What does this research mean for the field?

All non-trivial zeros of the Riemann zeta function lie exactly on the critical line Re(s) = 1/2, proven through a non-circular, unconditional approach. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.CHALLENGES_CONSENSUS.

February 24, 2026Open Access

Closing the Infinitesimal Gap in the Critical Auto-Duality Conjecture: Hierarchical Progressive Tightening, Proof Modifications, and New Mixed Bounds

Key Points

The aim is to analytically close the infinitesimal gap in the Critical Auto-Duality Conjecture related to non-trivial zeros.
Proved three new rigidity theorems driving δ(t) to zero for non-trivial zeros.
Established a Borg–Marchenko inverse spectral theorem adapted to specific potentials.
Derived an explicit quantitative unique continuation estimate for the bounds of δ(t).
Reformulated auto-duality as a homotopy equivalence between spectral models.
Achieved unconditional proof that all non-trivial zeros lie on the critical line Re(s) = 1/2.
Demonstrated that any non-zero δ disrupts the homotopy equivalence through logarithmic divergence of the Fredholm index.
Closed the program started by Coppi (2026), linking the Riemann Hypothesis to Schrödinger operators.

Abstract

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.

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