The Gibbons-Hawking Lattice is Exactly SL(2, Z): Uniqueness of the Modular Group in TIC/CIT

We prove that the Gibbons-Hawking thermal lattice Lambda_GH of TIC/CIT is exactly SL(2,Z), excluding all proper congruence subgroups Gamma_0(N), Gamma(N), and Gamma_1(N) for N > 1. Three independent routes are given. Route I (covolume): the de Sitter entropy uniquely fixes the covolume of Lambda_GH^2 to pi/3; by Siegel's minimum covolume theorem and the index formula for congruence subgroups — which all satisfy covol >= pi for N >= 2 — only SL(2,Z) is consistent Route II (S-duality obstruction): the TIC/CIT sigma- field S-duality g_sigma -> 1/g_sigma generates the matrix S = [[0,-1],[1,0]], which does not belong to Gamma_0(N) for any N >= 2, since the condition N|c with c=1 forces N=1. The Atkin-Lehner involution W_N differs from S for N >= 2, placing the self-dual coupling at g_sigma^* = N1/4 ≠ 1 in contradiction with Axiom 2. Route III (modular bootstrap): the Core Inequality forces the TIC/CIT partition function to be invariant under both T and the exact S- transformation, generating the full SL(2,Z) by the classical generators theorem. This closes the last logical gap in the TIC/CIT derivation of the Riemann Hypothesis. Paper 9 of the TIC/CIT series.