Lemma 13 ([Weighted maximum estimate]). Let G be coupled through level n with the growth bound of Lemma 11 ([Propagation of coupled growth]). Suppose that on the real ray |G(u)|≤C_N e^-NZ_n(u) (u≥u_N), every fixed N>0. The constants …

Correcting the weighted sources. Choose C greater than the fixed coefficient in the coupled growth estimate on safe columns and the side. Multiply G by the holomorphic gauge (HL_n-CZ_n). Its side modulus is bounded independently of H because x=0 ; its left boundary logarithm is at most H_ left x+O(1) . With constants decreased to absorb the fixed losses, its sources have modulus at most a finite sum of (Hx-cU) . Set K_H= 0,_ sources (Hx-cU/2) . The source-ratio lemma makes K_H finite for every fixed H . Crucially, keep half the negative cost: e^ Hx-cU ≤e^ K_H e^ -cU/2 . The last factor is bounded by a fixed multiple of e^ -u and is integrable on the bounded-width domain, because U/e^u→∞ . Lemma 10 ([Holomorphic proxies and Cauchy correction]) thus gives a Cauchy correction of norm at most C_1e^ K_H , where C_1 is independent of H . Let F_H be the resulting holomorphic function. By, C_1 is uniform for the controlled finite-request families as well; the bound uses their full forward domains, not just a window about the target. Their possibly different return times to smaller ordinary neighborhoods do not enter this constant. Boundedness for each fixed weight. For each fixed H , preliminary real decay makes the weighted function bounded on the real ray: choose N larger than the fixed multiple of H+C needed there, and include the finite interval before u_N in its bound. This bound can depend on H in an uncontrolled way; we do not use it in the final estimate. To see that F_H is bounded on each half-domain, write X+iY=Z_n . Its logarithmic growth is O_H(e^X) plus a path constant. On the upper half use the positive harmonic barrier δe^ aX (a(Y-π/4)), 1<a<2, and on the lower half use δe^ aX (a(Y+π/4)) . The side angles tend to ±π/2 , so a sufficiently far start makes both barriers positive on their respective half-domains. They dominate the infinity growth. The maximum principle on finite truncations and then δ0 proves boundedness on each half, hence on the full domain. Removing the preliminary constants. Now apply the bounded maximum principle to that full domain. The real ray is no longer a boundary. One can justify the infinite-domain principle by δe^ a'X (a'Y) , 0<a'<1 , before letting δ0 . Only the side and the fixed safe left column remain. After restoring the Cauchy correction their bounds give, with an additive constant depending on C_1 and the fixed data alone. This explains why the arbitrary constants in disappear. For n=1 , choose a fixed backstep so that the target's positive x_a is larger than _ left x . There is no source term, and gives |G(s_a)|≤C_0+CZ_1(s_a) -H(x_a-_ left x). Letting H→∞ proves G(s_a)=0 . Every sufficiently far real target permits this construction, giving the asserted identity on the real locus.