We prove that the Gibbons-Hawking thermal lattice LambdaGH of TIC/CIT is exactly SL (2, Z), excluding all proper congruence subgroups Gamma₀ (N), Gamma (N), and Gamma₁ (N) for N > 1. Three independent routes are given. Route I (covolume): the de Sitter entropy uniquely fixes the covolume of LambdaGH² 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 gₛigma -> 1/gₛigma generates the matrix S = [0, -1, 1, 0], which does not belong to Gamma₀ (N) for any N >= 2, since the condition N|c with c=1 forces N=1. The Atkin-Lehner involution WN differs from S for N >= 2, placing the self-dual coupling at gₛigma^* = N^1/4 ≠ 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.
Leandro de Oliveira (Thu,) studied this question.