forward domains, by the finite-request hypothesis. In particular supplies a common M e^ -u envelope for their corrected sources. Their Cauchy corrections therefore have norm at most CM e^ -u_b →0 , uniformly over the target family. On a single fixed path the compact-left and tail argument in Lemma 10 ([Holomorphic proxies and Cauchy correction]) gives the corresponding vanishing at infinity. Denote the corrected functions by G ; they are holomorphic and uniformly bounded. On the two outer collars their values approach, respectively, c_+(t) and c_-(t) , by the terminal transition. We give the argument that these two boundary limits cannot differ. Fix an ordinary value t_* in a sufficiently small real neighborhood and take a family of tests approaching it on every fixed u window about the varying anchor. Such tests can return to t_c at infinity: for example, on the needed right half-strip use t(s)=t_c+(t_*-t_c)e^ -δ_a(s-u_a) , δ_a0, with the fixed backward range and ordinary neighborhood chosen with slack. Its fixed positive derivatives tend to zero and it returns to t_c on every chosen path. Taking 0<δ_a≤δ_0 gives common finite-jet bounds on the entire forward domains, and δ_a→0 gives a common small-jet modulus and convergence to t_* on each fixed target window. The time of return to a smaller box may diverge; that time is used only in the preliminary real estimates, whose constants have already been removed by Lemma 13 ([Weighted maximum estimate]). Write X+iY=Z_m . For a fixed small η>0 , the boundary values on a window about the target are within η of the corresponding constants, once the anchor is sufficiently far. The X distances from the target to both ends of a fixed u window tend to infinity by Lemma 4 ([Real and angular path estimates]). On that window apply the logarithmic maximum principle to the difference from c_+(t_*) . Its upper boundary is bounded by η , its lower boundary by a fixed constant M , and its two ends by the bounded growth estimate. An affine function of Y interpolates the lateral logarithmic bounds. The actual angles tend to ±π/2 ; adding arbitrarily small relative angular slack and decaying cosine barriers at the two ends yields, at the real target, |G-c_+(t_*)| ≤12η+12 M. For completeness, the end barriers may be taken as fixed multiples of e^ -κ(X-X_-) (κY) and e^ -κ(X_+-X) (κY) with 0<κ<1 . Their values at the target tend to zero. First let the target go to infinity, then let the angular slack and η go to zero. The real-ray limit is c_+(t_*) . Applying the same argument from the lower boundary gives the limit c_-(t_*) . Thus these two values coincide for every such real t_* . The identity theorem for analytic germs makes the coefficients equal. With their common value on both lateral boundaries, apply the same maximum principle to their difference from G . The lateral bound is now η on both sides, so the estimate holds uniformly across the entire inner wedge, including targets approaching either boundary. Restoring the vanishing Cauchy correction changes nothing. Uniformity on smaller ordinary neighborhoods follows either from these estimates with common finite slack or by applying the construction to a putative sequence of violating ordinary inputs. The same argument uses the corresponding coupled determinations in the already used bands, and proves their asserted convergence.