We provide a complete analytical closure of the infinitesimal gap in the Critical Auto-Duality Conjecture (CADC). Starting from the hierarchical progressive framework developed in the companion extension, we prove three new rigidity theorems that together force δ (t) = 0 exactly for every non-trivial zero. First, we establish an exact Borg–Marchenko inverse spectral theorem adapted to the potential x²/4 + ∑ Re (x^ρ/ρ). Second, we derive a quantitative unique continuation estimate with explicit constant C₀ = 4 + 2C that controls the bound |δ (t) | ≤ 1/ (log log |t|) ^1/2. Third, we reformulate the classical auto-duality as a homotopy equivalence between spectral models and show that any non-zero δ breaks this equivalence via logarithmic divergence of the Fredholm index. Combining these results, we obtain the first non-circular, unconditional proof that all non-trivial zeros lie exactly on the critical line Re (s) = 1/2. The argument remains entirely within the non-circular discrete Hamiltonian framework and uses only unconditional arithmetic bounds (Huxley, Vinogradov–Korobov). This closes the program initiated in Coppi (2026) and transforms the Riemann Hypothesis into a rigidity statement for a specific Schrödinger operator.
Franck Coppi (Sun,) studied this question.