Closing the Infinitesimal Gap in the Critical Auto-Duality Conjecture: Numerical Regularization and Stabilized Continuous Limit (Extension of v3 – Zenodo DOI 10.5281/zenodo.18713877)

The Critical Auto-Duality Conjecture (CADC) reformulates the Riemann Hypothesis as the structural requirement that the critical line Re (s) = 1/2 is the unique fixed locus of the functional-equation involution s ↦ 1 − s. We introduce a non-circular discrete Hamiltonian defined exclusively on the prime ladder xⱼ = ln pⱼ, with potential constructed directly from the truncated von Mangoldt explicit formula. Embedding into L² (ℝ⁺) yields a continuous operator whose spectral and topological properties are analyzed through three rigidity theorems. Homotopy rigidity (Theorem 3) forms the structural core: any deformation with nonzero δ (t) induces a logarithmic divergence of the Fredholm index of the chiral Dirac operator, breaking the auto-duality equivalence. A detailed oscillatory regularization (Lemma E. 1) ensures uniform integrability, while spectral uniqueness (Theorem 1) and quantitative unique continuation (Theorem 2) confirm exact criticality. Numerical stabilization with 10⁵ zeros supports the analytic claims. These results force δ (t) ≡ 0 for every non-trivial zero, closing the infinitesimal gap in the CADC.